María-Luisa Rapún

Solving inhomogeneous inverse problems by topological derivative methods

We introduce new iterative schemes to reconstruct scatterers buried in a medium and their physical properties. The inverse scattering problem is reformulated as a constrained optimization problem involving transmission boundary value problems in heterogeneous media. Our first step consists in developing a reconstruction scheme assuming that the properties of the objects are known. In a second step, we combine iterations to reconstruct the objects with iterations to recover the material parameters. This hybrid method provides reasonable guesses of the parameter values and the number of scatterers, their location and size. Our schemes to reconstruct objects knowing their nature rely on an extended notion of topological derivative. Explicit expressions for the topological derivatives of the corresponding shape functionals are computed in general exterior domains. Small objects, shapes with cavities and poorly illuminated obstacles are easily recovered. To improve the predictions of the parameters in the successive guesses of the domains we use a gradient method.

Reduced order models based on local POD plus Galerkin projection

A method is presented to accelerate numerical simulations on parabolic problems using a numerical code and a Galerkin system (obtained via POD plus Galerkin projection) on a sequence of interspersed intervals. The lengths of these intervals are chosen according to several basic ideas that include an a priori estimate of the error of the Galerkin approximation. Several improvements are introduced that reduce computational complexity and deal with: (a) updating the POD manifold (instead of calculating it) at the end of each Galerkin interval; (b) using only a limited number of mesh points to calculate the right hand side of the Galerkin system; and (c) introducing a second error estimate based on a second Galerkin system to account for situations in which qualitative changes in the dynamics occur during the application of the Galerkin system. The resulting method, called local POD plus Galerkin projection method, turns out to be both robust and efficient. For illustration, we consider a time-dependent Fisher-like equation and a complex Ginzburg-Landau equation.

Topological Derivatives for Shape Reconstruction

Topological derivative methods are used to solve constrained optimization reformulations of inverse scattering problems. The constraints take the form of Helmholtz or elasticity problems with different boundary conditions at the interface between the surrounding medium and the scatterers. Formulae for the topological derivatives are found by first computing shape derivatives and then performing suitable asymptotic expansions in domains with vanishing holes. We discuss integral methods for the numerical approximation of the scatterers using topological derivatives and implement a fast iterative procedure to improve the description of their number, size, location and shape.

Detecting corrosion using thermal measurements

This paper deals with the inverse problem of detecting the level of corrosion at the interface of an inclusion given thermal measurements at the accessible boundary of the sample. This leads to a transmission problem for the heat equation with an unknown coefficient in a transmission condition. We consider both time-harmonic and delta-pulse excitations. In both cases we prove uniqueness results for the inverse problem. To reconstruct the unknown level of corrosion numerically, we study a non-iterative method and a regularized Newton method and compare their performances in a number of numerical experiments. Photothermal techniques are suitable means of inspecting composite materials with nondestructive tests. We are interested in a technique that consists in heating the accessible side of the material by a defocused laser beam. The goal is to reconstruct internal properties of the material (to detect structural defects, reconstruct the size, depth, orientation of the inclusions and/or physical properties of them) from measurements of the temperature at the side that has been thermically excited. Some recent papers on physical experiments with this kind of techniques are [12, 27, 33, 34]. In this work we study the detection of the level of corrosion at the interface separating the internal inhomogeneities from the matrix material. We explore two different kinds of thermal excitement, produced by time-harmonic incident heating fields and by delta-pulse excitations. In the first one we deal with a steady-state problem. In the corresponding forward problem one looks for time-harmonic solutions of an elliptic conductive-transmission problem. This kind of solution is commonly referred to as thermal waves. For a detailed study of these waves and their uses we refer to [3, 22–24] and references therein. The numerical solution of the forward

Hybrid topological derivative and gradient-based methods for electrical impedance tomography

We present a technique to reconstruct the electromagnetic properties of a medium or a set of objects buried inside it from boundary measurements when applying electric currents through a set of electrodes. The electromagnetic parameters may be recovered by means of a gradient method without a priori information on the background. The shape, location and size of objects, when present, are determined by a topological derivative-based iterative procedure. The combination of both strategies allows improved reconstructions of the objects and their properties, assuming a known background.

Domain reconstruction using photothermal techniques

A numerical method to detect objects buried in a medium by surface thermal measurements is presented. We propose a new approach combining the use of topological derivatives and Laplace transforms. The original optimization problem with time-dependent constraints is replaced by an equivalent problem with stationary constraints by means of Laplace transforms. The first step in the reconstruction scheme consists in discretizing the inversion formula to produce an approximate optimization problem with a finite set of constraints. Then, an explicit expression for the topological derivative of the approximate shape functional is given. This formula is evaluated at low cost using explicit expressions of the forward and adjoint fields involved. We apply this technique to a simple shape reconstruction problem set in a half space. Good approximations of the number, location and size of the obstacles are obtained. The description of their shapes can be improved by more expensive hybrid methods combining time averaging with topological derivative based iterative schemes.

Determining Planar Multiple Sound-Soft Obstacles from Scattered Acoustic Fields

An inverse problem is considered where the structure of multiple sound-soft planar obstacles is to be determined given the direction of the incoming acoustic field and knowledge of the corresponding total field on a curve located outside the obstacles. A local uniqueness result is given for this inverse problem suggesting that the reconstruction can be achieved by a single incident wave. A numerical procedure based on the concept of the topological derivative of an associated cost functional is used to produce images of the obstacles. No a priori assumption about the number of obstacles present is needed. Numerical results are included showing that accurate reconstructions can be obtained and that the proposed method is capable of finding both the shapes and the number of obstacles with one or a few incident waves.

An iterative method for parameter identification and shape reconstruction

An iterative strategy for the reconstruction of objects buried in a medium and the identification of their material parameters is analysed. The algorithm alternates guesses of the domains using topological derivatives with corrections of the parameters obtained by descent techniques. Numerical experiments in geometries with multiple scatterers show that our scheme predicts the number, location and shape of objects, together with their physical parameters, with reasonable accuracy in a few steps.

Defect Detection from Multi-frequency Limited Data via Topological Sensitivity

In this work we investigate the reconstruction of sound-hard obstacles buried in a bounded material medium by a non-iterative method based on the computation of topological derivatives. The main purpose is to suitably combine multi-frequency data in a fairly demanding measurement configuration: a very reduced number of aligned emitters and receivers are considered. The performance of the algorithm is illustrated for both single and multiple obstacle reconstructions.

LUPOD: Collocation in POD via LU decomposition

A collocation method is developed for the (truncated) POD of a set of snapshots. In other words, POD computations are performed using only a set of collocation points, whose number is comparable to the number of retained modes, in a similar fashion as in collocation spectral methods. Intending to rely on simple ideas which, moreover, are consistent with the essence of POD, collocation points are computed via the LU decomposition with pivoting of the snapshot matrix. The new method is illustrated in simple applications in which POD is used as a data-processing method. The performance of the method is tested in the computationally efficient construction of reduced order models based on POD plus Galerkin projection for the complex Ginzburg–Landau equation in one and two space dimensions.