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Events
Inverse and Spectral Problems for (Non)-Local Operators

MPI für Mathematik in den Naturwissenschaften Leipzig

E1 05 (Leibniz-Saal)

conference

10/09/2018 14/09/2018

Inverse and Spectral Problems for (Non)-Local Operators

Please see also the poster overview in the topic 'Poster session'.

Speakers

Pedro Caro

Basque Center for Applied Mathematics, Bilbao

Mihajlo Cekić

Max Planck Institute for Mathematics, Bonn

David dos Santos Ferreira

Université de Lorraine

Asma Hassannezhad

University of Bristol

Michael Levitin

University of Reading

Fabricio Maciá

Universidad Politécnica de Madrid

Lauri Oksanen

University College London

Gabriel Paternain

University of Cambridge

Program

08:30 - 08:50	Registration
08:50 - 09:00	Welcome Address
09:00 - 10:30	Minicourse: Gabriel Paternain (University of Cambridge) Geodesic X-ray transforms
10:30 - 11:00	Coffee
11:00 - 12:30	Minicourse: Michael Levitin (University of Reading) Spectral Properties of the Dirichlet-to-Neumann Map
12:30 - 14:30	Lunch Break
14:30 - 15:30	Lecture: Mihajlo Cekić (Max Planck Institute for Mathematics, Bonn) Calderon problem for Yang-Mills connections In the classical Calderon conjecture we want to recover a metric on a compact manifold, up to diffeomorphism fixing the boundary, from the Dirichlet-to-Neumann (DN) map of the metric Laplacian. This problem is routinely motivated by applications and is open in dimension 3 and higher. In this talk, we fix the metric and consider the DN map of the connection Laplacian for Yang-Mills connections on vector bundles. We sketch the proof of uniqueness up to gauges fixing the boundary and propose two approaches. In the first one we develop a new technique, involving degenerate unique continuation principles and an analysis of the zero set of solutions to an elliptic PDE. The second argument involves a Runge-type approximation along curves to recover holonomy and we are able to show uniqueness of both an arbitrary bundle and a Yang-Mills connection.
15:30 - 16:00	Coffee
16:00 - 17:00	Lecture: Maria Angeles Garcia-Ferrero (ICMAT, Madrid, Spain) Solutions to PDEs with prescribed properties Combining the construction of suitable local solutions to PDEs with appropriate global approximation theorems, we can obtain global solutions to PDEs with prescribed geometric or topological properties. This technique has been used in many contexts involving elliptic equations. In this talk I will focus on showing global approximation theorems for parabolic equations and their utilisation to prove the existence of solutions of the heat equation with local hot spots with prescribed behaviour, among other applications. This is a joint work with A. Enciso and D. Peralta-Salas.
17:00 - 17:30	Poster Blitz Filip Bosnic Bielefeld University, Germany Bobby Cheng University of Sussex, United KingdomQuantum Resonances in Relativistic Systems Significant amounts of research have been completed on mathematical quantum resonances in the non-relativistic setting. This is achieved by studying the spectra of the perturbed Schrödinger operator.However success in generalizing these results to the relativistic setting has been limited. In this poster I have outlined qualitatively some of the results of my research in this field thus far. Dimitrije Cicmilovic University of Bonn, GermanySymplectic non-squeezing theorem in Hilbert space Giovanni Covi Jyväskylän Yliopisto, FinlandInverse problems for a fractional conductivity equation Giovanni Covi Jyväskylän Yliopisto, FinlandInverse problems for a fractional conductivity equation We show global uniqueness in two inverse problems for a fractional conductivity equation: an unknown conductivity in a bounded