In computational physics, machine learning has now emerged as a powerful complementary tool to explore efficiently candidate designs in engineering studies. Outputs in such supervised problems are signals defined on meshes, and a natural question is the extension of general scalar output regression models to such complex outputs. Changes between input geometries in terms of both size and adjacency structure in particular make this transition non-trivial. In this work, we propose an innovative strategy for Gaussian process regression where inputs are large and sparse graphs with continuous node attributes and outputs are signals defined on the nodes of the associated inputs. The methodology relies on the combination of regularized optimal transport, dimension reduction techniques, and the use of Gaussian processes indexed by graphs. In addition to enabling signal prediction, the main point of our proposal is to come with confidence intervals on node values, which is crucial for uncertainty quantification and active learning. Numerical experiments highlight the efficiency of the method to solve real problems in fluid dynamics and solid mechanics.

Designing algorithms for solving high-dimensional Bayesian inverse problems directly in infinite‑dimensional function spaces – where such problems are naturally formulated – is crucial to ensure stability and convergence as the discretization of the underlying problem is refined. In this paper, we contribute to this line of work by analyzing a widely used sampler for linear inverse problems: Langevin dynamics driven by score‑based generative models (SGMs) acting as priors, formulated directly in function space. Building on the theoretical framework for SGMs in Hilbert spaces, we give a rigorous definition of this sampler in the infinite-dimensional setting and derive, for the first time, error estimates that explicitly depend on the approximation error of the score. As a consequence, we obtain sufficient conditions for global convergence in Kullback–Leibler divergence on the underlying function space. Preventing numerical instabilities requires preconditioning of the Langevin algorithm and we prove the existence and form of an optimal preconditioner. The preconditioner depends on both the score error and the forward operator and guarantees a uniform convergence rate across all posterior modes. Our analysis applies to both Gaussian and a general class of non‑Gaussian priors. Finally, we present examples that illustrate and validate our theoretical findings.

Abstract. 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 noniterative manner, from the sensor measurements, without knowledge of the medium and the snapshots. The ROM computation applies to any linear waves in lossless and nondispersive 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. We assess the performance of our inversion approach with numerical simulations and compare the results to those given by FWI.

Supervised learning has recently garnered significant attention in the field of computational physics due to its ability to effectively extract complex patterns for tasks like solving partial differential equations, or predicting material properties. Traditionally, such datasets consist of inputs given as meshes with a large number of nodes representing the problem geometry (seen as graphs), and corresponding outputs obtained with a numerical solver. This means the supervised learning model must be able to handle large and sparse graphs with continuous node attributes. In this work, we focus on Gaussian process regression, for which we introduce the Sliced Wasserstein Weisfeiler-Lehman (SWWL) graph kernel. In contrast to existing graph kernels, the proposed SWWL kernel enjoys positive definiteness and a drastic complexity reduction, which makes it possible to process datasets that were previously impossible to handle. The new kernel is first validated on graph classification for molecular datasets, where the input graphs have a few tens of nodes. The efficiency of the SWWL kernel is then illustrated on graph regression in computational fluid dynamics and solid mechanics, where the input graphs are made up of tens of thousands of nodes.

Abstract. We present an extension of the linear sampling method for solving the sound-soft inverse scattering problem in two dimensions with data generated by randomly distributed small scatterers. The theoretical justification of our novel sampling method is based on a rigorous asymptotic model, a modified Helmholtz–Kirchhoff identity, and our previous work on the linear sampling method for random sources. Our numerical implementation incorporates boundary elements, singular value decomposition, Tikhonov regularization, and Morozov’s discrepancy principle. We showcase the robustness and accuracy of our algorithms with a series of numerical experiments.

Conditional Average Treatment Effects (CATE) estimation is one of the main challenges in causal inference with observational data. In addition to Machine Learning based-models, nonparametric estimators called meta-learners have been developed to estimate the CATE with the main advantage of not restraining the estimation to a specific supervised learning method. This task becomes, however, more complicated when the treatment is not binary as some limitations of the naive extensions emerge. This paper looks into meta-learners for estimating the heterogeneous effects of multi-valued treatments. We consider different meta-learners, and we carry out a theoretical analysis of their error upper bounds as functions of important parameters such as the number of treatment levels, showing that the naive extensions do not always provide satisfactory results. We introduce and discuss meta-learners that perform well as the number of treatments increases. We empirically confirm the strengths and weaknesses of those methods with synthetic and semi-synthetic datasets.

