# Numerical Resolution for Inverse ProblemS

Date: Thu, Jan 8 - Fri, Jan 9 2015

Hour: 14:00

Location: BCAM

Speakers: Pedro Caro, Andrés Prieto, Joan Manuel Rayes, Anna Doubova, Carlos Castro, Ismael Rodrigo Bleyer, Matteo Santacesaria, María Luisa Rapán

**Numerical Resolution for Inverse Problems, BCAM, Bilbao, Spain, January 8-9, 2015****Scientific Organizers**

Luis VEGA, Juan Antonio BARCELÓ, Carlos CASTRO, Andoni GARCÍA**Program****January 8**

14:00-15:00 Pedro CARO

15:00-16:00 Andrés PRIETO

16:00-16:30 Coffee break

16:30-17:30 Joan Manuel REYES

17:30-18:30 Anna DOUBOVA**January 9**

9:30-10:30 Carlos CASTRO

10:30-11:00 Coffee break

11:00-12:00 Ismael Rodrigo BLEYER

12:00-13:00 Matteo SANTACESARIA

13:00-14:00 María Luisa RAPÁN**Speakers and abstracts**

Pedro CARO, ICMAT, CSIC

Stability: a motivation to study other inverse problems

First, we will introduce the Calderón problem consisting of recovering the electric properties of a medium by non-invasive measurements. This mathematical problem provides a theoretical model to detect significant changes in the conductivity of a medium. Unfortunately, the optimal stability to reconstruct these electric properties is very weak. In general terms, this means that this model present high contrast and low resolution.

After this discussion about the Calderón problem, we will present other inverse problems, with high construct and low resolution, for different physical frameworks, namely, electrodynamics, elasticity and fluid dynamics.

However, this kind of inverse problems does not seem to be the most powerful in applications since, despite the high contrast, they present a low resolution. Therefore, it seems appropriate to modify the mathematical problems to be able to preserve the high contrast and obtain higher resolution. Thus, the main goal of this talk will be present to different ways of carrying out such modifications.

Andrés PRIETO, Universidade da Coruña

A numerical approach to seabed characterization in coastal environments

The characterization of the seabed in coastal environments does not only involve the determination of the depth in the water column but also the geological classification of the bottom surface and the identification of the presence of marine vegetation. The physical exploration of these coastal environments are performed by means of side-scan sonars, which allow to measure the acoustic response of the elements on the seabed from the surface of the sea.

In this talk, the mathematical modelling of the coastal fluid media (highly heterogeneous and depending on several state variables) and the poroelastic models to govern the structural behaviour of the seabed will be revised. From a numerical point of view, since the fluid medium is assumed unbounded, it will be required the use of the Perfectly Matched Layer technique to truncate the computational domain. In addition, side-scan sonars typically involve measurements in a range from 1kHz to 200kHz and so efficient numerical procedures to approximate accurately the time-harmonic motion in hydro-acoustic problems at high-frequency regime will be considered.

Joan Manuel REYES, Cardiff University

Some techniques in 2D EIT and applications

The D-bar method of 2D Electrical Impedance Tomography (EIT) will be presented and empirically compared with the transport matrix reconstruction method of the Astala-Puivarinta theory on certain piecewise constant conductivity distributions. This was a joint work with Kari Astala, Lassi Puivarinta and Samuli Siltanen.

Finally, two applications of the algorithmic techniques for the D-bar method will be shown as follows:

i) Computing a numerical estimate of the Born approximation used in 2D inverse potential scattering. This is an ongoing research line together with Juan Antonio Barcellé and Carlos Castro.

ii) A new hybrid method combining EIT and Total Variation of image processing traditions. This work is in progress in collaboration with Sarah Hamilton, Samuli Siltanen and Xiaoqun Zhang.