domain is uniquely determined by measurements of solutions taken in arbitrary open, possibly disjoint subsets of the exterior. The results are based on a reduction from the fractional conductivity equation to the fractional Schrödingerequation. A NON LOCAL PROBLEM OF DIFFERENTIAL EQUATIONS VIA NON LOCAL OPERATORS OF ARBITRARY ORDER In the proposed poster, we are concerned with a nonlocal problemfor a class of differential equations using nonlocal fractional operators in Banachspaces. Some existence results of positive solutions are obtained. Then, thestability of the problem is discussed. Finally, to illustrate the effectiveness ofthe results obtained, some example are presented New mixed operators of Hadamard type with applications In the 2d poster, we establish recent results on the Hadamard operators. Some mixed integral operators, that generalize those of Hadamard type, are delivered and some of their properties are proved. Finally,some applications are discussed. Guy Fabrice Foghem Gounoue Bielefeld University, GermanyInhomogeneous Neumann Problem for the fractional Laplace operator Robin Görmer Universität Bremen, Germany3D Electrical Impedancetomography Andreea Grecu University of Bucharest, RomaniaThe Schrodinger equation on star-graphs under general coupling conditions Yulia Meshkova St. Petersburg State University, RussiaOperator error estimates for homogenization of elliptic and parabolic systems We consider a matrix strongly elliptic second order differential operator acting in a bounded domain with the Dirichlet boundary condition. The operator is self-adjoint. Coefficients are periodic and oscillate rapidly. We study the behavior of solutions of the corresponding elliptic and parabolic systems in the small period limit. The results can be written as approximations of the resolvent and the semigroup in L2→L2 and L2→H1 operator norms. So, the estimates of this type are called operator error estimates in homogenization theory. The talk is based on a joint work with T. A. Suslina. Mihai Nechita University College London, United KingdomUnique continuation for the Helmholtz equation. Stability estimates and numerical analysis We consider the unique continuation problem for the Helmholtz equation that arises, e.g. in inverse boundary value problems for the wave equation. Following previous work by Isakov we prove conditional Hölder stability estimates with constants independent of the wave number, when the solution is continued along a convex surface. The main tools we employ are Carleman estimates and semiclassical analysis.We then introduce a stabilized finite element method and prove convergence with the order given by the conditional stability, and with explicit dependence on the wave number. Andrea Nickel Bielefeld University, DeutschlandA nonlocal maximum principle Leyter Potenciano Machado University of Jyväskylä, FinlandA resolvent estimate for the magnetic Schrödinger operator in the presence of short and long-range Jesse Railo University of Helsinki & University of Jyväskylä, FinlandTensor tomography on Cartan–Hadamard manifolds We study the geodesic x-ray transform on Cartan–Hadamard manifolds, generalizing the x-ray transforms on Euclidean and hyperbolic spaces that arise in medical and seismic imaging. We prove solenoidal injectivity of this transform acting on functions and tensor fields of any order. The functions are assumed to be exponentially decaying if the sectional curvature is bounded, and polynomially decaying if the sectional curvature decays at infinity. Yafet Sanchez Sanchez Max Planck Institute-Bonn, GermanyPropagators on spacetimes of low regularity Vanja Wagner Universität Bielefeld, GermanyBoundary behaviour of certain pure-jump Markov processes on sets

