Spectral and Resonance Problems for Imaging, Seismology and Materials Science




Theme I - Microlocal analysis, scattering theory and resonances


  • Recent results on the approximation of resonances

    J. Ben-Artzi(Cardiff University)

    Abstract. This talk aims to address the following question: does there exist an algorithm for computing scattering resonances? This somewhat vague question (to which we give the answer 'yes, but...') leads to several new results. In these results we demonstrate that it is possible to approximate resonances (even with error bounds) by constructing explicit algorithms. The abstract structure underpinning this question is known as the Solvability Complexity Index (SCI) Hierarchy. The results on resonances are in collaboration with M. Marletta and F. Rösler, while the abstract SCI theory is joint work with M. Colbrook, A. Hansen, O. Nevanlinna and M. Seidel.

  • Inverse Regge poles problem on a warped ball

    T. Daudé(Université de Besançon)

    Abstract. In this talk, we shall study a new type of inverse problem on warped product Riemannian manifolds with connected boundary that we name warped balls. Using the symmetry of the geometry, we first define the set of Regge poles as the poles of the meromorphic continuation of the Dirichlet-to-Neumann map with respect to the complex angular momentum appearing in the separation of variables procedure. These Regge poles can also be viewed as the set of eigenvalues and resonances of a one-dimensional Schrödinger equation on the half-line, obtained after separation of variables. Secondly, we find a precise asymptotic localisation of the Regge poles in the complex plane and prove that they uniquely determine the warping function of the warped balls.

  • Global Carleman estimates in Lebesgue spaces

    D. Dos Santos Ferreira(IEC, Université de Lorraine)

    Abstract. Carleman estimates are weighted a priori estimates for differential operators (in our case the Laplace-Beltrami operator in a Riemannian compact manifold with boundary). One usually refers to these estimates as global when they involve boundary terms. Originally used to prove unique continuation results outside the analytic framework, they became a very useful tool in control theory and in the resolution of inverse problems. The L² theory based on integration by parts and commutator estimates is fairly well understood and advanced. The Lᵖ theory initiated by Jerison and Kenig usually requires parametrices constructions and the analysis of boundedness estimates for oscillatory integral operators. We aim to initiate an analysis involving boundary terms, and intend to apply it to the stabilization of the damped wave equation involving unbounded potentials. This is a joint work with Rémi Buffe.

  • Spectral decomposition of some non-self-adjoint operators

    J. Faupin(Université de Lorraine)

    Abstract. We will consider in this talk non-self-adjoint operators in Hilbert spaces given as relatively compact perturbations of a self-adjoint operator. Typical examples are Schrödinger operators with bounded, complex potentials vanishing at infinity. We will describe abstract conditions insuring that the Hilbert space admits a direct sum decomposition into H-invariant subspaces, generalizing the well-known spectral decomposition of self-adjoint operators in terms of their spectral measures. A central role in the talk will be played by spectral singularities, an abstract notion corresponding to that of real resonances for Schrödinger operators. We will also present a useful regularized functional calculus for non-self-adjoint operators.

  • Classical wave methods and modern gauge transforms: spectral asymptotics in the one dimensional case

    J. Galkowski(University College London)

    Abstract. The question of high energy asymptotics for the kernel of the spectral projector of the Laplacian in the context of compact manifolds is one of the most well studied areas of spectral theory since the early 1900s. In this talk, we discuss the analogous question for Schrodinger operators on the real line: What are the asymptotics for the spectral projector of a Schrodinger operator on ℝ? By combining the classical wave method, originally introduced by Levitan in the 1950s, with the periodic gauge transform technique, we are able to show that when the potential is bounded with all derivatives this kernel, known as the local density of states, has a full asymptotic expansion in powers of the spectral parameter. This proves a conjecture of Parnovski--Shterenberg in the one dimensional case.

    Based on joint work with Leonid Parnovski and Roman Shterenberg

  • Classically forbidden regions in the chiral model of twisted bilayer graphene

    M. Hitrik(UCLA)

    Abstract. Magic angles are a topic of current interest in condensed matter physics: when two sheets of graphene are twisted by those angles the resulting material becomes superconducting. In this talk, we shall discuss a simple operator describing the chiral limit of twisted bilayer graphene, whose spectral properties are thought to determine which angles are magical. It comes from a 2019 Physical Review Letter by Tarnopolsky--Kruchkov--Vishwanath. By adapting analytic hypoellipticity results of Kashiwara, Trepreau, Sjöstrand, and Himonas, we show that the corresponding eigenfunctions decay exponentially in suitable geometrically determined regions, as the angle of twisting decreases. This is joint work with Maciej Zworski.

