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Talk Abstracts
Geometric Methods in Inverse Problems and PDE Control


Paolo Albano (Department of Mathematics, University of Bologna) albano@dm.unibo.it 

Observability Estimates for a Parabolic-Hyperbolic Coupled System

http://ejde.math.swt.edu/Volumes/2000/22/abstr.html

http://ejde.math.swt.edu/Volumes/2000/53/abstr.html

We describe some (sufficient) conditions ensuring that a solution to a coupled parabolic-hyperbolic system can be observed from the boundary of a given cilinder. The main technical tool needed for such an analysis are Carleman estimates with singular (in time) weights. Moreover, we discuss the special case of operators with constant coefficients.

Joint work with D. Tataru (Northwestern University).

Giovanni Alessandrini (Scienze Matematiche, University of Trieste)

Size Estimates of Inclusions in an Elastic Body by Boundary Measurements    pdf

We consider the problem of determining an inclusion D in an elastic isotropic body made from boundary measurements of traction and displacement. We assume that the inclusion D is made of a different elastic material, either harder or softer, than the unperturbed specimen. We prove that the volume of D can be estimated, from above and below, by an easily expressed quantity related to work, only depending on the boundary traction and displacement.

Habib Ammari (Centre de Mathématiques Appliquées, CNRS UMR 7641 & Ecole Polytechnique, 91128 Palaiseau Cedex, France; email: ammari@cmapx.polytechnique.fr    fax: 33169333011; phone: 33169334565.

Reconstruction of Small Electromagnetic Inhomogeneities

We consider solutions to the (full) time-harmonic Maxwell's equations. Our first goal is to provide a rigorous derivation of the leading order boundary perturbations resulting from the presence of a finite number of interior inhomogeneities. Our second goal is to apply these asymptotic formulae for the purpose of identifying the location and certain properties of the shapes (polarization tensors) of the inhomogeneities from boundary measurements. In contrast to a boundary least squares fit to the measured data, we present a method based on appropriate averaging, using particular background solutions as weights. We also discuss the reconstruction of the small inhomogeneities in the time-dependent case and show how to generalize our approach to treat the case where we are only in possession of boundary measurements on part of the boundary. Our main idea for solving these inverse problems is to reduce them to calculations of inverse Fourier transforms.

George Avalos (Department of Mathematics and Statistics University of Nebraska-Lincoln)  gavalos@math.unl.edu

Exact Controllabilty of Structural Acoustic Interactions

Joint work with Irena Lasiecka.

In this paper, we work to discern exact controllability properties of two coupled wave equations, one of which holds on the interior of a bounded open domain  , and the other on a segment 0 of the boundary \partial . Moreover, the coupling is accomplished through terms on the boundary. Because of the particular physical application involved-the attenuation of acoustic waves within a chamber by means of active controllers on the chamber walls--control is to be implemented on the boundary only. We give here concise results of exact controllability for this system of interactions, with the control functions being applied through \partial . In particular, it is seen that for special geometries, control may be exerted on the boundary segment 0 only. We make use here of microlocal estimates derived for the Neumann-control of wave equations, as well as a special vector field which is now known to exist under certain geometrical situations.

Werner Ballmann (Rheinische Friedrich-Wilhelms-Universitaet Bonn)

Isospectral Manifolds    isospec.pdf   isospec.ps

I will talk about the construction of closed Riemannian manifolds whose Laplacians have the same spectrum. The ideas go back to Carolyn Gordon and Dorothee Schueth, I will present my interpretation of these ideas and will obtain isospectral metrics on S2 × S3.

James G. Berryman (Stanford Exploration Project, Lawrence Livermore National Laboratory)  berryman@kana.stanford.edu  http://sepwww.stanford.edu/sep/berryman/

Time-Reversal Acoustics and Maximum-Entropy Imaging

A common problem in acoustics and radar imaging is target location either from passive or active data collection and inversion. The passive case is source localization. The active case is reflection imaging. Time-reversal acoustics has the important characteristic that it provides a means of determining the eigenfunctions and eigenvalues of the scattering operator for either of these problems. Each eigenfunction may be approximately associated with an individual scatterer. The resulting decoupling of the scattered field from the various targets is a very important aid to locating the targets, and suggests a number of imaging and localization algorithms. One of these is maximum-entropy imaging.