Since their initial introduction, score-based diffusion models (SDMs) have been successfully applied to solve a variety of linear inverse problems in finite-dimensional vector spaces due to their ability to efficiently approximate the posterior distribution. However, using SDMs for inverse problems in infinite-dimensional function spaces has only been addressed recently, primarily through methods that learn the unconditional score. While this approach is advantageous for some inverse problems, it is mostly heuristic and involves numerous computationally costly forward operator evaluations during posterior sampling. To address these limitations, we propose a theoretically grounded method for sampling from the posterior of infinite-dimensional Bayesian linear inverse problems based on amortized conditional SDMs. In particular, we prove that one of the most successful approaches for estimating the conditional score in finite dimensions—the conditional denoising estimator—can also be applied in infinite dimensions. A significant part of our analysis is dedicated to demonstrating that extending infinite-dimensional SDMs to the conditional setting requires careful consideration, as the conditional score typically blows up for small times, contrarily to the unconditional score. We conclude by presenting stylized and large-scale numerical examples that validate our approach, offer additional insights, and demonstrate that our method enables large-scale, discretization-invariant Bayesian inference.

Abstract. We study an inverse problem for the wave equation, concerned with estimating the wave speed from data gathered by an array of sources and receivers that emit probing signals and measure the resulting waves. The typical approach to solving this problem is a nonlinear least squares minimization of the data misfit, over a search 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 wave speed 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 wave field at points inside the medium, also known as the “internal wave field.” This leads to improved performance of the inversion for a reasonable initial guess of the kinematics.

Abstract. We present an extension of the linear sampling method for solving the sound-soft inverse acoustic scattering problem with randomly distributed point sources. The theoretical justification of our sampling method is based on the Helmholtz–Kirchhoff identity, the cross-correlation between measurements, and the volume and imaginary near-field operators, which we introduce and analyze. Implementations in MATLAB using boundary elements, the SVD, Tikhonov regularization, and Morozov’s discrepancy principle are also discussed. We demonstrate the robustness and accuracy of our algorithms with several numerical experiments in two dimensions.

The number of complex infrastructures in an industrial setting is growing and is not immune to unexplained recurring events such as breakdowns or failure that can have an economic and environmental impact. To understand these phenomena, sensors have been placed on the different infrastructures to track, monitor, and control the dynamics of the systems. The causal study of these data allows predictive and prescriptive maintenance to be carried out. It helps to understand the appearance of a problem and find counterfactual outcomes to better operate and defuse the event. In this paper, we introduce a novel approach combining the case-crossover design which is used to investigate acute triggers of diseases in epidemiology, and the Apriori algorithm which is a data mining technique allowing to find relevant rules in a dataset. The resulting time series causal algorithm extracts interesting rules in our application case which is a non-linear time series dataset. In addition, a predictive rule-based algorithm demonstrates the potential of the proposed method.

This paper considers wave-based imaging through a heterogeneous (random) scattering medium. The goal is to estimate the support of the reflectivity function of a remote scene from measurements of the backscattered wave field. The proposed imaging methodology is based on the coherent interferometric (CINT) approach that exploits the local empirical cross correlations of the measurements of the wave field. The standard CINT images are known to be robust (statistically stable) with respect to the random medium, but the stability comes at the expense of a loss of resolution. This paper shows that a two-point CINT function contains the information needed to obtain statistically stable and high-resolution images. Different methods to build such images are presented, theoretically analyzed, and compared with the standard imaging approaches using numerical simulations. The first method involves a phase retrieval step to extract the reflectivity function from the modulus of its Fourier transform. The second method involves the evaluation of the leading eigenvector of the two-point CINT imaging function seen as the kernel of a linear operator. The third method uses an optimization step to extract the reflectivity function from some cross products of its Fourier transform. The presentation is for the synthetic aperture radar data acquisition setup, where a moving sensor probes the scene with signals emitted periodically and records the resulting backscattered wave. The generalization to other imaging setups, with passive or active arrays of sensors, is discussed briefly.

