Multilevel Monte Carlo research

Research groups

• Adelaide (Zivanovic) -- reliability analysis
• Angers (Panloup) -- invariant distributions
• Barcelona (Badia) -- high performance parallel MLMC software
• Basel (Harbrecht) -- elliptic SPDEs, sparse grids, multilevel QMC
• Bath (Heine, Müller, Shardlow, Yates) -- SDEs, Lévy-driven SDEs, stochastic reaction networks, quantum field theory
• BCAM (Croci) -- SPDEs, stochastic rounding in reduced-precision computing
• Chalmers (Lang, Malqvist) -- SPDEs, failure analysis
• Colorado, Boulder (Ketelson) -- MCMC, low rank methods
• Colorado, PNNL (Ganesh) -- moving-domain FEM approximation
• Columbia (Dong, Liu) -- fractional Brownian motion, elliptic PDEs
• Delft (Kirchner) -- Gaussian random fields
• Duisburg (Belomestny, Hutzenthaler) -- Bermudan and American options
• Edinburgh (Davie, Higham, Szpruch, Teckentrup, Zygalakis) -- SDEs, stochastic reaction networks, numerical analysis, inverse methods
• EPFL (Nobile) -- stochastic collocation, SPDEs, function approximation
• ESSEC (Jacob) -- unbiased MCMC
• ETH Zürich (Mishra, Jenny, Schwab) -- SPDEs, multilevel QMC
• Exeter (Dodwell) -- Bayesian inverse problems
• Florida State (Gunzburger, Mascagni) -- elliptic SPDEs, stochastic collocation, walk on spheres, applications to material science
• Fraunhofer ITWM (Iliev) -- SPDEs in engineering
• Heidelberg (Scheichl, Zech) -- elliptic SPDEs, MCMC, quantum field theory, forward and inverse UQ, sparse grids
• Heriott-Watt (Haji-Ali) -- multi-index Monte Carlo, risk estimation
• IIT Chicago (Hickernell) -- SDEs, infinite-dimensional integration, complexity analysis
• Imperial College (Cotter, Crisan, Luk) -- particle filters, ensemble forecasts, multi-index Monte Carlo, FPGA implementation
• Jyväskylä (Vihola) -- randomized MLMC
• Kaiserslautern (Heinrich, Korn, Ritter) -- finance, SDEs, parametric integration, complexity analysis
• KAUST (Jasra) -- sequential Monte Carlo
• KIT (Krumscheid) -- function approximation, complex computational models
• Lawrence Livermore National Lab (Fairbanks, Vassilevski) -- elliptic SPDEs, MCMC
• Leuven (Nuyens, Vandewalle) -- numerical analysis, uncertainty quantification, multilevel QMC, multi-index QMC
• LPMA (Frikha, Lemaire, Pagès) -- numerical analysis, multilevel extrapolation, finance applications
• Lugano (Krause) -- elliptic SPDEs
• Manchester (Cotter, Law) -- finance applications, elliptic SPDEs, stochastic collocation, inverse methods
• Mannheim (Neuenkirch) -- numerical analysis, fractional Brownian motion
• MIT (Marzouk) -- uncertainty quantification, SPDEs, Bayesian inverse problems
• Munich (Ullmann) -- uncertainty quantification, SPDEs
• Münster (Jentzen) -- machine learning, stochastic analysis, numerical analysis, PDEs
• NYU (Xiao) -- neuroscience
• Oden Institute (Webster, Willcox) -- elliptic SPDEs, inverse methods, multi-fidelity analysis
• Oldenburg (Chernov) -- numerical analysis, elliptic SPDEs, contact and obstacle problems, density estimation
• Oslo (Hoel) -- SDEs, SPDEs, Kalman filtering
• Osnabrück (Gnewuch) -- randomized multilevel QMC
• Oxford (Baker, Farrell, Giles, Hambly, Reisinger) -- SDEs, SPDEs, numerical analysis, finance applications, stochastic reaction networks, nested simulation
• Passau (Müller-Gronbach) -- infinite-dimensional integration, complexity analysis, stochastic approximation
• Princeton (Weinan E) -- machine learning
• Queensland (Dang) -- jump-diffusion models
• Rutgers (Wang) -- randomized MLMC, MCMC
• RWTH Aachen (Tempone) -- adaptive time-stepping, stochastic reaction networks, Multi-Index MC, sequential Monte Carlo
• Sandia National Lab (Eldred) -- multi-fidelity, optimisation under uncertainty
• Sorbonne (Kebaier) -- SDEs, numerical analysis, importance sampling
• Stanford (Blanchet, Glynn, Iaccarino, Tartakovsky) -- numerical analysis, randomized/debiased multilevel, engineering uncertainty quantification, groundwater flow
• Stuttgart (Barth) -- SPDEs
• Texas A&M (Efendiev) -- SPDEs in engineering
• Tokyo (Goda, Sato ) -- partial perfect information, expected value of sample information, machine learning
• UCL (Beskos, Briol, Guillas, Ni) -- Bayesian statistics, sequential design, UQ for natural hazards, machine learning, fractional Brownian motion
• UCLA (Caflisch) -- Coulomb collisions in physics
• UIUC (Chen) -- stochastic optimization
• UNSW (Dick, Kuo, Sloan) -- multilevel QMC
• UPC (Principe) -- elliptic SPDEs
• Utah (Kirby, Narayan) -- SPDEs
• Utrecht (Oosterlee) -- finance applications
• Vienna (Heitzinger) -- SPDEs and systems of SPDEs with applications in nanotechnology
• Warwick (Kyprianou, Mijatovic, Vollmer) -- Lévy processes, stochastic gradient methods in Big Data
• Waterloo (Ricardez-Sandoval) -- chemical engineering
• WIAS (Friz, Schoenmakers) -- rough paths, fractional Brownian motion, Bermudan options
• Wisconsin (Anderson) -- numerical analysis, stochastic reaction networks
• WWU (Dereich) -- Lévy-driven SDEs, stochastic approximation

Research areas

• Reviews
• Early research
• Key developments
• Other algorithmic developments
• Brownian SDEs -- finance applications
• Brownian SDEs -- other applications
• Lévy processes and Lévy-driven SDEs
• Complexity analysis
• Elliptic SPDEs
• Other SPDEs
• Multilevel QMC
• MCMC and other inverse methods
• Stochastic reaction networks
• Nested expectations
• McKean-Vlasov equations
• Computer science applications
• Density, distribution and function estimation
• Reliability and rare events
• Discontinuous functionals
• Importance sampling and other non-standard MLMC couplers
• Stochastic approximation and machine learning
• Multifidelity
• Numerical linear algebra
• Physics
• Other applications

Reviews

• M.B. Giles. `Multilevel Monte Carlo methods'. pp.83-103 in Monte Carlo and Quasi-Monte Carlo Methods 2012, Springer, 2013. link
• M.B. Giles. `Multilevel Monte Carlo methods'. Acta Numerica, 24:259-328, Cambridge University Press, 2015. link
• F.Y. Kuo, D. Nuyens. 'Application of Quasi-Monte Carlo methods to elliptic PDEs with random diffusion coefficients: a survey of analysis and implementation'. Foundations of Computational Mathematics, 16(6):1631-1696, 2016. link
• R. Schmidt, M. Voigt, M. Pisaroni, F. Nobile, P. Leyland, J. Pons-Prats, G. Bugeda. 'General introduction to Monte Carlo and multi-level Monte Carlo methods'. Uncertainty Management for Robust Industrial Design in Aeronautics, pp.265-278, 2019. link
• R. Tempone, S. Wolfers. 'Smolyak's algorithm: a powerful black box for the acceleration of scientific computations'. Sparse Grids and Applications, LNCSE Vol.123, Springer, pp.201-228, 2018. link

Early research

• A. Brandt, M. Galun, D. Ron. 'Optimal multigrid algorithms for calculating thermodynamic limits'. Journal of Statistical Physics, 74(1-2):313-348, 1994. link
• A. Brandt, V. Ilyin. 'Multilevel Monte Carlo methods for studying large scale phenomena in fluids'. Journal of Molecular Liquids, 105(2-3):245-248, 2003. link
• M. Emsermann. 'Variance reduction with quasi control variates'. PhD thesis, University of Colorado at Denver, 2000. link
• M. Emsermann, B. Simon. 'Improving simulation efficiency with quasi control variates'. Stochastic Models, 18(3):425-448, 2002. link
• S. Heinrich. 'Monte Carlo complexity of global solution of integral equations'. Journal of Complexity, 14(2):151-175, 1998. link
• S. Heinrich, E. Sindambiwe. 'Monte Carlo complexity of parametric integration'. Journal of Complexity, 15(3):317-341, 1999. link
• S. Heinrich. 'The multilevel method of dependent tests'. In Advances in Stochastic Simulation Methods, Springer, 2000. link
• S. Heinrich. 'Multilevel Monte Carlo methods'. Lecture Notes in Computer Science, 2179:58-67, 2001. link
• S. Heinrich. 'Monte Carlo approximation of weakly singular integral operators'. Journal of Complexity, 22(2):192-219, 2006. link
• A. Kebaier. 'Statistical Romberg extrapolation: a new variance reduction method and applications to option pricing'. Annals of Applied Probability, 15(4):2681-2705, 2005. link
• A. Keller. 'Hierarchical Monte Carlo image synthesis'. Mathematics and Computers in Simulation, 55(1-3):79-92, 2001. link
• Q. Li. 'N-dimension numerical solution of stochastic differential equations'. PhD thesis, Edinburgh University, 2007. link

Key developments

• M.B. Giles. `Multilevel Monte Carlo path simulation'. Operations Research, 56(3):607-617, 2008. link
• M.B. Giles, B.J. Waterhouse. 'Multilevel quasi-Monte Carlo path simulation'. pp.165-181 in Advanced Financial Modelling, in Radon Series on Computational and Applied Mathematics, de Gruyter, 2009. link
• A.L. Haji-Ali, F. Nobile, R. Tempone. 'Multi-index Monte Carlo: when sparsity meets sampling'. Numerische Mathematik, 132(4):767-806, 2016. link
• V. Lemaire, G. Pagès. 'Multilevel Richardson-Romberg extrapolation'. Bernoulli, 23(4A):2643-2692, 2017. link
• C.-H. Rhee, P.W. Glynn. 'Unbiased estimation with square root convergence for SDE models'. Operations Research, 63(5):1026-1043, 2015. link

Other algorithmic developments

• J.H. Blanchet, P.W. Glynn. 'Unbiased Monte Carlo for optimization and functions of expectations via multilevel randomization'. Proceedings of the 2015 Winter Simulation Conference. link
• R.R. Callens, M.G.R. Faes, D. Moens. 'Multilevel Quasi-Monte Carlo for interval analysis'. International Journal for Uncertainty Quantification, 12, 42022, 2022. link
• A. Chernov, E.M. Schetzke. 'A simple, bias-free approximation of covariance functions by the multilevel Monte Carlo method having nearly optimal complexity'. SIAM/ASA Journal on Uncertainty Quantification, 11(3):941-969, 2023. link
• N. Collier, A.-L. Haji-Ali, F. Nobile, E. von Schwerin, R. Tempone. 'A Continuation Multilevel Monte Carlo algorithm'. BIT Numerical Mathematics, 55(2):399-432, 2015. link
• M. Croci, K.E. Willcox, S.J. Wright. 'Multi-output multilevel best linear unbiased estimators via semidefinite programming'. Computer Methods in Applied Mechanics and Engineering, 413:116130, August 2023. link
• P.W. Glynn, C.-H. Rhee. 'Exact estimation for Markov chain equilibrium expectations'. Journal of Applied Probability, 51A:377-389, 2014. link
• A.-L. Haji-Ali, F. Nobile, E. von Schwerin, R. Tempone. 'Optimization of mesh hierarchies in multilevel Monte Carlo samplers'. Stochastics and PDEs: Analysis and Computations, 4(1):76-112, 2016. link
• D. Schaden, E. Ullmann. 'On multilevel best linear unbiased estimators'. SIAM/ASA Journal on Uncertainty Quantification, 8(2):601-635, 2020. link
• D. Schaden, E. Ullmann. 'Asymptotic analysis of multilevel best linear unbiased estimators'. SIAM/ASA Journal on Uncertainty Quantification, 9(3):953-978, 2021. link
• D. Schaden. 'Variance reduction with multilevel estimators'. PhD thesis, TU München, 2021. link
• M. Vihola. 'Unbiased estimators and multilevel Monte Carlo'. Operations Research, 66(2):448-462, 2018. link
• Z. Zheng, J. Blanchet, P. W. Glynn. 'Rates of convergence and CLTs for subcanonical debiased MLMC'. Monte Carlo and Quasi-Monte Carlo Methods. MCQMC 2016. Springer Proceedings in Mathematics & Statistics, vol 241, pp.465-479., 2018. link

