Special Session 19: New trends in inverse problems for partial differential equations

Localization of Point Scatterers via Sparse Optimization on Measures

Giovanni S. Alberti University of Genoa Italy Co-Author(s):    Romain Petit, Matteo Santacesaria

Abstract: We consider the inverse scattering problem for time-harmonic acoustic waves in a medium with pointwise inhomogeneities. In the Foldy--Lax model, the estimation of the scatterers` locations and intensities from far field measurements can be recast as the recovery of a discrete measure from nonlinear observations. We propose a ``linearize and locally optimize`` approach to perform this reconstruction. We first solve a convex program in the space of measures (known as the Beurling LASSO), which involves a linearization of the forward operator (the far field pattern in the Born approximation). Then, we locally minimize a second functional involving the nonlinear forward map, using the output of the first step as initialization. We provide guarantees that the output of the first step is close to the sought-after measure when the scatterers have small intensities and are sufficiently separated. We also provide numerical evidence that the second step still allows for accurate recovery in settings that are more involved.

Lipschitz-Stable Identification of Polyhedral Inclusions via Local Boundary Measurements

Andrea Aspri University of Milan Italy Co-Author(s):    Elena Beretta, Elisa Francini, Sergio Vessella

Abstract: In this talk, we address the nonlinear inverse problem of identifying polyhedral inclusions within a three-dimensional homogeneous isotropic conducting body using boundary measurements. Our focus is on the conductivity equation, where we derive a Lipschitz stability estimate for the Hausdorff distance between polyhedral inclusions, based on the local Dirichlet-to-Neumann (DtN) map. Additionally, we present a new uniqueness result in this general framework. This work is the result of a collaboration with Elena Beretta (NYU Abu Dhabi), Elisa Francini, and Sergio Vessella (University of Florence).

A Duality Between Scattering Poles and Interior Eigenvalues in Scattering Theory

Fioralba Cakoni Rutgers, The State University of New Jersey USA Co-Author(s):    Fioralba Cakoni and Dana Ziberberg

Abstract: The spectral properties of operators associated with scattering phenomena carry critical information about the scattering media. The theory of scattering resonances is a rich and elegant area of scattering theory. Although resonances are inherently dynamic in nature, they can be mathematically formulated as the poles of the meromorphic extension of the scattering operator. Scattering poles, which are complex with a negative imaginary part, encapsulate physical information: the real part of a pole corresponds to the rate of oscillation, while the imaginary part reflects the rate of decay. At a scattering pole, a non-zero scattered field exists even in the absence of an incident field. On the other side of this characterization, one could ask if there are frequencies for which an incident field does not scatter from the scattering object. For inhomogeneous media, this question leads to the concept of transmission eigenvalues, or interior eigenvalues in the case of obstacles. In this talk, we present a conceptually unified approach for characterizing and determining scattering poles and transmission eigenvalues for the scattering problem for inhomogeneous media. Our approach explores a duality that arises by interchanging the roles of incident and scattered fields. Both sets -the scattering poles and transmission eigenvalues- are connected to the kernel of the relative scattering operator, which maps incident fields to scattered fields. This operator corresponds to the exterior scattering problem for transmission eigenvalues and the interior scattering problem for scattering poles. We will conclude with numerical examples for the scattering by an obstacle as a proof of concept.

Uniqueness in the inverse boundary value problem for the weighted p-Laplacian in the plane

Catalin I Carstea National Yang Ming Chiao Tung University Taiwan Co-Author(s):    Ali Feizmohammadi

Abstract: In this talk I will show a proof of uniqueness in the inverse boundary value problem for the coefficient of the equation, in dimension 2. The approach involves the higher linearization procedure but, unlike the usual case, the linearization is done at a non-trivial solution. The question then reduces to an anisotropic Calderon problem.

Compressed sensing for photoacoustic tomography on the sphere

Alessandro Felisi University of Genoa Italy Co-Author(s):

Abstract: Photoacoustic Tomography (PAT) is an emerging medical imaging technology, distinguished as one of the most sophisticated hybrid modalities. Its primary aim is to map the high-contrast optical properties of biological tissues by leveraging high-resolution ultrasound measurements. Mathematically, this can be framed as an inverse source problem for the wave equation over a specific domain. In this talk, I will show how, by assuming signal sparsity, it is possible to establish stable recovery guarantees when the domain is spherical and the data collection is restricted to a limited portion of the boundary. The result is a consequence of a general compressed sensing framework for inverse problems developed with co-authors and stability estimates tailored to this specific problem.

Reconstructing early stages of prostate cancer growth

Matteo Fornoni University of Pavia Italy Co-Author(s):    Elena Beretta, Cecilia Cavaterra, Guillermo Lorenzo, Elisabetta Rocca

Abstract: To facilitate the estimation of disease dynamics and better guide ensuing clinical decisions, we investigate an inverse problem enabling the reconstruction of earlier tumour stages by using a single diagnostic MRI scan. We describe tumour dynamics with an Allen--Cahn phase-field model driven by a generic nutrient that follows reaction-diffusion dynamics. The model is completed with another reaction-diffusion equation for the local production of prostate-specific antigen, a key prostate cancer biomarker. We first improve previous well-posedness results by showing that the solution operator is continuously Fr\`{e}chet differentiable. We then analyse the backward inverse problem concerning the reconstruction of earlier tumour stages starting from measurements of the model variables at the final time. Since this problem is severely ill-posed, only very weak conditional stability of logarithmic type can be recovered from the terminal data. Nevertheless, by restricting the unknowns to a compact subset of a finite-dimensional subspace, we can derive an optimal Lipschitz stability estimate. Such results then lead to the development of a locally convergent iterative reconstruction algorithm based on the Landweber scheme. We finally show some numerical experiments validating the obtained theoretical results.

