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

Abstract

We generate data-driven reduced order models (ROMs) for inversion of the one and two dimensional Schrödinger equation in the spectral domain given boundary data at a few frequencies. The ROM is the Galerkin projection of the Schrödinger operator onto the space spanned by solutions at these sample frequencies. The ROM matrix is in general full, and not good for extracting the potential. However, using an orthogonal change of basis via Lanczos iteration, we can transform the ROM to a block triadiagonal form from which it is easier to extract $q$. In one dimension, the tridiagonal matrix corresponds to a three-point staggered finite-difference system for the Schrödinger operator discretized on a so-called spectrally matched grid which is almost independent of the medium. In higher dimensions, the orthogonalized basis functions play the role of the grid steps. The orthogonalized basis functions are localized and also depend only very weakly on the medium, and thus by embedding into the continuous problem, the reduced order model yields highly accurate internal solutions. That is to say, we can obtain, just from boundary data, very good approximations of the solution of the Schrödinger equation in the whole domain for a spectral interval that includes the sample frequencies. We present inversion experiments based on the internal solutions in one and two dimensions.

21 pages, 11 figures

References in corpus (5)

• A model reduction approach to numerical inversion for a parabolic partial differential equation
• Direct, nonlinear inversion algorithm for hyperbolic problems via projection-based model reduction
• A nonlinear method for imaging with acoustic waves via reduced order model backprojection
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Cited by in corpus (4)

• Lippmann-Schwinger-Lanczos algorithm for inverse scattering problems
• Reduced order model approach for imaging with waves
• On extension of the data driven ROM inverse scattering framework to partially nonreciprocal arrays
• Inverse scattering for Schrödinger equation in the frequency domain via data-driven reduced order modeling