On Multiresolution Methods in Numerical Analysis 4832

As a way to emphasize several distinct features of the mul-tiresolution methods based on wavelets, we describe connections between the multiresolution LU decomposition, multigrid and multiresolution re-duction/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 deened as that for a nite but arbitrary precision. We describe time evolution schemes for advection-diiusion equations, in particular the Navier-Stokes equation, based on using sparse operator-valued coeecients. We point out a signiicant improvement in the stability of such schemes.