Uniqueness and stability in anisotropic inverse problems.

Romina Gaburro University of Limerick Ireland Co-Author(s):

Abstract: In this talk we discuss the issues of uniqueness and stability in anisotropic inverse problems. As is well known, these are typically ill-posed and nonlinear problems. In the presence of anisotropy, there is a well-known fundamental obstruction to uniqueness due to Tartar: any diffeomorphism of the domain under investigation, which keeps the boundary fixed, modifies the physical properties under investigation (e.g. the conductivity of the medium in the celebrated Calder\`on`s inverse conductivity problem) but such change is not visible in the measurements of the inverse problem (e.g. in the Dirichlet-to-Neumann map in Calder\`on`s problem). In this talk we discuss recent advancements in anisotropic inverse problems, in particular regarding the issues of uniqueness and stability.

Direct sampling methods for elliptic inverse problems

Bangti Jin The Chinese University of Hong Kong Hong Kong Co-Author(s):

Abstract: Elliptic inverse problems represent a very broad range of applied problems. Traditionally these problems are solved using regularized least-squares methods, which however are computationally expensive. In this talk, we describe a novel direct sampling method for solving several model elliptic inverse problems, and present numerical experiments to illustrate the performance.

Calderon problem for fractional Schrodinger operators on closed Riemannian manifolds

Katya Krupchyk University of California, Irvine USA Co-Author(s):

Abstract: In this talk, we will discuss an analog of the anisotropic Calderon problem for fractional Schrodinger operators on closed Riemannian manifolds of dimension two and higher. We will demonstrate that the knowledge of a Cauchy data set of solutions to the fractional Schrodinger equation, given on an open nonempty subset of the manifold, determines both the Riemannian manifold up to an isometry and the potential up to the corresponding gauge transformation, under certain geometric assumptions on the manifold as well as the observation set. This is joint work with Ali Feizmohammadi and Gunther Uhlmann.

Construction of weakly neutral Inclusions via imperfect interfaces

Mikyoung Lim Korea Advanced Institute of Science and Technology Korea Co-Author(s):    Mikyoung Lim

Abstract: We consider the problem of planar conductivity inclusion with the imperfect interface condition, characterized by an interface parameter defined as a function on the inclusion boundary. When embedded in a medium with different conductivity, the inclusion causes the perturbation in the incident background field. Using the multipole expansion of this perturbation, we determine the interface parameter for a given inclusion of arbitrary shape that results in negligible perturbations under all uniform incident fields.

Direct and inverse problems for viscoelastic models of dislocations

Anna L Mazzucato Penn State University USA Co-Author(s):    Arum Lee

Abstract: We discuss both the forward as well as an inverse problem for viscoelastic models of dislocations that represent aseismic, creeping faults. We study a nonlocal linear slip rate-traction model and a nonlinear local slip rate-slip friction model. The inverse problem consists in determining the geometry of the dislocation surface as well as the slip vector from surface displacement measurements. This is joint work with PhD student Arum Lee, and extends prior results, joint work with Andrea Aspri, Elena Beretta, and Maarten de Hoop.

Nonlinearity helps convergence of the inverse Born series

Shari Moskow Drexel University USA Co-Author(s):    Nicholas Difilippis, John Schotland

Abstract: In previous work of the authors, we investigated the Born and inverse Born series for a scalar wave equation with linear and nonlinear terms, the nonlinearity being cubic of Kerr type. We reported conditions which guarantee convergence of the inverse Born series, enabling recovery of the coefficients of the linear and nonlinear terms. In this work, we show that if the coefficient of the linear term is known, an arbitrarily strong Kerr nonlinearity can be reconstructed, for sufficiently small data. Additionally, we show that similar convergence results hold for general polynomial nonlinearities. Our results are illustrated with numerical examples.

Optimality of stabilized finite element methods for elliptic unique continuation

Lauri Oksanen University of Helsinki Finland Co-Author(s):    Erik Burman, Mihai Nechita

Abstract: We consider finite element approximation in the context of the ill-posed elliptic unique continuation problem, and introduce a notion of optimal error estimates that includes convergence with respect to a mesh parameter and perturbations in data. The rate of convergence is determined by the conditional stability of the underlying continuous problem and the polynomial order of the finite element approximation space. We present a stabilized finite element method satisfying the optimal estimate, and discuss a proof showing that no finite element approximation can converge at a better rate.

Stable determination of the Winkler subgrade coefficient in a nanoplate

Eva Sincich University of Trieste Italy Co-Author(s):    G. Alessandrini, A. Morassi, E. Rosset, S. Vessella

Abstract: We study the inverse problem of determining the Winkler coefficient in a nanoplate resting on an elastic foundation and clamped at the boundary. The nanoplate is described within a simplified strain gradient elasticity theory for isotropic materials, under the Kirchhoff-Love kinematic assumptions in infinitesimal deformation. We prove a global H\{o}lder stability estimate of the subgrade coefficient by performing a single interior measurement of the transverse deflection of the nanoplate induced by a load concentrated at one point.

Passive manipulation of electromagnetic fields

Michael Vogelius Rutgers University USA Co-Author(s):

Abstract: In this talk I shall discuss some theoretical results concerning invisibility, non-scattering, and second harmonic generation. The results are in frequency domain and are formulated in the context of the Helmholtz equation.