Gerard Besson (Institut Fourier de Mathematiques, CNRS)  G.Besson@ujf-grenoble.fr

A Margulis Lemma Without Curvature    Slides

This is an attempt, in an elementary way, to prove compactness or precompactness theorems with as little curvature assumptions as possible. The curvature is being replaced by an asymptotic invariant : the entropy. The talk is intended to be elementary.

Dmitri Burago (Department of Mathematics, Penn State University)  burago@math.psu.edu

Volume/Distance Estimates, Gaussian Measures of Surfaces, and Ellipticity of Volume Functionals

The talk is based on a joint work with S. Ivanov.

We will discuss the following topics, which happen to be closely related:

1. Optimal fillings: metrics on a manifold (with boundary) that admit no volume-decreasing perturbations that do not decrease distances between boundary points. In other words, we will be looking for situations when an inequality for boundary distance functions of two metrics implies corresponding inequality for the volumes.

2. What are the restrictions on the Gaussian measures of closed surfaces (or surfaces with planar boundaries). For instance, under what conditions can one construct a polyhedron (for instance, a two-dimensional polyhedral surface in R4) with given areas and directions of faces and no boundary (or a planar boundary).

3. What is the relationship between different types of ellipticily for surface area functionals (an area functional is elliptic over Z (resp. R) if regions in affine planes are area minimizers among all Lipschits chains over Z(R) with the same boundary.

4. Asymptotic growth of volume for large balls in a periodic metric (a metric invariant under a co-compact action of an abelian group).

Jianguo Cao (University of Notre Dame)  jcao@nd.edu

The Spectrum of Non-compact Manifolds with Big Ends

In this lecture, we will discuss a new result on the bottom of the Laplace spectrum for non-compact manifolds with sufficiently large ends. We shall show that if such a manifold M satisfies a Gromov's length-area linear isoperimetric inequality then the bottom of the Laplace spectrum of such a manifold is positive. Furthermore, the Dirichlet problem at infinity is solvable for such an open space M. Examples of such an open space M above can be construed with prescribed ends. Therefore, the universal cover of such a space M is NOT necessarily to be diffeomorphic to the Euclidean space. For example, any Gromov-hyperbolic cone M over a (n-1)-dimensional space N with n > 1 is a Gromov-hyperbolic space with a big end.

Christopher B. Croke (University of Pennsylvania) ccroke@math.upenn.edu    

The Boundary Rigidity and Conjugacy Rigidty Problems    Slides

This overview talk will introduce the boundary rigidity problem "To What extent is the Riemannian metric of a compact Riemannian manifold with boundary determined by the distances between its boundary points?" After a brief survey of the early history of this problem it will then cover the relationship between the boundary rigidity problem and the conjugacy rigidity problem "To what extent is a Riemannian metric on a compact manifold without boundary determined by its geodesic flow." In particular one sees that conjugacy rigidity of closed manifolds implies the boundary rigidity of nice subdomains. The rest of the talk is devoted to discussing what is currently known about these problems. The hope is that this talk will lead into other talks where more of the techniques of proof will be discussed.

 

Patrick Eberlein (Department of Mathematics, University of North Carolina)  pbe@email.unc.edu

Nilpotent Groups and the Boundary Geometry of Negatively Curved Manifolds

Let X denote a complete, simply connected Riemannian manifold with sectional curvatures between two negative constants. U. Hamenstaedt has constructed a pseudometric on the geodesic boundary of X that is analogous to the Gromov pseudometric on the geodesic boundary of a simply connected space of nonpositive curvature. If X is a symmetric space, then this pseudometric is a metric that is isometric to a left invariant Carnot-Caratheodory metric on the horospheres of X. In this case the horospheres are isometric a single 2-step nilpotent, simply connected Lie group N with a left invariant metric of "Heisenberg type."

Conversely, a simply connected nilpotent Lie group N with an appropriate left invariant metric becomes a horosphere in a homogeneous space X defined by a canonical procedure that mimics the symmetric space situation.

