Wavelet approximation methods for pseudodifferential equations II: matrix compression and fast solution
- Dahmen, W.
- Prössdorf, Siegfried
- Schneider, R.
2010 Mathematics Subject Classification
- 65F35 65J10 65N30 65N35 65R20 47A20 47G30 45P05 41A25
- Periodic pseudodifferential equations, pre-wavelets, biorthogonal wavelets, generalized Petrov-Galerkin schemes, wavelet representation, atomic decomposition, Calderón-Zygmund operators, matrix compression, error analysis
This is the second part of two papers which are concerned with generalized Petrov-Galerkin schemes for elliptic periodic pseudodifferential equations in ℝn. This setting covers classical Galerkin methods, collocation, and quasiinterpolation. The numerical methods are based on a general framework of multiresolution analysis, i.e., of sequences of nested spaces which are generated by refinable functions. In this part we analyse compression techniques for the resulting stiffness matrices relative to wavelet type bases. We will show that, although these stiffness matrices are generally not sparse, the order of the overall computational work which is needed to realize a certain accuracy is of the form O(N(logN)b) where N is the number of unknowns and b ≥ 1 is some real number.