arXiv subjects

Darko Volkov

Publications and source records attributed to Darko Volkov.

15 recordsLinked to original sources

arXiv2025

Machine learning on manifolds for inverse scattering: Lipschitz stability analysis

Establishing Lipschitz stability estimates is crucial for ensuring the mathematical robustness of neural network (NN) approximations in machine learning (ML)-based parameter estimation, particularly in physics-informed settings. In this work, we derive such estimates for the inverse of a nonlinear map defined on a manifold that captures both unknown parameters and the nonlinear physical processes they influence. Our analysis is based on finite-dimensional, learnable representations of the manifold and provides Lipschitz stability estimates on the manifold-based subspaces, for a class of inverse maps associated with parameter dependent linear compact operators. Such operators model scattered and far-field data that can be used to detect structures such as cracks. We apply our theoretical ML manifold framework to inverse Helmholtz problems in unbounded regions exterior to cracks, addressing the scattered-field data-driven inverse problem while ensuring injectivity conditions on the manifold, a requirement for the Lipschitz stability. Our method accurately recovers crack-defining parameters without requiring prior knowledge of inputs such as incident wave types or external forces on the crack. Numerical experiments using NN approximations confirm the accuracy, efficiency, and robustness of the proposed approach.

math.NA↗

arXiv2024

An all-frequency stable integral system for Maxwell's equations in 3-D penetrable media: continuous and discrete model analysis

We introduce a new system of surface integral equations for Maxwell's transmission problem in three dimensions. This system has two remarkable features, both of which we prove. First, it is well-posed at all frequencies. Second, the underlying linear operator has a uniformly bounded inverse as the frequency approaches zero, ensuring that there is no low-frequency breakdown. The system is derived from a formulation we introduced in our previous work, which required additional integral constraints to ensure well -posedness across all frequencies. In this study, we eliminate those constraints and demonstrate that our new self adjoint, constraints-free linear system expressed in the desirable form of an identity plus a compact weakly-singular operator is stable for all frequencies. Furthermore, we propose and analyze a fully discrete numerical method for these systems and provide a proof of spectrally accurate convergence for the computational method. We also computationally demonstrate the high-order accuracy of the algorithm using benchmark scatterers with curved surfaces.

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arXiv2023

Optimal decay rates in Sobolev norms for singular values of integral operators

The regularity of integration kernels forces decay rates of singular values of associated integral operators. This is well-known for symmetric operators with kernels defined on $(a,b) \times (a,b)$, where $(a,b)$ is an interval. Over time, many authors have studied this case in detail. The case of spheres has also been resolved. A few authors have examined the higher dimensional case or the case of manifolds. Typically, these authors have provided decay estimates of singular values in $l^p$ norms, $p \geq 1$ or in case of faster decay due to regularity, $l^p$ quasi-norms, $0 < p < 1$. With that approach, it is straightforward to show that their estimates are optimal using periodic kernels obtained from Fourier series. Our new approach for deriving decay estimates of these singular values uses Weyl's asymptotic formula for Neumann eigenvalues that we combine to an appropriately defined inverse Laplacian. We obtain decay estimates in the form $n^α$ where for the $n$ -th singular value where $α$ depends on dimension and on the Sobolev regularity of the kernel. Since we are interested in optimal estimates in case of regular kernels, instead of writing an upper bound by a constant times $n^α$, we use the singular values of the kernel obtained by differentiation. While in \cite{birman1977estimates, delgado2021schatten} $l^p$ estimates were proven to be optimal by simply considering periodic Fourier series, these Fourier series do not provide sharp results for our estimates. Instead, series of Neumann eigenfunctions for the Laplacian that are specific to the domain of interest are used to prove that our decay estimates are optimal. Finally, we cover the case of real analytic kernels where we are also able to derive optimal estimates.

math.FA↗

arXiv2023

On the crack inverse problem for pressure waves in half-space

After formulating the pressure wave equation in half-space minus a crack with a zero Neumann condition on the top plane, we introduce a related inverse problem. That inverse problem consists of identifying the crack and the unknown forcing term on that crack from overdetermined boundary data on a relatively open set of the top plane. This inverse problem is not uniquely solvable unless some additional assumption is made. However, we show that we can differentiate two cracks $Γ_1$ and $Γ_2$ under the assumption that $\RR^3 \setminus \overline{Γ_1\cup Γ_2}$ is connected. If that is not the case we provide counterexamples that demonstrate non-uniqueness, even if $Γ_1$ and $Γ_2$ are smooth and "almost" flat. Finally, we show in the case where $\RR^3 \setminus \overline{Γ_1\cup Γ_2}$ is not necessarily connected that after excluding a discrete set of frequencies, $Γ_1$ and $Γ_2$ can again be differentiated from overdetermined boundary data.

math.AP↗

arXiv2023

Stability properties for a class of inverse problems

We establish Lipschitz stability properties for a class of inverse problems. In that class, the associated direct problem is formulated by an integral operator Am depending non-linearly on a parameter m and operating on a function u. In the inversion step both u and m are unknown but we are only interested in recovering m. We discuss examples of such inverse problems for the elasticity equation with applications to seismology and for the inverse scattering problem in electromagnetic theory. Assuming a few injectivity and regularity properties for Am, we prove that the inverse problem with a finite number of data points is solvable and that the solution is Lipschitz stable in the data. We show a reconstruction example illustrating the use of neural networks.

