arXiv subjects

Mikhail Zaslavsky

Publications and source records attributed to Mikhail Zaslavsky.

At least 19 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

Lippmann-Schwinger-Lanczos algorithm for inverse scattering problems with unknown reflectivity and loss distributions: One-dimensional Case

We consider one-dimensional inverse scattering in attenuating media where both the reflectivity and loss distributions are unknown. Mathematically, this corresponds to recovering the coefficients of a damped wave operator, or equivalently, a quadratic operator pencil in the frequency domain. The Lippmann-Schwinger equation maps the unknown reflectivity and loss distribution to the measured scattered data. This mapping is nonlinear, as it requires knowledge of the internal wavefield, which itself depends on the reflectivity and loss distribution. The Lippmann-Schwinger-Lanczos method addresses this nonlinearity by approximating the internal solutions through the lifting of states from a reduced-order model constructed directly from the measured data. In this work, we extend the method to dissipative problems, enabling the approximation of internal partial differential equation (PDE) solutions in media with both reflectivity and loss distributions. We present two complementary constructions of such internal solutions: one based on spectral data and another on frequency-domain measurements over a finite interval. This development establishes a direct link between data-driven reduced-order models for inverse problems and port-Hamiltonian dynamical systems, with reduced models obtained either from the associated spectral measure or via rational approximation. Compared to the Born approximation, which replaces the internal field with the background field, our approach yields more accurate internal reconstructions and enables faster and more robust recovery of the contrast as evidenced by our numerical experiments.

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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 Schrödinger 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 Schrödinger 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 Schrödinger 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↗

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

Distance preserving model order reduction of graph-Laplacians and cluster analysis

Graph-Laplacians and their spectral embeddings play an important role in multiple areas of machine learning. This paper is focused on graph-Laplacian dimension reduction for the spectral clustering of data as a primary application. Spectral embedding provides a low-dimensional parametrization of the data manifold which makes the subsequent task (e.g., clustering) much easier. However, despite reducing the dimensionality of data, the overall computational cost may still be prohibitive for large data sets due to two factors. First, computing the partial eigendecomposition of the graph-Laplacian typically requires a large Krylov subspace. Second, after the spectral embedding is complete, one still has to operate with the same number of data points. For example, clustering of the embedded data is typically performed with various relaxations of k-means which computational cost scales poorly with respect to the size of data set. In this work, we switch the focus from the entire data set to a subset of graph vertices (target subset). We develop two novel algorithms for such low-dimensional representation of the original graph that preserves important global distances between the nodes of the target subset. In particular, it allows to ensure that target subset clustering is consistent with the spectral clustering of the full data set if one would perform such. That is achieved by a properly parametrized reduced-order model (ROM) of the graph-Laplacian that approximates accurately the diffusion transfer function of the original graph for inputs and outputs restricted to the target subset. Working with a small target subset reduces greatly the required dimension of Krylov subspace and allows to exploit the conventional algorithms (like approximations of k-means) in the regimes when they are most robust and efficient.

cs.LG↗

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↗

arXiv2020

Reduced Order Model Approach to Inverse Scattering

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.

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↗

arXiv2019

Fast finite-difference convolution for 3D problems in layered media

We developed fast direct solver for 3D Helmholtz and Maxwell equations in layered medium. The algorithm is based on the ideas of cyclic reduction for separable matrices. For the grids with major uniform part (within the survey domain in the problems of geophysical prospecting, for example) and small non-uniform part (PML and coarsening to approximate problems in infinite domain) the computational cost of our approach is $O(N_xN_ylog(N_xN_y)N_z)$. For general non-uniform grids the cost is $O(N^{3/2}_xN^{3/2}_yN_z)$. The first asymptotics coincide with the cost of FFT-based methods, which can be applied for uniform gridding (in x and y) only. Our approach is significantly more efficient compared to the algorithms based on discrete Fourier transform which cost is $O(N^2_xN^2_yN_z)$. The algorithm can be easily extended for solving the elasticity problems as well.

math.NA↗

arXiv2018

Robust nonlinear processing of active array data in inverse scattering via truncated reduced order models

We introduce a novel algorithm for nonlinear processing of data gathered by an active array of sensors which probes a medium with pulses and measures the resulting waves. The algorithm is motivated by the application of array imaging. We describe it for a generic hyperbolic system that applies to acoustic, electromagnetic or elastic waves in a scattering medium modeled by an unknown coefficient called the reflectivity. The goal of imaging is to invert the nonlinear mapping from the reflectivity to the array data. Many existing imaging methodologies ignore the nonlinearity i.e., operate under the assumption that the Born (single scattering) approximation is accurate. This leads to image artifacts when multiple scattering is significant. Our algorithm seeks to transform the array data to those corresponding to the Born approximation, so it can be used as a pre-processing step for any linear inversion method. The nonlinear data transformation algorithm is based on a reduced order model defined by a proxy wave propagator operator that has four important properties. First, it is data driven, meaning that it is constructed from the data alone, with no knowledge of the medium. Second, it can be factorized in two operators that have an approximately affine dependence on the unknown reflectivity. This allows the computation of the Fr\'{e}chet derivative of the reflectivity to the data mapping which gives the Born approximation. Third, the algorithm involves regularization which balances numerical stability and data fitting with accuracy of the order of the standard deviation of additive data noise. Fourth, the algebraic nature of the algorithm makes it applicable to scalar (acoustic) and vectorial (elastic, electromagnetic) wave data without any specific modifications.

