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Shari Moskow

Publications and source records attributed to Shari Moskow.

13 recordsLinked to original sources
arXiv2026

Data generated internal solutions for the plasma wave equation: error bounds and numerical experiments in two dimensions

We consider the computation of internal solutions for a time domain plasma wave equation with an unknown potential $q$ from boundary response data. The internal solutions are computed by transforming known background snapshots using the Cholesky decomposition of the data-driven Gramian, or mass matrix. It was recently shown that in one dimension these data generated internal solutions converge in $L^2$ at order $\sqrt{\tau}$ for well chosen initial waves. Here we study the internal solution reconstruction in two dimensions, where a multiple input/multiple output (MIMO) setup and a block Gramian are needed, with the number of boundary sources increasing as the time sampling is refined. We show that the general error bound carries over to this setting: the distance between the data generated solutions and the best causal approximation from background snapshots is controlled by the best approximation mass matrix mismatch. Numerical refinement studies on a square domain measure the convergence of the data generated solutions alongside the best causal approximation from the background snapshots, with the relative errors appearing to go to zero at rate $\sqrt{\tau}$, and the absolute errors going to zero in $L^1$. We also show that the solution reconstructions remain accurate for high contrast composite media. Finally, since the mass matrix assembled from noisy response data can fail to be positive definite, regularization is needed; comparing a diagonal shift with an eigenvalue floor, we find the floor more robust, with reconstructions that remain more accurate than the unperturbed background field with noise that is up to ten percent of the root mean square data amplitude.

math.NA↗
arXiv2025

Asymptotic expansions for the transmission eigenvalues of periodic scatterers of bounded support

We consider the transmission eigenvalues for a bounded scatterer with a periodically varying index of refraction, and derive the first order corrections to the limiting transmission eigenvalues. We assume the scatterer contrast to be of one sign, in which case the transmission eigenvalue problem can be written in terms of operators corresponding to a fourth order PDE with periodic coefficients. We perform two-scale asymptotics for this biharmonic type homogenization problem and show convergence estimates which require a boundary corrector function, and this boundary corrector function appears in the formula for the transmission eigenvalues correction.

math.AP↗
arXiv2024

Nonlinearity helps the convergence of the inverse Born series

In previous work of the authors, we investigated the Born and inverse Born series for a scalar wave equation with linear and nonlinear terms, the nonlinearity being cubic of Kerr type [8]. We reported conditions which guarantee convergence of the inverse Born series, enabling recovery of the coefficients of the linear and nonlinear terms. In this work, we show that if the coefficient of the linear term is known, an arbitrarily strong Kerr nonlinearity can be reconstructed, for sufficiently small data. Additionally, we show that similar convergence results hold for general polynomial nonlinearities. Our results are illustrated with numerical examples.

math.NA↗
arXiv2023

Regularized Reduced Order Lippman-Schwinger-Lanczos Method for Inverse Scattering Problems in the Frequency Domain

Inverse scattering has a broad applicability in quantum mechanics, remote sensing, geophysical, and medical imaging. This paper presents a robust direct reduced order model (ROM) method for solving inverse scattering problems based on an efficient approximation of the resolvent operator regularizing the Lippmann-Schwinger-Lanczos (LSL) algorithm. We show that the efficiency of the method relies upon the weak dependence of the orthogonalized basis on the unknown potential in the Schr\"odinger equation by demonstrating that the Lanczos orthogonalization is equivalent to performing Gram-Schmidt on the ROM time snapshots. We then develop the LSL algorithm in the frequency domain with two levels of regularization. We show that the same procedure can be extended beyond the Schr\"odinger formulation to the Helmholtz equation, e.g., to imaging the conductivity using diffusive electromagnetic fields in conductive media with localized positive conductivity perturbations. Numerical experiments for Helmholtz and Schr\"odinger problems show that the proposed bi-level regularization scheme significantly improves the performance of the LSL algorithm, allowing for good reconstructions with noisy data and large data sets.

math.NA↗
arXiv2022

Born and inverse Born series for scattering problems with Kerr nonlinearities

We consider the Born and inverse Born series for scalar waves with a cubic nonlinearity of Kerr type. We find a recursive formula for the operators in the Born series and prove their boundedness. This result gives conditions which guarantee convergence of the Born series, and subsequently yields conditions which guarantee convergence of the inverse Born series. We also use fixed point theory to give alternate explicit conditions for convergence of the Born series. We illustrate our results with numerical experiments.

