Schedule for: 16w5083 - Dirichlet-to-Neumann Maps: Spectral Theory, Inverse Problems and Applications

A welcome drink will be served at the hotel.

The goal of this talk will be to review some of the recent results regarding the spectrum of the Dirichlet-to-Neumann operator (aka the Steklov spectrum). The emphasis will be on isoperimetry: I will discuss the general question to know how large a given Steklov eigenvalue can be in various settings: euclidean domains, compact surfaces and manifolds, conformal perturbations.

I will present a work in progress, in collaboration with A. El Soufi and A. Girouard. I consider a Riemannian manifold $(M,g)$ with boundary $\Sigma$ and I will discuss the two following questions : - what does occur with the spectrum of the Steklov operator $\mathcal{D}$ when we deform the Riemannian metric $g$ conformally on $M$, but let it fixed on the boundary $\Sigma$? - relations between the Steklov spectrum and the Laplace spectrum of the boundary $\Sigma$.

In this talk, we will consider the inverse spectral problem associated to the Dirichlet-to-Neumann operator on a manifold with boundary. Specifically, we will show how to recover the number of boundary components, as well as each of their lengths, from the spectrum of the Dirichlet-to-Neumann operator (also called the Steklov spectrum) on a smooth surface with boundary. The proof relies on surprisingly sharp spectral asymptotics which are interesting in their own right. This is joint work with A. Girouard (U. Laval), L. Parnovski (UCL), and I. Polterovich (U. Montreal).

We discuss a generalized index for certain meromorphic, unbounded, operator-valued functions that applies to energy parameter dependent Dirichlet-to-Neumann maps associated to uniformly elliptic partial differential operators, particularly, non-self-adjoint Schr\"odinger operators, on bounded Lipschitz domains. In particular, employing appropriate Krein-type resolvent formulas we prove an index formula that relates the difference of the algebraic multiplicities of the discrete eigenvalues of Robin and Dirichlet realizations of non-self-adjoint Schr\"odinger operators.

The high-accuracy computation of mixed Steklov-Neumann (sloshing) or Steklov-Dirichlet eigenvalues presents several challenges. In this talk, we present a strategy based on an integral-equation reformulation of the eigenvalue problem. The discretization strategy is based on the use of appropriately graded meshes. We demonstrate the high accuracy attainable by this strategy.

We consider finite volume Riemann surfaces with hyperbolic cusps. Such manifolds appear in number theory but have also been studied extensively in their own right. I will give a short introduction into the scattering and spectral theory of such surfaces. I will then show how the scattering matrix can be computed from the Dirichlet to Neumann map on a surface with boundary. This can be used in numerical computations to locate scattering resonances and investigate their behaviour under perturbations of the metric. Some numerical results will be presented. (joint work with Michael Levitin)

In this survey lecture we will discuss results which have been obtained on the eigenvalues of the Dirichlet to Neumann map for Riemannian manifolds with boundary. The discussion will include asymptotic behavior of the spectrum, geometric estimates on low eigenvalues, and problems on finding extremal metrics for the eigenvalues on surfaces.

There are several questions about the zero set of Laplace eigenfunctions that have proved to be extremely hard to deal with and remain unsolved. Among these are the study of the size of the zero set, the study of the number of connected components, and the study of the topology of such components. A natural approach is to randomize the problem and ask the same questions for the zero sets of random linear combinations of eigenfunctions. In this talk I will present some recent results in this direction.

By using real and complex geometrical optics solutions and a reduction to a vectorial Schroedinger equation we derive stability estimates for mentioned coefficients where logarithmic (unstable) part goes to zero as the energy/wave number grows. Optimizing with respect to the wave number we conclude that Hoelder type stability instead of logarithmic one can be achieved which promises a substantial increase in (numerical) resolution. We discuss possible further developments.

The anisotropic Calder/'{o}n problem consists in determining a Riemannian manifold with boundary from its Dirichlet-to-Neumann map. This problem arises as a model for electrical imaging in anisotropic media, and it is one of the fundamental inverse problems in a geometric setting. We discuss recent progress in this problem and its variants in three and higher dimensions.

In this talk, we give some simple counterexamples to uniqueness for the Calderon problem on Riemannian manifolds with boundary when the Dirichlet and Neumann data are measured on disjoint sets of the boundary. We provide counterexamples in the case of three dimensional Riemannian manifolds with boundary having the topology of toric cylinders. This is a work in collaboration with Thierry Daudé (Cergy-Pontoise) and Niky Kamran (McGill).

We consider the localized DN map on Lorentzian manifolds on a timelike set of the boundary. We show that we can recover the jet of the metric, the magnetic field and the potential on the boundary up to the natural gauge group of transformations. Next, we show that the DN map, properly localized, is an FIO associated to the lens relation and as such recovers the lens relation. We also show that the DN map recovers the light ray transform of the magnetic field and the potential in a stable way. This is a joint work with Plamen Stefanov.