Brownian SDEs -- finance applications

• M.B. Alaya, A. Kebaier. 'Multilevel Monte Carlo for Asian options and limit theorems'. Monte Carlo Methods and Applications, 20(3):181-194, 2014. link
• M. Altmayer, A. Neuenkirch. 'Multilevel Monte Carlo quadrature of discontinuous payoffs in the generalized Heston model using Malliavin integration by parts'. SIAM Journal on Financial Mathematics, 6(1):22-52, 2015. link
• R. Avikainen. 'On irregular functionals of SDEs and the Euler scheme'. Finance and Stochastics, 13(3):381-401, 2009. link
• D. Belomestny, F. Dickmann, T. Nagapetyan. 'Pricing Bermudan options via multilevel approximation methods'. SIAM Journal of Financial Mathematics, 6(1):448-466, 2015. link
• D. Belomestny, J. Schoenmakers, F. Dickmann. 'Multilevel dual approach for pricing American style derivatives'. Finance and Stochastics, 17(4):717-742, 2013. link
• F. Bourgey, S. De Marco. 'Multilevel Monte Carlo simulation for VIX options in the rough Bergomi model'. Journal of Computational Finance, 26(2):53-82, 2022 link
• S. Burgos, M.B. Giles. `Computing Greeks using multilevel path simulation'. pp.281-296 in Monte Carlo and Quasi-Monte Carlo Methods 2010, Springer, 2012. link
• S. Burgos. `The computation of Greeks with multilevel Monte Carlo'. PhD thesis, University of Oxford, 2014. link
• F. Dickmann. `Multilevel approach for Bermudan option pricing'. PhD thesis, Universitat Duisburg-Essen, 2015. link
• R. Gasparotto. 'Optimised Importance Sampling in Multilevel Monte Carlo'. MSc thesis, University of Oxford, 2015. link
• T. Gerstner, M. Noll. 'Randomized multilevel quasi-Monte Carlo path simulation'. In Recent Developments in Computational Finance, World Scientific / Imperial College Press, 2013. link
• M.B. Giles. `Multilevel Monte Carlo path simulation'. Operations Research, 56(3):607-617, 2008. link
• M.B. Giles. `Improved multilevel Monte Carlo convergence using the Milstein scheme'. pp.343-358, in Monte Carlo and Quasi-Monte Carlo Methods 2006, Springer, 2008. link
• M.B. Giles, D.J. Higham, X. Mao. 'Analysing multilevel Monte Carlo for options with non-globally Lipschitz payoff'. Finance and Stochastics, 13(3):403-413, 2009. link
• M.B. Giles, B.J. Waterhouse. 'Multilevel quasi-Monte Carlo path simulation'. pp.165-181 in Advanced Financial Modelling, in Radon Series on Computational and Applied Mathematics, de Gruyter, 2009. link
• M.B. Giles. `Multilevel Monte Carlo for basket options'. Proceedings of 2009 Winter Simulation Conference. link
• M.B. Giles, L. Szpruch. 'Multilevel Monte Carlo methods for applications in finance'. In Recent Developments in Computational Finance, World Scientific / Imperial College Press, 2013. link
• M.B. Giles, K. Debrabant, A. Roessler. 'Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation'. Discrete and Continuous Dynamical Systems -- series B, 24(8):3881-3903, 2019. link
• M.B. Giles, O. Sheridan-Methven. 'Analysis of nested multilevel Monte Carlo using approximate Normal random variables'. SIAM/ASA Journal on Uncertainty Quantification, 10(1):200-226, 2022. link
• M.B. Giles, A.-L. Haji-Ali. 'Multilevel path branching for digital options' Annals of Applied Probability, 34(5):4836-4862, 2024. link
• D. Giorgi, V. Lemaire, G. Pagès. 'Limit theorems for weighted and regular Multilevel estimators'. Monte Carlo Methods and Applications, 3(1):43-70, 2017. link
• D. Giorgi. 'Théorèmes limites pour estimateurs multilevel avec et sans poids. Comparaisons et applications'. PhD thesis, Université Pierre et Marie Curie, 2017. link
• Q. Han, S. Ji. 'Solving BSDEs based on novel multi-step schemes and multilevel Monte Carlo'. Journal of Computational and Applied Mathematics, 4171:114543, 2023. link
• N. Kahalé. 'General multilevel Monte Carlo methods for pricing discretely monitored Asian options'. European Journal of Operational Research, 287(2):739-748, 2020. link
• C. Mbaye, G. Pagès, F. Vrins. 'An antithetic approach of multilevel Richardson-Romberg extrapolation estimator for multidimensional SDES'. Numerical Analysis and its Applications, 2017. link
• C.X. Pang, X.J. Wang. 'Antithetic multilevel Monte Carlo method for approximations of SDEs with non-globally Lipschitz continuous coefficients'. Stochastic Processes and their Applications, 178:104467, 2024. link
• O. Sheridan-Methven. `Nested multilevel Monte Carlo methods and a modified Euler-Maruyama scheme utilising approximate Gaussian random variables suitable for vectorised hardware and low-precisions'. PhD thesis, University of Oxford, 2021. link
• A.L. Speight. 'A multilevel approach to control variates'. PhD thesis, Carnegie-Mellon University, 2007.
• A.L. Speight. 'A multilevel approach to control variates'. Journal of Computational Finance, 12(4):3-27, 2009. link
• A.L. Speight. 'Multigrid techniques in economics'. Operations Research, 58(4):1057-1078, 2010. link
• P. Turkedjiev. 'Numerical methods for backward stochastic differential equations of quadratic and locally Lipschitz type'. PhD thesis, Humboldt University, 2013. link
• A. Zeng, J. Xue. 'Multilevel Monte Carlo method for path-dependent barrier interest rate derivatives'. SIAM Journal on Financial Mathematics, 10(1):214-242, 2019. link
• C. Zheng. 'Multilevel Monte Carlo using approximate distributions of the CIR process'. BIT Numerical Mathematics, 63(2):38, 2023. link
• C. Zheng. 'Multilevel Monte Carlo simulation for the Heston stochastic volatility model'. Advances in Computational Mathematics, 49(6):81, 2023. link

Brownian SDEs -- other applications

• A. Abdulle, A. Blumenthal. 'Stabilized multilevel Monte Carlo method for stiff stochastic differential equations'. Journal of Computational Physics, 251:445-460, 2013. link
• A. Abdulle, A. Blumenthal. 'Improved stabilized multilevel Monte Carlo method for stiff stochastic differential equations'. Lecture Notes in Computational Science and Engineering, 103:537-545, 2015. link
• M.B. Alaya, A. Kebaier. 'Central limit theorem for the multilevel Monte Carlo Euler method'. Annals of Applied Probability, 25(1):211-234, 2015. link
• M.B. Alaya, A. Kebaier. T.B.T. Ngo. 'Central limit theorem for the antithetic multilevel Monte Carlo Euler method'. Annals of Applied Probability, 32(3):1970-2027, 2022. link
• H. Alzubaidi, T. Shardlow. 'Improved simulation techniques for first exit time of neural diffusion models'. Communications in Statistics - Simulation and Computation, 2013. link
• D.F. Anderson, D.J. Higham, Y. Sun. 'Multilevel Monte Carlo for stochastic differential equations with small noise'. SIAM Journal of Numerical Analysis, 54(2):505-529, 2016. link
• P. Andersson, A. Kohatsu-Higa. 'Unbiased simulation of stochastic differential equations using parametrix expansion'. Bernoulli, 23(3):2028-2057, 2017. link
• J. Blanchet, F. Zhang. 'Exact simulation for multivariate Ito diffusions'. Advances in Applied Probability 52(4):1003-1034, 2020. link
• K. Debrabant, A. Ghasemifard, N.C. Mattsson. 'Weak antithetic MLMC estimation of SDEs with the Milstein scheme for low-dimensional Wiener processes'. Applied Mathematics Letters, 91:22-27, 2019. link
• K. Debrabant, A. Roessler. 'On the acceleration of the multi-level Monte Carlo method'. Journal of Applied Probability 52(2):307-322, 2015. link
• F. Dickmann, N. Schweizer. 'Faster comparison of stopping times by nested conditional Monte Carlo'. Journal of Computational Finance 20(2):101-123, 2016. link
• W. Fang, M.B. Giles. 'Adaptive Euler-Maruyama method for SDEs with non-globally Lipschitz drift'. pp.217-234 in Monte Carlo and Quasi-Monte Carlo Methods 2016, Springer, 2018. link
• W. Fang, M.B. Giles. 'Multilevel Monte Carlo method for ergodic SDEs without contractivity'. Journal of Mathematical Analysis and Applications, 476(1):149-176, 2019. link
• W. Fang, M.B. Giles. 'Adaptive Euler-Maruyama method for SDEs with non-globally Lipschitz drift'. Annals of Applied Probability, 30(2):526-560, 2020. link
• W. Fang. 'Adaptive timestepping for SDEs with non-globally Lipschitz drift'. PhD thesis, University of Oxford, 2019. link
• A. Al Gerbi, B. Jourdain, E. Clément. 'Ninomiya-Victoir scheme: strong convergence, antithetic version and application to multilevel estimators'. Monte Carlo Methods and Applications 22(3):197-228, 2016. link
• A. Al Gerbi, B. Jourdain, E. Clément. 'Ninomiya-Victoir scheme: multilevel Monte Carlo estimators and discretization of the involved ordinary differential equations'. ESIAM Proceedings and Surveys 59:1-14, 2017. link
• T. Gerstner, S. Heinz. 'Dimension- and time-adaptive multilevel Monte Carlo methods'. pp.107-120 in Sparse Grids and Applications, Lecture Notes in Computational Science and Engineering, Volume 88, 2013. link
• A. Ghasemifard, M. Tahmasebi. 'Multilevel path simulation to jump-diffusion process with superlinear drift'. Applied Numerical Mathematics, 144:176-189, 2019. link
• M.B. Giles, L. Szpruch. 'Antithetic multilevel Monte Carlo estimation for multidimensional SDEs'. pp.367-384 in Monte Carlo and Quasi-Monte Carlo Methods 2012, Springer, 2013. link
• M.B. Giles, L. Szpruch. 'Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation'. Annals of Applied Probability, 24(4):1585-1620, 2014. link
• M.B. Giles, C. Lester, J. Whittle. 'Non-nested adaptive timesteps in multilevel Monte Carlo computations'. Monte Carlo and Quasi-Monte Carlo Methods 2014, Springer, 2015. link
• M.B. Giles, F. Bernal. 'Multilevel estimation of expected exit times and other functionals of stopped diffusions'. SIAM/ASA Journal on Uncertainty Quantification, 6(4):1454-1474, 2018. link
• M.B. Giles, M.B. Majka, L. Szpruch, S.J. Vollmer, K.C. Zygalakis. 'Multi-level Monte Carlo methods for the approximation of invariant measures of stochastic differential equations'. Statistics and Computing, 30:507-524, 2020. link
• P.W. Glynn. 'Exact simulation vs. exact estimation'. Winter Simulation Conference, 2016. link
• Q. Guo, W. Liu, X. Mao, W. Zhan. 'Multi-level Monte Carlo methods with the truncated Euler-Maruyama scheme for stochastic differential equations'. International Journal of Computer Mathematics, 95(9):1715-1726, 2019. link
• D.J. Higham, X. Mao, M. Roj, Q. Song, G. Yin. 'Mean exit times and the multilevel Monte Carlo method'. SIAM Journal on Uncertainty Quantification, 1(1):2-18, 2013. link
• D.J. Higham, M. Roj. 'Computing mean first exit times for stochastic processes using multi-level Monte Carlo'. Proceedings of 2012 Winter Simulation Conference, 2013. link
• H. Hoel, E. von Schwerin, A. Szepessy, R. Tempone. 'Adaptive multilevel Monte Carlo simulation'. Numerical Analysis of Multiscale Computations, 82:217-234, 2012. link
• H. Hoel, E. von Schwerin, A. Szepessy, R. Tempone. 'Implementation and analysis of an adaptive multilevel Monte Carlo algorithm'. Monte Carlo Methods and Applications, 20(1):1-41, 2014. link
• M. Hutzenthaler, A. Jentzen, P. Kloeden. 'Divergence of the multilevel Monte Carlo method for nonlinear stochastic differential equations'. Annals of Applied Probability, 23(5):1913-1966, 2013. link
• O. Iliev, T. Nagapetyan, K. Ritter. 'Monte Carlo simulation of asymmetric flow field flow fractionation'. pp.115-123 in Monte Carlo Methods and Applications: Proceedings of the 8th IMACS Seminar on Monte Carlo methods, de Gruyter, 2013.
• B. Jourdain, A. Kebaier. 'Non-asymptotic error bounds for the multilevel Monte Carlo Euler method applied to SDEs with constant diffusion coefficient'. Electronic Journal of Probability, 24,12, 2019. link
• G. Katsiolides, E.H. Müller, R. Scheichl, T. Shardlow, M.B. Giles, D.J. Thomson. 'Multilevel Monte Carlo and improved timestepping methods in atmospheric dispersion modelling'. Journal of Computational Physics, 354(1):320-343, 2018. link
• S. Mancini, F. Bernal, J.A. Acebron. 'An efficient algorithm for accelerating Monte Carlo approximations of the solution to boundary value problems'. Journal of Scientific Computing, 66(2):577-597, 2016. link
• C. Mbaye, G. Pagès, F. Vrins. 'An antithetic approach of multilevel Richardson-Romberg extrapolation estimator for multidimensional SDES'. In Numerical Analysis and Its Applications (NAA 2016), Lecture Notes in Computer Science, vol 10187. Springer, 2017. link
• E.H. Müller, R. Scheichl, T. Shardlow. 'Improving multilevel Monte Carlo for stochastic differential equations with application to the Langevin equation'. Royal Society Proceedings A, 2015. link
• A. Neuenkirch, L. Szpruch. 'First order strong approximations of scalar SDEs defined in a domain'. Numerische Mathematik, 128(1):103-136, 2014. link
• G. Pagès, F. Panloup. 'Weighted multilevel Langevin simulation of invariant measures'. Annals of Applied Probability, 28(6):3358-3417, 2018. link
• S. Pauli, R.N. Gantner, P. Arbenz, A. Adelmann. 'Multilevel Monte Carlo for the Feynman-Kac formula for the Laplace equation'. BIT Numerical Mathematics, 55(4):1125-1143, 2015. link
• T. Primozic. 'Estimating expected first passage times using multilevel Monte Carlo algorithm'. MSc dissertation, 2011. link
• C.-H. Rhee, P.W. Glynn. 'Unbiased estimation with square root convergence for SDE models'. Operations Research, 63(5):1026-1043, 2015. link
• J. Su, C. Hou, Y. Ma, Y. Wang. 'Multilevel Monte Carlo method for the Brownian configuration field of polymer fluids'. AIP Advances, 10:095013, 2020. link

Lévy processes and Lévy-driven SDEs

• M.B. Alaya, K. Hajji, A. Kebaier. 'Importance sampling and statistical Romberg method for Lévy processes'. Stochastic Processes and their Applications, 126(7):1901-1931, 2016. link
• D. Belomestny, T. Nagapetyan. 'Multilevel path simulation for weak approximation schemes with application to Lévy-driven SDEs'. Bernoulli, 23(2):927-950, 2017. link
• J.I.G. Cázares , A. Mijatovic, G.U. Bravo. 'Geometrically Convergent Simulation of the Extrema of Lévy Processes'. Mathematics of Operations Research, November 2021. link
• J.I.G. Cázares, A. Mijatovic. 'Simulation of the drawdown and its duration in Lévy models via stick-breaking Gaussian approximation'. Finance and Stochastics, 26(4):671-732, 2022. link
• J.I.G. Cázares, A. Mijatovic. 'A Monte Carlo algorithm for the extrema of tempered stable processes'. Advances in Applied Probability, 55(4):1362-1389, 2023. link
• D.-M. Dang. 'A multi-level dimension reduction Monte-Carlo method for jump-diffusion models'. Journal of Computational and Applied Mathematics, 324:49-71, 2017. link
• S. Dereich. 'Multilevel Monte Carlo algorithms for Lévy-driven SDEs with Gaussian correction'. Annals of Applied Probability, 21(1):283-311, 2011. link
• S. Dereich, F. Heidenreich. 'A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations'. Stochastic Processes and their Applications, 121(7):1565-1587, 2011. link
• S. Dereich, S. Li. 'Multilevel Monte Carlo for Lévy-driven SDEs: Central limit theorems for adaptive Euler schemes'. Annals of Applied Probability, 26(1):136-185, 2016. link
• S. Dereich, S. Li. 'Multilevel Monte Carlo Implementation for SDEs Driven by Truncated Stable Processes'. Monte Carlo and Quasi-Monte Carlo Methods, 2016 link
• A. Ferreiro-Castilla, A.E. Kyprianou, R. Scheichl, G. Suryanarayana. 'Multi-level Monte-Carlo simulation for Lévy processes based on the Wiener-Hopf factorisation'. Stochastic Processes and their Applications, 124(2):985-1010, 2014. link
• A. Ferreiro-Castilla, K. van Schaik. 'Applying the Wiener-Hopf Monte Carlo simulation technique for Lévy processes to path functionals'. Journal of Applied Probability, 52(1):129-148, 2015. link
• M.B. Giles, Y. Xia. 'Multilevel Monte Carlo For Exponential Lévy Models'. Finance and Stochastics, 21(4):995-1026, 2017. link
• H. Marxen. 'The Multilevel Monte Carlo method used on a Lévy driven SDE'. Monte Carlo Methods and Applications, 16(2):167-190, 2010. link
• A. Mijatovic, R. Palfray. 'A weak MLMC scheme for Lévy-copula-driven SDEs with applications to the pricing of credit, equity and interest rate derivatives'. Applied Mathematical Finance, online Sept 2024. link
Received 26 Oct 2022, Accepted 31 Aug 2024, Published online: 23 Sep 2024
• M. Mohamed. 'Solving some stochastic partial differential equations driven by Lévy noise using two SDEs' Stochastics, 2022. link
• Y. Xia, M.B. Giles. `Multilevel path simulation for jump-diffusion SDEs'. pp.695-708 in Monte Carlo and Quasi-Monte Carlo Methods 2010, Springer, 2012. link
• Y. Xia. `Multilevel Monte Carlo for jump processes'. PhD thesis, University of Oxford, 2014. link

