Multiscale S-Fraction Reduced-Order Models for Massive Wavefield Simulations

Druskin, Vladimir; Mamonov, A.; Zaslavsky, Mikhail

doi:10.1137/16m1072103

Cited by 6 publications

(8 citation statements)

References 28 publications

(35 reference statements)

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“…We describe the computation of the block-Cholesky factor L q using an approach outlined in [21]. As mentioned in section 4.5, the block Cholesky factorization is not uniquely defined.…”

Section: Appendix B Computation Of the Block-bidiagonal L Qmentioning

confidence: 99%

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

et al. 2018

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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.

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“…We describe the computation of the block-Cholesky factor L q using an approach outlined in [21]. As mentioned in section 4.5, the block Cholesky factorization is not uniquely defined.…”

Section: Appendix B Computation Of the Block-bidiagonal L Qmentioning

confidence: 99%

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

et al. 2018

Self Cite

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“…Alternatively, when the number of points is required to be small, one may use Chebyshev points (or nodes) which are the roots of the Chebyshev polynomial of the first kind. The Chebyshev points are particularly beneficial when utilizing Chebyshev polynomials as the basis functions due to the discrete orthogonality (24).…”

Section: Bisection Methods and Convex Feasibility Problemmentioning

confidence: 99%

“…Otherwise, there are several standard methods to define the abovementioned lifting see, e.g., [35, Chapter 1]. Matrix functions have drawn attention in recent years, see e.g., [1,20,26,36,46,70], and proved to be an efficient tool in applications such as reduced order models [24,28], solving ODEs [45], engineering models [21], image denoising [49] and graph neural network [44], just to name a few.…”

Section: Introductionmentioning

confidence: 99%

Flexible rational approximation for matrix functions

Sharon¹,

Peiris²,

Sukhorukova³

et al. 2021

Preprint

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A rational approximation is a powerful method for estimating functions using rational polynomial functions. Motivated by the importance of matrix function in modern applications and its wide potential, we propose a unique optimization approach to construct rational approximations for matrix function evaluation. In particular, we study the minimax rational approximation of a real function and observe that it leads to a series of quasiconvex problems. This observation opens the door for a flexible method that calculates the minimax while incorporating constraints that may enhance the quality of approximation and its properties. Furthermore, the various properties, such as denominator bounds, positivity, and more, make the output approximation more suitable for matrix function tasks. Specifically, they can guarantee the condition number of the matrix, which one needs to invert for evaluating the rational matrix function. Finally, we demonstrate the efficiency of our approach on several applications of matrix functions based on direct spectrum filtering.

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“…This would require the solution of a discretized system with O(N k ) state variables, O(N k−1 ) sources and receivers, and O(N ) frequencies or time steps [22]. Model-order reduction aims to reduce the complexity and computational burden of large-scale problems and here we target all three of these factors.Recently, promising results were obtained in the time-domain via multiscale model reduction [8,10]. The time-domain multiscale algorithms can be efficiently parallelized via domain-decomposition, but time stepping still needs to be carried out sequentially, while frequency-domain problems can be solved in parallel for different frequencies.…”

mentioning

confidence: 99%

“…Recently, promising results were obtained in the time-domain via multiscale model reduction [8,10]. The time-domain multiscale algorithms can be efficiently parallelized via domain-decomposition, but time stepping still needs to be carried out sequentially, while frequency-domain problems can be solved in parallel for different frequencies.…”

mentioning

confidence: 99%

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

Druskin¹,

Remis²,

Zaslavsky³

et al. 2018

Multiscale Model. Simul.

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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.AMS subject classifications. 65M60, 65M80, 93B11, 37M05, 78A05 1. Introduction. Numerical modeling of wave propagation is fundamental to many applications in design optimization and wavefield imaging. In the oil and gas industry, for instance, the solution of the Maxwell equations is required to invert electromagnetic measurements, while in seismic imaging the solution to the elastodynamic wave equation is needed to ultimately image the subsurface of the Earth.Finite difference discretization of the governing wave equations leads to large-scale linear systems, whose solution is computationally intense. Imaging and optimization often use multiple frequencies, sources, and receivers, which leads to systems that need to be evaluated for multiple right-hand sides, time-steps or frequencies, depending on whether the problem is solved in the time-or frequency-domain. Therefore, these so-called multiple-input/multiple-output (MIMO) systems have a high demand on memory and computational power, causing long runtimes. To be more specific, let us consider a surface seismic imaging problem in a k dimensional space (1 ≤ k ≤ 3), with maximal propagation distance of N wavelengths. This would require the solution of a discretized system with O(N k ) state variables, O(N k−1 ) sources and receivers, and O(N ) frequencies or time steps [22]. Model-order reduction aims to reduce the complexity and computational burden of large-scale pro...

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Multiscale S-Fraction Reduced-Order Models for Massive Wavefield Simulations

Cited by 6 publications

References 28 publications

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

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

Flexible rational approximation for matrix functions

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

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