This will be a survey talk on how the Dirichlet-to-Neumann map and its analogs arises in several inverse problems. We will also formulate some open problems.

We shall discuss uniform $L^p$ resolvent estimates for elliptic operators. Originally obtained by Kenig, Ruiz, and Sogge in the case of the Euclidean space, they have been established by Dos Santos Ferreira, Kenig, and Salo for compact manifolds, in the case of the Laplacian. We shall discuss an extension to the case of higher order self-adjoint operators, as well as to some weakly non-selfadjoint operators, such as the stationary damped wave operator. Our approach is based on the techniques of semiclassical Strichartz estimates. Applications to spectral theory for periodic Schrodinger operators as well as to inverse boundary problems for elliptic operators with coefficients of low regularity will be presented. This talk is based on joint works with Gunther Uhlmann and with Nicolas Burq and David Dos Santos Ferreira.

We prove sharp upper and lower bounds for the nodal length of Steklov eigenfunctions on real-analytic Riemannian surfaces with boundary. The argument involves frequency function methods for harmonic functions in the interior of the surface as well as the construction of exponentially accurate approximations for the Steklov eigenfunctions near the boundary. This is joint work with Iosif Polterovich and David Sher.

The main focus of this talk would be on the inverse Steklov problem on compact 2-dimensional Riemannian orbifolds. This study is inspired by the work of Girouard, Parnovski, Polterovich and Sher on the Steklov spectrum of a Riemannian surface. We obtain a precise asymptotic for Steklov eigenvalues of a 2-dimensional orbifold. We also discuss whether an orbifold and a smooth surface can be distinguished by their Steklov spectra, and whether the number of singular and non-singular boundary components are audible. This is a work in progress joint with T. Arias-Marco, E. Dryden, C. Gordon, A. Ray and E. Stanhope.

We give an overview of the appearance of Dirichlet-to-Neumann maps, and Neumann-to-Dirichlet maps, as the data in geophysical inverse problems, specifically, in electrostatic (DC), elastostatic, low-frequency electromagnetic (MT and CSEM), and time-harmonic (vibroseis) and hyperbolic (passive and active source) seismic inverse problems. We discuss uniqueness and conditional Lipschitz stability results, and open problems, with a view to the solid earth, real-world geomaterial properties, heterogeneities, and interior structures.

We discuss the inverse problem of determining the, possibly anisotropic, conductivity of a body $\Omega\subset\mathbb{R}^{n}$ when the so-called Neumann-to-Dirichlet map is locally given on a non empty curved portion $\Sigma$ of the boundary $\partial\Omega$. We prove that anisotropic conductivities that are \textit{a-priori} known to be piecewise constant matrices on a given partition of $\Omega$ with curved interfaces can be uniquely determined in the interior from the knowledge of the local Neumann-to-Dirichlet map. This is joint work with Giovanni Alessandrini and Maarten V. de Hoop.

"The inverse source problem for the Helmholtz equation (time harmonic wave equation) seeks to recover information about a radiating source from remote observations of a monochromatic (single frequency) radiated wave measured far from the source (the far field). The two properties of far fields that we use to deduce information about shape and location of sources depend on the physical phenomenon of evanescence, which limits imaging resolution to the size of a wavelength, and the formula for calculating how a far field changes when the source is translated. We will show how adaptations of ""uncertainty principles"", as described by Donoho and Stark [1] provide a very useful and simple tool for this kind of analysis. This is joint work with Roland Griesmaier. [1] Donoho and Stark -- Uncertainty Principles and Signal Recovery SJAP 1989"

"We review the variational approach to the inverse conductivity problem, in the case of discontinuous conductivities. We discuss several difficulties that arise, in particular instability. We present a variational approach that combines, simultaneously, regularisation and discretisation of the inverse problem. We show that the corresponding discrete regularised solutions are a good approximation of the solution to the inverse problem. The method provides a clear indication on the regularisation parameter and on the mesh size of the discretisation that should be used when solving numerically the inverse problem."

