DOCUMENTA MATHEMATICA, Extra Volume ICM III (1998), 481-490

Gregory Beylkin

Title: On Multiresolution Methods in Numerical Analysis

As a way to emphasize several distinct features of the multiresolution methods based on wavelets, we describe connections between the multiresolution LU decomposition, multigrid and multiresolution reduction/homogenization for self-adjoint, strictly elliptic operators. We point out that the multiresolution LU decomposition resembles a direct multigrid method (without W-cycles) and that the algorithm scales properly in higher dimensions. Also, the exponential of these operators is sparse where sparsity is defined as that for a finite but arbitrary precision. We describe time evolution schemes for advection-diffusion equations, in particular the Navier-Stokes equation, based on using sparse operator-valued coefficients. We point out a significant improvement in the stability of such schemes.

1991 Mathematics Subject Classification: 65M55, 65M99, 65F05, 65F50, 65R20, 35J, 76D05

Keywords and Phrases: multigrid methods, fast multipole method, wavelet bases, multiresolution analysis, multiresolution LU decomposition, time evolution schemes, exponential of operators, advection-diffusion equations

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