09:00 - 10:30	Minicourse: Gabriel Paternain (University of Cambridge) Geodesic X-ray transforms
10:30 - 11:00	Coffee
11:00 - 12:30	Minicourse: Michael Levitin (University of Reading) Spectral Properties of the Dirichlet-to-Neumann Map
12:30 - 13:30	Lunch at MPI combined with Poster Session Filip Bosnic Bielefeld University, Germany Bobby Cheng University of Sussex, United KingdomQuantum Resonances in Relativistic Systems Significant amounts of research have been completed on mathematical quantum resonances in the non-relativistic setting. This is achieved by studying the spectra of the perturbed Schrödinger operator.However success in generalizing these results to the relativistic setting has been limited. In this poster I have outlined qualitatively some of the results of my research in this field thus far. Dimitrije Cicmilovic University of Bonn, GermanySymplectic non-squeezing theorem in Hilbert space Giovanni Covi Jyväskylän Yliopisto, FinlandInverse problems for a fractional conductivity equation Giovanni Covi Jyväskylän Yliopisto, FinlandInverse problems for a fractional conductivity equation We show global uniqueness in two inverse problems for a fractional conductivity equation: an unknown conductivity in a bounded domain is uniquely determined by measurements of solutions taken in arbitrary open, possibly disjoint subsets of the exterior. The results are based on a reduction from the fractional conductivity equation to the fractional Schrödingerequation. A NON LOCAL PROBLEM OF DIFFERENTIAL EQUATIONS VIA NON LOCAL OPERATORS OF ARBITRARY ORDER In the proposed poster, we are concerned with a nonlocal problemfor a class of differential equations using nonlocal fractional operators in Banachspaces. Some existence results of positive solutions are obtained. Then, thestability of the problem is discussed. Finally, to illustrate the effectiveness ofthe results obtained, some example are presented New mixed operators of Hadamard type with applications In the 2d poster, we establish recent results on the Hadamard operators. Some mixed integral operators, that generalize those of Hadamard type, are delivered and some of their properties are proved. Finally,some applications are discussed. Guy Fabrice Foghem Gounoue Bielefeld University, GermanyInhomogeneous Neumann Problem for the fractional Laplace operator Robin Görmer Universität Bremen, Germany3D Electrical Impedancetomography Andreea Grecu University of Bucharest, RomaniaThe Schrodinger equation on star-graphs under general coupling conditions Yulia Meshkova St. Petersburg State University, RussiaOperator error estimates for homogenization of elliptic and parabolic systems We consider a matrix strongly elliptic second order differential operator acting in a bounded domain with the Dirichlet boundary condition. The operator is self-adjoint. Coefficients are periodic and oscillate rapidly. We study the behavior of solutions of the corresponding elliptic and parabolic systems in the small period limit. The results can be written as approximations of the resolvent and the semigroup in L2→L2 and L2→H1 operator norms. So, the estimates of this type are called operator error estimates in homogenization theory. The talk is based on a joint work with T. A. Suslina. Mihai Nechita University College London, United KingdomUnique continuation for the Helmholtz equation. Stability estimates and numerical analysis We consider the unique continuation problem for the Helmholtz equation that arises, e.g. in inverse boundary value problems for the wave equation. Following previous work by Isakov we prove conditional Hölder stability estimates with constants independent of the wave number, when the solution is continued along a convex surface. The main tools we employ are Carleman estimates and semiclassical analysis.We then introduce a stabilized finite element method and prove convergence with the order given by the conditional stability, and with explicit dependence on the wave number. Andrea Nickel Bielefeld University, DeutschlandA nonlocal maximum principle Leyter Potenciano Machado University of Jyväskylä, FinlandA resolvent estimate for the magnetic Schrödinger operator in the presence of short and long-range Jesse Railo University of Helsinki & University of Jyväskylä, FinlandTensor tomography on Cartan–Hadamard manifolds We study the geodesic x-ray transform on Cartan–Hadamard manifolds, generalizing the x-ray transforms on Euclidean and hyperbolic spaces that arise in medical and seismic imaging. We prove solenoidal injectivity of this transform acting on functions and tensor fields of any order. The functions are assumed to be exponentially decaying if the sectional curvature is bounded, and polynomially decaying if the sectional curvature decays at infinity. Yafet Sanchez Sanchez Max Planck Institute-Bonn, GermanyPropagators on spacetimes of low regularity Vanja Wagner Universität Bielefeld, GermanyBoundary behaviour of certain pure-jump Markov processes on sets
14:30 - 15:30	Lecture: Pedro Caro (Basque Center for Applied Mathematics, Bilbao) The Calderón problem with corrupted data I will talk about the inverse Calderón problem, which consists of determining the conductivity inside a medium by electrical measurements on its surface. Ideally, these measurements determine the Dirichlet-to-Neumann map and, therefore, one usually assumes the data to be given by such a map. This situation corresponds to having access to infinite-precision measurements, which is totally unrealistic. In this lecture, we will discuss the Calderón problem assuming the data to contain measurement errors and provide formulas to reconstruct the conductivity and its normal derivative on the surface.
15:30 - 16:00	Coffee
16:00 - 17:30	Minicourse: Fabricio Maciá (Universidad Politécnica de Madrid) Modes and quasimodes: concentration and non-concentration In this course we will present basic tools for the study of high-frequency eigenfunctions or quasimodes of Schrödinger-type elliptic operators on a compact manifold. Our mail goal will be to give a quantitative description of concentration and oscillation effects that can be developed by a sequence of quasimodes, and in particular to show how this issue is related the global geometry of the manifold. We will give applications to unique continuation-type estimates and to the Calderón problem. We will mainly use tools of from the theory of pseudodifferential operators and semiclassical measures that will be reviewed during the course.

