Adaptive wavelet techniques in Numerical Simulation

This chapter highlights recent developments concerning adaptive wavelet methods for time dependent and stationary problems. The first problem class focusses on hyperbolic conservation laws where 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 where 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, works for nonlinear problems and where the sparsity of both, functions and operators, is exploited. key words: Conservation laws, boundary integral equations, elliptic problems, saddle point problems, mixed formulations, nonlinear problems, matrix compression, adaptive application of operators, best N-term approximation, tree approximation, convergence rates, complexity estimates