For a general X we consider relations between the geometry of its boundary and the geometry of its horospheres. We give particular attention to the Hamenstaedt pseudometric and the case that X admits a finite volume quotient X/G, where G is a noncocompact lattice in the isometry group of X. In the latter case G determines cocompact lattices G1,G2, ... Gk in horospheres N1, N2, ... Nk that correspond to the cusps of X/G. The problem of geodesic conjugacy rigidity for X/G suggests comparison to the problems of geodesic conjugacy and marked length spectrum rigidity for the compact nilmanifolds G1/N1, G2/N2, ... Gk/Nk. We discuss these two rigidity problems for 2-step nilmanifolds G\N, where N has a left invariant Riemannian metric.

Matthias M. Eller (Department of Mathematics, Georgetown University)   mme4@georgetown.edu

Unique Continuation for Solutions to Systems of Partial Differential Equations with Non-Analytic Coefficients

The classical result of unique continuation for solutions to a PDE with non-analytic coefficients is Hörmander's theorem (1963). This theorem guarantees unique continuation across strongly pseudo-convex surfaces. It provides optimal uniqueness results for second order elliptic equations but is not as useful for second order hyperbolic equations and for higher order equations or systems of equations. Hörmander's theorem is based on Carleman estimates. These are weighted energy estimates that carry a large parameter.

In 1995 Tataru managed to relax the strong pseudo-convexity condition proving a new uniqueness result. This result applies to solutions to the wave equation with time independent coefficients gives unique continuation across non-characteristic surfaces. In other words it yields the same conclusion than Holmgren's theorem. The proof of Tataru's result relies on a new type of Carleman estimate.

Later in the 1990s the method of Carleman estimates was applied to certain systems of mathematical physics. Using the special structure of e.g. the isotropic Maxwell system as well as the robustness of Carleman estimates with respect to lower order terms uniqueness results were proved. A similar result was proved for the system of elasticity. A common feature of these systems is the fact that they are both hyperbolic and that they can be transformed into second order system which is coupled only through lower order terms.

A more challenging problem is the system of thermo-elasticity which combines hyperbolic and parabolic feature. However, Isakov was able to prove a uniqueness result when he developed a new type of Carleman estimate carrying two large parameters. Finally, through a combination of differential geometry and Carleman estimates one can show uniqueness for solutions to an anisotropic Maxwell system, i.e. the case where the coefficients are matrices.

Michael Galbraith (School of Mathematics, University of Minnesota)  galbrait@math.umn.edu

A Geometric-Optics Proof of a Theorem on Boundary Control Given a Convex Function   pdf    postscript

Peter Gibson (Institut für Mechanik II, Technische Universität Darmstadt)  gibson@ag2.mechanik.tu-darmstadt.de

Discretized Inverse Problems and "Splitting" of Distributions    Slides

Tridiagonal matrices are the discrete analog of a fundamental class of objects, namely second order (one-dimensional) differential operators. As such, the spectral analysis of tridiagonal matrices, both finite and infinite, relates naturally to a range of physical inverse problems. In particular, the so-called interior point problem for tridiagonal matrices is motivated by the inverse analysis of certain physical systems. The interior point problem is ill-posed in the sense that it's solution is not unique. In studying the geometry of the solution set, one encounters a curious phenomenon, "splitting" of distributions, which will be described in this talk.

Allan Greenleaf (Department of Mathematics, University of Rochester)  allan@math.rochester.edu

Global Uniqueness in the Calderon problem for Conormal Conductivities

In dimensions greater than or equal to three, we consider the Calderon problem for conductivities that have conormal singularities along a hypersurface, H, of order less than -2. This allows the conductivities to be differentiable of order 1 + \epsilon but no better. Assuming the hypersurfaces satisfy a geometric condition, we show that two such conductivities for which the corresponding operators have the same Cauchy data must be equal.

This is joint work with M. Lassas and G. Uhlmann.

Robert Gulliver (School of Mathematics, University of Minnesota)  gulliver@math.umn.edu

Boundary Control of Wave Equations via Geometry

Consider a hyperbolic PDE with smooth, time-independent coefficients in M x [0,T], where M is a smooth, relatively compact domain in real n-space or is a smooth n-dimensional manifold with boundary bd M: utt = aijuij + lower-order terms. We use the coefficients aij to define a Riemannian metric ds2 = gij dxi dxj, where for each x in M, (gij(x)) is the inverse of the matrix (aij(x)). Modulo first-order terms, the right-hand side of the PDE is the Riemannian Laplace operator. We use concepts such as convexity and geodesic curvature in their Riemannian, and hence coordinate-independent, versions.

