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# Wavelet-Based Computational Modeling and Operator Compression for 3D Elliptic Problems

### Kevin Amaratunga

MIT

###
February 25, 1999

102 Bradley Hall, 4 pm

Tea 3:30 pm, Math Lounge

**Abstract: ** Wavelet representations have previously been used
with reasonable success in the computer simulation of physical
processes. A common approach is to use a Galerkin formulation, which
leads to multilevel schemes in which the rate of convergence improves
proportionately with the number of zeros at $\pi$. One of the primary
limitations of the classical wavelet representations, however, is that
they are restricted to regular grids. This places wavelets at a
serious disadvantage with respect to other computational methods such
as the Finite Element Method and Boundary Element Method, which offer
greater flexibility for modeling domains with complex geometry.

In this talk, we develop a wavelet approach for efficiently modeling
problems with fairly complex 3D geometry. The approach is based on
the use of second generation wavelets, in which the strict shift
invariance and scale invariance laws usually associated with classical
wavelets are relaxed. We are thus able to derive the benefits of
multiresolution without losing flexibility in the description of the
geometry. The wavelet approach opens the possibility for very
significant compression of the operator matrix when modeling boundary
integral equations in 3D. We demonstrate this by considering an
example of a potential problem described by a surface mesh with over
ten-thousand degrees of freedom. Compression factors of over 100 can
easily be achieved with only a modest loss in computational accuracy.
Consequently, the wavelet approach offers considerable savings in
computer processing power and memory over the Boundary Element Method.

This talk will be accessible to graduate students.