Nonlinear approximation and solution adaptivity

Peter Oswald
Bell Labs
Lucent Technologies

Abstract: I will give some examples of nonlinear approximation problems arising in connection with adaptive algorithms for operator equations as well as for image and signal processing. Many of them can be cast into the form of nonlinear N-term approximation: For a given ``function'' f, find a linear combination of at most N functions from a given ``dictionary'' such that a certain error measure is optimized. For larger, redundant dictionaries, there are a lot of open theoretical and algorithmical problems associated with N-term approximation. What is well-understood is a case of multivariate N-term approximation from B-spline and wavelet dictionaries in function spaces that models isotropic h-refinement in the finite element method. I show another example that analyzes also certain anisotropic refinement.