The goal of synthetic aperture imaging is to estimate the reflectivity of a remote region of interest by processing data gathered with a moving sensor which emits periodically a signal and records the backscattered wave. We introduce and analyze a high-resolution interferometric method for synthetic aperture imaging through an unknown scattering medium which distorts the wave. The method builds on the coherent interferometric approach which uses empirical cross-correlations of the measurements to mitigate the distortion, at the expense of a loss of resolution of the image. The new method shows that, while mitigating the wave distortion, it is possible to obtain a robust and sharp estimate of the modulus of the Fourier transform of the reflectivity function. A high-resolution image can then be obtained by a phase retrieval algorithm.

In this paper we introduce and analyze a passive synthetic aperture radar system motivated by space surveillance radar networks for detecting, tracking, imaging, and identifying small debris (aka target) in low-earth orbit. We propose a system with a powerful transmitter on the ground and one or several flying receiver platforms. Each platform can separate the direct signals from the source, coming from below, and the reflected signals, coming from above. In the case of a single receiver platform, the image is formed by cross correlating the Doppler-compensated direct and reflected signals. We show that its resolution is described by the ideal Rayleigh resolution formulas. That is, range resolution is proportional to the transmitter pulse width, and the cross-range resolutions are determined by the synthetic angular cones determined by the target and receiver trajectories. However, the one-receiver imaging modality is not capable of determining both the target location and velocity. In the case of multiple receiver platforms, the image is computed by cross correlating the Doppler-compensated reflected signals. In this case, the image focusing relies on differences of travel times between the moving target and the receivers, and we show that two well separated pairs of receiver platforms are sufficient for determining the target location and velocity. The resolution formulas for this imaging modality are new and different from the Rayleigh resolution limits. They are derived analytically from first principles and are validated with detailed numerical simulations.

In this paper we consider the problem of imaging a fast moving small object. The imaging system consists of a powerful emitter and several passive receivers located on the ground. Our aim is to compare the well-known matched-filter imaging method with correlation-based imaging. Imaging with correlations has the advantage of not requiring any knowledge about the probing pulse and the emitter position, both assumed known with high accuracy in the case of matched-filter imaging. But correlation-based imaging requires recording fully resolved signals without down-ramping while matched-filer imaging does not. To account for the fast moving target's velocity, Doppler compensation is necessary for both imaging methods. Our resolution analysis, from first principles, shows that with the two methods we have similar resolution in the cross-range direction, for both the location and the velocity of the moving target. In the range direction, matched-filter imaging has better resolution mainly because it benefits from the signal bandwidth, which is not true for correlation-based imaging that relies on travel time differences. We also analyze the role of the number of receivers and show that a small number of them sparsely distributed provides an image with resolution close to the one obtained with a dense array of comparable overall size.

We consider passive synthetic aperture imaging where a single moving receiver antenna records signals that are generated by distant unknown noise sources and backscattered by one or several reflectors. The reflectors can be imaged by migrating the autocorrelation functions of the received signals. We compare this passive synthetic aperture imaging with the usual, active synthetic aperture imaging. In the usual synthetic aperture imaging the moving receiver antenna is also a transmitter, and imaging is done by application of a matched filter to the recorded signals. We show that image resolution is the same in both active and passive synthetic aperture imaging, provided that the illumination in the passive case is rich enough.

We consider correlation-based imaging of a reflector located on one side of a passive array where the medium is homogeneous. On the other side of the array the illumination by remote impulsive sources goes through a strongly scattering medium. It has been shown in [J. Garnier and G. Papanicolaou, Inverse Problems, 28 (2012), 075002] that migrating the cross correlations of the passive array gives an image whose resolution is as good as if the array was active and the array response matrix was that of a homogeneous medium. In this paper we study the signal to noise ratio (SNR) of the image as a function of statistical properties of the strongly scattering medium, the signal bandwidth, and the source and passive receiver array characteristics. Using a Kronecker model for the strongly scattering medium we show that image resolution is as expected and that the SNR can be computed in an essentially explicit way. We show with direct numerical simulations using full wave propagation solvers in random media that the theoretical predictions based on the Kronecker model are accurate.