Complexity analysis

• J. Baldeaux, M. Gnewuch. 'Optimal randomized multilevel algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition'. SIAM Journal of Numerical Analysis, 52(3):1128-1155, 2014. link
• J. Creutzig, S. Dereich, T. Müller-Gronbach, K. Ritter. 'Infinite-dimensional quadrature and approximation of distributions'. Foundations of Computational Mathematics, 9(4):391-429, 2009. link
• T. Daun, S. Heinrich. 'Complexity of Banach space valued and parametric integration'. pp.297-316 in Monte Carlo and Quasi-Monte Carlo Methods 2012, Springer, 2013. link
• T. Daun, S. Heinrich. 'Complexity of parametric initial value problems in Banach spaces'. Journal of Complexity, 30(4):392-429, 2014. link
• T. Daun, S. Heinrich. 'Complexity of parametric integration in various smoothness classes'. Journal of Complexity, 30(6):750-766, 2014. link
• T. Daun, S. Heinrich. 'Complexity of Banach space valued and parametric stochastic Ito integration'. Journal of Complexity, 40:100-122, 2017. link
• J. Dick, M. Gnewuch. 'Infinite-dimensional integration in weighted Hilbert spaces: anchored decompositions, optimal deterministic algorithms, and higher order convergence'. Foundations of Computational Mathematics, 14(5):1027-1077, 2014. link
• J. Dick, M. Gnewuch. 'Optimal randomized changing dimension algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition'. Journal of Approximation Theory, 184:111-145, 2014. link
• M.B. Giles, M. Hefter, L. Mayer, K. Ritter 'Random bit quadrature and approximation of distributions on Hilbert spaces'. Foundations of Computational Mathematics, 19(1):205-238, 2019. link
• M.B. Giles, M. Hefter, L. Mayer, K. Ritter. 'Random bit multilevel algorithms for stochastic differential equations'. Journal of Complexity, 54:101395, 2019. link
• M.B. Giles, M. Hefter, L. Mayer, K. Ritter. 'An adaptive random bit multilevel algorithm for SDEs'. in Multivariate Algorithms and Information-Based Complexity, De Gruyter, 2020. link
• M. Gnewuch. 'Infinite-dimensional integration on weighted Hilbert spaces'. Mathematics of Computation, 81(280):2175-2205, 2012. link
• M. Gnewuch. 'Lower error bounds for randomized multilevel and changing dimension algorithms'. pp.399-415 in Monte Carlo and Quasi-Monte Carlo Methods 2012, Springer, 2013. link
• F.J. Hickernell, T. Müller-Gronbach, B. Niu, K. Ritter. 'Multi-level Monte Carlo algorithms for infinite-dimensional integration on ℝ^ℕ'. Journal of Complexity, 26(3):229-254, 2010. link
• M. Hutzenthaler, A. Jentzen, T. Kruse. 'Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities'. Foundations of Computational Mathematics, 2021. link
• T. Müller-Gronbach, K. Ritter. 'Variable subspace sampling and multi-level algorithms'. pp.131-156 in Monte Carlo and Quasi-Monte Carlo Methods 2008, Springer, 2009. link
• B. Niu, F.J. Hickernell, T. Müller-Gronbach and K. Ritter. 'Deterministic multi-level algorithms for infinite-dimensional integration on ℝ^ℕ'. Journal of Complexity, 27(3-4):331-351, 2010. link
• B. Niu. 'Monte Carlo simulation of infinite-dimensional integrals'. PhD thesis, Illinois Institute of Technology, 2011. link

Elliptic SPDEs

• A. Abdulle, A. Barth, Ch. Schwab. 'Multilevel Monte Carlo methods for stochastic elliptic multiscale PDEs'. Multiscale Modeling and Simulation, 11(4):1033-1070, 2013. link
• S. Badia, J. Hampton, J. Principe. 'Embedded multilevel monte carlo for uncertainty quantification in random domains'. International Journal for Uncertainty Quantification, 11(1):119-142, 2021. link
• A. Barth, Ch. Schwab, N. Zollinger. 'Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients'. Numerische Mathematik, 119(1):123-161, 2011. link
• J. Beck, Y. Liu, E. von Schwerin, R. Tempone. 'Goal-oriented adaptive finite element multilevel Monte Carlo with convergence rates'. Computer Methods in Applied Mechanics and Engineering, 4021:115582, 2022. link
• A. Bespalov, D. Praetorius, M. Ruggeri. 'Convergence and rate optimality of adaptive multilevel stochastic Galerkin FEM'. IMA Journal of Numerical Analysis, 2021. link
• C. Bierig, A. Chernov. 'Convergence analysis of multilevel Monte Carlo variance estimators and application for random obstacle problems'. Numerische Mathematik, 130(4):579-613, 2015. link
• N.K. Chada, H. Hoel, A. Jasra, G.E. Zouraris. 'Improved efficiency of multilevel Monte Carlo for stochastic PDE through strong pairwise coupling'. Journal of Scientific Computing, 93(3):62, 2022. link
• J. Charrier, R. Scheichl, A. Teckentrup. 'Finite element error analysis of elliptic PDEs with random coefficients and its application to multilevel Monte Carlo methods'. SIAM Journal on Numerical Analysis, 51(1):322-352, 2013. link
• K.A. Cliffe, M.B. Giles, R. Scheichl, A.L. Teckentrup. 'Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients'. Computing and Visualization in Science, 14(1):3-15, 2011. link
• D. Crisan, P. Del Moral, J. Houssineau, A. Jasra. 'Unbiased multi-index Monte Carlo'. Stochastic Analysis and Applications, 36(2):257-273, 2018. link
• M. Croci. 'Multilevel Monte Carlo methods for uncertainty quantification in brain simulations'. PhD thesis, University of Oxford, 2021 link
• M. Croci, M.B. Giles, M.E. Rognes, P.E. Farrell. 'Efficient white noise sampling and coupling for multilevel Monte Carlo with non-nested meshes'. SIAM/ASA Journal on Uncertainty Quantification, 6(4):1454-1474, 2018. link
• M. Croci, M.B. Giles, P.E. Farrell. 'Multilevel quasi Monte Carlo methods for elliptic PDEs with random field coefficients via fast white noise sampling'. SIAM Journal on Scientific Computing, 43(4), A2840-A2868, 2021. link
• M. Croci, V. Vinje, M.E. Rognes. 'Fast uncertainty quantification of tracer distribution in the brain interstitial fluid with multilevel and quasi Monte Carlo'. International Journal for Numerical Methods in Biomedical Engineering, 37(1),e3412, 2021. link
• G. Detommaso, T. Dodwell, R. Scheichl. 'Continuous level Monte Carlo and sample-adaptive model hierarchies' SIAM Journal on Uncertainty Quantification, 7(1):93-116, 2019. link
• J. Dick, F.Y. Kuo, Q.T. Le Gia, and Ch. Schwab. 'Multilevel higher order QMC Petrov-Galerkin discretization for affine parametric operator equations'. SIAM Journal on Numerical Analysis, 54(4):2541-2568, 2016. link
• J. Dick, M. Feischl, Ch. Schwab. 'Improved efficiency of a multi-index FEM for computational uncertainty quantification'. SIAM Journal on Numerical Analysis, 57(4):1744-1769, 2019. link
• Y. Efendiev, C. Kronsbein, F. Legoli. 'Multi-level Monte Carlo approaches for numerical homogenization'. Multiscale Modeling & Simulation, 13(4):1107-1135, 2015. link
• M. Eigel, C. Merdon, J. Neumann. 'An adaptive multilevel Monte Carlo method with stochastic bounds for quantities of interest with uncertain data'. SIAM/ASA Journal on Uncertainty Quantification, 4(1):1219-1245, 2016. link
• F. Fagerlund, F. Hellman, A. Malqvist, A. Niemi. 'Multilevel Monte Carlo methods for computing failure probability of porous media flow systems'. Advances in Water Resources, 94:498-509, 2016. link
• H.R. Fairbanks, S. Osborn, P.S. Vassilevski. 'Estimating posterior quantity of interest expectations in a multilevel scalable framework'. Numerical Linear Algebra with Applications, 28(3):e2352, 2021. link
• A. Galetzka, Z. Bontinck, U. Romer, S. Schops. 'Multilevel Monte Carlo simulation of the eddy current problem with random parameters'. IEEE Symposium on Applied Computational Electromagnetics (ACES), 2017. link
• A. Galetzka, Z. Bontinck, U. Romer, S. Schops. 'A multilevel Monte Carlo method for high-dimensional uncertainty quantification of low-frequency electromagnetic devices'. IEEE Transactions on Magnetics, 55(8):8705673, 2019. link
• I.G. Graham, R. Scheichl, E. Ullmann. 'Mixed finite element analysis of lognormal diffusion and multilevel Monte Carlo methods'. Stochastic Partial Differential Equations: Analysis and Computations, June 2015. link
• H. Harbrecht, M. Peters, and M. Siebenmorgen. 'On multilevel quadrature for elliptic stochastic partial differential equations'. pp.161-179 in Sparse Grids and Applications, Lecture Notes in Computational Science and Engineering, Volume 88, 2013. link
• H. Harbrecht, M. Peters, and M. Siebenmorgen. 'Multilevel Accelerated Quadrature for PDEs with Log-Normally Distributed Diffusion Coefficient'. SIAM/ASA Journal on Uncertainty Quantification, 4(1):520-551, 2016. link
• L. Herrmann. 'Strong convergence analysis of iterative solvers for random operator equations'. Calcolo, 56(4):46, 2019. link
• M. Icardi, G. Boccardo, R. Tempone. 'On the predictivity of pore-scale simulations: estimating uncertainties with multilevel Monte Carlo'. Advances in Water Resources, 95:46-60, 2016. link
• O. Iliev, J. Mohring, N. Shegunov. 'Renormalization based MLMC method for scalar elliptic SPDE'. Lecture Notes in Computer Science, 10665:295-303, 2018. link
• R. Kornhuber, Ch. Schwab, M.-W. Wolf. 'Multilevel Monte Carlo finite element methods for stochastic elliptic variational inequalities'. SIAM Journal on Numerical Analysis, 52(3):1243-1268, 2014. link
• C. Kronsbein. 'On selected efficient numerical methods for multiscale problems with stochastic coefficients'. PhD thesis, TU Kaiserslautern, 2012. link
• P. Kumar, C.W. Oosterlee, R.P. Dwight. 'A multigrid multilevel Monte Carlo method using high-order finite-volume scheme for lognormal diffusion problems'. International Journal for Uncertainty Quantification, 2017. link
• P. Kumar, C. Rodrigo, F.J. Gaspar, C.W. Oosterlee. 'A parametric acceleration of multilevel Monte Carlo convergence for nonlinear variably saturated flow'. Computational Geosciences, 24:311-331, 2020. link
• F.Y. Kuo, D. Nuyens. 'Application of Quasi-Monte Carlo methods to elliptic PDEs with random diffusion coefficients: a survey of analysis and implementation'. Foundations of Computational Mathematics, 16(6):1631-1696, 2016. link
• F.Y. Kuo, R. Scheichl, Ch. Schwab, I. Sloan, E. Ullmann. 'Multi-level quasi-Monte Carlo methods for lognormal diffusion problems'. Mathematics of Computation, 86:2827-2860, 2017. link
• F.Y. Kuo, Ch. Schwab, I. Sloan. 'Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients'. Foundations of Computational Mathematics, 15(2):411-449, 2015. link
• J. Lang, R. Scheichl, D. Silvester. 'A fully adaptive multilevel stochastic collocation strategy for solving elliptic PDEs with random data'. Journal of Computational Physics, 419:, 2020. link
• M. Li, X. Luo. 'An MLMCE-HDG method for the convection diffusion equation with random diffusivity'. Computers and Mathematics with Applications, 127:127-143, 2022. link
• X. Li, J. Liu, S. Xu. 'A multilevel approach towards unbiased sampling of random elliptic partial differential equations'. Advances in Applied Probability, 50(4):1007-1031, 2018. link
• J. Martínek. 'Mixed precision in uncertainty quantification methods'. Master's thesis, Charles University, 2023. link
• R. Merkle, A. Barth. 'Multilevel Monte Carlo estimators for elliptic PDEs with Lévy-type diffusion coefficient'. BIT Numerical Mathematics, 2022. link
• J. Mohring, R. Milk, A. Ngo, O. Klein, O. Iliev, M. Ohlberger, P. Bastian. 'Uncertainty quantification for porous media flow using multilevel Monte Carlo'. Large-Scale Scientific Computing, LNCS 9374, pp.145-152, 2015. link
• F. Nobile, F. Tesei. 'A multilevel Monte Carlo method with control variate for elliptic PDEs with log-normal coefficients'. Stochastic Partial Differential Equations: Analysis and Computations, 3(3):398-444, 2015. link
• M. Neumüller and A. Thalhammer. 'A fully parallelizable space-time multilevel Monte Carlo method for stochastic differential equations with additive noise'. SIAM Journal on Scientific Computing, 40(3):C388-C400, 2018. link
• S. Osborn, P. Zulian, T. Benson, U. Villa, R. Krause, P.S. Vassilevski. 'Scalable hierarchical PDE sampler for generating spatially correlated random fields using nonmatching meshes'. Numerical Linear Algebra with Applications, 25(3):e2146, 2018. link
• V. Rey, S. Krumscheid, F. Nobile. 'Quantifying uncertainties in contact mechanics of rough surfaces using the multilevel Monte Carlo method'. International Journal of Engineering Science, 138:50-64, 2019. link
• P. Robbe, D. Nuyens, S. Vandewalle. 'A dimension-adaptive Multi-Index Monte Carlo method applied to a model of a heat exchanger'. pp.429-445 in Monte Carlo and Quasi-Monte Carlo Methods, 2016, Springer, 2018. link
• P. Robbe, D. Nuyens, S. Vandewalle. 'Recycling samples in the multigrid multilevel (quasi-)Monte Carlo method'. SIAM Journal on Scientific Computing, 41(5):S37-S60, 2019. link
• P. Robbe. 'Multilevel uncertainty quantification methods for robust design of industrial applications'. PhD thesis, KU Leuven, 2019. link
• M. Spetlik, J. Brezina. 'Groundwater contaminant transport solved by Monte Carlo methods accelerated by Deep Learning meta-model' Applied Sciences, 12,7382, 2022. link
• M. Spetlik, J. Brezina, E. Laloy. 'Deep learning surrogate for predicting hydraulic conductivity tensors from stochastic discrete fracture-matrix models'. Computational Geosciences, 2024. link
• A.L. Teckentrup, R. Scheichl, M.B. Giles, E. Ullmann. 'Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients'. Numerische Mathematik, 125(3):569-600, 2013. link
• A.L. Teckentrup. 'Multilevel Monte Carlo methods for highly heterogeneous media'. Proceedings of 2012 Winter Simulation Conference. link
• A.L. Teckentrup. 'Multilevel Monte Carlo methods and uncertainty quantification'. PhD thesis, University of Bath, 2013. link
• F. Vidal-Codina, N.-C. Nguyen, M.B. Giles, J. Peraire. 'A model and variance reduction method for computing statistical outputs of stochastic elliptic partial differential equations'. Journal of Computational Physics, 297:700-720, 2015. link