Anna DOUBOVA, Universidad de Sevilla

Numerical solution of some geometric inverse problems

We consider some geometric inverse problems for the wave and the Lamé equations motivated by Elastography. We present several recent results and open questions concerning the numerical reconstruction of the unknown domain where the equations evolve. In the numerical experiments, we solve appropriate optimization problems. This needs the numerical solution of the PDE´s, that is performed with FreeFem++. The routines on the ff-NLopt package, that provide an interface to a free/open-source library for nonlinear optimization, are also required. We present some numerical results in the 2D and 3D cases. This is joint work with E. Fernández-Cara (Dpto. E.D.A.N., Universidad de Sevilla, cara@us.es).

Carlos CASTRO, ETSI Caminos, Canales y Puertos, Universidad Politécnica de Madrid

Numerical approximation of the inverse scattering problem

The Hemholtz equation in R2 with unknown compactly supported potential V (x) is considered. This work is aimed at recovering the potential V (x) from scattering data . One of the most successful approaches in this field is to recover the Born approximation VB(x) that is obtained from a suitable inversion of the far field pattern. It is well known that in many cases the Born approximation shares the same discontinuities as the potential V (x). Thus, if the potential is the characteristic function of a bounded, open set the Born approximation allows to recover V (x) completely from the discontinuities of VB(x).

In this talk, error estimates of the numerical approximation of the Born approximation are presented. The following situations are considered: a) fixed energy, b) backscattering and c) fixed incident wave direction. The numerical simulations for the different Born approximations are compared.

Ismael Rodrigo BLEYER, University of Helsinki

Digital speech: an application of the dbl-RTLS method for solving GIF problem

"Digital Speech Processing" refers to the study of a speech signal. Namely, these signals are processed in a digital representation, as for example, synthesis, analysis, enhancement, compression and recognition may refers to this process.

In this talk we are interested on solving the core problem known as "Glottal Inverse Filtering" (GIF). Commonly this problem can be modelled by convolving a pressure function (input signal) with an impulse response function (filter).

Our approach is done in a deterministic setup based on the dbl-RTLS (double regularised total least squares). Therefore our second goal is to give an overview on this novel method, algorithm and its numerical implementation based on an alternating minimisation procedure.

Keywords: ill-posed problems, noisy operator, noisy right hand side, reg- ularised total least squares, double regularisation, alternating minimisation, glottal inverse filtering, digital speech.

Matteo SANTACESARIA, University of Helsinki

New numerical results for the Gel'fand-Calderon problem.

In this talk we present new numerical reconstruction results for the Gel'fand-Calderon inverse problem in two dimensions. This is the problem of the recovery of a potential in the Schrodinger equation from boundary data (Dirichlet-to-Neumann map) at fixed energy. Practical applications include ocean acoustic tomography and seismic imaging. We will review global uniqueness, reconstruction and stability results, and in particular the Lipschitz stable approximate reconstruction algorithm developed in collaboration with R. Novikov. The numerical results are obtained with this algorithm, and they clearly show how the resolution increases with the energy. The algorithm is based on the theory of inverse quantum scattering, a topic which will also be covered in the talk. This is a joint work with S. Siltanen and M. Lassas.

María Luisa RAPÁN, ETSI Aeronáuticos, Universidad Politécnica de Madrid

Shape reconstruction using multifrequency topological derivatives

In this work we propose a non-iterative method based on topological derivative computations to detect structural defects or inclusions (determine their location, size, shape, orientation) by combining multifrequency near-field observations of the total acoustic wave at a few observation points. We will show some numerical tests illustrating the viability of this technique in different situations including the simultaneous reconstruction of objects of different sizes and the identification of poorly illuminated defects.

## Organizers:

Universidad Politécnica de Madrid, University of Helsinki, ICMAT, Madrid, Universidade da Coruña, Cardiff University, Universidad de Sevilla, University of Helsinki

## Confirmed speakers:

Pedro Caro, Andrés Prieto, Joan Manuel Rayes, Anna Doubova, Carlos Castro, Ismael Rodrigo Bleyer, Matteo Santacesaria, María Luisa Rapán

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