In this paper, the detection of a small reflector in a randomly heterogeneous medium using second-harmonic generation is investigated. The medium is illuminated by a time-harmonic plane wave at frequency $\omega$. It is assumed that the reflector has a nonzero second-order nonlinear susceptibility, and thus emits a wave at frequency $2\omega$ in addition to the fundamental frequency linear scattering. It is shown how the fundamental frequency signal and the second-harmonic signal propagate in the medium. A statistical study of the images obtained by migrating the boundary data is performed. It is proved that the second-harmonic image is more stable with respect to medium noise than the one obtained with the fundamental signal. Moreover, the signal-to-noise ratio for the second-harmonic image does not depend either on the second-order susceptibility tensor or on the volume of the particle.

We consider imaging in a scattering medium where the illumination goes through this medium but there is also an auxiliary, passive receiver array that is near the object to be imaged. Instead of imaging with the source-receiver array on the far side of the object, we image with the data of the passive array on the near side of the object. The imaging is done with travel time migration using the cross correlations of the passive array data. We showed in [J. Garnier and G. Papanicolaou, Inverse Problems, 28 (2012), 075002] that if (i) the source array is infinite, (ii) the scattering medium is modeled by either an isotropic random medium in the paraxial regime or a randomly layered medium, and (iii) the medium between the auxiliary array and the object to be imaged is homogeneous, then imaging with cross correlations completely eliminates the effects of the random medium. It is as if we imaged with an active array, instead of a passive one, near the object. The purpose of this paper is to analyze the resolution of the image when both the source array and the passive receiver array are finite. We show with a detailed analysis that for isotropic random media in the paraxial regime, not only is imaging not affected by the inhomogeneities, but the resolution can in fact be enhanced. This is because the random medium can increase the diversity of the illumination. We also show analytically that this will not happen in a randomly layered medium, and there may be some loss of resolution in this case.

In this paper, we provide a mathematical model for the electrolocation in weakly electric fishes. We first investigate the forward complex conductivity problem and derive the approximate boundary conditions on the skin of the fish. Then we provide a dipole approximation for small targets away from the fish. Based on this approximation, we obtain a noniterative location search algorithm using multifrequency measurements. We present numerical experiments to illustrate the performance and the stability of the proposed multifrequency location search algorithm. Finally, in the case of disk- and ellipse-shaped targets, we provide a method for reconstructing separately the conductivity, the permittivity, and the size of the targets from multifrequency measurements.

The focus of this work is on rigorous mathematical analysis of the topological derivative based detection algorithms for the localization of an elastic inclusion of vanishing characteristic size. A filtered quadratic misfit is considered, and the performance of the topological derivative imaging functional resulting therefrom is analyzed. Our analysis reveals that the imaging functional may not attain its maximum at the location of the inclusion. Moreover, the resolution of the image is below the diffraction limit. Both phenomena are due to the coupling of pressure and shear waves propagating with different wave speeds and polarization directions. A novel imaging functional based on the weighted Helmholtz decomposition of the topological derivative is, therefore, introduced. It is thereby substantiated that the maximum of the imaging functional is attained at the location of the inclusion and the resolution is enhanced and proves to be the diffraction limit. Finally, we investigate the stability of the proposed imaging functionals with respect to measurement and medium noises.

In this paper we consider the inverse conductivity problem in two dimensions. We apply an extended Kalman filter to track both the location and the orientation of a small target in motion from multistatic response measurements. We also analyze the effect of the limited-view aspect on the stability and the efficiency of our tracking approach. Our algorithm is based on the use of the generalized polarization tensors, which can be reconstructed from the multistatic response measurements by simply solving a linear system. The reconstruction problem of generalized polarization tensors from multistatic response measurements has the remarkable property that low order generalized polarization tensors are not affected by the error caused by the instability of higher orders in the presence of measurement noise.

We consider imaging with passive arrays of sensors using as illumination ambient noise sources. The first step for imaging under such circumstances is the computation of the cross correlations of the recorded signals, which have attracted a lot of attention recently because of their numerous applications in seismic imaging, volcano monitoring, and petroleum prospecting. Here, we use these cross correlations for imaging reflectors with travel-time migration. While the resolution of the image obtained this way has been studied in detail, an analysis of the signal-to-noise ratio (SNR) is presented in this paper along with numerical simulations that support the theoretical results. It is shown that the SNR of the image inherits the SNR of the computed cross correlations and therefore is proportional to the square root of the bandwidth of the noise sources times the recording time. Moreover, the SNR of the image is proportional to the array size. This means that the image can be stabilized by increasing the size of the array when the recorded signals are not of long duration, which is important in applications such as nondestructive testing.

