arXiv subjects

Alexander V. Mamonov

Publications and source records attributed to Alexander V. Mamonov.

At least 19 recordsLinked to original sources

arXiv2026

Model order reduction of parametric dynamical systems by slice sampling tensor completion

Recent studies have demonstrated the great potential of reduced order modeling for parametric dynamical systems using low-rank tensor decompositions (LRTD). In particular, within the framework of interpolatory tensorial reduced order models (ROM), LRTD is computed for tensors composed of snapshots of the system's solutions, where each parameter corresponds to a distinct tensor mode. This approach requires full sampling of the parameter domain on a tensor product grid, which suffers from the curse of dimensionality, making it practical only for systems with a small number of parameters. To overcome this limitation, we propose a sparse sampling of the parameter domain, followed by a low-rank tensor completion. The resulting specialized tensor completion problem is formulated for a tensor of order $C + D$, where $C$ fully sampled modes correspond to the snapshot degrees of freedom, and $D$ partially sampled modes correspond to the system's parameters. To address this non-standard tensor completion problem, we introduce a low-rank tensor format called the hybrid tensor train. Completion in this format is then integrated into an interpolatory tensorial ROM. We demonstrate the effectiveness of both the completion method and the ROM on several examples of dynamical systems derived from finite element discretizations of parabolic partial differential equations with parameter-dependent coefficients or boundary conditions.

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arXiv2026

Quantitative synthetic aperture radar inversion

We study an inverse scattering problem for monostatic synthetic aperture radar (SAR): Estimate the wave speed in a heterogeneous, isotropic and nonmagnetic medium probed by waves emitted and measured by a moving antenna. The forward map, from the wave speed to the measurements, is derived from Maxwell's equations. It is a nonlinear map that accounts for multiple scattering and it is very oscillatory at high frequencies. This makes the standard, nonlinear least squares data fitting formulation of the inverse problem difficult to solve. We introduce an alternative, two-step approach: The first step computes the nonlinear map from the measurements to an approximation of the electric field inside the unknown medium aka, the internal wave. This is done for each antenna location in a non-iterative manner. The internal wave fits the data by construction, but it does not solve Maxwell's equations. The second step uses optimization to minimize the discrepancy between the internal wave and the solution of Maxwell's equations, for all antenna locations. The optimization is iterative. The first step defines an imaging function whose computational cost is comparable to that of standard SAR imaging, but it gives a better estimate of the support of targets. Further iterations improve the quantitative estimation of the wave speed. We assess the performance of the method with numerical simulations and compare the results with those of standard inversion.

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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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arXiv2025

A variational Lippmann-Schwinger-type approach for the Helmholtz impedance problem on bounded domains

Recently, reduced order modeling methods have been applied to solving inverse boundary value problems arising in frequency domain scattering theory. A key step in projection-based reduced order model methods is the use of a sesquilinear form associated with the forward boundary value problem. However, in contrast to scattering problems posed in $\mathbb{R}^d$, boundary value formulations lose certain structural properties, most notably the classical Lippmann-Schwinger integral equation is no longer available. In this paper we derive a Lippmann-Schwinger type equation aimed at studying the solution of a Helmholtz boundary value problem with a variable refractive index and impedance boundary conditions. In particular, we start from the variational formulation of the boundary value problem and we obtain an equivalent operator equation which can be viewed as a bounded domain analogue of the classical Lippmann-Schwinger equation. We first establish analytical properties of our variational Lippmann-Schwinger type operator. Based on these results, we then show that the parameter-to-state map, which maps a refractive index to the corresponding wavefield, maps weakly convergent sequences to strongly convergent ones when restricted to refractive indices in Lebesgue spaces with exponent greater than 2. Finally, we use the derived weak to strong sequential continuity to show existence of minimizers for a reduced order model based optimization methods aimed at solving the inverse boundary value problem as well as for a conventional data misfit based waveform inversion method.