The boundary control problem is whether, for any initial conditions at t = 0, there is a choice of (say) Dirichlet boundary values on (bd M) x [0,T] so that the solution of the PDE vanishes for t >= T. Write T0 for the minimum value of T for which this is possible. It has been shown (under strong smoothness hypotheses: e.g. Bardos--Lebeau--Rauch) that T0 is the maximum length of geodesics in M.

We will outline a few recent results: (1) If there is a convex function v:M --> [0,K] with Hessian matrix greater than 2c ds2, then T0 <= 2 sqrt(K/c); (2) If bd M is locally convex, and if the minimizing geodesic joining any two points of bd M is unique and nondegenerate, then T0 is the maximum of the distance between boundary points; and (3) If n = 2, bd M is locally convex and there are no closed geodesics in M, then T0 is finite, with an estimate. We will emphasise result (3), which uses Grayson's work on the flow of curves by curvature.

(1) is due to Lasiecka-Triggiani-Yao, with a new proof by Michael Galbraith; (2) is joint work with Walter Littman; and (3) is joint work with Littman and Santiago Betelú.

Soenke Hansen (Universität Paderborn)

Propagation of Polarization in Elastodynamics with Residual Stress and Travel Times

The inverse problem for the hyperbolic system of elastodynamics is to recover the density, the Lame parameters, and the residual stress tensor from measurements at the boundary. We show that from such measurements, encoded in the hyperbolic DN map, one can recover, separately, the travel times of shear and pressure waves. This allows us to use known solutions about inverse kinematic problems in the solution of inverse problems of elastodynamics. We obtain our results using the methods of microlocal analysis. In particular, we study the propagation of polarization in elastodynamics boundary problems. The travel times and, more generally, the canonical boundary relations associated with S and P waves are shown to be determined by boundary measurements of high-frequency waves even when caustics develop along rays.

This is joint work with G. Uhlmann.

Victor Isakov (Department of Mathematics and Statistics Wichita State University)  victor.isakov@wichita.edu

Uniqueness of the Continuation, Control and Inverse Problems for the Dynamical Lame System

We consider the classical dynamical Lame system of the elasticity theory in a (bounded) three-dimensional space domain. We give uniqueness of the continuation across any noncharacteristic surface when the coefficients of this system are time-independent and finitely smooth (joint work with Eller, Nakamura, and Tataru) and derive from it approximate controllability at the final moment of time by boundary (displacement) data. Combining these results, the boundary control method of Belishev, and the results of Nakamura and Uhlmann on identification of stationary Lame system we obtain uniqueness of time-independent elastic parameters when observation time is large.

By imposing additional pseudo-convexity type constraints (implying absence of trapper rays) we derive Carleman type estimates and exact boundary controllability (joint work with Cheng and Yamamoto).

A common tool is a reduction to a principally diagonal hyperbolic system of second order and use of available results for hyperbolic equations.

Chris Judge (Indiana University) cjudge@indiana.edu 

Behavior of the Laplace Spectrum of Degenerating Riemannian Manifolds

I will discuss the limiting behavior of eigenfunctions/eigenvalues of the Laplacian of a family of Riemannian metrics that degenerates on a hypersurface. In particular, I will describe the transition from purely discrete to continuous spectrum and give sufficient conditions for eigenvalue branches to have limits. The results generalize earlier work concerning the classical degeneration of hyperbolic surfaces.

Hyeonbae Kang (Department of Mathematics, Seoul National University)  hkang@math.snu.ac.kr

Boundary Determination of Anisotropic Conductivity via the Dirichlet to Numann Map

Suppose that is a Cm-smooth anisotropic conductivity. We show that the derivatives of up to order m-1 can be recovered on the boundary of the given domain via the Dirichlet to Numann map (up to isometry).

Vladimir V. Kryzhniy (Kuban State University of Technology, Krasnodar, Russia)  kryzhniy@usa.net

Regularizing Algorithm of Numerical Inversion of Laplace Transform usage in Expansion of Exponential Signal Into Partial Fractions

The problem of exponential signal expansion into partial fraction is characteristic for many Physical systems relaxing from exited into a normal state, in particular for systems exited with the help of Nuclear Magnetic Resonance. For solving this problem the usage of regularized algorithm of Laplace Transform Numerical Inversion is suggested. It turns out to be that experimental measurement could have multiple usages for exponential sum transformation into the sum of other functions. The Laplace Transform Numerical Inversion algorithm could be taken as a start point for solving this problem.