Other SPDEs

• A. Alkhatib, M. Babael. 'Applying the multilevel Monte Carlo method for heterogeneity-induced uncertainty quantification of surfactant/polymer flooding'. SPE Journal, 21(4), 2016. link
• M. Ballesio, J. Beck, A. Pandey, L. Parisi, E. von Schwerin, R. Tempone. 'Multilevel Monte Carlo acceleration of seismic wave propagation under uncertainty'. International Journal on Geomathematics, 10(1):22, 2019. link
• J. Badwaik, C. Klingenberg, N.H. Risebro, A.M. Ruf. 'Multilevel Monte Carlo finite volume methods for random conservation laws with discontinuous flux' ESAIM: Mathematical Modelling and Numerical Analysis, 55(3):1039-1065, 2021. link
• A. Barth, A. Lang. 'Multilevel Monte Carlo method with applications to stochastic partial differential equations'. Int Journal of Computer Mathematics, 89(18):2479-2498, 2012. link
• A. Barth, A. Lang, Ch. Schwab. 'Multilevel Monte Carlo method for parabolic stochastic partial differential equations'. BIT Numerical Mathematics, 53(1):3-27, 2013. link
• N. Baumgarten, C. Wieners. 'The parallel finite element system M++ with integrated multilevel preconditioning and multilevel Monte Carlo methods' Computers and Mathematics with Applications, 81:391-406, 2021. link
• N. Baumgarten, S. Krumscheid, C. Wieners. 'A fully parallelized and budgeted multilevel Monte Carlo method and the application to acoustic waves' SIAM/ASA Journal on Uncertainty Quantification, 12(3):23M1588354, 2024. link
• S. Ben Bader, P. Benedusia, A. Quaglino, P. Zulian, R. Krause. 'Space-time multilevel Monte Carlo methods and their application to cardiac electrophysiology'. Journal of Computational Physics, 433:110164, 2021. link
• S. Ben Bader, H. Harbrecht, R.Kraus, M.L. Multerer, A. Quaglino, M. Schmidlin. 'Space-time multilevel quadrature methods and their application for cardiac electrophysiology'. SIAM-ASA Journal on Uncertainty Quantifications, 11(4):1329-1356, 2024. link
• P. Blondeel, P. Robbe, C.V. Hoorickx, S. François, G. Lombaert, S. Vandewalle. 'p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element methods with applications in civil engineering' Algorithms, 13(5):110, 2020. link
• G. Bugeda, J. Pons-Prats. 'Multilevel Monte-Carlo methods applied to the stochastic analysis of aerodynamic problems'. Congresso de Métodos Numéricos em Engenharia 2015. link
• K. Bujok, C. Reisinger. 'Numerical valuation of basket credit derivatives in structural jump-diffusion models'. Journal of Computational Finance, 15(4):115-158, 2012. link
• R. Butler, T.J. Dodwell, T. Kim, S. Kynaston, R. Scheichl, R.T. Haftka, N.H. Kim. 'Uncertainty quantification of composite structures with defects using multilevel Monte Carlo simulations'. Proceedings of the 17th AIAA Non-Deterministic Approaches Conference, 2014. link
• Q. Chen, J. Ming. 'The multilevel Monte Carlo method for simulations of turbulent flows'. Monthly Weather Review, 146(9):2933-2947, 2018. link
• T. Cui, H. De Sterck, A.D. Gilbert, S. Polishchuk, R. Scheichl. 'Multilevel Monte Carlo methods for stochastic convection–diffusion eigenvalue problems'. Journal of Scientific Computing, 99(3):77, 2024. link
• T.J. Dodwell, S. Kynaston, R. Butler, R.T. Haftka, N.H. Kim, R. Scheichl. 'Multilevel Monte Carlo simulations of composite structures with uncertain manufacturing defects'. Probabilistic Engineering Mechanics, 63:103116, 2021. link
• J. Dölz, H. Harbrecht, C. Jerez-Hanckes, M. Multerer. 'Isogeometric multilevel quadrature for forward and inverse random acoustic scattering'. Computer Methods in Applied Mechanics and Engineering, 388:114242, 2022. link
• Weinan E, M. Hutzenthaler, A. Jentzen, T. Kruse. 'On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations'. Journal of Scientific Computing, 79(3):1534-1571, 2019. link
• Weinan E, M. Hutzenthaler, A. Jentzen, T. Kruse. 'Multilevel Picard iterations for solving smooth semilinear parabolic heat equations'. Partial Differential Equations and Applications, 2(6):80, 2021. link
• Y. Efendiev, O. Iliev, C. Kronsbein. 'Multi-level Monte Carlo methods using ensemble level mixed MsFEM for two-phase flow and transport simulations'. Computational Geosciences, 17(5):833-850, 2013. link
• H.C. Elman, J. Liang, T. Sànchez-Vizuet. 'Multilevel Monte Carlo methods for the Grad-Shafranov free boundary problem' Computer Physics Communications, 298:109099, 2024. link
• H. Fairbanks, A. Doostan, C. Ketelsen, G. Iaccarino. 'A low-rank control variate for multilevel Monte Carlo simulation of high-dimensional uncertain systems'. Journal of Computational Physics, 341:121-139, 2017. link
• M. Ganesh, B.C. Reyes, A. Purkayastha. 'An FEM-MLMC algorithm for a moving shutter diffraction in time stochastic model'. Discrete and Continuous Dynamical Systems - Series B, 24(1):257-272, 2019. link
• M. Ganesh, B. Reyes. 'An efficient multi-level high-order algorithm for simulation of a class of Allen-Cahn stochastic systems'. Journal of Computational and Applied Mathematics, 401:113765, 2022. link
• M.B. Giles. 'Multilevel Monte Carlo simulation'. 2009. link
• M.B. Giles, C. Reisinger. 'Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance'. SIAM Journal of Financial Mathematics, 3(1):572-592, 2012. link
• C.J. Gittelson, J. Konno, Ch. Schwab and R. Stenberg. 'The multi-level Monte Carlo finite element method for a stochastic Brinkman problem'. Numerische Mathematik, 125:347-386, 2013. link
• C.J. Gittelson. 'Convergence rates of multilevel and sparse tensor approximations for a random elliptic PDE'. SIAM Journal on Numerical Analysis, 51(4):2426-2447, 2013. link
• S. Graubner. 'Multi-level Monte Carlo Methoden für stochastische partielle Differentialgleichungen'. Diplomarbeit, TU Darmstadt, 2008. link
• K.M. Hamdia, H. Ghasem, X. Zhuang, T. Rabczuk. 'Multilevel Monte Carlo method for topology optimization of flexoelectric composites with uncertain material properties'. Engineering Analysis with Boundary Elements, 134:412-418, 2022. link
• Y. Hao, X. Wang, K. Zhang. 'Multi-level Monte Carlo weak Galerkin method with nested meshes for stochastic Brinkman problem'. Journal of Computational and Applied Mathematics, 330:214-227, 2018. link
• J. Hu, S. Jin, J. Li, L. Zhang. 'On multilevel Monte Carlo methods for deterministic and uncertain hyperbolic systems'. Journal of Computational Physics, 47515:111847, 2023. link
• M. Hutzenthaler, T. Kruse. 'Multilevel Picard approximations of high-dimensional semilinear parabolic differential equations with gradient-dependent nonlinearities'. SIAM Journal on Numerical Analysis, 58(2):929-961, 2020. link
• M. Hutzenthaler, T.A. Nguyen. 'Multilevel Picard approximations of high-dimensional semilinear parabolic differential equations with locally monotone coefficient functions'. Applied Numerical Mathematics 181:151-175, 2022. link
• V. Keshavarzzadeh, R.M. Kirby, A. Narayan. 'Multilevel designed quadrature for partial differential equations with random inputs'. SIAM Journal on Scientific Computing, 43(2):A1412-A1440, 2021. link
• A. Khodadadian, L. Taghizadeh, C. Heitzinger. 'Three-dimensional optimal multi-level Monte-Carlo approximation of the stochastic drift-diffusion-Poisson system in nanoscale devices'. Journal of Computational Electronics, 17(1):76-89, 2018. link
• A. Khodadadian, M. Parvizi, M. Abbaszadeh, M. Dehghan, C. Heitzinger. 'A multilevel Monte Carlo finite element method for the stochastic Cahn-Hilliard-Cook equation'. Computational Mechanics, 64:937-949, 2019. link
• G. Kimaev, L.A. Ricardez-Sandoval. 'Multilevel Monte Carlo applied to chemical engineering systems subject to uncertainty'. AIChE Journal, 64(5):1651-1661, 2018. link
• G. Kimaev, L. Ricardez-Sandoval. 'Multilevel Monte Carlo for noise estimation in stochastic multiscale systems'. Chemical Engineering Research and Design, 140:33-43, 2018. link
• G. Kimaev, D. Chaffart, L.A. Ricardez-Sandoval. 'Multilevel Monte Carlo applied for uncertainty quantification in stochastic multiscale systems'. AIChE Journal, 66(8),e16262, 2020. link
• P. Kumar, M. Schmelzer, R.P. Dwight. 'Stochastic turbulence modeling in RANS simulations via multilevel Monte Carlo' Computers and Fluids, 201:104420, 2020. link
• A. Lang, A. Petersson. 'Monte Carlo versus multilevel Monte Carlo in weak error simulations of SPDE approximations'. Mathematics and Computers in Simulation, 143:99-113, 2018. link
• F. Leonardi, S. Mishra, Ch. Schwab 'Numerical approximation of statistical solutions of planar incompressible flows'. Mathematical Models and Methods in Applied Sciences, 26(13):2471-2523, 2016. link
• A. Litvinenko, A.C. Yucel, H. Bagci, J. Oppelstrup, E. Michielssen, R. Tempone. 'Computation of electromagnetic fields scattered from objects with uncertain shapes using multilevel Monte Carlo method'. IEEE Journal on Multiscale and Multiphysics Computational Techniques, 4:37-50, 2019. link
• D. Logashenko, A. Litvinenko, R. Tempone, E. Vasilyeva, G. Wittum. 'Uncertainty quantification in the Henry problem using the multilevel Monte Carlo method'. Journal of Computational Physics, 503:112854, 2024. link
• Y. Luo, Z. Wang. 'A multilevel Monte Carlo ensemble scheme for random parabolic PDEs'. SIAM Journal on Scientific Computing, 41(1):A622-A642, 2019. link
• S. Mishra, Ch. Schwab, J. Sukys. 'Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions'. Journal of Computational Physics, 231(8):3365-3388, 2012. link
• S. Mishra, Ch. Schwab, J. Sukys. 'Multi-level Monte Carlo finite volume methods for shallow water equations with uncertain topography in multi-dimensions'. SIAM Journal on Scientific Computing, 34(6):761-784, 2012. link
• S. Mishra, Ch. Schwab, J. Sukys. 'Multi-level Monte Carlo finite volume methods for uncertainty quantification of acoustic wave propagation in random heterogeneous layered medium'. Journal of Computational Physics, 312(5):192-217, 2016. link
• N. Miska, D. Balzani. 'Quantification of uncertain macroscopic material properties resulting from variations of microstructure morphology based on statistically similar volume elements: application to dual-phase steel microstructures'. Computational Mechanics, 64(6):1621-1637, 2019. link
• F. Müller, P. Jenny, D.W. Meyer. 'Multilevel Monte Carlo for two phase flow and Buckley-Leverett transport in random heterogeneous porous media'. Journal of Computational Physics, 250:685-702, 2013. link
• F. Müller, P. Jenny, D.W. Meyer. 'Parallel multilevel Monte Carlo for two phase flow and transport in random heterogeneous porous media With sampling-error and discretization-error balancing'. SPE Journal, 21(6):2027-2037, 2016. link
• F. Müller, D.W. Meyer, P. Jenny. 'Solver-based vs. grid-based multilevel Monte Carlo for two phase flow and transport in random heterogeneous porous media'. Journal of Computational Physics, 268:39-50, 2014. link
• M. Pisaroni, F. Nobile, P. Leyland. 'A Continuation Multi Level Monte Carlo (C-MLMC) method for uncertainty quantification in compressible inviscid aerodynamics'. Computer Methods in Applied Mechanics, 326:20-50, 2017. link
• M. Pisaroni, F. Nobile, P. Leyland. 'Continuation multilevel Monte Carlo'. Uncertainty Management for Robust Industrial Design in Aeronautics, pp.305-325, 2019. link
• M. Pisaroni, F. Nobile, P. Leyland. 'Continuation multilevel Monte Carlo evolutionary algorithm for robust aerodynamic shape design'. Journal of Aircraft, 56(2):771-786, 2019. link
• C. Reisinger, Z. Wang. 'Analysis of Multi-Index Mone Carlo estimators for a Zakai SPDE'. Journal of Computational Mathematics, 36(2):202-236, 2018.
• C. Reisinger and Z. Wang. 'Analysis of sparse grid multilevel estimators for multi-dimensional Zakai equations'. Sparse Grids and Applications, 205-228, Springer Lectures Notes in Computational Science and Engineering, 2021. link
• H.M. Ruzayqat, A. Jasra. 'Unbiased estimation of the solution to Zakai's equation'. Monte Carlo Methods and Applications, 26(2):113-129, 2020. link
• C. Sanchez-Linares, M. de la Asuncion, M. Castro, J. M. Gonzalez-Vida, J. Macias, S. Mishra. 'Uncertainty quantification in tsunami modeling using multi-level Monte Carlo finite volume method'. Journal of Mathematics in Industry, 6:5, 2016. link
• L. Taghizadeh, A. Khodadadian, C. Heitzinger. 'The optimal multilevel Monte-Carlo approximation of the stochastic drift-diffusion-Poisson system'. Computer Methods in Applied Mechanics and Engineering, 318:739-761, 2017. link
• S. Taverniers, S.B.M. Bosma, D.M. Tartakovsky. 'Accelerated multilevel Monte Carlo with kernel-based smoothing and latinized stratification'. Water Researces Research, 56(9):e2019WR026984, 2020. link
• F. Vidal-Codina, N.-C. Nguyen, M.B. Giles, J. Peraire. 'An empirical interpolation and model-variance reduction method for computing statistical outputs of parametrized stochastic partial differential equations'. SIAM/ASA Journal on Uncertainty Quantification, 4(1):244-265, 2016. link
• F. Vidal-Codina, J. Saa-Seoane, N.-C. Nguyen, J. Peraire, J. 'A multiscale continuous Galerkin method for stochastic simulation and robust design of photonic crystals'. Journal of Computational Physics: X, 2:100016, 2019. link
• Z. Yang, J. Ming, C. Qiu, M. Li, X. He. 'A multigrid multilevel Monte Carlo method for Stokes-Darcy model with random hydraulic conductivity and Beavers-Joseph condition'. Journal of Scientific Computing, 90(2):68, 2022. link
• J. Zech, D. Dung, Ch. Schwab. 'Multilevel approximation of parametric and stochastic PDES'. Mathematical Models and Methods in Applied Sciences, 29(9):S0218202519500349, 2019. link
• A. Zeng, J. Xue. 'Multilevel Monte Carlo method for path-dependent barrier interest rate derivatives' SIAM Journal on Financial Mathematics, 10(1):214-242, 2019. link