  • Quasinormal modes as eigenvalues of non-selfadjoint operators: the stability vs the definition problem

    J.L. Jaramillo(Université de Bourgogne)

    Abstract. Adopting a hyperboloidal approach to linear wave equations subject to outgoing boundary conditions, quasinormal modes can be cast in terms of the eigenvalue problem of a non-selfadjoint operator. Two different but closely related problems are posed. Firstly, the very definition of quasinormal modes by identifying the appropriate Hilbert space and, secondly, the study of the (in)stability properties of quasinormal modes under small (e.g. random) perturbations of the operator. Dwelling in particular in the setting of black hole spacetimes, whereas the "definition problem" has received significant attention from analysts in recent times, the "stability problem" (of pressing nature for physicists working in gravitational wave physics) has received only limited mathematical attention. In this talk, we present exploratory numerical results aiming at bringing the attention of analysts to such instability problem of black hole quasinormal modes, with the ultimate goal of developing a sound mathematical assessment.

  • Resonance-free regions and structural optimization of scattering poles

    I. Karabash(IAM, Universität Bonn)

    Abstract. Fabrication and numerical experiments for high-Q optical cavities led to new analytical and computational problems connected with optimization of resonances. The talk is devoted to the problem how to design an open resonator that has a scattering pole as close as possible to the real line for a certain engineeringly relevant family of functions describing the cavity structure. From the perspective of spectral theory, this question is related to the description of the largest possible resonance-free region. It is planned to explain the rigorous analytical formulation of the problem and, in particular, why the Pareto optimization settings are important for the existence of optimizers. If time permits, the recent optimal control approach developed jointly with Herbert Koch and Ievgen Verbytskyi (doi:10.1016/j.matpur.2020.02.005) will be discussed in connection with the resulting Hamilton-Jacobi-Bellman PDEs and the computational extremal synthesis.

  • Inverse resonance scattering on rotationally symmetric manifolds

    E. Korotyaev(AAIS, Northeast Normal University, Changchun, China)

    Abstract.We discuss inverse resonance scattering for the Laplacian on a rotationally symmetric manifold M = (0, ∞) × Y whose rotation radius is constant outside some compact interval. The Laplacian on M is unitarily equivalent to a direct sum of one-dimensional Schrödinger operators with compactly supported potentials on the half-line. We prove

    • Asymptotics of counting function of resonances at large radius.
    • rotation radius is uniquely determined by its eigenvalues and resonances.
    • There exists an algorithm to recover the rotation radius from its eigenvalues and resonances.

    The proof is based on some non-linear real analytic isomorphism between two Hilbert spaces. It is a joint result with Hiroshi Isozaki.


Theme II - Seismology and general relativity


  • Seismology and geometry of gas giants

    M. De Hoop(Rice University)

    Abstract. I will touch upon a tomography problem but the focus will be on the geometry.

  • Inverse problems in surface-wave tomography with spectral and resonance data

    A. Iantchenko(Malmö University)

    Abstract. Semiclassical analysis can be employed to describe surface waves in an elastic half space which is quasi-stratified near its boundary. In the case of an isotropic medium, the surface wave decouples up to principal parts into Love and Rayleigh waves associated to scalar and matrix spectral problems, respectively. Since the mathematical features (such as spectrum, resonances) of these problems can be extracted from the seismograms, we are interested in recovering the Lamé parameters from these data. We generalize spectral methods for Schrödinger operators to the Rayleigh problem, which is essentially not of Schrödinger type; and give comprehensive analysis of the wavenumber resonances, known in seismology as leaking modes.

