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Dans le cadre de l’étude de la stabilité des barrages, la connaissance de l’interface entre le barrage et la roche revêt une grande importance. Le recours à des techniques géophysiques peut apporter des informations complémentaires par rapport aux mesures géotechniques. Nous proposons ici une méthode de traitement des mesures sismiques, l’objectif étant d'obtenir une image de l'interface entre le béton du barrage et le rocher de la fondation avec une résolution métrique. Il s’agit d’une technique de type « Full Waveform Inversion » avec optimisation de forme. Des résultats numériques utilisant des mesures synthétiques montrent la capacité de la méthode à retrouver l'interface avec une précision satisfaisante, pour un nombre limité de points de mesure et en présence de bruit.

When an inverse problem is solved by a gradient-based optimization algorithm, the corresponding forward and adjoint problems, which are introduced to compute the gradient, can be also solved iteratively. The idea of iterating at the same time on the inverse problem unknown and on the forward and adjoint problem solutions yields the concept of one-shot inversion methods. We are especially interested in the case where the inner iterations for the direct and adjoint problems are incomplete, that is, stopped before achieving a high accuracy on their solutions. Here, we focus on general linear inverse problems and generic fixed-point iterations for the associated forward problem. We analyze variants of the so-called multistep one-shot methods, in particular semi-implicit schemes with a regularization parameter. We establish sufficient conditions on the descent step for convergence by studying the eigenvalues of the block matrix of the coupled iterations. Several numerical experiments are provided to illustrate the convergence of these methods in comparison with the classical gradient descent, where the forward and adjoint problems are solved exactly by a direct solver instead. We observe that very few inner iterations are enough to guarantee good convergence of the inversion algorithm, even in the presence of noisy data. [ABSTRACT FROM AUTHOR]

We present algorithms for computing strongly singular and near-singular surface integrals over curved triangular patches, based on singularity subtraction, the continuation approach, and transplanted Gauss quadrature. We demonstrate the accuracy and robustness of our method for quadratic basis functions and quadratic triangles by integrating it into a boundary element code and solving several scattering problems in three dimensions. We also give numerical evidence that the utilization of curved boundary elements enhances computational efficiency compared to conventional planar elements. [ABSTRACT FROM AUTHOR]

We present an algorithm for the computation of scattering poles for an impenetrable obstacle with Dirichlet or Robin boundary conditions in acoustic scattering. This paper builds upon the previous work of Cakoni et al. (2020) titled 'A duality between scattering poles and transmission eigenvalues in scattering theory' (Cakoni et al. 2020 Proc. A.476, 20200612 (doi:10.1098/rspa.2020.0612)), where the authors developed a conceptually unified approach for characterizing the scattering poles and interior eigenvalues corresponding to a scattering problem. This approach views scattering poles as dual to interior eigenvalues by interchanging the roles of incident and scattered fields. In this framework, both sets are linked to the kernel of the relative scattering operator that maps incident fields to scattered fields. This mapping corresponds to the exterior scattering problem for the interior eigenvalues and the interior scattering problem for scattering poles. Leveraging this dual characterization and inspired by the generalized linear sampling method for computing the interior eigenvalues, we present a novel numerical algorithm for computing scattering poles without relying on an iterative scheme. Preliminary numerical examples showcase the effectiveness of this computational approach. [ABSTRACT FROM AUTHOR]

We consider the inverse geometrical problem of identifying the discontinuity curve of an electrical conductivity from boundary measurements. This standard inverse problem is used as a model to introduce and study a combined inversion algorithm coupling a gradient descent on the Kohn-Vogelius cost functional with a domain decomposition method that includes the unknown curve in the domain partitioning. We prove the local convergence of the method in a simplified case and numerically show its efficiency for some two dimensional experiments. [ABSTRACT FROM AUTHOR]

We introduce a new spectrum associated with the scattering from penetrable objects at a fixed frequency by employing the modified background technique. This spectrum can be used as qualitative indicator on the material properties of the medium. We prove that, as opposed to the case of classical transmission eigenvalues, the inside-outside duality method allows to reconstruct this spectrum from full aperture far field measurements at a fixed frequency. We derive a necessary and sufficient condition that characterizes the modified transmission eigenvalues using the phase of a modified far field operator constructed form measured data. The main ingredients in the analysis are the factorization of the modified far field operator and the asymptotic expansion of the solution in the neighborhood of the eigenvalues. We also prove that one can obtain an approximation of the associated incident waves. We then propose a numerical implementation of the method and validated its efficiency on some synthetic two dimensional examples. [ABSTRACT FROM AUTHOR]

We derive conditional stability estimates for inverse scattering problems related to the time harmonic magnetic Schrödinger equation. We prove logarithmic-type estimates for retrieving the magnetic (up to a gradient) and electric potentials from near field or far field maps. Our approach combines techniques from similar results obtained in the literature for inhomogeneous inverse scattering problems based on the use of geometrical optics solutions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We revisit the differential sampling method introduced in (Haddar and Nguyen 2017 Comput. Math. Appl. 74 2831–55) for the identification of a periodic domain and some local perturbation. We provide a theoretical justification of the method that avoids assuming that the local perturbation is also periodic. Our theoretical framework uses functional spaces with continuous dependence with respect to the Floquet–Bloch variable. The corner stone of the analysis is the justification of the generalized linear sampling method in this setting for a single Floquet–Bloch mode. [ABSTRACT FROM AUTHOR]