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It is known that cross correlation of waves generated by noise sources, propagating in an unknown medium and recorded by a sensor array, can provide information about the medium. In this paper the medium is a three-dimensional small-scale randomly layered medium with slow macroscopic variations. The main objective here is to set forth a framework for analysis of cross correlations of waves generated by noise sources and propagating in such a medium and, moreover, to use this framework to design estimators for macroscale medium features. The noise sources are located at the bottom of a random medium slab and generate a random wave field that is scattered by the rapid random fluctuations of the medium and then recorded at the surface. Taking into account the pressure release boundary conditions at the surface, this situation corresponds to the so-called daylight configuration. The analysis is carried out in the asymptotic framework where the typical wavelength is small compared to the scale of the macroscopic variations of the background medium and large compared to the decoherence length of the microscopic random fluctuations of the medium. It is shown that the cross correlation of the waves recorded at the surface contains statistically stable information about the macroscopic background medium.

It is well known that the travel time or even the full Green's function between two passive sensors can be estimated from the cross correlation of recorded signal amplitudes generated by ambient noise sources. It is also known that the direction of the energy flux from the noise sources affects the estimation of the travel time. Using the stationary phase method, we show here that the travel time can be effectively estimated when the ray joining the two sensors continues into the noise source region. We extend this analysis to passive sensor imaging of reflectors with different ambient noise source configurations by suitably migrating the cross correlations. If in addition there is multiple scattering in the medium, then reflectors can be imaged with passive sensor networks or arrays by migrating suitable fourth-order cross correlations. Fourth-order cross correlations can also be used with auxiliary passive sensors in order to enhance travel time estimation in a scattering medium.

The first contribution of this paper is a theoretical investigation of combinatorial optimization problems. Their landscapes are specified by the set of neighborhoods of all points of the search space. The aim of the paper consists of the estimation of the number N of local optima and the distributions of the sizes $(\alpha_j)$ of their attraction basins. For different types of landscapes we give precise estimates of the size of the random sample that ensures that at least one point lies in each attraction basin. A practical methodology is then proposed for identifying these quantities (N and $(\alpha_j)$ distributions) for an unknown landscape, given a random sample of starting points and a local steepest ascent search. This methodology can be applied to any landscape specified with a modification operator and provides bounds on search complexity to detect all local optima. Experiments demonstrate the efficiency of this methodology for guiding the choice of modification operators, eventually leading to the design of problem-dependent optimization heuristics.

The behavior of a (1+1)-ES process on Rudolph's binary long k paths is investigated extensively in the asymptotic framework with respect to string length l. First, the case of k=l/sup /spl alpha// is addressed. For /spl alpha//spl ges/1/2, we prove that the long k path is a long path for the (1+1)-ES in the sense that the process follows the entire path with no shortcuts, resulting in an exponential expected convergence time. For /spl alpha/<1/2, the expected convergence time is also exponential, but some shortcuts occur in the meantime that speed up the process. Next, in the case of constant k, the statistical distribution of convergence time is calculated, and the influence of population size is investigated for different (/spl mu/+/spl lambda/)-ES. The histogram of the first hitting time of the solution shows an anomalous peak close to zero, which corresponds to an exceptional set of events that speed up the expected convergence time with a factor of l/sup 2/. A direct consequence of this exceptional set is that performing independent (1+1)-ES processes proves to be more advantageous than any population-based (/spl mu/+/spl lambda/)-ES.

In the binary evolutionary optimization framework, two mutation operators are theoretically investigated. For both the standard mutation, in which all bits are flipped independently with the same probability, and the 1-bit-flip mutation, which flips exactly one bit per bitstring, the statistical distribution of the first hitting times of the target are thoroughly computed (expectation and variance) up to terms of order l (the size of the bitstrings) in two distinct situations: without any selection, or with the deterministic (1 + l)-ES selection on the OneMax problem. In both cases, the 1-bit-flip mutation convergence time is smaller by a constant (in terms of l) multiplicative factor. These results extend to the case of multiple independent optimizers.