ABSTRACT:
Facing the challenge to extract and efficiently represent information
inherent in masses of data or complex systems, approaches based on
multiscale representations are playing a dominant role in current
scientific research in many areas of science. The leading paradigm
consists in spending a minimal amount of degrees of freedom and work
while extracting and representing the maximal amount of information.
I want to discuss two classes of problems for which adaptive
multilevel schemes can be developed along this line. The first class
concerns explicitly given information and discusses the problem of
fitting nonuniformly distributed data to approximate surfaces. The
second class deals with approximating optimization problems for
operator equations, specifically, control problems constrained by
elliptic partial differential equations. Here the information - the
state and control of a system - is contained implicitly. Both
applications have in common that the formulation of the solution
method is based on minimizing a quadratic functional and that the
concept of adaptivity in a coarse-to-fine fashion combined with
thresholding plays a central role. In order to develop most efficient
numerical schemes, also issues like conditioning linear systems and
fast iterative solvers have to be dealt with.
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