Yaroslav Kurylev (Department of Mathematical Sciences, Loughborough University)

Geometric Convergence for Manifolds with Boundary and Reconstruction of a Riemannian Manifold    Slides

Joint work with A. Katsuda, M. Lassas.

We consider the problem of a Riemannian manifold reconstruction. The inverse data used is the boundary spectral data of the Neumann Laplacian or its finite approximation. We describe (pre)compacts in Gromov-Hausdorff toplogy which provide stable identification of a Riemannian manifold. Under additional conditions on geometry, we describe a stable algorithm to find a metric approximation to the unknown manifold. Bibliography;

[1] Belishev, M. An approach to multidimensional inverse problems for the wave equation. Dokl. Akad. Nauk SSSR 297 (1987), 524-527.

[2] Belishev, M., Kurylev, Y. To the reconstruction of a Riemannian manifold via its spectral data (BC-method). Comm. Part. Diff. Eq. 17 (1992), no. 5-6, 767--804.

[3] Katsuda A., Kurylev Y. and Lassas M. Stability and reconstruction in the inverse boundary spectral problem. to appear.

[4] Katchalov A., Kurylev Y. and Lassas M. Inverse boundary spectral problems, Chapman/CRC (2001), xx+290 pp.

[5] Kurylev, Y. Multidimensional Gel'fand inverse problem and boundary distance map. in: Inv. Probl. related to Geom. (ed. H.Soga), (1997), 1-15.

Matti Lassas (Department of Mathematics, University of Helsinki)  lassas@cc.helsinki.fi

Inverse Boundary Spectral Problems and Gauge Transformations    pdf    postscript    dvi

Joint work with A. Katchalov and Y. Kurylev.

We consider the inverse boundary spectral problem for second order elliptic differential operators on a compact manifold and the inverse problems for the corresponding hyperbolic equations. The objective of these problems is to reconstruct the unknown manifold and the operator on it from the boundary spectral data (the boundary, the eigenvalues and the boundary values of the eigenfunctions), or for the wave equation, from the hyperbolic Dirichlet-to-Neumann map or the energy flux through the boundary. It turns out that all these boundary data determine equivalence class of the boundary spectral data in the gauge transformations $u(x)\mapsto \kappa(x)u(x)$. To analyse these inverse problems simultaneously, we consider the problem of reconstruction of the gauge-equivalence class of the operator from the gauge-equivalence class of the boundary data. We also describe some methods to solve this problem for selfadjoint [1-4], and non-selfadjoint operators [5].

[1] Belishev, M. An appropch to mult dimensional inverse problems for the wave equation. Dokl. Akad. Nauk SSSR 297(1987), 524-527.

[2] Belishev, M., Kurylev, Y. A nonstationary inverse problem for the multidimenional wave equation "in the large." Zap. Nauk. Sem. LOMI 165(1987), 21-30.

[3] Belishev, M., Kurylev, Y. To the reconstruction of a Riemannian manifold via its spectral data (BC-method). Comm. Part. Diff. Eq. 17(1992), 767-804.

[4] Katchalov A., Kurylev Y. and Lassas M. Inverse boundary spectral problems, Chapman/CRC, in print, 290 pp.

[5] Kurylev, Y., Lassas, M. Gelf'and inverse problem for a quadratic operator pencil. J. Funct. Anal. 176(2000), 247-263.

Walter Littman (School of Mathematics, University of Minnesota)  littman@math.umn.edu

The Scope of the "SUR" Method in Boundary Control

A decade ago a method of boundary control briefly described by "(local)Smoothing + Uniqueness + Reversibility => controllability" (or "SUR => controllability") was devised by S. Taylor and the speaker. It essentially consists of reducing the Control question to a Fredholm equation in the initial manifold t = 0. The method will be illustrated by examples and its advantages and disadvantages will be compared with those of other methods.