The application motivating this work is related to the identification of deposits inside nuclear power plant steam generators using eddy-current probes. We consider a realistic experimental process that relies on the scan of a domain by sweeping along a tube axis a probe made out of coils, playing the role of the sources/receivers. Solving the inverse shape problem associated with these measurements using a least squares method requires solutions to the eddy-current and the adjoint problems for a large number of right-hand sides at each gradient-descent iteration. Additional cost in the forward solver comes from the use of a potential formulation of the problem that has the advantage of being independent from the topology of the conductive media (that may vary during iterations). We use a level-set approach to avoid remeshing and handle unknown topologies. The crucial ingredient in our algorithm is an optimized way of handling high numbers of right-hand sides for iterative solvers of large-scale problems. We first benchmark various block Krylov methods, block GMRES, and block BGCRODR to test their effectiveness compared to their standard counterpart, i.e., GMRES and GCRODR. Then we propose for BGCRODR a new implementation for recycling information from previously generated Krylov bases that scales better than traditional approaches. This part is independent from the practical inverse problem at hand. The efficiency of the overall inversion procedure is finally demonstrated on realistic synthetic 3D examples. [ABSTRACT FROM AUTHOR]

We present algorithms for computing weakly singular and near-singular integrals arising when solving the 3D Helmholtz equation with curved boundary elements. These are based on the computation of the preimage of the singularity in the reference element's space using Newton's method, singularity subtraction, the continuation approach, and transplanted Gauss quadrature. We demonstrate the accuracy of our method for quadratic basis functions and quadratic triangles with several numerical experiments, including the scattering by two half-spheres. [ABSTRACT FROM AUTHOR]

We propose a rapid and robust iterative algorithm to solve inverse acoustic scattering problems formulated as a PDE constrained shape optimization problem. We use a level-set method to represent the obstacle geometry and propose a new scheme for updating the geometry based on an adaptation of accelerated gradient descent methods. The resulting algorithm aims at reducing the number of iterations and improving the accuracy of reconstructions. To cope with regularization issues, we propose a smoothing to the shape gradient using a single layer potential associated with ik where k is the wave number. Numerical experiments are given for several data types (full aperture, backscattering, phaseless, multiple frequencies) and show that our method outperforms a nonaccelerated approach in terms of convergence speed, accuracy, and sensitivity to initial guesses. [ABSTRACT FROM AUTHOR]

We consider the inverse acoustic time domain scattering problem for absorbing scatterers modeled by impedance boundary conditions. We present and analyze a factorization method for reconstructing the obstacle boundary from far field measurements. The analysis is based on using the Laplace transform and proving the coercivity of the solution operator in suitable weighted spaces in time. This leads us to consider a modified far field operator parameterized by the imaginary part of Laplace variable which is different but can be arbitrarily close to the original far field operator. As a proof of concept we also provide some preliminary numerical examples to test and discuss the effectiveness of the resulting inversion method when applied to the original far field operator. [ABSTRACT FROM AUTHOR]

Differential evolution indicators are introduced for 3D spatiotemporal imaging of micromechanical processes in elastic solids where progressive variations due to manufacturing and/or aging are housed in a highly scattering background of a priori unknown or uncertain structure. In this vein, a threetier imaging platform is established where (1) the domain is periodically (or continuously) subject to illumination and sensing in an arbitrary configuration; (2) sequential sets of measured data are deployed to distill far-field signatures of the domain's internal structure through carefully constructed, noniterative solutions to the scattering equation; and (3) the resulting solution sequence is then used to rigorously construct an imaging functional carrying appropriate invariance with respect to the unknown stationary components of the background, e.g., pre-existing interstitial boundaries. This gives birth to differential indicators that specifically recover the 3D support of evolution within a network of unknown scatterers. The direct scattering problem is formulated in the frequency domain where the background consists of a random distribution of monolithic fragments. The constituents are connected through highly heterogeneous interfaces of unknown elasticity and dissipation spanning from perfectly bonded to traction-free contacts which are subject to evolution in time and space. The support of interfacial boundaries is periodically illuminated by a set of incident waves and thus-induced scattered fields are captured over a generic observation surface. The performance of the proposed imaging indicator is illustrated through a set of numerical experiments for sequential reconstruction of evolving damage zones featuring randomly distributed cracks and bubbles. [ABSTRACT FROM AUTHOR]

We propose and study a data completion algorithm for recovering missing data from the knowledge of Cauchy data on parts of the same boundary. The algorithm is based on surface representation of the solution and is presented for the Helmholtz equation. This work is an extension of the data completion algorithm proposed by the two last authors where the case of data available of a closed boundary was studied. The proposed method is a direct inversion method robust with respect to noisy incompatible data. Classical regularization methods with discrepancy selection principles can be employed and automatically lead to a convergent schemes as the noise level goes to zero. We conduct 3D numerical investigations to validate our method on various synthetic examples. [ABSTRACT FROM AUTHOR]