Multilevel QMC

• D. Crevillén-Garcia, H. Power. 'Multilevel and quasi-Monte Carlo methods for uncertainty quantification in particle travel times through random heterogeneous porous media'. Royal Society Open Science, August 2017. link
• M. Croci, M.B. Giles, P.E. Farrell. 'Multilevel quasi Monte Carlo methods for elliptic PDEs with random field coefficients via fast white noise sampling'. SIAM Journal on Scientific Computing, 43(4), A2840-A2868, 2021. link
• J. Dick, F.Y. Kuo, Q.T. Le Gia, and Ch. Schwab. 'Multilevel higher order QMC Petrov-Galerkin discretization for affine parametric operator equations'. SIAM Journal on Numerical Analysis, 54(4):2541-2568, 2016. link
• M. Ganesh, B. Reyes. 'An efficient multi-level high-order algorithm for simulation of a class of Allen-Cahn stochastic systems'. Journal of Computational and Applied Mathematics, 401:113765, 2022. link
• T. Gerstner, M. Noll. 'Randomized multilevel quasi-Monte Carlo path simulation'. In Recent Developments in Computational Finance, World Scientific / Imperial College Press, 2013. link
• A.D. Gilbert, R. Scheichl. 'Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems I: regularity and analysis'. IMA Journal of Numerical Analysis, 44(1):466-503, 2024. link
• A.D. Gilbert, R. Scheichl. 'Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems II: efficient algorithms and numerical results'. IMA Journal of Numerical Analysis, 44(1):504-535, 2024. link
• M.B. Giles, B.J. Waterhouse. 'Multilevel quasi-Monte Carlo path simulation'. pp.165-181 in Advanced Financial Modelling, in Radon Series on Computational and Applied Mathematics, de Gruyter, 2009. link
• M.B. Giles, F.Y. Kuo, I.H. Sloan. 'Combining sparse grids, multilevel MC and QMC for elliptic PDEs with random coefficients'. pp.265-281 in Monte Carlo and Quasi-Monte Carlo Methods, 2016, Springer, 2018. link
• A. Khodadadian, L. Taghizadeh, C. Heitzinger. 'Optimal multilevel randomized quasi-Monte-Carlo method for the stochastic drift-diffusion-Poisson system'. Computer Methods in Applied Mechanics and Engineering, 329:480-497, 2018. link
• F.Y. Kuo, D. Nuyens. 'Application of Quasi-Monte Carlo methods to elliptic PDEs with random diffusion coefficients: a survey of analysis and implementation'. Foundations of Computational Mathematics, 16(6):1631-1696, 2016. link
• F.Y. Kuo, Ch. Schwab, I. Sloan. 'Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients'. Foundations of Computational Mathematics, 15(2):411-449, 2015. link
• F.Y. Kuo, R. Scheichl, Ch. Schwab, I. Sloan, E. Ullmann. 'Multi-level quasi-Monte Carlo methods for lognormal diffusion problems'. Mathematics of Computation, 86:2827-2860, 2017. link
• S.A. Nagy, M.A. El-Beltagy, M. Wafa. 'Multilevel Monte Carlo by using the Halton sequence'. Monte Carlo Methods and Applications, 26(3):193-203, 2020. link
• H.J. Unwin. 'Uncertainty quantification of engineering systems using the multilevel Monte Carlo method'. PhD thesis, University of Cambridge, 2018. link

MCMC and other inverse methods

• A.A. Ali, E. Ullmann and M. Hinze. 'Multilevel Monte Carlo analysis for optimal control of elliptic PDEs with random coefficients'. SIAM/ASA Journal on Uncertainty Quantification, 5(1):466-492, 2017. link
• T. Alsup, T. Hartland, B. Peherstorfer, N. Petra, N. 'Further analysis of multilevel Stein variational gradient descent with an application to the Bayesian inference of glacier ice models'. Advances in Computational Mathematics, 50(4):65, 2024. link
• M. Ballesio, A. Jasra, E. von Schwerin, R. Tempone. 'A Wasserstein coupled particle filter for multilevel estimation'. Stochastic Analysis and Applications, 41(5):820-859, 2023. link
• J. Beck, B. Mansour Dia, L. Espath, R. Tempone. 'Multilevel double loop Monte Carlo and stochastic collocation methods with importance sampling for Bayesian optimal experimental design' International Journal for Numerical Methods in Engineering, 121(1):3482-3503, 2020. link
• A. Beskos, A. Jasra, K.J.H. Law, R. Tempone, Y. Zhou. 'Multilevel sequential Monte Carlo samplers'. Stochastic Processes and their Applications, 127(5):1417-1440, 2017. link
• A. Beskos, A. Jasra, K.J.H. Law, Y. Marzouk, Y. Zhou. 'Multilevel sequential Monte Carlo with dimension-independent likelihood-informed proposals'. SIAM/ASA Journal on Uncertainty Quantification, 6(2):762-786, 2018. link
• N.K. Chada, J. Franks, A. Jasra, K.J. Law, M. Vihola. 'Unbiased inference for discretely observed hidden Markov model diffusions'. SIAM/ASA J. Uncertainty Quantification, 9(2):763-787, 2021. link
• A. Chernov, H. Hoel, K.J.H Law, F. Nobile, R. Tempone. 'Multilevel ensemble Kalman filtering for spatio-temporal processes'. Numerische Mathematik, 147(1), 71-125, 2021.
• T.A. Cui, G. Detommaso, R. Scheichl. 'Multilevel dimension-independent likelihood-informed MCMC for large-scale inverse problems'. Inverse Problems, 40(3):035005, 2024.
• T.J. Dodwell, C. Ketelsen, R. Scheichl, A.L. Teckentrup. 'A hierarchical multilevel Markov Chain Monte Carlo algorithm with applications to uncertainty quantification in subsurface flow'. SIAM/ASA Journal on Uncertainty Quantification, 3(1):1075-1108, 2015. link
• R. Douc, P.E. Jacob, A. Lee, D. Vats. 'Solving the Poisson equation using coupled Markov chains'. arXiv pre-print, 2023 link
• X. Du, H. Wang. 'Efficient estimation of expected information gain in Bayesian experimental design with multi-index Monte Carlo'. Statistics and Computing, 34(6):200, 2024. link
• Y. Efendiev, B. Jin, M. Presho, X. Tan. 'Multilevel Markov Chain Monte Carlo method for high-contrast single-phase flow problems'. Communications in Computational Physics, 17(1):259-286, 2015. link
• H. Fairbanks, U. Villa, P.S. Vassilevski. 'Multilevel hierarchical decomposition of finite element white noise with application to multilevel Markov Chain Monte Carlo'. SIAM Journal on Scientific Computing, online June 2021. link
• K. Fossum, T. Mannseth, A.S. Stordal. 'Assessment of multilevel ensemble-based data assimilation for reservoir history matching' Computational Geosciences, 24(1):217-239, 2020. link
• R.N. Gantner, C. Schillings, Ch. Schwab. 'Binned multilevel Monte Carlo for Bayesian inverse problems with large data'. pp.511-519 in Domain Decomposition Methods in Science and Engineering XXII, Volume 104, Lecture Notes in Computational Science and Engineering, Springer, 2016. link
• T. Goda, T. Hironaka, W. Kitade, A. Foster. 'Unbiased MLMC stochastic gradient-based optimization of Bayesian experimental designs'. SIAM Journal on Scientific Computing 44(1):20M1338848, 2022. link
• A. Gregory, C.J. Cotter, S. Reich. 'Multilevel ensemble transform particle filtering'. SIAM Journal on Scientific Computing, 38(3):1317-1338, 2016. link
• A. Gregory, C.J. Cotter. 'A seamless multilevel ensemble transform particle filter'. SIAM Journal on Scientific Computing, 39(6):2684-2701, 2017. link
• D. Gruhlke. 'Convergence of Multilevel MCMC methods on path spaces'. PhD thesis, Universitat Bonn, 2014. link
• N. Guha, X. Tan. 'Multilevel approximate Bayesian approaches for flows in highly heterogeneous porous media and their applications' Journal of Computational and Applied Mathematics, 317:700-717, 2017. link
• Z. He, Z. Xu, X. Wang. 'Unbiased MLMC-based variational Bayes for likelihood-free inference'. SIAM Journal on Scientific Computing, 44(4), A1884-A1910, 2022. link
• K. Heine, D. Burrows. 'Multilevel bootstrap particle filter'. Bernoulli, 29(1):551-579, 2023. link
• V.H. Hoang, Ch. Schwab, A.M. Stuart. 'Complexity analysis of accelerated MCMC methods for Bayesian inversion'. Inverse Problems, 29(8), 2013. link
• H. Hoel, K.J.H. Law, R. Tempone. 'Multilevel ensemble Kalman filtering'. SIAM Journal of Numerical Analysis, 54(3):1813-1839, 2016. link
• P.E. Jacob, J. O'Leary, Y.F. Atchadé. 'Unbiased Markov chain Monte Carlo methods with couplings'. Journal of the Royal Statistical Society. Series B: Statistical Methodology, 82(3):543-600, 2020. link
• A. Jasra, S. Jo, D. Nott, C. Shoemaker, R. Tempone. 'Multilevel Monte Carlo in approximate Bayesian computation'. Stochastic Analysis and Applications,37(3):346-360, 2019. link
• A. Jasra, K. Kamatani, K.J.H. Law, Y. Zhou. 'A multi-index Markov chain Monte Carlo method'. International Journal for Uncertainty Quantification, 8(1):61-73, 2018. link
• A. Jasra, K. Kamatani, K.J.H. Law, Y. Zhou. 'Bayesian static parameter estimation for partially observed diffusions via multilevel Monte Carlo'. SIAM Journal on Scientific Computing, 40(2):A887-A902, 2018. link
• A. Jasra, K. Kamatani, P.P. Osei, Y. Zhou. 'Multilevel particle filters: normalizing constant estimation'. Statistics and Computing, 28(1):47-60, 2019. link
• A. Jasra, K.J.H. Law, P.P. Osei. 'Multilevel particle filters for Lévy-driven stochastic differential equations'. Statistics and Computing, 29(4):775-789, 2019. link
• A. Jasra, K.J.H. Law, D. Lu. 'Unbiased estimation of the gradient of the log-likelihood in inverse problems'. Statistics and Computing, 31(21), 2021. link
• A. Jasra, K.J.H. Law, N. Walton, S.D. Yang. 'Multi-index sequential Monte Carlo ratio estimators for Bayesian inverse problems'. Foundations of Computational Mathematics, May 2023. link
• A. Jasra, M. Maama, H. Ombao. 'Multi-index sequential Monte Carlo ratio estimators for Bayesian inverse problems'. Advances in Applied Probability, 2024. link
• K. Kirchner, Ch. Schwab. 'Monte Carlo convergence rates for kth moments in Banach spaces'. Journal of Functional Analysis, 286(3):11021, 2024. link
• J. Latz, I. Papaioannou, E. Ullmann. 'Multilevel Sequential Monte Carlo for Bayesian inverse problems'. Journal of Computational Physics, 368:154-178, 2018. link
• K. Li, D. Giles, T. Karvonen, S. Guillas, F.-X. Briol. 'Multilevel Bayesian Quadrature'. arXiv preprint, 2022. link
• D. Lu, D. Ricciuto, K. Evans. 'An efficient Bayesian data-worth analysis using a multilevel Monte Carlo method'. Advances in Water Resources, 113:223-235, 2018. link
• M.B. Lykkegaard, T.J. Dodwell, C. Fox, G. Mingas, R. Scheichl. 'Multilevel delayed acceptance MCMC'. SIAM/ASA Journal on Uncertainty Quantification, 11(1):1-30, 2023. link
• J.P. Madrigal-Cianci, F. Nobile, R. Tempone. 'Analysis of a class of multilevel Markov Chain Monte Carlo algorithms based on independent Metropolis-Hastings'. SIAM/ASA Journal on Uncertainty Quantification, 11(1):91-138, 2023. link
• T. Mannseth, K. Fossum, S.I. Aanonsen. 'Calculating Bayesian model evidence for porous-media flow using a multilevel estimator'. Journal of Computational Physics, 514:113209, 2024. link
• L. Middleton, G. Deligiannidis, A. Doucet, P.E. Jacob. 'Unbiased Markov chain Monte Carlo for intractable target distributions'. Electronic Journal of Statistics, 14(2):2842-2891, 2020. link
• P. Del Moral, A. Jasra, K.J.H. Law. 'Multilevel sequential Monte Carlo: Mean square error bounds under verifiable conditions'. Stochastic Analysis and Applications, 35(3):478-498, 2017. link
• P. Del Moral, A. Jasra, K.J.H Law, Y. Zhou. 'Multilevel sequential Monte Carlo samplers for normalizing constants'. ACM Transactions on Modelling and Computer Simulation, 27:3, 2017. link
• T.P. Prescott, R.E. Baker. 'Multifidelity approximate Bayesian computation'. SIAM-ASA Journal on Uncertainty Quantification, 8(1):114-138, 2020. link
• T.P. Prescott, D.J. Warne, R.E. Baker. 'Efficient multifidelity likelihood-free Bayesian inference with adaptive computational resource allocation'. Journal of Computational Physics, 496:112577, 2024. link
• R. Scheichl, A.M. Stuart, A. Teckentrup. 'Quasi-Monte Carlo and multilevel Monte Carlo methods for computing posterior expectations in elliptic inverse problems'. SIAM/ASA Journal on Uncertainty Quantification, 5(1):493-518, 2017. link
• G. Shaimerdenova, H. Hoel, R. Tempone. 'Multi-index ensemble Kalman filtering'. Journal of Computational Physics, 470:111561, 2022. link
• T. Wang, G. Wang. 'Unbiased Multilevel Monte Carlo methods for intractable distributions: MLMC meets MCMC'. Journal of Machine Learning Research (JMLR), 22:1-40, 2023 link
• D.J. Warne, R.E. Baker, M.J. Simpson. 'Multilevel rejection sampling for approximate Bayesian computation'. Computational Statistics & Data Analysis, 124:71-86, 2018. link
• D.J. Warne, T.P. Prescott, R.E. Baker, M.J. Simpson. 'Multifidelity multilevel Monte Carlo to accelerate approximate Bayesian parameter inference for partially observed stochastic processes'. Journal of Computational Physics, 469,111543, 2022. link
• S. Yang, V. Zankin, M. Balandat, S. Scherer, K.T. Carlberg, N. Walton, K.J.H. Law. 'Accelerating look-ahead in Bayesian optimization: multilevel Monte Carlo is all you need'. Proceedings of the 41st International Conference on Machine Learning (ICML), 2024. link