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arXiv2022

A parallel sampling algorithm for inverse problems with linear and nonlinear unknowns

We derive a parallel sampling algorithm for computational inverse problems that present an unknown linear forcing term and a vector of nonlinear parameters to be recovered. It is assumed that the data is noisy and that the linear part of the problem is ill-posed. The vector of nonlinear parameters m is modeled as a random variable. A dilation parameter alpha is used to scale the regularity of the linear unknown and is also modeled as a random variable. A posterior probability distribution for (m; alpha) is derived following an approach related to the maximum likelihood regularization parameter selection [5]. A major difference in our approach is that, unlike in [5], we do not limit ourselves to the maximum likelihood value of alpha. We then derive a parallel sampling algorithm where we alternate computing proposals in parallel and combining proposals to accept or reject them as in [4]. This algorithm is well-suited to problems where proposals are expensive to compute. We then apply it to an inverse problem in seismology. We show how our results compare favorably to those obtained from the Maximum Likelihood (ML), the Generalized Cross Validation (GCV), and the Constrained Least Squares (CLS) algorithms.

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arXiv2021

Stability properties of a crack inverse problem in half space

We show in this paper a Lipschitz stability result for a crack inverse problem in half space. The direct problem is a Laplace equation with zero Neumann condition on the top boundary. The forcing term is a discontinuity across the crack. This formulation can be related to geological faults in elastic media or to irrotational incompressible flows in a half space minus an inner wall. The direct problem is well posed in an appropriate functional space. We study the related inverse problem where the jump across the crack is unknown, and more importantly, the geometry and the location of the crack are unknown. The data for the inverse problem is of Dirichlet type over a portion of the top boundary. We prove that this inverse problem is uniquely solvable under some assumptions on the geometry of the crack. The highlight of this paper is showing a stability result for this inverse problem. Assuming that the crack is planar, we show that reconstructing the plane containing the crack is Lipschitz stable despite the fact that the forcing term for the underlying PDE is unknown. This uniform stability result holds under the assumption that the forcing term is bounded above and the Dirichlet data is bounded below away from zero in appropriate norms.

math.AP↗

arXiv2021

A stochastic algorithm for fault inverse problems in elastic half space with proof of convergence

A general stochastic algorithm for solving mixed linear and nonlinear problems was introduced in [11]. We show in this paper how it can be used to solve the fault inverse problem, where a planar fault in elastic half-space and a slip on that fault have to be reconstructed from noisy surface displacement measurements. With the parameter giving the plane containing the fault denoted by m and the regularization parameter for the linear part of the inverse problem denoted by C, both modeled as random variables, we derive a formula for the posterior marginal of m. Modeling C as a random variable allows to sweep through a wide range of possible values which was shown to be superior to selecting a fixed value [11]. We prove that this posterior marginal of m is convergent as the number of measurement points and the dimension of the space for discretizing slips increase. Simply put, our proof only assumes that the regularized discrete error functional for processing measurements relates to an order 1 quadrature rule and that the union of the finite-dimensional spaces for discretizing slips is dense. Our proof relies on trace class operator theory to show that an adequate sequence of determinants is uniformly bounded. We also explain how our proof can be extended to a whole class of inverse problems, as long as some basic requirements are met. Finally, we show numerical simulations that illustrate the numerical convergence of our algorithm.

math.NA↗

arXiv2019

Stochastic solutions to mixed linear and nonlinear inverse problems

We derive an efficient stochastic algorithm for inverse problems that present an unknown linear forcing term and a set of nonlinear parameters to be recovered. It is assumed that the data is noisy and that the linear part of the problem is ill-posed. The vector of nonlinear parameters to be recovered is modeled as a random variable. This random vector is augmented by a random regularization parameter for the linear part. A probability distribution function for this augmented random vector knowing the measurements is derived. The derivation is based on the maximum likelihood regularization parameter selection which we generalize to the case where the underlying linear operator is rectangular and depends on a nonlinear parameter. Unlike in previous studies, we do not limit ourselves to the most likely regularization parameter, instead we show that due to the dependence of the problem on the nonlinear parameter there is a great advantage in exploring all positive values of the regularization parameter. Based on our new probability distribution function, we construct a propose and accept or reject algorithm to compute the posterior expected value and covariance of the nonlinear parameter. This algorithm is greatly accelerated by using a parallel platform where we alternate computing proposals in parallel and combining proposals to accept or reject them. Finally, our new algorithm is illustrated by solving an inverse problem in seismology. We show that the results obtained by our algorithm are more accurate than those found using Generalized Cross Validation or using the discrepancy principle, and that our algorithm has the capability to quantify uncertainty.