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arXiv2025

Waveform inversion with a data driven estimate of the internal wave

We study an inverse problem for the wave equation, concerned with estimating the wave speed, aka velocity, from data gathered by an array of sources and receivers that emit probing signals and measure the resulting waves. The typical mathematical formulation of velocity estimation is a nonlinear least squares minimization of the data misfit, over a search velocity space. There are two main impediments to this approach, which manifest as multiple local minima of the objective function: The nonlinearity of the mapping from the velocity to the data, which accounts for multiple scattering effects, and poor knowledge of the kinematics (smooth part of the wave speed) which causes cycle-skipping. We show that the nonlinearity can be mitigated using a data driven estimate of the internal wave field. This leads to improved performance of the inversion for a reasonable initial guess of the kinematics.

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arXiv2025

Inverse scattering for Schrödinger equation in the frequency domain via data-driven reduced order modeling

In this paper we develop a numerical method for solving an inverse scattering problem of estimating the scattering potential in a Schrödinger equation from frequency domain measurements based on reduced order models (ROM). The ROM is a projection of Schrödinger operator onto a subspace spanned by its solution snapshots at certain wavenumbers. Provided the measurements are performed at these wavenumbers, the ROM can be constructed in a data-driven manner from the measurements on a surface surrounding the scatterers. Once the ROM is computed, the scattering potential can be estimated using non-linear optimization that minimizes the ROM misfit. Such an approach typically outperforms the conventional methods based on data misfit minimization. We develop two variants of ROM-based algorithms for inverse scattering and test them on a synthetic example in two spatial dimensions.

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arXiv2025

Reduced Order Modeling for First Order Hyperbolic Systems with Application to Multiparameter Acoustic Waveform Inversion

Waveform inversion seeks to estimate an inaccessible heterogeneous medium from data gathered by sensors that emit probing signals and measure the generated waves. It is an inverse problem for a second order wave equation or a first order hyperbolic system, with the sensor excitation modeled as a forcing term and the heterogeneous medium described by unknown, spatially variable coefficients. The traditional ``full waveform inversion" (FWI) formulation estimates the unknown coefficients via minimization of the nonlinear, least squares data fitting objective function. For typical band-limited and high frequency data, this objective function has spurious local minima near and far from the true coefficients. Thus, FWI implemented with gradient based optimization algorithms may fail, even for good initial guesses. Recently, it was shown that it is possible to obtain a better behaved objective function for wave speed estimation, using data driven reduced order models (ROMs) that capture the propagation of pressure waves, governed by the classic second order wave equation. Here we introduce ROMs for vectorial waves, satisfying a general first order hyperbolic system. They are defined via Galerkin projection on the space spanned by the wave snapshots, evaluated on a uniform time grid with appropriately chosen time step. Our ROMs are data driven: They are computed in an efficient and non-iterative manner, from the sensor measurements, without knowledge of the medium and the snapshots. The ROM computation applies to any linear waves in lossless and non-dispersive media. For the inverse problem we focus attention on acoustic waves in a medium with unknown variable wave speed and density. We show that these can be determined via minimization of an objective function that uses a ROM based approximation of the vectorial wave field inside the inaccessible medium.

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arXiv2025

Autoencoded UMAP-Enhanced Clustering for Unsupervised Learning

We propose a novel approach to unsupervised learning by constructing a non-linear embedding of the data into a low-dimensional space followed by any conventional clustering algorithm. The embedding promotes clusterability of the data and is comprised of two mappings: the encoder of an autoencoder neural network and the output of UMAP algorithm. The autoencoder is trained with a composite loss function that incorporates both a conventional data reconstruction as a regularization component and a clustering-promoting component built using the spectral graph theory. The two embeddings and the subsequent clustering are integrated into a three-stage unsupervised learning framework, referred to as Autoencoded UMAP-Enhanced Clustering (AUEC). When applied to MNIST data, AUEC significantly outperforms the state-of-the-art techniques in terms of clustering accuracy.