Stochastic reaction networks

• D.F. Anderson, D.J. Higham. 'Multi-level Monte Carlo for continuous time Markov chains, with applications in biochemical kinetics'. SIAM Multiscale Modelling and Simulation, 10(1):146-179, 2012. link
• D.F. Anderson, D.J. Higham, Y. Sun. 'Complexity of multilevel Monte Carlo tau-leaping'. SIAM Journal on Numerical Analysis, 52(6):3106-3127, 2014. link
• D.F. Anderson, M. Koyama. 'An asymptotic relationship between coupling methods for stochastically modeled population processes'. IMA Journal of Numerical Analysis, 35(4):1757-1778, 2014. link
• D.F. Anderson, T.G. Kurtz. Stochastic Analysis of Biochemical Systems, Springer, 2017. link
• D.F. Anderson, C. Yuan. 'Low variance couplings for stochastic models of intracellular processes with time-dependent rate functions'. Bulletin of Mathematical Biology, 81(8):2902-2930, 2019. link
• C. Ben Hammouda, A. Moraes, R. Tempone. 'Multilevel hybrid split-step implicit tau-leap'. Numerical Algorithms, 74(2):527-560, 2017. link
• C. Ben Hammouda, N. Ben Rached, R. Tempone. 'Importance sampling for a robust and efficient multilevel Monte Carlo estimator for stochastic reaction networks'. Statistics and Computing, August 2020. link
• M.B. Giles, C. Lester, J. Whittle. 'Non-nested adaptive timesteps in multilevel Monte Carlo computations'. Monte Carlo and Quasi-Monte Carlo Methods 2014, Springer, 2016. link
• C. Lester. 'Efficient simulation techniques for biochemical reaction networks'. PhD thesis, University of Oxford, 2017. link
• C. Lester, C.A. Yates, M.B. Giles, R.E. Baker. 'An adaptive multi-level simulation algorithm for stochastic biological systems'. Journal of Chemical Physics, 142(2), 2015. link
• C. Lester, R.E. Baker, M.B. Giles, C.A. Yates. 'Extending the multi-level method for the simulation of stochastic biological systems'. Bulletin of Mathematical Biology, 78(8):1640-1677, 2016. link
• C. Lester, C.A. Yates, R.E. Baker. 'Robustly simulating biochemical reaction kinetics using multi-level Monte Carlo approaches'. Journal of Computational Physics, 375:1401-1423, 2018. link
• A. Moraes, R. Tempone, P. Vilanova. 'Multilevel hybrid Chernoff tau-leap'. SIAM Multiscale Modeling & Simulation, 12(2):581-615, 2014. link
• A. Moraes, R. Tempone, P. Vilanova. 'A multilevel adaptive reaction-splitting simulation method for stochastic reaction networks'. SIAM Journal on Scientific Computing, 38(4):A2091-A2117, 2016. link
• D.J. Warne, R.E. Baker, M.J. Simpson. 'Simulation and inference algorithms for stochastic biochemical reaction networks: From basic concepts to state-of-the-art'. Journal of the Royal Society Interface, 16(151):71-86, 2018. link
• D. Wilson, R.E. Baker. 'Multi-level methods and approximating distribution functions'. AIP Advances, 6(7), 2016. link

Nested expectation

• A. Alfonsi, A. Cherchali, J.A. Infante Acevedo. 'Multilevel Monte-Carlo for computing the SCR with the standard formula and other stress tests'. Insurance: Mathematics and Economics, 100:234-260, 2021. link
• D. Belomestny, M. Ladkau, J. Schoenmakers. 'Multilevel simulation based policy iteration for optimal stopping -- convergence and complexity'. SIAM/ASA Journal on Uncertainty Quantification, 3(1):140958463, 2015. link
• F. Bourgey, S. De Marco, E. Gobet, A. Zhou. 'Multilevel Monte Carlo methods and lower-upper bounds in initial margin computations' Monte Carlo Methods and Applications, 26(2):131-161, 2020. link
• K. Bujok, C. Reisinger. 'Numerical valuation of basket credit derivatives in structural jump-diffusion models'. Journal of Computational Finance, 15(4):115-158, 2012. link
• K. Bujok, B. Hambly, C. Reisinger. 'Multilevel simulation of functionals of Bernoulli random variables with application to basket credit derivatives'. Methodology and Computing in Applied Probability, 17(3):579-604, 2015. link
• W. Fang, Z. Wang, M.B. Giles, C.H. Jackson, N.J. Welton, C. Andrieu, H. Thom. 'Multilevel and quasi Monte Carlo methods for the calculation of the expected value of partial perfect information'. Medical Decision Making, 42(2):168-181, 2022. link
• S. Ganesh, F. Nobile. 'Gradient-based optimisation of the conditional-value-at-risk using the multi-level Monte Carlo method'. Journal of Computational Physics, 495:112523, 2023. link
• M.B. Giles. `Multilevel Monte Carlo methods'. Acta Numerica, 24:259-328, Cambridge University Press, 2015. link
• M.B. Giles. 'MLMC for Nested Expectations'. pp.425-442 in Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan, Springer, 2018. link
• M.B. Giles, T. Goda. 'Decision-making under uncertainty: using MLMC for efficient estimation of EVPPI'. Statistics and Computing, 29(4):739-751, 2019. link
• M.B. Giles, A.-L. Haji-Ali. 'Multilevel nested simulation for efficient risk estimation'. SIAM/ASA Journal on Uncertainty Quantification, 7(2):497-525, 2019. link
• D. Giorgi, V. Lemaire, G. Pagès. 'Limit theorems for weighted and regular Multilevel estimators'. Monte Carlo Methods and Applications, 3(1):43-70, 2017. link
• D. Giorgi, V. Lemaire, G. Pagès. 'Weak error for nested multilevel Monte Carlo' Methodology and Computing in Applied Probability, 22(3):1325-1348, 2020. link
• T. Goda, T. Hironaka, T. Iwamoto. 'Multilevel Monte Carlo estimation of expected information gains'. Stochastic Analysis and Applications, 38(4):581-600, 2020. link
• T. Goda, T. Hironaka, W. Kitade, A. Foster. 'Unbiased MLMC stochastic gradient-based optimization of Bayesian experimental designs'. SIAM Journal on Scientific Computing 44(1):A286-A311, 2022. link
• T. Goda, D. Murakami, K. Tanaka, K. Sato. 'Decision-theoretic sensitivity analysis for reservoir development under uncertainty using multilevel quasi-Monte Carlo methods'. Computational Geosciences, 22(4):1009-1020 2018. link
• W. Gou. `Estimating Value-at-Risk using Multilevel Monte Carlo Maximum Entropy method'. MSc thesis, University of Oxford, 2016. link
• A.L. Haji-Ali, M.B. Giles. 'Sub-sampling and other considerations for efficient risk estimation in large portfolios'. Journal of Computational Finance, 26(1):113-140, 2022. link.
• T. Hironaka, M.B. Giles, T. Goda, H. Thom. 'Multilevel Monte Carlo estimation of the expected value of sample information'. SIAM/ASA Journal on Uncertainty Quantification, 8(3):1236-1259, 2020. link
• D. Sinha, S.P. Chakrabarty. 'A review of efficient Multilevel Monte Carlo algorithms for derivative pricing and risk management'. MethodsX, 10:102078, 2023. link
• Y. Syed, G. Wang. 'Optimal randomized multilevel Monte Carlo for repeatedly nested expectations'. Proceedings of Machine Learning Research, 202:33343-33364, 2023. link
• Z.H. Xu, Z.J. He, X.Q. Wang. 'Efficient risk estimation via nested multilevel quasi-Monte Carlo simulation'. Journal of Computational and Applied Mathematics, 443:115745, 2024. link

McKean-Vlasov equations

• J.H. Bao, C. Reisinger, P.P. Ren, W. Stockinger. 'Milstein schemes and antithetic multilevel Monte Carlo sampling for delay McKean-Vlasov equations and interacting particle systems'. IMA Journal of Numerical Analysis, 44(4): 2437-2479, 2024. link
• U. Botija-Munoz, C.G. Yuan. 'Multilevel Monte Carlo EM scheme for MV-SDEs with small noise'. Numerical Algebra Control and Optimizations, April 2024. link
• A.M. Dimits, B.I. Cohen, R.E. Caflisch, M.S. Rosin, L.F. Ricketson. 'Higher-order time integration of Coulomb collisions in a plasma using Langevin equations'. Journal of Computational Physics, 242:561-580, 2013. link
• A.L. Haji-Ali. 'Pedestrian flow in the mean-field limit'. MSc thesis, KAUST, 2012. link
• A.L. Haji-Ali, R. Tempone. 'Multilevel and multi-index Monte Carlo methods for the McKean-Vlasov equation'. Statistics and Computing, 28(4):923-935, 2018. link
• M. Hutzenthaler, T. Kruse, T.A. Nguyen. 'Multilevel Picard approximations for McKean-Vlasov stochastic differential equations' Journal of Mathematical Analysis and Applications, 507(11):125761, 2022. link
• L.F. Ricketson. 'Two approaches to accelerated Monte Carlo simulation of Coulomb collisions'. PhD thesis, UCLA, 2014. link
• M.S. Rosin, L.F. Ricketson, A.M. Dimits, R.E. Caflisch, B.I. Cohen. 'Multilevel Monte Carlo simulation of Coulomb collisions'. Journal of Computational Physics, 274:140-157, 2014. link
• L. Szpruch, S. Tan, A. Tse. 'Iterative multilevel particle approximation for McKean-Vlasov SDEs'. Annals of Applied Probability, 29(4):2230-2265, 2019. link
• L. Szpruch, A. Tse. 'Antithetic multilevel sampling method for nonlinear functionals of measure'. Annals of Applied Probability, 31(3): 1100-1139, 2021. link

Computer science applications

• C. Adcock, Y. Ye, L. Jofre, G. Iaccarino. 'Multilevel Monte Carlo sampling on heterogeneous computer architectures'. International Journal for Uncertainty Quantification, 10(6):575-594, 2020. link
• D. An, N. Linden, J.-P. Liu, A. Montanaro, C. Shao, J. Wang. 'Quantum-accelerated multilevel Monte Carlo methods for stochastic differential equations in mathematical finance' Quantum, 5,481, 2021. link
• S. Badia, J. Hampton, J. Principe. 'A massively parallel implementation of multilevel Monte Carlo for finite element models'. Mathematics and Computers in Simulation, 213:18-39, 2023. link
• N. Baumgarten, C. Wieners. 'The parallel finite element system M++ with integrated multilevel preconditioning and multilevel Monte Carlo methods' Computers and Mathematics with Applications, 81:391-406, 2021. link
• N. Baumgarten, S. Krumscheid, C. Wieners. 'A fully parallelized and budgeted multilevel Monte Carlo method and the application to acoustic waves' SIAM/ASA Journal on Uncertainty Quantification, 12(3):23M1588354, 2024. link
• C. Brugger, C. de Schryver, N. Wehn, S. Omland, M. Hefter, K. Ritter, A. Kostiuk, R. Korn. 'Mixed precision multilevel Monte Carlo on hybrid computing systems'. pp.215-222 in proceedings of IEEE Conference on Computational Intelligence for Financial Engineering & Economics (CIFEr), 2014. link
• G.C.T. Chow, A.H.T. Tse, Q. Jin, W. Luk, P.H.W. Leong, D.B. Thomas. 'A mixed precision Monte Carlo methodology for reconfigurable accelerator systems'. pp.57-66 in Proceedings of the ACM/SIGDA international symposium on Field Programmable Gate Arrays, 2012. link
• D. Drzisga, B. Gmeiner, U. Rüde, R. Scheichl, B. Wohlmuth. 'Scheduling massively parallel multigrid for multilevel Monte Carlo methods' SIAM Journal on Scientific Computing, 39(5):S837-S897, 2017. link
• M.B. Giles, O. Sheridan-Methven. 'Analysis of nested multilevel Monte Carlo using approximate Normal random variables'. SIAM/ASA Journal on Uncertainty Quantification, 10(1):200-226, 2022. link
• O. Iliev, N. Shegunov, P. Armyanov, A. Semerdzhiev, I. Christov. 'On parallel MLMC for stationary single phase flow problem'. LSSC 2021: Large-Scale Scientific Computing pp 464-471, 2022. link
• H.A. Lay, Z. Colgin, V. Reshniak, A.Q.M. Khaliq. 'On the implementation of multilevel Monte Carlo simulation of the stochastic volatility and interest rate model using multi-GPU clusters'. Monte Carlo Methods and Applications, 24(4):309-321, 2018. link
• L. Lin, Y. Tong. 'Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers' PRX QUANTUM, 3:010318, 2022. link
• B. Lindsey, M. Leslie, W. Luk. 'A Domain Specific Language for accelerated Multilevel Monte Carlo simulations' IEEE 27th International Conference on Application-specific Systems, Architectures and Processors, 2016. link
• J. Martínek. 'Mixed precision in uncertainty quantification methods'. Master's thesis, Charles University, 2023. link
• M. Neumüller, A. Thalhammer. 'A fully parallelizable space-time multilevel Monte Carlo method for stochastic differential equations with additive noise'. SIAM Journal on Scientific Computing, 40(3):C388-C400, 2018. link
• I.-B. Nimerenco. 'A nested Multilevel Monte Carlo framework for efficient simulations on FPGAs'. MSc thesis, University of Oxford, 2024. link
• S. Omland, M. Hefter, K. Ritter, C. Brugger, C. de Schryver, N. Wehn, A. Kostiuk. 'Exploiting mixed-precision arithmetic in a multilevel Monte Carlo approach on FPGAs'. pp.191-220 in FPGA Based Accelerators for Financial Applications, Springer, 2015. link
• S. Pauli, P. Arbenz, Ch. Schwab. 'Intrinsic fault tolerance of multilevel Monte Carlo methods'. Journal of Parallel and Distributed Computing, 84:24-36, 2015. link
• S. Pauli, P. Arbenz. 'Determining optimal multilevel Monte Carlo parameters with application to fault tolerance'. Computers & Mathematics with Applications, 70(11):2638-2651, 2015. link
• C. de Schryver, P. Torruella, N. Wehn. 'A multi-level Monte Carlo FPGA accelerator for option pricing in the Heston model'. Proceedings - Design, Automation and Test in Europe, 248-253, 2013. link
• O. Sheridan-Methven. `Nested multilevel Monte Carlo methods and a modified Euler-Maruyama scheme utilising approximate Gaussian random variables suitable for vectorised hardware and low-precisions'. PhD thesis, University of Oxford, 2021. link
• J. Sukys, S. Mishra, Ch. Schwab. 'Static load balancing for multi-level Monte Carlo finite volume solvers'. International Conference on Parallel Processing and Applied Mathematics (PPAM). pp.245-254, 2011. link
• R. Tosi, R. Amela, R.M. Badia, R. Rossi. 'A parallel dynamic asynchronous framework for uncertainty quantification by hierarchical Monte Carlo algorithms'. Journal of Scientific Computing, 89(10)1O:28, 2021. link

