Special Session 129: Inverse problems for nonlocal / nonlinear PDEs

Inverse Scattering Problem for fractional Schrodinger equation

Tuhin Ghosh Harish-Chandra Research Institute India Co-Author(s):

Abstract: In this talk, we will introduce the direct and the inverse scattering problem for the fractional Schrodinger equation. Restricted by the resolvent estimates, in some regime of the nonlocality, we will solve the Born approximation problem.

Inverse problems for subdiffusion with an unknown terminal time

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

Abstract: Inverse problems of recovering space-dependent parameters, e.g., initial condition, space-dependent source, or potential coefficient in a subdiffusion model from the terminal observation are classical. However, all existing studies have assumed that the terminal time at which one takes the observation is exactly known. In this talk, we present uniqueness and stability results for three canonical inverse problems, e.g., backward problem, inverse source, and inverse potential problems from the terminal observation at an unknown time. We show the uniqueness and stability of the inverse problems and also present numerical illustrations of the behavior of the inverse problem.

Inverse problems for parabolic and pseudo-parabolic equations with p-Laplacian diffusion and damping

Khonatbek Khompysh Institute of Mathematics and Mathematical Modeling Kazakhstan Co-Author(s):    Kenzhebai Kh.

Abstract: In this talk, we discuss on uniquely solvability of inverse problems for parabolic and pseudo-parabolic equations perturbed by p-Laplacian diffusion and damping. Inverse problems consist of recovering a time dependent source/potential under the measurement in the integral form over the space domain. Under the suitable assumptions on the data, we establish existence and uniqueness of weak solutions posed inverse problems. This work supported by the grants no AP19676624 and AP23486218 Ministry of Science and Higher Education of the Republic of Kazakhstan (Kazakhstan)

An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions

Mokhtar KIRANE Khalifa University United Arab Emirates Co-Author(s):    M. Al-Gwaiz, M. Kirane, S.A. Malik

Abstract: We consider the inverse source problem for a time fractional diffusion equation. The unknown source term is independent of the time variable, and the problem is considered in two dimensions. A bi-orthogonal system of functions consisting of two Riesz bases of the space \(L^2((0,1)\times (0,1))\), obtained from eigenfunctions and associated functions of the spectral problem and its adjoint problem, is used to represent the solution of the inverse problem. Using the properties of the bi-orthogonal system of functions, we show the existence and uniqueness of the solution of the inverse problem and its continuous dependence on the data.

Inverse problems for semilinear Schrodinger equations on Riemannian manifolds at large frequency

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

Abstract: In this talk, we will discuss inverse boundary problems for semilinear Schrodinger equations on smooth compact Riemannian manifolds of dimension two and higher with smooth boundary, at a large fixed frequency. We will demonstrate that certain classes of cubic nonlinearities are uniquely determined from the knowledge of the nonlinear Dirichlet-to-Neumann map at a large fixed frequency on quite general Riemannian manifolds. In particular, in contrast to the previous results available, here the manifolds need not satisfy any product structure, may have trapped geodesics, and the geodesic ray transform need not be injective. Only a mild assumption about the geometry of intersecting geodesics is required. This is joint work with Shiqi Ma, Suman Kumar Sahoo, Mikko Salo, and Simon St-Amant.

Fixed angle inverse scattering and rigidity of the Minkowski spacetime

Lauri Oksanen University of Helsinki Finland Co-Author(s):    Rakesh, Mikko Salo

Abstract: An acoustic medium occupying a compact domain with non-constant sound speed is probed by an impulsive plane wave, and the far-field response is measured in all directions for all frequencies. A longstanding open problem, called the fixed angle scattering inverse problem, is the recovery of the sound speed from this far-field response. In some situations, the acoustic properties of the medium are modeled by a Lorentzian metric and then the goal is the recovery of this metric from the far field measurements corresponding to a finite number of incoming plane waves. We consider a time domain, near field version of this problem and show that natural fixed angle measurements distinguish between a constant velocity (the Minkowski metric) medium and a non-constant velocity (a general Lorentzian metric) medium.

