Seminar in Numerical Analysis: Matthias Bollhöfer (Technische Universität Braunschweig)
Hierarchical matrix approximations have become an attractive numerical approach in solving partial differential equations whenever the analytic solution can be represented by a kernel functions that allows for approximate local separable representations. The philosophy of a hierarchical matrix approximation consists of borrowing a matrix partition from an admissibility condition of the underlying analytic model and working with blocks that are expected to be of low rank. While the existence of hierarchical matrix approximations is relatively well understood, the concrete way of numerically computing a suitable approximation still raises some open questions such as the h-independent convergence of the computed approximation.
In this talk we present a new technique to locally preserve constraints inside the hierarchical matrix approximation. Numerical experiments indicate that imposing these local constraints leads to constant number of iteration steps when solving elliptic partial differential equations of second order while without preserving these constraints the number of iteration steps grow as h → 0. We will further discuss this approach from the theoretical point of view and will sketch why our approximate hierarchical LU decomposition leads to a spectral equivalent approximation.
This is joint work with Mario Bebendorf and Michael Bratsch from the University of Bonn.
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iCal