"We study the geometric Whitney problem on how a Riemannian manifold $(M,g)$ can be constructed to approximate a metric space $(X,d_X)$. This problem is closely related to manifold interpolation (or manifold learning) where a smooth $n$-dimensional surface $S\subset {\mathbb R}^m$, $m>n$ needs to be constructed to approximate a point cloud in ${\mathbb R}^m$. These questions are encountered in differential geometry, machine learning, and in many inverse problems encountered in applications. The determination of a Riemannian manifold includes the construction of its topology, differentiable structure, and metric. We give constructive solutions to the above problems. Moreover, we characterize the metric spaces that can be approximated, by Riemannian manifolds with bounded geometry: We give sufficient conditions to ensure that a metric space can be approximated, in the Gromov-Hausdorff or quasi-isometric sense, by a Riemannian manifold of a fixed dimension and with bounded diameter, sectional curvature, and injectivity radius. Also, we show that similar conditions, with modified values of parameters, are necessary. Moreover, we characterise the subsets of Euclidean spaces that can be approximated in the Hausdorff metric by submanifolds of a fixed dimension and with bounded principal curvatures and normal injectivity radius. The above interpolation problems are also studied for unbounded metric sets and manifolds. The results for Riemannian manifolds are based on a generalisation of the Whitney embedding construction where approximative coordinate charts are embedded in ${\mathbb R}^m$ and interpolated to a smooth surface. We also give algorithms that solve the problems for finite data. The results are done in collaboration with C. Fefferman, S. Ivanov, Y. Kurylev, and H. Narayanan. References: [1] C. Fefferman, S. Ivanov, Y. Kurylev, M. Lassas, H. Narayanan: Reconstruction and interpolation of manifolds I: The geometric Whitney problem. ArXiv:1508.00674"

In this joint work with B.M. Brown and J. Reyes, I discuss recovery of the coefficients in a non-selfadjoint Maxwell system from boundary measurements on a part of the boundary. Using an abstract reformulation of a trick of Ammari and Uhlmann (2004) in terms of operator resolvents, together with a transformation of the Maxwell system which has been used by several authors including Caro, we are able prove a uniqueness result which does not require hypotheses about the geometry of the boundary where measurements cannot be made.

In this talk we consider the lens rigidity problem with partial data for conformal metrics, in the presence of a magnetic field, on a compact manifold with boundary. We show that one can uniquely determine the conformal factor and the magnetic field near a strictly convex (with respect to magnetic geodesics) boundary point where the lens data is accessible. The local result also leads to a global lens rigidity result under some global foliation assumption.

We prove a quantum Sabine law for the location of resonances in transmission problems. In this talk, our main applications are to scattering by strictly convex, smooth, transparent obstacles and highly frequency dependent delta potentials. In each case, we give a sharp characterization of the resonance free regions in terms of dynamical quantities. In particular, we relate the imaginary part of resonances to the chord lengths and reflectivity coefficients for the ray dynamics and hence give a quantum version of the Sabine law from acoustics

In 1975 Hersch, Payne and Schiffer used the concept of conjugate harmonic functions on the complex plane to prove a sharp upper bound for Steklov eigenvalues on simply connected domains. In this talk we will discuss a higher dimensional version of this concept defined for an arbitrary Riemannian manifold with boundary - conjugate harmonic forms. As a result, an inequality relating Steklov eigenvalues of the manifold with the Laplace eigenvalues of the boundary is obtained. This inequality is reduced to Hersch-Payne-Schiffer inequality in the case of simply connected domains and yields improved upper bounds even in a two-dimensional case.

In this talk I present Carleman estimates for fractional Schrödinger equations by means of the Caffarelli-Silvestre extension. I discuss how these imply the strong unique continuation principle even in the presence of rough potentials. Furthermore, I explain how they can be exploited to derive quantitative unique continuation results and upper bounds on the size of nodal domains of eigenfunctions associated with the corresponding Dirichlet-to-Neumann map.

" In inverse scattering theory the aim is to determine a scattering potential $q$ from appropriate measurements. In many applications the scatterer is non-smooth and vastly complicated. For such scatterers, the inverse problem is not so much to recover the exact micro-structure of an object but merely to determine the parameters or functions describing the properties of the micro-structure. An example of such a parameter is the correlation length of the medium which is related to the typical size of ``particles'' inside the scatterer. In mathematical terms, the potential $q$ is assumed to be a Gaussian random function whose covariance operator is a classical pseudo-differential operator. We show that the backscattered field, obtained from a single realization of the random potential $q$, determines uniquely the principal symbol of the covariance operator. This is a joint work with Tapio Helin and Matti Lassas. "

A well known inverse problem is whether one can determine the potential in a Schr\”odinger equation from the Dirichlet-to-Neumann map. The problem with partial data restricts to the case where the Dirichlet-to-Neumann map is only measured on possibly very small subset of the boundary. This problem was solved in dimension two by Imanuvilov, Uhlmann and Yamamoto but remains open in higher dimensions. The best results so far in dimension higher than three are due to Kenig, Sj\”ostrand and Uhlmann who prove uniqueness of the potential from the partial Dirichlet-to-Neumann map measured on arbitrary subset of, say, a strictly convex open set. We are interested in the quantitative counterpart of this result, and in the stability estimates that one can obtain on the potentials from the Dirichlet-to-Neumann maps. This is a joint work with Pedro Caro and Alberto Ruiz.