math.NA↗

arXiv2017

Compressing Large-Scale Wave Propagation Models via Phase-Preconditioned Rational Krylov Subspaces

Rational Krylov subspace (RKS) techniques are well-established and powerful tools for projection-based model reduction of time-invariant dynamic systems. For hyperbolic wavefield problems, such techniques perform well in configurations where only a few modes contribute to the field. RKS methods, however, are fundamentally limited by the Nyquist-Shannon sampling rate, making them unsuitable for the approximation of wavefields in configuration characterized by large travel times and propagation distances, since wavefield responses in such configurations are highly oscillatory in the frequency-domain. To overcome this limitation, we propose to precondition the RKSs by factoring out the rapidly varying frequency-domain field oscillations. The remaining amplitude functions are generally slowly varying functions of source position and spatial coordinate and allow for a significant compression of the approximation subspace. Our one-dimensional analysis together with numerical experiments for large scale 2D acoustic models show superior approximation properties of preconditioned RKS compared with the standard RKS model-order reduction. The preconditioned RKS results in a reduction of the frequency sampling well below the Nyquist-Shannon rate, a weak dependence of the RKS size on the number of inputs and outputs for multiple-input/multiple-output (MIMO) problems, and, most importantly, in a significant coarsening of the finite-difference grid used to generate the RKS. A prototype implementation indicates that the preconditioned RKS algorithm is competitive in the modern high performance computing environment.

math.NA↗

arXiv2017

Untangling the nonlinearity in inverse scattering with data-driven reduced order models

The motivation of this work is an inverse problem for the acoustic wave equation, where an array of sensors probes an unknown medium with pulses and measures the scattered waves. The goal of the inversion is to determine from these measurements the structure of the scattering medium, modeled by a spatially varying acoustic impedance function. Many inversion algorithms assume that the mapping from the unknown impedance to the scattered waves is approximately linear. The linearization, known as the Born approximation, is not accurate in strongly scattering media, where the waves undergo multiple reflections before they reach the sensors in the array. Thus, the reconstructions of the impedance have numerous artifacts. The main result of the paper is a novel, linear-algebraic algorithm that uses a reduced order model (ROM) to map the data to those corresponding to the single scattering (Born) model. The ROM construction is based only on the measurements at the sensors in the array. The ROM is a proxy for the wave propagator operator, that propagates the wave in the unknown medium over the duration of the time sampling interval. The output of the algorithm can be input into any off-the-shelf inversion software that incorporates state of the art linear inversion algorithms to reconstruct the unknown acoustic impedance.

math.NA↗

arXiv2017

A nonlinear method for imaging with acoustic waves via reduced order model backprojection

We introduce a novel nonlinear imaging method for the acoustic wave equation based on data-driven model order reduction. The objective is to image the discontinuities of the acoustic velocity, a coefficient of the scalar wave equation from the discretely sampled time domain data measured at an array of transducers that can act as both sources and receivers. We treat the wave equation along with transducer functionals as a dynamical system. A reduced order model (ROM) for the propagator of such system can be computed so that it interpolates exactly the measured time domain data. The resulting ROM is an orthogonal projection of the propagator on the subspace of the snapshots of solutions of the acoustic wave equation. While the wavefield snapshots are unknown, the projection ROM can be computed entirely from the measured data, thus we refer to such ROM as data-driven. The image is obtained by backprojecting the ROM. Since the basis functions for the projection subspace are not known, we replace them with the ones computed for a known smooth kinematic velocity model. A crucial step of ROM construction is an implicit orthogonalization of solution snapshots. It is a nonlinear procedure that differentiates our approach from the conventional linear imaging methods (Kirchhoff migration and reverse time migration - RTM). It resolves all dynamical behavior captured by the data, so the error from the imperfect knowledge of the velocity model is purely kinematic. This allows for almost complete removal of multiple reflection artifacts, while simultaneously improving the resolution in the range direction compared to conventional RTM.

