# Computational Mathematics and Scientific Computing Seminar

## Spectrally Accurate Direct Solution of Variable-media Scattering Problems Via Impedance Maps

**Speaker:**
Alex Barnett, Dartmouth College

**Location:**
Warren Weaver Hall 1302

**Date:**
March 7, 2014, 10 a.m.

**Synopsis:**

Time-harmonic scattering from a bounded region of smoothly varying wave speed has applications in ultrasound imaging, underwater acoustics, and quantum mechanics. Yet, at high frequency, iterative methods become intolerably slow. We present an efficient direct numerical algorithm which builds the interior Dirichlet-to-Neumann (DtN) map for the region via bottom-up recursive merges of impedance-to-impedance (ItI) maps on a quadtree of boxes. The method is robust since the ItI map is unitary. On the smallest (leaf) boxes spectral collocation is used. Coupling to the radiation condition gives a provably 2nd-kind integral equation. Problems 100 wavelengths in size can be solved to 9 digit accuracy in a few minutes on a workstation; each new incident wave then requires around one second.

Joint work with Adrianna Gillman (Dartmouth) and Gunnar Martinsson (CU Boulder)