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arXiv2018

Stability estimates for the fault inverse problem

We study in this paper stability estimates for the fault inverse problem. In this problem, faults are assumed to be planar open surfaces in a half space elastic medium with known Lamé coefficients. A traction free condition is imposed on the boundary of the half space. Displacement fields present jumps across faults, called slips, while traction derivatives are continuous. It was proved in \cite{volkov2017reconstruction} that if the displacement field is known on an open set on the boundary of the half space, then the fault and the slip are uniquely determined. In this present paper, we study the stability of this uniqueness result with regard to the coefficients of the equation of the plane containing the fault. If the slip field is known we state and prove a Lipschitz stability result. In the more interesting case where the slip field is unknown, we state and prove another Lipschitz stability result under the additional assumption, which is still physically relevant, that the slip field is one directional.

math.AP↗

arXiv2018

A stochastic approach to reconstruction of faults in elastic half space

We introduce in this study an algorithm for the imaging of faults and of slip fields on those faults. The physics of this problem are modeled using the equations of linear elasticity. We define a regularized functional to be minimized for building the image. We first prove that the minimum of that functional converges to the unique solution of the related fault inverse problem. Due to inherent uncertainties in measurements, rather than seeking a deterministic solution to the fault inverse problem, we then consider a Bayesian approach. In this approach the geometry of the fault is assumed to be planar, it can thus be modeled by a three dimensional random variable whose probability density has to be determined knowing surface measurements. The randomness involved in the unknown slip is teased out by assuming independence of the priors, and we show how the regularized error functional introduced earlier can be used to recover the probability density of the geometry parameter. The advantage of the Bayesian approach is that we obtain a way of quantifying uncertainties as part of our final answer. On the downside, this approach leads to a very large computation since the slip is unknown. To contend with the size of this computation we developed an algorithm for the numerical solution to the stochastic minimization problem which can be easily implemented on a parallel multi-core platform and we discuss techniques aimed at saving on computational time. After showing how this algorithm performs on simulated data, we apply it to measured data. The data was recorded during a slow slip event in Guerrero, Mexico.

math.AP↗

arXiv2014

Detection and classification from electromagnetic induction data

In this paper we introduce an efficient algorithm for identifying conductive objects using induction data derived from eddy currents. Our method consists of first extracting geometric features from the induction data and then matching them to precomputed data for known objects from a given dictionary. The matching step relies on fundamental properties of conductive polarization tensors and new invariants introduced in this paper. A new shape identification scheme is introduced and studied. We test it numerically in the presence of measurement noise. Stability and resolution capabilities of the proposed identification algorithm are quantified in numerical simulations.

math.AP↗

arXiv2014

Preferred Frequencies for Coupling of Seismic Waves and Vibrating Tall Buildings

We study a model for the so called "city effect" in which an earthquake can be locally enhanced by the collective response of tall buildings in a large city. We use a set of equations coupling vibrations in buildings to motion under the ground. These equations were previously studied exclusively in the case of a finite set of identical, equally spaced, buildings. These two restrictions are lifted in this paper. We may now simulate geometries involving infinitely many buildings as long as an initial pattern of buildings is repeated. Our new method using periodic domains and periodic Green's functions yields much faster computations. This is the main reason why we are now able to study systems of buildings of variable height, mass, and rigidity. We show how solving for the wavenumber in a non-linear equation involving the integral of a function solution to an adequate integral equation, we are able to find resonant frequencies coupling seismic waves and vibrating tall buildings. Interestingly, in the case of non identical buildings, our simulations indicate that the response to this coupling phenomenon may differ drastically from one building to another.

math.AP↗

arXiv2014

Existence of Frequency Modes Coupling Seismic Waves and Vibrating Tall Buildings

We prove in this paper an existence result for frequency modes coupling seismic waves and vibrating tall buildings. The derivation from physical principles of a set of equations modeling this phenomenon was done in previous studies. In this model all vibrations are assumed to be anti plane and time harmonic so the two dimensional Helmholtz equation can be used. A coupling frequency mode is obtained once we can determine a wavenumber such that the solution of the corresponding Helmholtz equation in the lower half plane with relevant Neumann and Dirichlet at the interface satisfies a specific integral equation at the base of an idealized tall building. Although numerical simulations suggest that such wavenumbers should exist, as far as we know, to date, there is no theoretical proof of existence. This is what this present study offers to provide.

math.AP↗

arXiv2013

Target Detection and Characterization from Electromagnetic Induction Data

The goal of this paper is to contribute to the field of nondestructive testing by eddy currents. We provide a mathematical analysis and a numerical framework for simulating the imaging of arbitrarily shaped small volume conductive inclusions from electromagnetic induction data. We derive, with proof, a small-volume expansion of the eddy current data measured away from the conductive inclusion. The formula involves two polarization tensors: one associated with the magnetic contrast and the second with the conductivity of the inclusion. Based on this new formula, we design a location search algorithm. We include in this paper a discussion on data sampling, noise reduction, and on probability of detection. We provide numerical examples that support our findings.

math.AP↗