Shari Moskow (Department of Mathematics, University of Florida)  moskow@math.ufl.edu

Identification of Conductivity Imperfections of Small Diameter

We show asymptotic formulae for a voltage potential in the presence of small inhomogeneities. We then discuss several possible techniques using these formulae to determine the location, size, and/or conductivities of these imperfections.

Joint work with Habib Ammari and Michael Vogelius.

Gen Nakamura (Department of Mathematics, Hokkaido University)  gnaka@math.sci.hokudai.ac.jp

Discretization of Dirichlet to Neumann Map and Inverse Boundary Value Problem    Slides

The Dirichlet to Neumann map has been used in extensive studies for inverse boundary value problems. These studies have shown the importance of Dirichlet to Neumann map for the theoretical studies of the inverse boundary value problems. However, the Dirichlet to Neumann map is the observation data by infinitely many measurements and it is really impractical observation data. If there is a systematic way to discretize the Dirichlet to Neumann map and possibility to use the discretized Dirichlet to Neumann map for identifying the unknown approximately, all of these theoretical studies which looked quite impractical become more meaningful. In my talk, a general principle of discretizing the Dirichlet to Neumann will be presented and this principle will be proved for the inverse boundary value problem for identifying the potential of the Schrodinger equation. Also, the rate of the convergence and the accuracy of the Tikhonov regularized solution to the true solution will be discussed when we take a discretized Diriclet to Neumann map with some noise as observation data. This is a joint work with Jin Cheng.

Jean-Pierre Puel (Laboratoire de Mathematiques Appliquees, Universite de Versailles St Quentin) jppuel@cmapx.polytechnique.fr

Some Methods for Studying Exact Controllability to Trajectories in Parabolic Evolution Equations and Applications

We will present the general problem of exact controllability to trajectories for parabolic dissipative problems with applications to various situations for the heat equation and possibly to Navier Stokes equations. We will give a general method which shows how global Carleman inequalities are involved in the resolution of this problem and we will give some complete results following Imanuvilov's method. As an indirect application we will comment upon the question of data assimilation for the corresponding systems.

Lizabeth V. Rachele (Department of Mathematics, Tufts University)  lrachele@math.purdue.edu

Uniqueness in Inverse Problems for Elastic Media   pdf    postscript

Rakesh (Department of Mathematical Sciences, University of Delaware)

Inverse Problems for Hyperbolic PDE    Slides

We discuss results for one space dimensional problems motivated by formally determined multidimentsional problems. We prove injectivity of the forward map (the "uniqueness"), characterize its range, and construct its inverse, for some problems. These are partly based on work done with Paul Sacks.

Fadil Santosa (IMA and MCIM)    santosa@math.umn.edu

Level Set Method for Inverse Problems and Optimal Design
pdf    postscript

We consider problems in which the desired unknown is the description of a region. Examples arising in inverse problems include determination of scattering obstacles, aperture reconstruction, and electrical impedance imaging. In optimal design, typical problems are ones where we seek a geometry which optimizes a certain design objective subject to some constraints. Both types of problems can be viewed as optimization problem where the unknown is parameterized by a level set function. In this talk, the speaker will give an overview of an approach based on the level set method. It will be followed by examples from different applications.

Jin Keun Seo (Yonsei University, Seoul, Korea)  jkseo@math.umn.edu

A Real Time Algorithm for the Location Search of Discontinuous Conductivities with One Measurement

We consider an inverse problem for finding the anomaly of discontinuous electrical conductivity by one current-voltage observation. We develop a real time algorithm for determining the location of the anomaly. This new idea is based on the observation of the pattern of a simple weighted combination of the input current and the output voltage. Combined with the size estimation result, this algorithm gives a good initial guess for Newton-type schemes. We give the rigorous proof for the location search algorithm. Both the mathematical analysis and its numerical implementation indicate our location search algorithm is very fast, stable and efficient.

This is joint work with Ohin Kwon and Jeong-Rock Yoon.

 

Vladimir Sharafutdinov (Sobolev Institute of Mathematics, Novosibirsk, Russia)  sharaf@math.nsc.ru

Deformation Boundary Rigidity and Other Applications of the Ray Transform to Inverse Problems    pdf  postscript    dvi

The ray transform of symmetric tensor fields of rank 2 first arises in the linearization of the boundary rigidity problem. It tuns out to be also useful in many other tomography problems. We discuss the recent progress in deformation boundary rigidity as well as applications of the ray transform to anisotropic inverse problems for electrodynamic and elastic waves.

