2019/09/13 by L Borcea, Liliana Borcea, V Druskin +7 · 16 citations
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Basis (linear algebra) #Basis function #Discretization #Electromagnetic Simulation and Numerical Methods #Embedding #Galerkin method #Matrix (chemical analysis) #Model Reduction and Neural Networks #Numerical methods in engineering #Operator (biology) #Tridiagonal matrix #cs.NA #math.NA #msc:41A20 #msc:65M32
paper · pdf · doi:10.1088/1361-6420/ab750b
published in Inverse Problems 36(5), 055010 (IOP Publishing) · 21 pages, 11 figures
arxiv created 2019/09/13 · openalex created_date 2019/09/19 · openalex publication_date 2020/02/11 · arxiv updated 2020/06/24 · openalex updated_date 2026/08/06
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 tridiagonal 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.