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arXiv2024

When data driven reduced order modeling meets full waveform inversion

Waveform inversion is concerned with estimating a heterogeneous medium, modeled by variable coefficients of wave equations, using sources that emit probing signals and receivers that record the generated waves. It is an old and intensively studied inverse problem with a wide range of applications, but the existing inversion methodologies are still far from satisfactory. The typical mathematical formulation is a nonlinear least squares data fit optimization and the difficulty stems from the non-convexity of the objective function that displays numerous local minima at which local optimization approaches stagnate. This pathological behavior has at least three unavoidable causes: (1) The mapping from the unknown coefficients to the wave field is nonlinear and complicated. (2) The sources and receivers typically lie on a single side of the medium, so only backscattered waves are measured. (3) The probing signals are band limited and with high frequency content. There is a lot of activity in the computational science and engineering communities that seeks to mitigate the difficulty of estimating the medium by data fitting. In this paper we present a different point of view, based on reduced order models (ROMs) of two operators that control the wave propagation. The ROMs are called data driven because they are computed directly from the measurements, without any knowledge of the wave field inside the inaccessible medium. This computation is non-iterative and uses standard numerical linear algebra methods. The resulting ROMs capture features of the physics of wave propagation in a complementary way and have surprisingly good approximation properties that facilitate waveform inversion. In this arxiv version two important typos are corrected when compared to the published version. The typo was in the second equation in Theorem 3 and carried over into Corollary 1. The proofs are correct.

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arXiv2024

A priori analysis of a tensor ROM for parameter dependent parabolic problems

A space-time-parameters structure of parametric parabolic PDEs motivates the application of tensor methods to define reduced order models (ROMs). Within a tensor-based ROM framework, the matrix SVD - a traditional dimension reduction technique - yields to a low-rank tensor decomposition (LRTD). Such tensor extension of the Galerkin proper orthogonal decomposition ROMs (POD-ROMs) benefits both the practical efficiency of the ROM and its amenability for rigorous error analysis when applied to parametric PDEs. The paper addresses the error analysis of the Galerkin LRTD-ROM for an abstract linear parabolic problem that depends on multiple physical parameters. An error estimate for the LRTD-ROM solution is proved, which is uniform with respect to problem parameters and extends to parameter values not in a sampling/training set. The estimate is given in terms of discretization and sampling mesh properties, and LRTD accuracy. The estimate depends on the local smoothness rather than on the Kolmogorov n-widths of the parameterized manifold of solutions. Theoretical results are illustrated with several numerical experiments.

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arXiv2023

Tensorial parametric model order reduction of nonlinear dynamical systems

For a nonlinear dynamical system that depends on parameters, the paper introduces a novel tensorial reduced-order model (TROM). The reduced model is projection-based, and for systems with no parameters involved, it resembles proper orthogonal decomposition (POD) combined with the discrete empirical interpolation method (DEIM). For parametric systems, TROM employs low-rank tensor approximations in place of truncated SVD, a key dimension-reduction technique in POD with DEIM. Three popular low-rank tensor compression formats are considered for this purpose: canonical polyadic, Tucker, and tensor train. The use of multilinear algebra tools allows the incorporation of information about the parameter dependence of the system into the reduced model and leads to a POD-DEIM type ROM that (i) is parameter-specific (localized) and predicts the system dynamics for out-of-training set (unseen) parameter values, (ii) mitigates the adverse effects of high parameter space dimension, (iii) has online computational costs that depend only on tensor compression ranks but not on the full-order model size, and (iv) achieves lower reduced space dimensions compared to the conventional POD-DEIM ROM. The paper explains the method, analyzes its prediction power, and assesses its performance for two specific parameter-dependent nonlinear dynamical systems.