  • Inertia-gravity waves in geophysical vortices

    J. Vidal(CNRS - Université Grenoble Alpes (UGA))

    Abstract.Rotating stratified flows often exhibit (almost) isolated pancake-like vortices, whose lifetime may depend on small-scale bulk turbulence due to nonlinear couplings of wave motions. Motivated by such applications, we investigate the inertia-gravity waves that can exist in geophysical vortices [1] (e.g. in Mediterranean eddies or Jupiter's vortices). We consider a fluid enclosed within a triaxial ellipsoid, which is stratified in density with a constant Brunt-Väisälä frequency (using the Boussinesq approximation) and uniformly rotating along a (possibly) tilted axis with respect to gravity. The wave problem is then governed by a mixed hyperbolic-elliptic equation for the velocity. As in the rotating non-stratified case [2], we find that the wave spectrum is pure point in ellipsoids (i.e. only consists of eigenvalues) with smooth polynomial eigenvectors. Then, we further characterise the spectrum using numerical computations (obtained with a bespoke Galerkin method) and microlocal analysis. In particular, we uncover the existence of low-frequency waves, which are absent in unbounded fluids but are reminiscent of coastal Kelvin waves. These waves owe their existence to rotation and stratification, and their mathematical origin is clarified.

    • [1] Vidal J. & Colin de Verdière C., 2023, Inertia-gravity waves in geophysical vortices, Submitted
    • [2] Colin de Verdière C. & Vidal J., 2023, The spectrum of the Poincaré operator in an ellipsoid, Submitted, arXiv:2305.01369

    This work received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 847433).

  • Jupiter's Interior Structure and the Origin of Saturn's Ring

    B. Militzer(Berkley)

    Abstract.This talk will discuss the recent gravity measurements of Saturn and Jupiter by the Cassini and Juno spacecrafts. The interpretation of these measurements and the construction of interior models for these giant planets relies to a large degree on results from ab initio computer simulations of hydrogen, helium, and heavier elements at megabar pressures. Here we will review the uncertainties of the computed equations of state and identify conditions where the interior models are particularly sensitive to. Then this talk will discuss recent findings of the Juno mission and explain why the unexpectedly low magnitudes of the gravity coefficients J4 and J6 imply that Jupiter has a dilute core at its center [1,2] instead having of a traditional compact core that is composed to 100% of heavy elements. Saturn stands out among the planets in our solar system for two reasons: It has a prominent set of rings and its spin axis is tilted by 27o. Both facts are difficult to reconcile with the well-established picture of planet formation, which assumes the planets emerged from a protoplanetary disk. Furthermore recent gravity measurements by the Cassini spacecraft implied an unexpected young age for Saturn's rings of only ~100 million years, which rules out that the rings are primordial. Here we reconcile all these observations by constructing models for Saturn's interior structure and by performing dynamical simulations of all relevant solar system objects. We present a dynamical scenario that explains how Saturn's rings formed and how its spin axis was tilted [3].

    • [1] B. Militzer, "Study of Jupiter's Interior with Quadratic Monte Carlo Simulations", Astrophysical J. 953 (2023) 111, doi:10.3847/1538-4357/ace1f1
    • [2] B. Militzer, et al., "Juno Spacecraft Measurements of Jupiter's Gravity Imply a Dilute Core", Planet. Sci. J. 3 (2022) 185, doi:10.3847/PSJ/ac7ec8
    • [3] J. Wisdom, R. Dbouk, B. Militzer, W. B. Hubbard, F. Nimmo, B. Downey, R. French, "Loss of a satellite could explain Saturn's obliquity and young rings", Science 377 (2022) 1285, doi:10.1126/science.abn1234
  • On the inverse problem for Love waves in a layered, elastic half-space

    J. Ricaud(CMAP, École polytechnique)

    Abstract.In this talk, I will present recent results on Love waves in a layered, elastic half-space, with the goal of recovering the parameters of the medium from the empirical knowledge of the frequency–wavenumber couples of the Love waves. To that end, I will first present the direct problem and the characterization of Love waves' existence through the dispersion relation, then I will address the inverse problem and show how one can recover parameters of the medium. This is joint work with Maarten de Hoop, Josselin Garnier, and Alexei Iantchenko, and is based on [1].

