Direct, Nonlinear Inversion Algorithm for Hyperbolic Problems via Projection-Based Model Reduction

Druskin, Vladimir; Mamonov, A.; Thaler, Andrew E.; Zaslavsky, Mikhail

doi:10.1137/15m1039432

Cited by 25 publications

(73 citation statements)

References 43 publications

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“…Here we recall the calculation of mass and stiffness matrices from the data introduced in [5,12] that we repeat for the convenience of the reader. Let us rewrite the data model (1.8) in terms of the propagator, using equation (2.8)…”

Section: Data-driven Rommentioning

confidence: 99%

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Reduced Order Model Approach to Inverse Scattering

Borcea¹,

Druskin²,

Mamonov³

et al. 2020

SIAM J. Imaging Sci.

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We study an inverse scattering problem for a generic hyperbolic system of equations with an unknown coefficient called the reflectivity. The solution of the system models waves (sound, electromagnetic or elastic), and the reflectivity models unknown scatterers embedded in a smooth and known medium. The inverse problem is to determine the reflectivity from the time resolved scattering matrix (the data) measured by an array of sensors. We introduce a novel inversion method, based on a reduced order model (ROM) of an operator called wave propagator, because it maps the wave from one time instant to the next, at interval corresponding to the discrete time sampling of the data. The wave propagator is unknown in the inverse problem, but the ROM can be computed directly from the data. By construction, the ROM inherits key properties of the wave propagator, which facilitate the estimation of the reflectivity. The ROM was introduced previously and was used for two purposes: (1) to map the scattering matrix to that corresponding to the single scattering (Born) approximation and (2) to image i.e., obtain a qualitative estimate of the support of the reflectivity. Here we study further the ROM and show that it corresponds to a Galerkin projection of the wave propagator. The Galerkin framework is useful for proving properties of the ROM that are used in the new inversion method which seeks a quantitative estimate of the reflectivity. * One can also consider truncation of the whole space, as long as the medium is known and non-scattering on one side of the array of sensors. P(q) = cos τ L(q)L(q) T .( 1.9) The ROM is defined by a pair of matrices P ROM (q) ∈ R nm×nm and b ROM ∈ R nm×m , which are proxies of P(q) and the initial wave b(x). These matrices are calculated from the data (1.8) (i.e., the ROM is data-driven) and they capture physical aspects of the wave propagation that are needed for inversion. The ROM was introduced in [12,6] and was used in [13] for imaging, and in [5] for transforming the data (1.8) to that corresponding to the single scattering (Born) approximation. The new results in this paper are:1. We show that the ROM propagator P ROM (q) is a Galerkin projection of the operator (1.9), and use the Galerkin framework to prove properties of the ROM that facilitate the solution of the inverse scattering problem. † The kinematic model (smooth wave speed) appears in the coefficients of the operators L(q) and L(q) T (see [6] and sections 3-4). We suppress the dependence on the known kinematic model in our notation.3

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Section: Data-driven Rommentioning

confidence: 99%

“…where L(q) is the square root of the self-adjoint, non-negative definite operator 12) and I denotes the identity. Equations (2.9-2.11) are an exact time stepping scheme for the hyperbolic problem (1.1-1.3), with boundary conditions taken into account in the definition of L(q) and L(q) T .…”

Section: Introductionmentioning

confidence: 99%

Reduced Order Model Approach to Inverse Scattering

Borcea¹,

Druskin²,

Mamonov³

et al. 2020

SIAM J. Imaging Sci.

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“…When q = 0, the γ j ,γ j are primary and dual finite difference grid steps. For nonzero q, finite difference scheme (15) can be transformed to a discrete Schrödinger form with a discrete Liouville transform [12]. For simplicity, we won't do that here, since we don't use the finite difference framework for reconstructions.…”

Section: 2mentioning

confidence: 99%

“…A relatively new class of approaches employ ideas from model reduction theory to do direct inversion using reduced order forward models [9], [15], [10], [11], [16] with impressive numerical results. These works have grown out of earlier work on spectrally matched finite difference grids [14], where special grid steps were chosen so that the numerical solution matched functionals of the solution at the boundary in the spectral domain.…”

Section: Introductionmentioning

confidence: 99%

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

Borcea¹,

Druskin²,

Mamonov³

et al. 2020

Inverse Problems

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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.

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