Density, distribution and function estimation

• Q. Ayoul-Guilmard, S. Ganesh, S. Krumscheid, F. Nobile. 'Quantifying uncertain system outputs via the multilevel Monte Carlo method -- distribution and robustness measures'. International Journal for Uncertainty Quantification, 13(5):61-98, 2023. link
• C. Bayer, C. Ben Hammouda, R. Tempone. 'Numerical smoothing and hierarchical approximations for efficient option pricing and density estimation'. arXiv preprint 2003.05708, 2020. link
• C. Bierig, A. Chernov. 'Approximation of probability density functions by the Multilevel Monte Carlo Maximum Entropy method'. Journal of Computational Physics, 314:661-681, 2016. link
• C. Bierig, A. Chernov. 'Estimation of arbitrary order central statistical moments by the multilevel Monte Carlo method'. Stochastics and PDE Analysis and Computations, 4(1):3-40, 2016. link
• J. Blanchet, Z. Liu. 'Malliavin-based multilevel Monte Carlo estimators for densities of max-stable processes'. pp.75-97 in Monte Carlo and Quasi-Monte Carlo Methods 2016, Springer, 2018. link
• M. Egéa. '(Non)-penalized multilevel methods for non-uniformly log-concave distributions'. Electronic Journal of Probability, 29:40, 2024. link
• M. Egéa, F. Panloup. 'Multilevel Langevin pathwise average for Gibbs approximation'. Mathematics of Operations Research, online, 2024. link
• M. Giles, T. Nagapetyan, K. Ritter. 'Multi-Level Monte Carlo approximation of distribution functions and densities'. SIAM/ASA Journal on Uncertainty Quantification, 3:267-295, 2015. link
• W. Gou. `Estimating Value-at-Risk using Multilevel Monte Carlo Maximum Entropy method'. MSc thesis, University of Oxford, 2016. link
• A.-L. Haji-Ali, F. Nobile, R. Tempone, S. Wolfers. 'Multilevel weighted least squares polynomial approximation'. ESIAM: Mathematical Modelling and Numerical Analysis, 54(2):649-677, 2020. link
• S. Heinrich, E. Sindambiwe. 'Monte Carlo complexity of parametric integration'. Journal of Complexity, 15(3):317-341, 1999. link
• S. Heinrich. 'Multilevel Monte Carlo methods'. Lecture Notes in Computer Science, 2179:58-67, 2001. link
• A. Kebaier, A. Kohatsu-Higa. 'An optimal control variance reduction method for density estimation'. Stochastic Processes and their Applications, 118(12):2143-2180, 2008. link
• S. Krumscheid, F. Nobile. 'Multilevel Monte Carlo approximation of functions'. SIAM/ASA Journal on Uncertainty Quantification, 6(3):1256-1293, 2018. link
• S. Krumscheid, F. Nobile, M. Pisaroni. 'Quantifying uncertain system outputs via the multilevel Monte Carlo method -- Part I: Central moment estimation'. Journal of Computational Physics, 414:109466, 2020. link
• D. Lu, G. Zhang, C. Webster, C. Barbier. 'An improved multilevel Monte Carlo method for estimating probability distribution functions in stochastic oil reservoir simulations'. Water Resources Research, 52(12):9642-9660, 2016. link
• S. Taverniers, S.B.M. Bosma, D.M. Tartakovsky. 'Accelerated multilevel Monte Carlo with kernel-based smoothing and latinized stratification'. Water Researces Research, 56(9):e2019WR026984, 2020. link
• S. Taverniers, D.M. Tartakovsky. 'Estimation of distributions via multilevel Monte Carlo with stratified sampling'. Journal of Computational Physics, 419:109572, 2020. link
• R. Tempone, S. Wolfers. 'Smolyak's algorithm: a powerful black box for the acceleration of scientific computations'. Sparse Grids and Applications, LNCSE Vol.123, Springer, pp.201-228, 2018. link
• D. Wilson, R.E. Baker. 'Multi-level methods and approximating distribution functions'. AIP Advances, 6(7), 2016. link
• Y. Xu, A. Narayan. 'Budget-limited distribution learning in multifidelity problems'. Numerische Mathematik, 153(1):171-212, 2023. link
• X.J. Zhu, L. Di Rienzo, X.K. Ma, L. Codecasa. 'Enhanced multilevel Monte Carlo Method applied to FDTD for probability distribution estimation'. IEEE Transactions on Antennas and Propagation, 71(10):8390-8395, 2023. link

Reliability and rare events

• L.J.M. Aslett, T. Nagapetyan, S.J. Vollmer. 'Multilevel Monte Carlo for reliability theory'. Reliability Engineering & System Safety, 165:188-196, 2017. link
• D. Elfverson, F. Hellman, A. Malqvist. 'A multilevel Monte Carlo method for computing failure probabilities'. SIAM/ASA Journal on Uncertainty Quantification, 4(1):312-330, 2016. link
• F. Fagerlund, F. Hellman, A. Malqvist, A. Niemi. 'Multilevel Monte Carlo methods for computing failure probability of porous media flow systems'. Advances in Water Resources, 94:498-509, 2016. link
• K.M. Hamdia, H. Ghasemi. 'Reliability analysis of the stress intensity factor using multilevel Monte Carlo methods'. Probabilistic Engineering Mechanics, 74:103497, 2023. link
• A.S.N. Huda, R. Zivanovic. 'Improving distribution system reliability calculation efficiency using multilevel Monte Carlo method'. International Transactions on Electrical Energy Systems, 2017. link
• A.S.N. Huda, R. Zivanovic. 'Multilevel Monte Carlo simulation applied to distribution systems reliability evaluation'. IEEE PowerTech conference, 2017. link
• A.S.N. Huda, R. Zivanovic. 'An efficient method with tunable accuracy for estimating expected interruption cost of distribution systems'. International Journal of Electrical Power and Energy Systems, 105:98-109, 2019. link
• A.S.N. Huda, R. Zivanovic. 'Study effect of components availability on distribution system reliability through multilevel Monte Carlo method'. IEEE Transactions on Industrial Informatics, 15(6):3133-3142, 2019. link
• A.S.N. Huda, R. Zivanovic. 'Estimating sensitivity of interrupted energy and outage costs for customers in government, institutions and office buildings due to distribution grid failures using multilevel Monte Carlo technique'. Energy and Buildings, 323:114766, 2024. link
• C. Proppe. 'A multilevel moving particles method for reliability estimation'. Probabilistic Engineering Mechanics, 59:103018, 2020. link
• E. Sharifnia, S. Tindemans. 'Multilevel Monte Carlo with surrogate models for resource adequacy assessment'. 17th International Conference on Probabilistic Methods Applied to Power Systems (PMAPS), 2022. link
• S. Tindemans, G. Strbac. 'Accelerating system adequacy assessment using the multilevel Monte Carlo approach'. Electric Power Systems Research, 189:106740, 2020. link
• E. Ullmann, I. Papaioannou. 'Multilevel estimation of rare events'. SIAM/ASA Journal of Uncertainty Quantification, 3(1):922-953, 2015. link
• F. Wagner, J. Latz, I. Papaioannou, E. Ullmann. 'Multilevel sequential importance sampling for rare event estimation'. SIAM Journal on Scientific Computing, 42(4):A2062-A2087, 2020. link
• F. Wagner, J. Latz, I. Papaioannou, E. Ullmann. 'Error analysis for probabilities of rare events with approximate models'. SIAM Journal on Numerical Analysis, 59(4):1948-1975, 2021. link
• J. Wang, X. Gao, R. Cao, Z. Sun. 'A multilevel Monte Carlo method for performing time-variant reliability analysis'. IEEE Access, 9:31773-31781, 2021. link

Discontinuous functionals

• M. Altmayer, A. Neuenkirch. 'Multilevel Monte Carlo quadrature of discontinuous payoffs in the generalized Heston model using Malliavin integration by parts'. SIAM Journal on Financial Mathematics, 6(1):22-52, 2015. link
• C. Bayer, C. Ben Hammouda, R. Tempone. 'Multilevel Monte Carlo combined with numerical smoothing for robust and efficient option pricing and density estimation'. arXiv preprint 2003.05708, 2020. SIAM Journal on Scientific Computing, 46(3), A1514-A1548, 2024. link
• S. Burgos, M.B. Giles. `Computing Greeks using multilevel path simulation'. pp.281-296 in Monte Carlo and Quasi-Monte Carlo Methods 2010, Springer, 2012. link
• S. Burgos. `The computation of Greeks with multilevel Monte Carlo'. PhD thesis, University of Oxford, 2014. link
• D. Elfverson, F. Hellman, A. Malqvist. 'A multilevel Monte Carlo method for computing failure probabilities'. SIAM/ASA Journal on Uncertainty Quantification, 4(1):312-330, 2016. link
• M.B. Giles. `Improved multilevel Monte Carlo convergence using the Milstein scheme'. pp.343-358, in Monte Carlo and Quasi-Monte Carlo Methods 2006, Springer, 2008. link
• M.B. Giles. `Multilevel Monte Carlo methods'. Acta Numerica, 24:259-328, Cambridge University Press, 2015. link
• M.B. Giles. 'MLMC techniques for discontinuous functions'. pp.33-47 in Monte Carlo and Quasi-Monte Carlo Methods 2022, Springer, 2024. link
• M.B. Giles, F. Bernal. 'Multilevel estimation of expected exit times and other functionals of stopped diffusions'. SIAM/ASA Journal on Uncertainty Quantification, 6(4):1454-1474, 2018. link
• M.B. Giles, K. Debrabant, A. Roessler. 'Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation'. Discrete and Continuous Dynamical Systems -- series B, 24(8):3881-3903, 2019. link
• M.B. Giles, A.-L. Haji-Ali. 'Multilevel nested simulation for efficient risk estimation'. SIAM/ASA Journal on Uncertainty Quantification, 7(2):497-525, 2019. link
• M. Giles, T. Nagapetyan, K. Ritter. 'Multi-Level Monte Carlo approximation of distribution functions and densities'. SIAM/ASA Journal on Uncertainty Quantification, 3:267-295, 2015. link
• A.-L. Haji-Ali, J. Spence, A. Teckentrup. 'Adaptive multilevel Monte Carlo for probabilities'. SIAM Journal on Numerical Analysis, 60(4):2125-2149, 2022. link
• S. Krumscheid, F. Nobile. 'Multilevel Monte Carlo approximation of functions'. SIAM/ASA Journal on Uncertainty Quantification, 6(3):1256-1293, 2018. link
• Y. Xia, M.B. Giles. `Multilevel path simulation for jump-diffusion SDEs'. pp.695-708 in Monte Carlo and Quasi-Monte Carlo Methods 2010, Springer, 2012. link
• Y. Xia. `Multilevel Monte Carlo for jump processes'. PhD thesis, University of Oxford, 2014. link

Importance sampling and other non-standard MLMC couplers

• J. Beck, B. Mansour Dia, L. Espath, R. Tempone. 'Multilevel double loop Monte Carlo and stochastic collocation methods with importance sampling for Bayesian optimal experimental design' International Journal for Numerical Methods in Engineering, 121(1):3482-3503, 2020. link
• M. Ben Alaya, K. Hajji, A. Kebaier. 'Importance sampling and statistical Romberg method for Lévy processes'. Stochastic Processes and their Applications, 126(7):1901-1931, 2016. link
• M. Ben Alaya, K. Hajji, A. Kebaier. 'Adaptive importance sampling for multilevel Monte Carlo Euler method'. Stochastics, 95(2):303-327, 2023. link
• C. Ben Hammouda, N. Ben Rached, R. Tempone. 'Importance sampling for a robust and efficient multilevel Monte Carlo estimator for stochastic reaction networks'. Statistics and Computing, August 2020. link
• W. Fang, M.B. Giles. 'Multilevel Monte Carlo method for ergodic SDEs without contractivity'. Journal of Mathematical Analysis and Applications, 476(1):149-176, 2019. link
• W. Fang, M.B. Giles. 'Importance Sampling for Pathwise Sensitivity of Stochastic Chaotic Systems'. SIAM/ASA Journal on Uncertainty Quantification, 9(3):1217-1241, 2021. link.
• W. Fang. 'Adaptive timestepping for SDEs with non-globally Lipschitz drift'. PhD thesis, University of Oxford, 2019. link
• R. Gasparotto. 'Optimised Importance Sampling in Multilevel Monte Carlo'. MSc thesis, University of Oxford, 2015. link
• M.B. Giles. `Multilevel Monte Carlo methods'. Acta Numerica, 24:259-328, Cambridge University Press, 2015. link
• A. Kebaier, J. Lelong. 'Coupling importance sampling and multilevel Monte Carlo using sample average approximation'. Methodology and Computing in Applied Probability, 20(2):611-641, 2018. link
• J. Latz, I. Papaioannou, E. Ullmann. 'Multilevel sequential Monte Carlo for Bayesian inverse problems'. Journal of Computational Physics, 368:154-178, 2018. link
• V. Lemaire, M. Thieullen, N. Thomas. 'Thinning and multilevel Monte Carlo methods for piecewise deterministic (Markov) processes with an application to a stochastic Morris-Lecar model' Advances in Applied Probability, 52(1):138-172, 2020. link
• F. Wagner, J. Latz, I. Papaioannou, E. Ullmann. 'Multilevel sequential importance sampling for rare event estimation'. SIAM Journal on Scientific Computing, 42(4):A2062-A2087, 2020. link
• Y. Xia, M.B. Giles. `Multilevel path simulation for jump-diffusion SDEs'. pp.695-708 in Monte Carlo and Quasi-Monte Carlo Methods 2010, Springer, 2012. link
• Y. Xia. `Multilevel Monte Carlo for jump processes'. PhD thesis, University of Oxford, 2014. link

