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Reduced order models for spectral domain inversion: Embedding into the continuous problem and generation of internal data

Banff International Research Station for Mathematical Innovation and Discovery

We generate reduced order Galerkin models for inversion of the Schr\"odinger equation given boundary data in the spectral domain for one and two dimensional problems. We show that in one dimension, after Lanczos orthogonalization, the Galerkin system is precisely the same as the three point staggered finite difference system on the corresponding spectrally matched grid. The orthogonalized basis functions depend only very weakly on the medium, and thus by embedding into the continuous problem, the reduced order model yields highly accurate internal solutions. In higher dimensions, the orthogonalized basis functions play the role of the grid steps, and highly accurate internal solutions are still obtained. We present inversion experiments based on the internal solutions in one and two dimensions. This is joint with: L. Borcea, V. Druskin, A. Mamonov, M. Zaslavsky.

Mathematics; Functional analysis; Calculus of variations and optimal control; optimization; Numerical analysis

video/mp4

Author affiliation: Drexel University

2019-12-25

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/72998

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/