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arXiv2022

Waveform inversion via reduced order modeling

We introduce a novel approach to waveform inversion, based on a data driven reduced order model (ROM) of the wave operator. The presentation is for the acoustic wave equation, but the approach can be extended to elastic or electromagnetic waves. The data are time resolved measurements of the pressure wave gathered by an acquisition system which probes the unknown medium with pulses and measures the generated waves. We propose to solve the inverse problem of velocity estimation by minimizing the square misfit between the ROM computed from the recorded data and the ROM computed from the modeled data, at the current guess of the velocity. We give the step by step computation of the ROM, which depends nonlinearly on the data and yet can be obtained from them in a non-iterative fashion, using efficient methods from linear algebra. We also explain how to make the ROM robust to data inaccuracy. The ROM computation requires the full array response matrix gathered with collocated sources and receivers. However, we show that the computation can deal with an approximation of this matrix, obtained from towed-streamer data using interpolation and reciprocity on-the-fly. While the full-waveform inversion approach of nonlinear least-squares data fitting is challenging without low frequency information, due to multiple minima of the data fit objective function, we show that the ROM misfit objective function has a better behavior, even for a poor initial guess. We also show by an explicit computation of the objective functions in a simple setting that the ROM misfit objective function has convexity properties, whereas the least squares data fit objective function displays multiple local minima.

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arXiv2022

Velocity estimation via model order reduction

A novel approach to full waveform inversion (FWI), based on a data driven reduced order model (ROM) of the wave equation operator is introduced. The unknown medium is probed with pulses and the time domain pressure waveform data is recorded on an active array of sensors. The ROM, a projection of the wave equation operator is constructed from the data via a nonlinear process and is used for efficient velocity estimation. While the conventional FWI via nonlinear least-squares data fitting is challenging without low frequency information, and prone to getting stuck in local minima (cycle skipping), minimization of ROM misfit is behaved much better, even for a poor initial guess. For low-dimensional parametrizations of the unknown velocity the ROM misfit function is close to convex. The proposed approach consistently outperforms conventional FWI in standard synthetic tests.

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arXiv2022

Interpolatory tensorial reduced order models for parametric dynamical systems

The paper introduces a reduced order model (ROM) for numerical integration of a dynamical system which depends on multiple parameters. The ROM is a projection of the dynamical system on a low dimensional space that is both problem-dependent and parameter-specific. The ROM exploits compressed tensor formats to find a low rank representation for a sample of high-fidelity snapshots of the system state. This tensorial representation provides ROM with an orthogonal basis in a universal space of all snapshots and encodes information about the state variation in parameter domain. During the online phase and for any incoming parameter, this information is used to find a reduced basis that spans a parameter-specific subspace in the universal space. The computational cost of the online phase then depends only on tensor compression ranks, but not on space or time resolution of high-fidelity computations. Moreover, certain compressed tensor formats enable to avoid the adverse effect of parameter space dimension on the online costs (known as the curse of dimension). The analysis of the approach includes an estimate for the representation power of the acquired ROM basis. We illustrate the performance and prediction properties of the ROM with several numerical experiments, where tensorial ROM's complexity and accuracy is compared to those of conventional POD-ROM.

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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

Reduced order model approach for imaging with waves

We introduce a novel, computationally inexpensive approach for imaging with an active array of sensors, which probe an unknown medium with a pulse and measure the resulting waves. The imaging function uses a data driven estimate of the "internal wave" originating from the vicinity of the imaging point and propagating to the sensors through the unknown medium. We explain how this estimate can be obtained using a reduced order model (ROM) for the wave propagation. We analyze the imaging function, connect it to the time reversal process and describe how its resolution depends on the aperture of the array, the bandwidth of the probing pulse and the medium through which the waves propagate. We also show how the internal wave can be used for selective focusing of waves at points in the imaging region. This can be implemented experimentally and can be used for pixel scanning imaging. We assess the performance of the imaging methods with numerical simulations and compare them to the conventional reverse-time migration method and the "backprojection" method introduced recently as an application of the same ROM.

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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.

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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.

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