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MRADecompositionMatrix.m

Usage : MRADecompositionMatrix(sig)
   
Input : an \( L\times L\times 4\times 3\times \) array, where
   
  \( (i,j,k,l) \) is the kth Haar wavelet coefficient of the
   
  \( l \)-th diffusion coefficient:
   
  \( l=1 \) corresponds to (x,x)
   
  \( l=2 \) corresponds to (y,y)
   
  \( l=3 \) corresponds to (x,y)
   
  at the point \( (i,j) \).


The output is an \( L\times L\times 3\times 6\times 16\times 16 \) array, corresponding to the decomposition of the operator according to the multi-resolution analysis induced by the multiwavelet used.


\begin{displaymath}
\frac{d}{dx}a^{x}(x,y)\frac{d}{dx}+\frac{d}{dy}a^{y}(x,y)\fr...
...}(i,j) & \\
\ldots & & \ldots
\end{array}\right]
\end{array}\end{displaymath}


where \( \textrm{ER}(i,j)=\textrm{E}_{1}(i,j)+\textrm{E}_{2}(i,j)\frac{d}{dx}+\textrm{...
...\textrm{E}_{5}(i,j)\frac{d^{2}}{dy^{2}}+\textrm{E}_{6}(i,j)\frac{d^{2}}{dxdy} \).


There is a degree of freedom in the choice of the difference operator (see [1]), which will be referred to as the differentiation parameter, \( \theta \). A differentiation parameter \( \theta =1 \) corresponds to a standard forward-backward scheme, comparable to that used for the Haar basis -in that case, there is also a differentiation parameter but the result does not depend on the choice of this parameter. A differentiation parameter \( \theta =\frac{1}{2} \) corresponds to a centered difference scheme. The Gaussian reduction we perform cannot be used in that case. This procedure provided the array ER in the case \( \theta =\frac{1}{2}+\epsilon \), with the coefficients in \( \textrm{E}_{1}\ldots \textrm{E}_{6} \) limited to the second order in epsilon:

\begin{displaymath}
\textrm{E}=\textrm{E}^{0}+\epsilon \textrm{E}^{1}+\epsilon ^{2}\textrm{E}^{2}.\end{displaymath}


We therefore have ER of dimension

\begin{displaymath}
\begin{array}[b]{ccccccccccc}
\underbrace{L} & \times & \und...
...,\frac{d}{dx},\ldots \frac{d^{2}}{dxdy} & & i & & j
\end{array}\end{displaymath}


See also MWSmoothing (Paragraph A.9) and AsymptoticRationalReduction (Paragraph A.2).

  1. B. Alpert, G. Beylkin, D. Gines, L. Vozovoi, Adaptive Solution of Partial Differential Equations in Multiwavelet Bases, May 1999.


next up previous
Next: MWSmoothing.m Up: Haar and Multiwavelet homogenization Previous: HaarSmoothing.m
Yves Capdeboscq 2002-01-15