    • [1] M. V. de Hoop, J. Garnier, A. Iantchenko, J. R., _Inverse problem for Love waves in a layered, elastic half-space_, e-prints (2023), hal-03994654


Theme III - Inverse Problems and Imaging


  • Numerical resolution of the inverse source problem for EEG

    M. Darbas(Paris XIII)

    Abstract. In the talk, I will present a numerical method for solving the EEG inverse source problem which is able to take into account the heterogeneity of the head tissues. One of the applications is to consider fontanels in the skull layer for neonates. The approach consists in firstly transmitting the recorded data from the scalp to the cortex using the quasi-reversibility method. The second stage solves the inverse source problem within the brain by applying the method developed in the software tool FindSources3D. Numerical simulations in the case of the multi-layered spherical head model illustrate both the promising and limiting features of the approach. This is a joint work with Juliette Leblond (Team FACTAS, Inria Sophia-Antipolis), Jean-Paul Marmorat (CMA MinesParis) and Pierre-Henri Tournie (LJLL, Sorbonne Université).

  • Acoustic nonlinearity parameter tomography as an inverse coefficient problem for a nonlinear wave equation

    B. Kaltenbacher(Universität Klagenfurt)

    Abstract. We consider an undetermined coefficient inverse problem for a nonlinear partial differential equation occuring in high intensity ultrasound propagation as used in acoustic tomography. In particular, we investigate the recovery of the nonlinearity coefficient commonly labeled as B/A in the literature, which is part of a space dependent coefficient κ in the Westervelt equation governing nonlinear acoustics. Corresponding to the typical measurement setup, the overposed data consists of time trace measurements on some zero or one dimensional set Σ representing the receiving transducer array. In this talk, we will dwell on topics like modeling, uniqueness, numerical reconstruction schemes as well as simultaneous reconstruction of κ and further space depenent coefficients. In particular, these uniqueness proofs benefit from inverse spectral results. If time permits, we will also show some recent results pertaining to the formulation of this problem in frequency domain and numerical reconstruction of piecewise constant coefficients in two space dimensions. This is joint work with Bill Rundell, Texas A&M University.

  • Asteroid tomography by radar scattering with laboratory analogue: what can be told about the interior?

    S. Pursiainen(Tampere University).

    Abstract. In this presentation, I will discuss recent experimental results on reconstructing the internal structure of a 3D printed asteroid scale model with the external shape of 25143 Itokawa based on experimental microwave scattering data measured in the Common Microwave Resource Center of Institut Fresnel, Aix-Marseille University, and École Centrale de Marseille, Marseille, France. These results enlighten the capability and limitations of future in-situ radar investigations, such as the Juventas Radar (JuRa) experiment of ESA's coming HERA mission. The results cover a homogeneous and detailed internal permittivity distribution designed upon different hypotheses of asteroids' internal composition.


Theme IV - Subwavelenght resonances and applications


  • Band structure and Dirac points of real-space quantum optics in periodic media

    E. Hiltunen(Yale University)

    Abstract. The field of photonic crystals is almost exclusively based on a Maxwell model of light. To capture light-matter interactions, it natural to study such systems under a quantum-mechanical photon model instead. In the real-space parametrization, interacting photon-atom systems are governed by a system of nonlocal partial differential equations. In this talk, we study resonant phenomena of such systems. Using integral equations, we phrase the resonant problem as a nonlinear eigenvalue problem. In a setting of high-contrast atom inclusions, we obtain fully explicit characterizations of resonances, band structure, and Dirac cones. Additionally, we present a strikingly simple relation between the Green's function of the nonlocal equation and that of the local (Helmholtz) equation. In particular, we generalize existing lattice-summation methods to the nonlocal case. Based on this, we are able to achieve efficient numerical calculations of band structures of interacting photon-atom systems.

  • On Foldy-Lax models for time-domain scattering by multiple small particles

    M. Kachanovska(POEMS INRIA)

    Abstract. Foldy-Lax models describe wave scattering by multiple obstacles, in the regime where their diameter is small compared to the wavelength of the incident wave. While frequency-domain Foldy-Lax models have been well-studied (see, for example, [Martin 2006; Challa and Sini 2014; 2015; 2016]), their time-domain counterparts have only recently garnered attention, cf. e.g. [Barucq et al. 2021; Sini et al. 2021]. This presentation is dedicated to the derivation, stability, and convergence analysis of the time-domain Foldy-Lax model for sound-soft scattering by multiple scatterers. We will construct these models through a Galerkin spatial semi-discretization of a single-layer boundary integral formulation, using a carefully selected Galerkin space. The convergence of these semi-discretizations is ensured not by increasing the number of basis functions, as is common in numerical analysis, but by allowing the asymptotic parameter, namely the diameter of the particles, to tend to zero. The resulting asymptotic formulation can be shown to be stable in time, meaning it exhibits at most polynomial time growth of solutions, independently of the geometric configuration. We will discuss how to choose an appropriate Galerkin space, first for the case of circular particles, then for particles of arbitrary shapes, and present some convergence estimates. Our findings will be illustrated through numerical experiments. This is a joint work with Adrian Savchuk.