09:00 - 10:00	Minicourse: Gabriel Paternain (University of Cambridge) Geodesic X-ray transforms
10:00 - 10:30	Coffee
10:30 - 12:00	Minicourse: Lauri Oksanen (University College London) Inverse problems on hyperbolic PDEs
12:00 - 13:00	Minicourse: Michael Levitin (University of Reading) Spectral Properties of the Dirichlet-to-Neumann Map
13:00 - 00:00	Discussion/Talk: Free afternoon discussions

09:00 - 10:30	Minicourse: Fabricio Maciá (Universidad Politécnica de Madrid) Modes and quasimodes: concentration and non-concentration In this course we will present basic tools for the study of high-frequency eigenfunctions or quasimodes of Schrödinger-type elliptic operators on a compact manifold. Our mail goal will be to give a quantitative description of concentration and oscillation effects that can be developed by a sequence of quasimodes, and in particular to show how this issue is related the global geometry of the manifold. We will give applications to unique continuation-type estimates and to the Calderón problem. We will mainly use tools of from the theory of pseudodifferential operators and semiclassical measures that will be reviewed during the course.
10:30 - 11:00	Coffee
11:00 - 12:30	Minicourse: Lauri Oksanen (University College London) Inverse problems on hyperbolic PDEs
12:30 - 14:30	Lunch Break
14:30 - 15:30	Lecture: Asma Hassannezhad (University of Bristol) Bounds on the Riesz means of mixed Steklov eigenvalues The goal of this talk is to study bounds on the Riesz means of mixed Steklov problems. The Riesz mean is a convex function of eigenvalues and has an important role and connection with other spectral quantities. We recall the results known in this direction for the Laplace eigenvalues. Then we introduce the mixed Steklov problem and state the main results. We also discuss some key ideas of the proof. This is joint work with Ari Laptev.
15:30 - 16:00	Coffee
16:00 - 17:00	Lecture: David dos Santos Ferreira (Université de Lorraine) A Borg-Levinson theorem for magnetic Schrödinger operators on a Riemannian manifold This talk is concerned with the inverse spectral problem of recovering the magnetic field and the electric potential in a Riemannian manifold from some asymptotic knowledge of the boundary spectral data of the corresponding Schrödinger operator with Dirichlet (or Neumann) boundary conditions. It combines a representation formula coming from Isozaki as well as a construction of solutions to the Schrödinger equation in simple manifolds already used by Bellassoued and myself in the quantitative study of inverse problems on the Dirichlet-to-Neumann map associated with evolution equations (also inspired by the study of the anisotropic Calderón problem by Kenig, Salo, Uhlmann and myself). This is a joint work with Mourad Bellassoued, Mourad Choulli, Yavar Kian and Plamen Stefanov.
19:00 - 00:00	Dinner

09:00 - 10:00	Minicourse: Fabricio Maciá (Universidad Politécnica de Madrid) Modes and quasimodes: concentration and non-concentration In this course we will present basic tools for the study of high-frequency eigenfunctions or quasimodes of Schrödinger-type elliptic operators on a compact manifold. Our mail goal will be to give a quantitative description of concentration and oscillation effects that can be developed by a sequence of quasimodes, and in particular to show how this issue is related the global geometry of the manifold. We will give applications to unique continuation-type estimates and to the Calderón problem. We will mainly use tools of from the theory of pseudodifferential operators and semiclassical measures that will be reviewed during the course.
10:00 - 10:30	Coffee
10:30 - 11:30	Minicourse: Lauri Oksanen (University College London) Inverse problems on hyperbolic PDEs

Participants

Musawa Almusawa

University of Sussex

Clemens Bombach

Technische Universität Chemnitz

Filip Bosnic

Bielefeld University

Pedro Caro

Basque Center for Applied Mathematics, Bilbao

Mihajlo Cekić

Max Planck Institute for Mathematics, Bonn

Bobby Cheng

University of Sussex

Dimitrije Cicmilovic

University of Bonn

Giovanni Covi

Jyväskylän Yliopisto

Francesco Della Porta

Oxford University

David dos Santos Ferreira

Université de Lorraine

Guy Fabrice Foghem Gounoue

Bielefeld University

Maria Angeles Garcia-Ferrero

ICMAT

Tuhin Ghosh

University of Washington

Robin Görmer

Universität Bremen

Andreea Grecu

University of Bucharest

Waleed hamali

university of sussex

Asma Hassannezhad

University of Bristol

Onirban Islam

University of Leipzig

Moritz Kassmann

Bielefeld University

Dimitra Kyriakopoulou

University College London

Michael Levitin

University of Reading

Yi-Hsuan Lin

Hong Kong University Science and Technology

Boya Liu

Univeristy of California, Irvine

Pan Liu

The university of Cambridge

Fabricio Maciá

Universidad Politécnica de Madrid

Yulia Meshkova

St. Petersburg State University

Jean Daniel Mukam

Chemnitz University of Technology

Adama Ndoye

TU Chemnitz

Mihai Nechita

University College London

Andrea Nickel

Bielefeld University

Lauri Oksanen

University College London

Gabriel Paternain

University of Cambridge

Jan-Frederik Pietschmann

TU Chemnitz

Leyter Potenciano Machado

University of Jyväskylä

Arseny Raiko

Lomonosov Moscow State Univercity

Jesse Railo

University of Helsinki & University of Jyväskylä

Angkana Rüland

Max-Planck-Institut für Mathematik in den Naturwissenschaften

Yafet Sanchez Sanchez

Max Planck Institute-Bonn

Amir Vig

University of California, Irvine

Vanja Wagner

Universität Bielefeld

Lili Yan

University of California Irvine

Konstantinos Zemas

Max Planck Institute for Mathematics in the Sciences

Scientific Organizer

Angkana Rüland

Max-Planck-Institut für Mathematik in den Naturwissenschaften

Administrative Contact

Katja Heid

Contact via Mail