We analyze the Factorization method to reconstruct the geometry of a local defect in a periodic absorbing layer using almost only incident plane waves at a fixed frequency. A crucial part of our analysis relies on the consideration of the range of a carefully designed far field operator, which characterizes the geometry of the defect. We further provide some validating numerical results in a two dimensional setting. [ABSTRACT FROM AUTHOR]

We develop a factorization method to obtain an explicit characterization of a (possibly nonconvex) Dirichlet scattering object from measurements of time-dependent causal scattered waves in the far field regime. In particular, we prove that far fields of solutions to the wave equation due to particularly modified incident waves characterize the obstacle by a range criterion involving the square root of the time derivative of the corresponding far field operator. Our analysis makes essential use of a coercivity property of the solution of the Dirichlet initial boundary value problem for the wave equation in the Laplace domain. This forces us to consider this particular modification of the far field operator. The latter in fact can be chosen arbitrarily close to the true far field operator given in terms of physical measurements. [ABSTRACT FROM AUTHOR]

The Bloch−Torrey Partial Differential Equation (PDE) can be used to model the diffusion Magnetic Resonance Imaging (dMRI) signal in biological tissue. In this paper, we derive an Anisotropic Diffusion Transmission Condition (ADTC) for the Bloch−Torrey PDE that accounts for anisotropic diffusion inside thin layers. Such diffusion occurs, for example, in the myelin sheath surrounding the axons of neurons. This ADTC can be interpreted as an asymptotic model of order two with respect to the layer thickness and accounts for water diffusion in the normal direction that is low compared to the tangential direction. We prove uniform stability of the asymptotic model with respect to the layer thickness and a mass conservation property. We also prove the theoretical quadratic accuracy of the ADTC. Finally, numerical tests validate these results and show that our model gives a better approximation of the dMRI signal than a simple transmission condition that assumes isotropic diffusion in the layers. [ABSTRACT FROM AUTHOR]

We extend the so-called generalized linear sampling method (GLSM) to the case of limited aperture data at a fixed frequency. In this case the factorization of the sampling operator does not obey the symmetry required in the justification of the GLSM introduced by Audibert and Haddar [Inverse Probl., 30 (2004), p. 035011]. We propose a new formulation by adding an extra penalty term that asymptotically corrects the nonsymmetry of the GLSM original penalty term. The analysis of the new formulation is first presented in an abstract framework. We then show how to apply our setting to the scalar problem with far field measurements or near field measurements on a limited aperture. We finally validate the method through some numerical tests in two dimensions and for far field measurements. [ABSTRACT FROM AUTHOR]

We investigate the scattering problem for the case of locally perturbed periodic layers in,. Using the Floquet–Bloch transform in the periodicity direction, we reformulate this scattering problem as an equivalent system of coupled volume integral equations. We then apply a spectral method to discretize the obtained system after periodization in the direction orthogonal to the periodicity directions of the medium. The convergence of this method is established and validating numerical results are provided. [ABSTRACT FROM AUTHOR]

The non-destructive control of steam generators is an essential task for the safe and failure-free operation of nuclear power plants. Due to magnetite particles in the cooling water of the plants, a frequent source for failures are magnetic deposits in the cooling loop of steam generators. From eddy current signals measured inside a U-tube in the steam generator, we propose and analyze a regularized shap optimization algorithm to identify magnetic deposits outside the U-tube with either known or unknown physical properties. Motivated by the cylindrical geometry of the U-tubes we assume an axisymmetric problem setting, reducing Maxwell’s equations to a 2-D elliptic eddy current problem. The feasibility of the proposed algorithms is illustrated via numerical examples demonstrating in particular the stability of the method under noise. [ABSTRACT FROM PUBLISHER]

Diffusion magnetic resonance imaging (dMRI) is an imaging modality that probes the diffusion characteristics of a sample via the application of magnetic field gradient pulses. If the imaging voxel can be divided into different Gaussian diffusion compartments with inter-compartment exchange governed by linear kinetics, then the dMRI signal can be described by the Kärger model, which is a well-known model in Nuclear Magnetic Resonance. However, the Kärger model is limited to the case when the duration of the diffusion-encoding gradient pulses is short compared to the time delay between the start of the pulses. Under this assumption, the time at which to evaluate the Kärger model to obtain the dMRI signal is unambiguously the delay between the pulses. Recently, a new model of the dMRI signal, the Finite-Pulse Kärger (FPK) model, was derived for arbitrary diffusion gradient profiles. Relying on the FPK model, we show that when the duration of the gradient pulses is not short, the time at which to evaluate the Kärger model should be the time delay between the start of the pulses, shortened by one third of the pulse duration. With this choice, we show the sixth order convergence of the Kärger model to the FPK model in the non-dimensionalized pulse duration. [ABSTRACT FROM AUTHOR]

We propose a new numerical method for the solution of the Bernoulli free boundary value problem for harmonic functions in a doubly connected domain D in [ABSTRACT FROM AUTHOR]