Adaptive Wavelet Techniques in Numerical Simulation
Albert Cohen, Wolfgang Dahmen, Ronald DeVore · 2004
Abstract This chapter highlights recent developments concerning adaptive wavelet methods for time dependent and stationary problems. The first problem class focuses on hyperbolic conservation laws in which wavelet concepts exploit sparse representations of the conserved variables. Regarding the second problem class, we begin with matrix compression in the context of boundary integral equations in which the key issue is now to obtain sparse representations of (global) operators like singular integral operators in wavelet coordinates. In the remainder of the chapter, a new fully adaptive algorithmic paradigm along with some analysis concepts are outlined, which, in particular, work for nonlinear problems and where the sparsity of both, functions and operators, is exploited.