John Sylvester (Mathematics Department , Dale Winebrenner, Applied Physics Laboratory, University of Washington)

Inverse Theory and an Experiment

We will discuss a linear inverse scattering calculation that will explain the difficulties with a conventional electro-magnetic remote sensing experiment and suggest how to design a better one.

The experiment (circular intensity differential scattering) deals with determining the presence of chiral material in a solution using a a single source (a green laser) and a single receiver. The model is Maxwell's equations and we will show how to use the Born approximation and a little geometry to decide where to put the source and the receiver and how to polarize the beam, so as to best distinguish the small chiral effect from the much larger effects due to changes in permittivity. The result suggests that the prevailing experimental technique for sensing chirality can be improved upon.

Takashi Takiguchi (Department of Mathematics, National Defense Academy) takashi@cc.nda.ac.jp 

A Generalization of Helgason's Support Theorem    pdf    postscript

We disscuss a gengeralization of Helgason's support theorem for the Radon transform. In this theorem, the assumption of rapid decay of functions is essential. We restrict this rapid decay condition to an open cone and give a generalization. We also mention that our generalization is not possible with no global decay condition, to prove which we construct a counterexample.

Michael E. Taylor (Department of Mathematics, University of North Carolina)  met@email.unc.edu

The Wave Equation: Analytical Subject and Analytical Tool    Slides

Since it was produced to model vibrations of various objects, from membranes to electromagnetic fields, the wave equation has been a topic to which many tools from analysis have been brought to bear. These tools include energy identities, methods of harmonic analysis, geometrical optics, and symplectic geometry, to name a few. In turn the study of wave equations has given back to analysis, particularly the rest of PDE, many dividends. For example, many sharp results on the Laplace operator fall out of the study of the wave equation. This talk will survey some of this interplay, both classical and recent.

Roberto Triggiani (Department of Mathematics University of Virginia)   rt7u@virginia.edu

Differential Geometric Methods in the Control of PDEs (Overview)

Over the past 4-5 years, differential (Riemann) geometric methods have emerged as a powerful new line of research to obtain general inverse-type, a-priori inequalities of interest in boundary control theory (continuous observability/stabilization inequalities) for various classes of PDEs. Their range of applicability now includes: second order hyperbolic equations; Schrodinger-type equations; various plate-like equations; systems of elasticity; very complicated shell models described more below, etc, all with variable coefficients, where the 'classical' energy methods of the early/mid-eighties proved inadequate. In all of these PDEs classes, main features of these differential geometric methods are:

1. they apply to operators with principal part which is allowed to have variable coefficients (in space) with low regularity, C1;

2. they tolerate energy level terms which are both space- and time-dependent, and only in L-infinity in time and space;

3. they yield rather general and verifiable sufficient conditions, which may serve for the construction of many complicated, variable coefficient examples, as well as for counter-examples (say, in the hyperbolic case, in dimension greater than 2), even when the control acts on the whole boudary;

4. they provide a good estimate (for some classes, optimal estimate) of the minimal time for observability in the hyperbolic case, and arbitrary short time when there is no finite speed of propagation;

5. they combine well with microlocal analysis methods needed for sharp trace estimates and for shifting topologies, thus producing at the end very general observability/stabilization results, with variable coefficients and with no geometric conditions on the observed (controlled) portion of the boundary;

6. ultimately, and with the same effort, they apply to these classes of PDEs defined on Riemann manifolds.

In addition, differential geometric methods have recently provided the intrinsic language for: (i) modelling the motion of dynamic shells far beyond the classical approach (rooted in classical geometry), and (ii) performing observability/stabilization energy methods on their very complicated equations, for which the classical setting based on Christoffel symbols appears to be unfeasible. A shell is a curved geometric object which can be modeled as a system of two PDEs both of hyperbolic type with strong coupling depending on the curvature: an 'elastic wave-type' equation ('curved system of elasticity') in the in-plane displacement; and a 'curved Kirchhoff plate-like equation' for the vertical displacement.