  • Quasi-normal modes and the resonant response of open phononic systems

    V. Laude  (FEMTO-ST, Besançon)

    Abstract. Resonant structures supporting elastic waves attached to a substrate suffer from radiation loss. As a result, instead of the normal modes of closed systems having purely real eigenfrequencies, open systems possess quasi-normal modes (QNMs) characterized by complex valued eigenfrequencies and diverging power at infinity. Because of the eigen-expansion theorem, that states that the response of a system to a given excitation is a superposition of all eigenmodes, there is strong interest in obtaining all quasi-normal modes. Quasi-normal modes have been widely discussed in photonics, but less for acoustic and elastic waves. Of special interest are the determination of complex eigenvalues and eigenmodes, and the definition of an adequate modal volume and elastic equivalent of the Purcell effect. We make use of the unconjugated form of the reciprocity relation for elastic waves in order to obtain a relation between the solution to the time-harmonic elastodynamic wave equation and the discrete set of quasi-normal modes. Of significance is the fact that the total energy of QNMs is unbounded, so that usual normalization relations for normal modes can not be employed. Instead, a multipole expansion with complex coefficients is naturally obtained. In passing, a complex modal volume is defined. We then consider a practical way to obtain all quasi-normal modes of an elastic resonating structure. Our technique relies on a stochastic excitation method to obtain all possible resonances from the response of the system to a random force and extends it to the quasi-normal mode problem. The technique yields poles in the response that can be continued in the complex frequency plane. Starting from an initial guess along the real frequency axis, a fast iterative algorithm based on the power method for eigenvectors is implemented. Once the quasi-normal modes have been obtained, the response of the system to a given excitation can be computed efficiently from the expansion theorem, by forming the projection of the excitation on each quasi-normal mode as a function of frequency. We implement an equivalent of the photonic techniques to phononic structures, including perfectly matched layers to transform the semi-infinite domain to a finite, but complex-valued, domain. We obtain numerically the modal volume of open elastic systems. As a test system, we consider a nanopillar on a surface and then a pair of coupled nanopillars forming a kind of tuning fork, resonating in the hundreds of MHz to a few GHz range. We obtain the quasi-normal modes in both cases and verify that the response of the system to an excitation of the nanopillars is well accounted for by the superposition of the first quasi-normal-modes. As will be discussed, the same strategy can be followed to obtain quasi-normal modes of open acoustic systems, as well as of coupled acoustic and elastic systems.

  • On the origin of Minnaert resonances

    A. Posilicano(Universitá dell'Insubria, Como)

    Abstract. We consider the appearance of what are called "MInnaert resonances" in the scattering of sound waves in a medium with a small inhomogeneity enjoying a high contrast of both its mass density and bulk modulus. This phenomenon is explained in terms of the behavior, as the size of the inhomogeneity decreases to zero, of the norm resolvent limit of a related frequency-dependent Schroedinger operator, the limit being not trivial if and only if the frequency coincides with that of Minnaert. Joint work with Andrea Mantile and Mourad Sini.

  • Resonant Materials in Inverse Problems

    M. Sini(RICAM, Linz)

    Abstract. In the recent years, we witness an increase of interest in drugs delivery, imaging and therapy modalities using contrast agents. Such agents are small-scaled particles but enjoy extreme contrasts as compared to the usual media. These properties allow them to resonate at special frequencies. In this talk, we describe an approach that allows us to use this resonant behaviour for imaging. In particular, we will show how contrasting the remotely measured waves, generated before and after injecting the small particles, provides us with the travel time function (for time-domain imaging) or the dispersion function (for time-harmonic imaging). Then, we extract the values of the needed coefficients, as the speed of propagation, from the reconstructed travel time (or dispersion) function. As a test example, we will discuss the ultrasound imaging modality using gas bubbles (or liquid droplets) as contrast agents. Finally, we will show how this approach can go beyond these imaging modalities.

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