math.NA↗

arXiv2016

Multi-scale S-fraction reduced-order models for massive wavefield simulations

We developed a novel reduced-order multi-scale method for solving large time-domain wavefield simulation problems. Our algorithm consists of two main stages. During the first "off-line" stage the fine-grid operator (of the graph Laplacian type} is partitioned on coarse cells (subdomains). Then projection-type multi-scale reduced order models (ROMs) are computed for the coarse cell operators. The off-line stage is embarrassingly parallel as ROM computations for the subdomains are independent of each other. It also does not depend on the number of simulated sources (inputs) and it is performed just once before the entire time-domain simulation. At the second "on-line" stage the time-domain simulation is performed within the obtained multi-scale ROM framework. The crucial feature of our formulation is the representation of the ROMs in terms of matrix Stieltjes continued fractions (S-fractions). The layered structure of the S-fraction introduces several hidden layers in the ROM representation, that results in the block-tridiagonal dynamic system within each coarse cell. This allows us to sparsify the obtained multi-scale subdomain operator ROMs and to reduce the communications between the adjacent subdomains which is highly beneficial for a parallel implementation of the on-line stage. Our approach suits perfectly the high performance computing architectures, however in this paper we present rather promising numerical results for a serial computing implementation only. These results include 3D acoustic and multi-phase anisotropic elastic problems.

math.NA↗

arXiv2015

Direct, nonlinear inversion algorithm for hyperbolic problems via projection-based model reduction

We estimate the wave speed in the acoustic wave equation from boundary measurements by constructing a reduced-order model (ROM) matching discrete time-domain data. The state-variable representation of the ROM can be equivalently viewed as a Galerkin projection onto the Krylov subspace spanned by the snapshots of the time-domain solution. The success of our algorithm hinges on the data-driven Gram--Schmidt orthogonalization of the snapshots that suppresses multiple reflections and can be viewed as a discrete form of the Marchenko--Gel'fand--Levitan--Krein algorithm. In particular, the orthogonalized snapshots are localized functions, the (squared) norms of which are essentially weighted averages of the wave speed. The centers of mass of the squared orthogonalized snapshots provide us with the grid on which we reconstruct the velocity. This grid is weakly dependent on the wave speed in traveltime coordinates, so the grid points may be approximated by the centers of mass of the analogous set of squared orthogonalized snapshots generated by a known reference velocity. We present results of inversion experiments for one- and two-dimensional synthetic models.

math.NA↗

arXiv2015

Nonlinear seismic imaging via reduced order model backprojection

We introduce a novel nonlinear seismic imaging method based on model order reduction. The reduced order model (ROM) is an orthogonal projection of the wave equation propagator operator on the subspace of the snapshots of the solutions of the wave equation. It can be computed entirely from the knowledge of the measured time domain seismic data. The image is a backprojection of the ROM using the subspace basis for the known smooth kinematic velocity model. The implicit orthogonalization of solution snapshots is a nonlinear procedure that differentiates our approach from the conventional linear methods (Kirchhoff, RTM). It allows for the removal of multiple reflection artifacts. It also enables us to estimate the magnitude of the reflectors similarly to the true amplitude migration algorithms.

math.NA↗

arXiv2014

S-fraction multiscale finite-volume method for spectrally accurate wave propagation

We develop a method for numerical time-domain wave propagation based on the model order reduction approach. The method is built with high-performance computing (HPC) implementation in mind that implies a high level of parallelism and greatly reduced communication requirements compared to the traditional high-order finite-difference time-domain (FDTD) methods. The approach is inherently multiscale, with a reference fine grid model being split into subdomains. For each subdomain the coarse scale reduced order models (ROMs) are precomputed off-line in a parallel manner. The ROMs approximate the Neumann-to-Dirichlet (NtD) maps with high (spectral) accuracy and are used to couple the adjacent subdomains on the shared boundaries. The on-line part of the method is an explicit time stepping with the coupled ROMs. To lower the on-line computation cost the reduced order spatial operator is sparsified by transforming to a matrix Stieltjes continued fraction (S-fraction) form. The on-line communication costs are also reduced due to the ROM NtD map approximation properties. Another source of performance improvement is the time step length. Properly chosen ROMs substantially improve the Courant-Friedrichs-Lewy (CFL) condition. This allows the CFL time step to approach the Nyquist limit, which is typically unattainable with traditional schemes that have the CFL time step much smaller than the Nyquist sampling rate.

math.NA↗

arXiv2014

An Extended Krylov Subspace Model-Order Reduction Technique to Simulate Wave Propagation in Unbounded Domains

In this paper we present a novel extended Krylov subspace reduced-order modeling technique to efficiently simulate time- and frequency-domain wavefields in open complex structures. To simulate the extension to infinity, we use an optimal complex-scaling method which is equivalent to an optimized perfectly matched layer in which the frequency is fixed. Wavefields propagating in strongly inhomogeneous open domains can now be modeled as a non-entire function of the complex-scaled wave operator. Since this function contains a square root singularity, we apply an extended Krylov subspace technique to construct fast converging reduced-order models. Specifically, we use a modified version of the extended Krylov subspace algorithm as proposed by Jagels and Reichel [Linear Algebra Appl., Vol. 434, pp. 1716 - 1732, 2011], since this algorithm allows us to balance the computational costs associated with computing powers of the wave operator and its inverse. Numerical experiments from electromagnetics and acoustics illustrate the performance of the method.

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