Stochastic approximation and machine learning

• H. Asi, Y. Carmon, A. Jambulapati, Y. Jin, A. Sidford. 'Stochastic bias-reduced gradient methods'. Advances in Neural Information Processing Systems (NeurIPS), 34, 2021. link
• S. Dereich, T. Müller-Gronbach. 'General multilevel adaptations for stochastic approximation algorithms of Robbins-Munro and Polyak-Ruppert type'. Numerische Mathematik, 142(2):279-328, 2019. link
• S. Dereich. 'General multilevel adaptations for stochastic approximation algorithms II: CLTs'. Stochastic Processes and their Applications, 132:226-260, 2021. link
• R. Dorfman, K.Y. Levy. 'Adapting to mixing time in stochastic optimization with Markovian data'. International Conference on Machine Learning, Vol 162, 2022. link
• Weinan E., J. Han, A. Jentzen. 'Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning'. Nonlinearity, 35(1):278-310, 2022. link
• N. Frikha. 'Multi-level stochastic approximation algorithms'. Annals of Applied Probability, 26(2):933-985, 2016. link
• M. Fujisawa, I. Sato. 'Multilevel Monte Carlo variational inference'. Journal of Machine Learning Research, 22, 2021. link
• T. Gerstner, B. Harrach, D. Roth, M. Simon. 'Multilevel Monte Carlo learning'. arXiv pre-print, 2021. link
• M.B. Giles, M.B. Majka, L. Szpruch, S.J. Vollmer, K.C. Zygalakis. 'Multi-level Monte Carlo methods for the approximation of invariant measures of stochastic differential equations'. Statistics and Computing, 30:507-524, 2020. link
• T. Goda, T. Hironaka, W. Kitade, A. Foster. 'Unbiased MLMC stochastic gradient-based optimization of Bayesian experimental designs'. SIAM Journal on Scientific Computing 44(1):20M1338848, 2022. link
• T. Goda, W. Kitade. 'Constructing unbiased gradient estimators with finite variance for conditional stochastic optimization'. Mathematics and Computers in Simulation, 204:743-763, 2023. link
• P.A. Guth, A. Van Barel. 'Multilevel quasi-Monte Carlo for optimization under uncertainty'. Numerische Mathematik, 154:443-484, 2023. link
• Y. Hu, X. Chen, N. He. 'On the bias-variance-cost tradeoff of stochastic optimization'. Advances in Neural Information Processing Systems (NeurIPS), 2021. link
• K. Ishikawa, T. Goda. 'Efficient debiased evidence estimation by multilevel Monte Carlo sampling'. Proceedings of Machine Learning Research (PMLR), 161:34-43, 2021. link
• A. Jasra, K.J. Law, A. Tarakanov, F. Yu. 'Randomized multilevel Monte Carlo for embarrassingly parallel inference'. Communications in Computer and Information Science, 1512:3-21, 2022. link
• D. Levy, Y. Carmon, J.C. Duchi, A. Sidford. 'Large-scale methods for distributionally robust optimization'. Advances in Neural Information Processing Systems (NeurIPS), 33, 2020. link
• K.O. Lye, S. Mishra, R. Molinaro. 'A multi-level procedure for enhancing accuracy of machine learning algorithms'. European Journal of Applied Mathematics, 32(3):436-469, 2021. link
• M.B. Majka, M. Sabate-Vidales, L. Szpruch. 'Multi-index antithetic stochastic gradient algorithm'. Statistics and Computing, 33:49, 2023. link
• S.A. Menchon, H.J. Kappen. 'Learning effective state-feedback controllers through efficient multilevel importance samplers'. International Journal of Control, 92(12):2776-2783, 2019. link
• T. Nagapetyan, L. Szpruch, S.J. Vollmer, K. Zygalakis. 'Multilevel Monte Carlo for scalable Bayesian computations'. 29th Conference on Neural Information Processing Systems (NIPS), 2016. link
• Y. Shi, R. Cornish. 'On multilevel Monte Carlo unbiased gradient estimation for deep latent variable models'. Proceedings of Machine Learning Research (PMLR), 130:3925-3933, 2021. link
• S. Yang, V. Zankin, M. Balandat, S. Scherer, K.T. Carlberg, N. Walton, K.J.H. Law. 'Accelerating look-ahead in Bayesian optimization: multilevel Monte Carlo is all you need'. Proceedings of the 41st International Conference on Machine Learning (ICML), 2024. link

Multifidelity

• C.M. Fleeter, G. Geraci, D.E. Schiavazzi, A. M. Kahn, A.L. Marsden. 'Multilevel and multifidelity uncertainty quantification for cardiovascular hemodynamics' Computer Methods in Applied Mechanics and Engineering, 365:113030, 2020. link
• F. Law, A. Cerfon, B. Peherstorfer, F. Wechsung. 'Meta variance reduction for Monte Carlo estimation of energetic particle confinement during stellarator optimization'. Journal of Computational Physics, 112524, 2023. link
• P. Mohanamuraly, J.-D. Müller. 'An adjoint-assisted multilevel multifidelity method for uncertainty quantification and its application to turbomachinery manufacturing variability'. International Journal for Numerical Methods in Engineering, 122(9):2179-2204, 2020. link
• D. Patsiali, A.A. Taflanidis. 'Multi-fidelity Monte Carlo for seismic risk assessment applications'. Structural Safety, 93:102129, 2021. link
• B. Peherstorfer. 'Multifidelity Monte Carlo Estimation with Adaptive Low-Fidelity Models' SIAM Journal on Uncertainty Quantification, 7(2):579-603, 2019. link
• B. Peherstorfer, M. Gunzburger, K. Willcox. 'Convergence analysis of multifidelity Monte Carlo estimation'. Numerische Mathematik, 139(3):683-707, 2019. link
• I. Rosenbaum, J. Staum. 'Multilevel Monte Carlo metamodeling'. Operations Research, 65(4):1062-1077, 2017. link
• L. Scarabosio, B. Wohlmuth, J.T. Oden, D. Faghihi. 'Goal-oriented adaptive modeling of random heterogeneous media and model-based multilevel Monte Carlo methods'. Computers and Mathematics with Applications, 78(8):2700-2718, 2019. link
• E. Sharifnia, S. Tindemans. 'Multilevel Monte Carlo with surrogate models for resource adequacy assessment'. 17th International Conference on Probabilistic Methods Applied to Power Systems (PMAPS), 2022 link
• A. Tran, P. Robbe, H. Lim. 'Multi-fidelity microstructure-induced uncertainty quantification by advanced Monte Carlo methods'. Materialia, 27:101705, 2023. link

Numerical linear algebra

• J.A. Acebron. 'A probabilistic linear solver based on a multilevel Monte Carlo method' Journal of Scientific Computing, 82(3):65, 2020. link
• J.A. Acebron, J.R. Herrero, J. Monteiro. 'A highly parallel algorithm for computing the action of a matrix exponential on a vector based on a multilevel Monte Carlo method' Computers and Mathematics with Applications, 79(12):3495-3515, 2020. link
• A. Frommer, M.N. Khalil, G. Ramirez-Hidalgo. 'A multilevel approach to variance reduction in the stochastic estimation of the trace of a matrix'. SIAM Journal on Scientific Computing, 44(4):A2536-A2556, 2022. link
• A. Frommer, M.N. Khalil. 'MGMLMC++ as a variance reduction method for estimating the trace of a matrix inverse'. Proceedings of Science, 430:017, 2023. link
• E. Hallman, D. Troester. 'A multilevel approach to stochastic trace estimation'. Linear Algebra and Its Applications, 638:125-1491, April 2022. link
• P. Lashomb, R.B. Morgan, T. Whyte, W. Wilcox. 'Multipolynomial Monte Carlo for trace estimation in lattice QCD'. Computer Physics Communications, 300:109163, 2024. link
• Y. Wu, N. Polydorides. 'A multilevel Monte Carlo estimator for matrix multiplication'. SIAM Journal of Scientific Computing, 42(5):A2731-A2749, 2020. link

Physics

• K. Jansen, E.H. Müller, R. Scheichl. 'Multilevel Monte Carlo algorithm for quantum mechanics on a lattice'. Physical Review D, 102, 114512, 2020. link
• P. Lashomb, R.B. Morgan, T. Whyte, W. Wilcox. 'Multipolynomial Monte Carlo for trace estimation in lattice QCD'. Computer Physics Communications, 300:109163, 2024. link
• X. Lei, Z. Zhou. 'Multi-level Monte Carlo path integral molecular dynamics for thermal average calculation in the nonadiabatic regime'. Numerical Mathematics, 14(2):321-354, 2021. link
• E. Løvbak, G. Samaey, S. Vandewalle. 'A multilevel Monte Carlo method for asymptotic-preserving particle schemes in the diffusive limit'. Numerische Mathematik, 2021 link
• E. Løvbak. 'Multilevel and adjoint Monte Carlo methods for plasma edge neutral particle models'. PhD thesis, KU Leuven, 2024. link
• B. Mortier, P. Robbe, M. Baelmans, G. Samaey. 'Multilevel asymptotic-preserving Monte Carlo for kinetic-diffusive particle simulations of the Boltzmann-BGK equation' Journal of Computational Physics, 4501:110736, 2022. link
• P.B. Rohrbach, H. Kobayashi, R. Scheichl, N.B. Wilding, R.L. Jack. 'Multilevel simulation of hard-sphere mixtures'. Journal of Chemical Physics, 157(12),124109, 2022. link
• T. Whyte, A. Stathopoulos, E. Romero, K. Orginos. 'Interpolation as a means of shift selection for multilevel Monte Carlo with lattice displacements'. Proceedings of Science 4306:255, 2023. link
• T. Whyte, A. Stathopoulos, E. Romero, K. Orginos. 'Optimizing shift selection in multilevel Monte Carlo for disconnected diagrams in lattice QCD'. Computer Physics Communications, 294:108928, 2024. link

Other applications

• G. Aitken, L. Beevers, M.A. Christie. 'Multi-level Monte Carlo models for flood inundation uncertainty quantification'. Water Resources Research, 58(11):e2022WR032599, 2022. link
• S. Baynes, S.L. Cotter, P.T. Russell, E.M. Ryan, T.W. Waite. 'Efficient forecasting and uncertainty quantification for large-scale account level Monte Carlo models of debt recovery'. Journal of RSS, Series C -- Applied Statistics, 2023. link
• C. Bayer, P.K. Friz, S. Riedel, J. Schoenmakers. 'From rough path estimates to multilevel Monte Carlo'. SIAM Journal of Numerical Analysis, 54(3):1449-1483, 2016. link
• F. Beiser, H.H. Holm, K.O. Lye, J. Eidsvik. 'Multi-level data assimilation for simplified ocean models'. Nonlinear Processes in Geophysics, January 2024. link
• S. Berrone, C. Canuto, S. Pieraccini, S. Scialo. 'Uncertainty quantification in discrete fracture network models: stochastic geometry'. Water Resources Research, 54(2):1338-1352, 2018. link
• N.-Y. Chiang, Y. Lin, Q. Long. 'Efficient propagation of uncertainties in manufacturing supply chains: Time buckets, L-leap, and multilevel Monte Carlo methods' Operations Research Perspective, 7:100144, 2020. link
• M.C.A. Clare, M.D. Piggott, C.J. Cotter. 'Assessing erosion and flood risk in the coastal zone through the application of multilevel Monte Carlo methods'. Coastal Engineering, 174:104118, 2022. link
• M.C.A. Clare, T.W.B. Leijnse, R.T. Mccall, F.L.M. Diermanse, C.J. Cotter, M.D. Piggott. 'Multilevel multifidelity Monte Carlo methods for assessing uncertainty in coastal flooding'. Natural Hazards and Earth System Sciences, 22(8):2491-2515, 2022. link
• A. Ehara, S. Guillas. 'An adaptive strategy for sequential designs of multilevel computer experiments'. International Journal for Uncertainty Quantification, 13(4):61-98, 2023. link
• I.G. Graham, M.J. Parkinson, R. Scheichl. 'Error analysis and uncertainty quantification for the heterogeneous transport equation in slab geometry'. IMA Journal of Numerical Analysis 41(4):2331-2361, 2021. link
• K.M. Hamdia, H. Ghasemi, X. Zhuang, T. Rabczuk. 'Multilevel Monte Carlo method for topology optimization of flexoelectric composites with uncertain material properties'. Engineering Analysis with Boundary Elements, 134:412-418, 2022. link
• A.-L. Haji-Ali, F. Nobile, L. Tamellini, R. Tempone. 'Multi-index stochastic collocation convergence rates for random PDEs with parametric regularity'. Foundations of Computational Mathematics,16(6):1555-1605, 2016. link
• A.-L. Haji-Ali, F. Nobile, L. Tamellini, R. Tempone. 'Multi-index stochastic collocation for random PDEs' Computer Methods in Applied Mechanics and Engineering, 306:95-122, 2016. link
• M. Hobbs, H. Rappel, T. Dodwell. 'A probabilistic peridynamic framework with an application to the study of the statistical size effect'. Applied Mathematical Modelling, 128:137-153, 2024. link
• M. Hoerger, H. Kurniawati, A. Elfes. 'Multilevel Monte-Carlo for solving POMDPs online'. Springer Proceedings in Advanced Robotics, 20:174-190, 2022. link
• A. Jasra, J. Heng, Y. Xu, A.N. Bishop. 'A multilevel approach for stochastic nonlinear optimal control'. International Journal of Control, 95(5):1290-1304, 2022. link
• N. Kahalé. 'On the effective dimension and multilevel Monte Carlo'. Operations Research Letters, 50(4):415-421, 2022. link
• N. Kahalé. 'Unbiased time-average estimators for Markov chains'. Mathematics of Operations Research, Oct 2023. link
• P. Kloeden, A. Neuenkirch, R. Pavani. 'Multilevel Monte Carlo for stochastic differential equations with additive fractional noise'. Annals of Operations Research, 189(1):255-276, 2011. link
• F. Menhorn, G. Geraci, D.T. Seidl, Y.M. Marzouk, M.S. Eldred, H.-J. Bungartz. 'Multilevel Monte Carlo estimators for derivative-free optimization under uncertainty'. International Journal for Uncertainty Quantification, 14(3):21-65, 2024. link
• P. Mycek, M. de Lozzo. 'Multilevel Monte Carlo covariance estimation for the computation of Sobol indices'. SIAM/ASA Journal on Uncertainty Quantification, 7(4):1323-1348, 2019. link
• S. Pieraccini. 'Uncertainty quantification analysis in discrete fracture network flow simulations'. International Journal on Geomathematics, 11(1):12, 2020. link
• A. Richard, X. Tan, F. Yang. 'Discrete-time simulation of Stochastic Volterra equations'. Stochastic Processes and their Applications, 141:109-138, 2021. link
• M.G. Rudolph, T. Wohling, T. Wagener, A. Hartmann. 'Extending GLUE with multilevel methods to accelerate statistical inversion of hydrological models'. Water Resources Research, 60(10):e2024WR037735, 2024. link
• M. Sobieraj. 'A multilevel Monte Carlo algorithm for stochastic differential equations driven by countably dimensional Wiener process and Poisson random measure'. Applied Numerical Mathematics, 206:141-160, 2024. link
• J. Sukys, U. Rasthofer, F. Wermelinger, P. Hadjidoukas, P. Koumoutsakos. 'Multilevel control variates for uncertainty quantification in simulations of cloud cavitation'. SIAM Journal on Scientific Computing, 40(5):B1361-B1390, 2018. link
• A.L. Teckentrup, P. Jantsch, C.G. Webster, M. Gunzburger. 'A multilevel stochastic collocation method for partial differential equations with random input data'. SIAM/ASA Journal on Uncertainty Quantification, 3(1):1046-1074, 2015. link
• P. Vanmechelen, G. Lombaert, G. Samaey. 'Multilevel MCMC with level-dependent data in a model case of structural damage assessment'. pp.643-658 in Monte Carlo and Quasi-Monte Carlo Methods 2022, Springer, 2024. link
• F. Watson, Ø. Klemetsdal, K.O. Lye. 'Multifidelity multilevel Monte Carlo method for rapid uncertainty quantification in CO2 storage applications'. ECMOR, Sept 2024. link
• Z.-C. Xiao, K.K. Lin. 'Multilevel Monte Carlo for cortical circuit models'. Journal of Computational Neuroscience, 50(1):9-15, 2022. link
• C. Ye, X.A. Luo. 'Multilevel Monte Carlo approach for a stochastic optimal control problem based on the gradient projection method'. Applied Math, 3:98-116, 2023. link
• Z. Zhou, G. Wang, J.H. Blanchet, P.W. Glynn. 'Unbiased optimal stopping via the MUSE'. Stochastic Processes and their Applications, 166:104088, 2023. link
• X.J. Zhu, L. Di Rienzo, X.K. Ma, L. Codecasa. 'Multilevel Monte Carlo FDTD Method for Uncertainty Quantification'. IEEE Antennas and Wireless Propagation Letters, 21(10):2030-2034, 2022. link
• X.J. Zhu, L. Di Rienzo, X.K. Ma, L. Codecasa. 'Enhanced multilevel Monte Carlo Method applied to FDTD for probability distribution estimation'. IEEE Transactions on Antennas and Propagation, 71(10):8390-8395, 2023. link

BiBTeX file

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