In this talk we shall review a recent Riemann geometric line of research for PDEs with variable coefficients as above, or else on manifolds, yielding general Carleman-type estimates and hence observability/stabilization estimates. They are obtained by using energy methods (multipliers) in a corresponding natural Riemann metric. These multipliers may be viewed as extending the 'Carleman multipliers' in the Euclidean setting with constant principal part (and variable energy level terms). In addition, the combination of these Riemann methods with microlocal sharp trace estimates will be given. Three canonical cases are chosen:

(i) the control of general second order hyperbolic equations with purely Neumann BC on both the controlled and the uncontrolled portion of the boundary;

(ii) the control of a general plate equation with third order derivatives on the displacement and first order derivative on the velocity;

(iii) boundary stabilization of a shell by a non-linear natural feedback in the moment and strains, with no geometric conditions on the controlled part of the boundary.

Gunther Uhlmann (University of Washington)   gunther@math.washington.edu

Inside-Out: Inverse Boundary Problems    slides

In this talk we will survey some of the significant progress obtained in the last 20 years or so on inverse boundary problems (IBP). The type of IBP we will discuss concern with the determination of the coefficient(s) of a partial differential equation on a bounded domain of Euclidean space or, more generally, a compact Riemannian manifold with boundary, by measuring the associated Dirichlet to Neumann map.

Nicolas Valdivia (Department of Mathematics, Wichita State University)  valdivia@math.twsu.edu

Uniqueness in Inverse Obstacle Scattering with General Conductive Boundary Condition

We give uniqueness results for the inverse scattering problem where the unknown scatterer D is a bounded open set and some coefficients of an elliptic equation are unknown as well. On the boundary of D conductivity conditions are prescribed, so we consider a penetrable medium. Our data is the scattering amplitude A given by one frequency. The proof is based on the method of singular solutions which is constructive and can be used for numerical methods.

Judith Vancostenoble (Laboratoire de Mathematiques Emile Picard, Universite Paul Sabatier)  vancoste@mip.ups-tlse.fr

Optimality of Energy Estimates for the Wave Equation with Nonlinear Boundary Velocity Feedbacks

Joint work with Patrick Martinez.

We consider the wave equation damped by a nonlinear boundary velocity feedback q(ut). We consider the case where q has a linear growth at infinity. We prove that the usual decay rate estimates proved by M. Nakao, A. Haraux, F. Conrad et al., E. Zuazua, V. Komornik when q has a polynomial behavior at zero and by the second author in the general case are in fact optimal in one space dimension.

More generally, we prove that the energy decays exactly like the solution of an explicit and simple ordinary differential equation.

Masahiro Yamamoto (Graduate School of Mathematical Sciences, The University of Tokyo) myama@ms.u-tokyo.ac.jp

Inverse Problems for Hyperbolic Systemsby Carleman Estimates 

We consider inverse problems of determining source terms in evolutional systems in mathematical physics (e.g. the isotropic Lame system, Maxwell's systems, some fluid dynamics). The inverse problem is described as follows; utt = Lu + R(x,t)f(x), where u is a state variable (e.g., displacement) and L is a partial differential operator (e.g., the isotropic Lame operator), R is a given matrix-valued function and f = f(x) is unknown to be determined from boundary measurements. We mainly discuss the global uniqueness and the stability in this inverse problem, by means of Carleman estimates. This kind of inverse problems have been studied by Bukhgeim, Isakov, Klibanov and others. Here we mainly take systems whose principal parts are coupled and we will show a new way for establishing the uniqueness and stability. This work is a joint paper with Oleg Imanuvilov (Iowa State University, Ames).

Stephen Zelditch (Department of Mathematics, Johns Hopkins University) szelditch@jhu.edu

Inverse Spectral and Resonance problems for Analytic Plane Domains

We will give some inverse spectral results for analytic plane domains with one symmetry. First, we consider bounded simply connected analytic plane domains with one isometry that reverses a bouncing ball orbit of a fixed length L. One may think of the domain as obtained by flipping the graph of a function with zeros only at two endpoints around the x- axis. Among such domains, we prove that the Dirichlet (or Neumann) spectrum determines the domain. Second, we consider the plane minus a mirror symmetric pair of convex analytic obstacles. We prove that the resonances of the exterior domain in a logarithmic neighborhood of the real axis determines the obstacles. The method is based on an exact trace formula suggested by Balian-Bloch and on an analysis of Feynman diagrams and their amplitudes.

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