math.NA↗
arXiv2022

Stability and reconstruction of a special type of anisotropic conductivity in magneto-acoustic tomography with magnetic induction

We consider the issues of stability and reconstruction of the electrical anisotropic conductivity of biological tissues in a domain $\Omega\subset\mathbb{R}^3$ by means of the hybrid inverse problem of magneto-acoustic tomography with magnetic induction (MAT-MI). The class of anisotropic conductivities considered here is of type $\sigma(\cdot)=A(\cdot,\gamma(\cdot))$ in $\Omega$, where $[\lambda^{-1}, \lambda]\ni t\mapsto A(\cdot, t)$ is a one-parameter family of matrix-valued functions which are \textit{a-priori} known to be $C^{1,\beta}$, allowing us to stably reconstruct $\gamma$ in $\Omega$ in terms of an internal functional $F(\sigma)$. Our results also extend previous results in MAT-MI where $\sigma(\cdot) = \gamma(\cdot) D(\cdot)$, with $D$ an \textit{a-priori} known matrix-valued function on $\Omega$ to a more general anisotropic structure which depends non-linearly on the scalar function $\gamma$ to be reconstructed.

math.AP↗
arXiv2021

On extension of the data driven ROM inverse scattering framework to partially nonreciprocal arrays

Data-driven reduced order models (ROMs) recently emerged as powerful tool for the solution of inverse scattering problems. The main drawback of this approach is that it was limited to the measurement arrays with reciprocally collocated transmitters and receivers, that is, square symmetric matrix (data) transfer functions. To relax this limitation, we use our previous work [14], where the ROMs were combined with the Lippmann-Schwinger integral equation to produce a direct nonlinear inversion method. In this work we extend this approach to more general transfer functions, including those that are non-symmetric, e.g., obtained by adding only receivers or sources. The ROM is constructed based on the symmetric subset of the data and is used to construct all internal solutions. Remaining receivers are then used directly in the Lippmann-Schwinger equation. We demonstrate the new approach on a number of 1D and 2D examples with non-reciprocal arrays, including a single input/multiple outputs (SIMO) inverse problem, where the data is given by just a single-row matrix transfer function.

math.NA↗
arXiv2021

Lippmann-Schwinger-Lanczos algorithm for inverse scattering problems

Data-driven reduced order models (ROMs) are combined with the Lippmann-Schwinger integral equation to produce a direct nonlinear inversion method. The ROM is viewed as a Galerkin projection and is sparse due to Lanczos orthogonalization. Embedding into the continuous problem, a data-driven internal solution is produced. This internal solution is then used in the Lippmann-Schwinger equation, thus making further iterative updates unnecessary. We show numerical experiments for spectral domain domain data for which our inversion is far superior to the Born inversion and works as well as when the true internal solution is known.

math.NA↗
arXiv2019

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

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.

math.NA↗
arXiv2017

The Imaging of Small Perturbations in an Anisotropic Media

In this paper, we employ asymptotic analysis to determine information about small volume defects in a known anisotropic scattering medium from far field scattering data. The location of the defects is reconstructed via the MUSIC algorithm from the range of the multi-static response matrix derived from the asymptotic expansion of the far field pattern in the presence of small defects. Since the same data determines the transmission eigenvalues corre- sponding to the perturbed media, we investigate how the presence of the defects changes the transmission eigenvalues and use this information to recover the strength of the small defects. We provide convergence results on transmission eigenvalues as the size of the defects tends to zero as well as derive the first correction term in the asymptotic expansion of the simple transmission eigen- values. Numerical examples are presented to show the viability of our imaging method.

math.AP↗
arXiv2014

Nonlinear eigenvalue approximation for compact operators

In \cite{Os} a general spectral approximation theory was developed for compact operators on a Banach space which does not require that the operators be self-adjoint and also provides a first order correction term. Here we extend some of the results of that paper to nonlinear eigenvalue problems. We present examples of its application that arise in electromagnetics.

math-ph↗
arXiv2013

Local Inversions in Ultrasound Modulated Optical Tomography

Ultrasound modulated optical tomography, also called acousto-optics tomography, is a hybrid imaging modality that aims to combine the high contrast of optical waves with the high resolution of ultrasound. We follow the model of the influence of ultrasound modulation on the light intensity measurements developed in [Bal Schotland PRL 2010]. We present sufficient conditions ensuring that the absorption and diffusion coefficients modeling light propagation can locally be uniquely and stably reconstructed from the corresponding available information. We present an iterative procedure to solve such a problem based on the analysis of linear elliptic systems of redundant partial differential equations.

math.AP↗