Project proposal for INRIA Saclay Ile de France Shape Reconstruction and Identification (D´ etermination de Formes et Identification) (DeFI) http://www.cmap.polytechnique.fr/~defi 01/06/2008 Project coordinator • Houssem Haddar (Directeur de Recherche (DR2), INRIA) Members • Gr´ egoire Allaire (Prof. de Math. Appl., Ecole polytechnique) • Olivier Pantz (Prof. charg´ e de cours, Ecole Polytechnique) PostDoc • Ozgur Ozdemir (2007-2008), Armin Lechleiter (2008-2009), Alex Kelly (2008-2009) PhD students • Ridha Mdimagh (2005), Yosra Boukari (2007), Anne Cossonni` ere (2008) Associated members 1 • Laurent Bourgeois (Enseignant-Chercheur, ENSTA) • Antonin Chambolle (Directeur de Recherche, CNRS, CMAP) PhD students co-supervised outside the group • Berang` ere Delourme (2007, POems, INRIA-Rocquencourt), 1 Associated members are expected to spend 20% of their research activity within the DeFI team
An Overall Quick Presentation The research activity of our team is dedicated to the design, analysis and implementa- tion of efficient numerical methods to solve inverse and/or shape and topological optimiza- tion problems in connection with acoustics, electromagnetism, elastodynamics, and waves in general. Sought practical applications include radar and sonar applications, bio-medical imaging techniques, non-destructive testing, structural design, composite materials, ... Roughly speaking, the model problem consists in determining information on, or opti- mizing the geometry (topology) and/or the physical properties of unknown targets from given constraints or measurements, for instance measurements of diffracted waves. In general this kind of problems is non linear. The inverse ones are also severely ill-posed and therefore require special attention from regularization point of view, and non trivial adaptations of classical optimization methods. Our scientific research interests are three-fold: • Theoretical understanding and analysis of the forward and inverse mathematical models, including in particular the development of simplified models for adequate asymptotic configurations. • The design of efficient numerical optimization/inversion methods which are quick and robust with respect to noise. Special attention will be paid to algorithms capable of treating large scale problems (e.g. 3-D problems) and/or suited for real-time imaging. • Development of prototype softwares for precise applications or tutorial toolboxes. Our team is born after the association of two groups: the inverse problem component of the POems project ( INRIA ) and the shape and topology optimization group OPTOPO at the CMAP (Ecole Polytechnique). As it will be developed in the presentation of our scientific issues and research themes, this association is a natural consequence of several scientific convergences, that we can quickly summarize in 1) the similarity and complementarity of developed tools: Asymptotic methods, Level Set methods, Sampling methods, 2) common problematics: the need of special regularization strategies, retrieving forms without a priori knowledge of its topology, the application to problems in connection with waves, and 3) the perspective of coupling methods to efficiently treat large scale problems and the desire of addressing challenging time-domain problems. The DeFI group is hosted by the department of applied mathematics ( CMAP ) at Ecole Polytechnique. 1
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Contents 1 Introduction: general presentation 5 2 Scientific issues through a model problem 8 2.1 Mathematical settings for the inverse problem . . . . . . . . . . . . . . . . 9 2.2 On general theoretical issues . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Approximate models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 The optimization point of view . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5 Special sampling methods for multi-static data . . . . . . . . . . . . . . . . 15 2.6 Some potential applications . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.6.1 Imaging of urban infrastructures from multi-static data . . . . . . . 17 2.6.2 Bio-medical Applications . . . . . . . . . . . . . . . . . . . . . . . . 18 2.6.3 Innovative methods in non destructive testing and parameter iden- tification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 Description of research themes 20 3.1 Real-time algorithms for imaging at a fixed frequency in the non-linear regime 20 3.2 The conformal mapping method for EIT problems. . . . . . . . . . . . . . 21 3.3 Identifictaion/Invisibility of anisotropic media . . . . . . . . . . . . . . . . 22 3.4 Regularization and stability issues . . . . . . . . . . . . . . . . . . . . . . . 23 3.5 Shape and topological optimization methods . . . . . . . . . . . . . . . . . 24 3.6 Asymptotic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.7 Hybrid inversion methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.8 Sampling and optimization methods for time dependent problems . . . . . 27 4 Softwares 28 5 Results Dissemination 28 6 Collaborators 29 6.1 Academic (National and International) . . . . . . . . . . . . . . . . . . . . 29 6.2 Positioning within INRIA . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 6.3 Industrial grants and contacts . . . . . . . . . . . . . . . . . . . . . . . . . 32 3
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1 Introduction: general presentation The research activity of our team will be dedicated to the design, analysis and imple- mentation of efficient numerical methods to solve inverse and/or shape and topological optimization problems in connection with acoustics, electromagnetism, elastodynamics, and waves in general. We are particularly interested in the development of fast methods that are suited for real-time imaging and/or large scale problems. These goals require to work on both the physical and the mathematical models, and indeed require a solid expertise in related numerical solvers. This introduction intends to give a general overview of our research interests and themes. We choose to present them through the specific example of inverse scattering problems (from inhomogeneities), which will be central in most of foreseen developments. The practical problem would be to identify an inclusion from measurements of diffracted waves generated after sending some incident waves into the probed medium. Typical ap- plications include biomedical imaging where using micro-waves one would like to probe the presence of pathological cells, non destructive testing of (composite) materials where one would like to identify the presence of faults from measurements of ultrasonic waves or also imaging of urban infrastructures where using ground penetrating radars (GPR) one is interested in finding the location of buried facilities such as pipelines, waste deposits, . . . . This kind of applications requires in particular fast and reliable algorithms. By “imaging” we shall refer to the inverse problem where the concern is only the location and the shape of the inclusion, while in the “identification” problem one would also be interested in getting information on the inclusion physical parameters. Both problems (imaging and identification) are non linear and ill-posed (lack of stability with respect to measurements errors if some careful constrains are not added). Moreover, the unique determination of the geometry or the physical parameters is not guaranteed in general if sufficient measurements are not available. As an example, in the case of anisotropic inclusions, one can show that an appropriate set of data uniquely determine the geometry but not the material properties. These theoretical considerations are in gen- eral very difficult to address and often relies on ad-hoc techniques, especially in the case of inverse vectorial problems (e.g. the electromagnetic problem modeled by Maxwell’s equa- tions). More developed theories, based on the so-called Carleman estimates, are available for the scalar case. Even though not central to our research activity, our group will pursue some investigations in this direction through the study of the Cauchy problem associated with the Laplace or the Helmholtz operator in domains with singularities. We are also in- terested in applying these results to quantify the stability properties of impedance inverse boundary value problems or impedance tomography problems (see Section 3.4). These theoretical results (uniqueness, stability) are not only important in understanding the mathematical properties of the inverse problem, but are also useful for qualitative interpretation of numerical results (which information can be stably reconstructed) and the choice of appropriate regularization techniques. The latter point will also be discussed in 5
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