Mathematical models for nonlinear ultrasound contrast imaging with microbubbles

Teresa Rauscher University of Klagenfurt Austria Co-Author(s):    Vanja Nikolic

Abstract: Ultrasound contrast imaging is a specialized imaging technique that applies microbubble contrast agents to traditional medical sonography providing real-time visualization of blood flow and vessels. Gas filled microbubbles are injected into the body where they undergo compression and rarefaction and interact nonlinearly with the ultrasound waves. Therefore, the propagation of sound through bubbly liquid is a strongly nonlinear problem that can be modeled by a nonlinear acoustic wave equation for the propagation of the pressure waves coupled with an ordinary differential equation for the bubble dynamics. We start by deriving different models and then focus on the coupling of the Westervelt equation and the Rayleigh-Plesset equation, where we show well-posedness locally in time under suitable conditions on the initial data. Finally, we present numerical experiments on the single bubble dynamics and the interaction of the microbubbles and ultrasound waves.

A policy iteration method for inverse mean field games

Kui Ren Columbia University USA Co-Author(s):

Abstract: We propose a policy iteration method to solve an inverse problem for a mean-field game (MFG) model, specifically to reconstruct the obstacle function in the game from the partial observation data of value functions, which represent the optimal costs for agents. The proposed approach decouples this complex inverse problem, which is an optimization problem constrained by a coupled nonlinear forward and backward PDE system in the MFG, into several iterations of solving linear PDEs and linear inverse problems. This method can also be viewed as a fixed-point iteration that simultaneously solves the MFG system and inversion. We prove its linear rate of convergence. In addition, numerical examples in 1D and 2D, along with performance comparisons to a direct least-squares method, demonstrate the superior efficiency and accuracy of the proposed method for solving inverse MFGs. This is a joint work with Nathan Soedjak and Shahyin Tong of Columbia University.

Inverse problems for some attenuated wave equations

Cong Shi University of Vienna Austria Co-Author(s):    Barbara Kaltenbacher and Otmar Scherzer

Abstract: In this talk we will introduce a general attenuated wave equation in the frequency domain using pseudo differential operator, which can be transformed into fractionally damped wave equations in the time domain. We will first explore how this general model relates to various fractionally damped wave equations and illustrate the connection among the initial conditions derived from physical principle. Next, we will establish the uniqueness of both direct and inverse problems associated with this framework. Additionally, we will discuss the ill-posedness of the inverse problem by analyzing the asymptotic behavior of the singular values of the forward operator.

Inverse problems for nonlinear parabolic equations in domains with moving boundaries

Madi Yergaliyev Institute of Mathematics and Mathematical Modeling Kazakhstan Co-Author(s):    Muvasharkhan Jenaliyev, Medina Kassen

Abstract: We will explore inverse problems for nonlinear parabolic equations in degenerate domains and domains with moving boundaries. It is important to note that a significant characteristic of such inverse problems is that they are studied in degenerate domains, which, in turn, leads to additional solvability conditions. For example, for one inverse problem, the conditions for unique solvability are derived as a connection between a known multiplier on the right-hand side of the equation and the functions governing the changes in the boundaries of the nonlinearly degenerate domain.

Numerical Reconstruction of Potential and Initial Data in Subdiffusion using Observations at Two Time Levels

Zhi Zhou The Hong Kong Polytechnic University Hong Kong Co-Author(s):    Xu Wu

Abstract: In this talk, we will discuss the numerical recovery of both initial data and a spatially dependent potential in time-fractional subdiffusion models using observations at two different time levels. This problem is challenging because the coefficients-to-state map is more complex than for a single coefficient, and the decoupling method remains unclear. Our investigation addresses critical aspects of the numerical treatment and analysis of this inverse problem, including proving conditional stability, developing an efficient solver, and designing a discrete numerical scheme with a provable error estimate. We develop a fixed-point iterative algorithm to recover the initial data and potential together. By establishing novel \textsl{a priori} estimates for the discrete direct problem, we demonstrate the contraction mapping property of the fixed-point iteration, leading to both convergence of the iteration and error estimates for the fully discrete reconstruction.

The Calder\`{o}n problem for nonlocal wave equations with polyhomogeneous nonlinearities

Philipp Zimmermann Universitat de Barcelona Switzerland Co-Author(s):

Abstract: The main purpose of this talk is to present recent results on the Calder\`{o}n problem for nonlocal wave equations with polyhomogeneous nonlinearities. We start by discussing the unique determination of homogeneous nonlinearities from the Dirichlet to Neumann map. Then, we explain how this approach can be generalized to recover polyhomogeneous nonlinearities. On the way, we discuss an optimal Runge approximation result, which in turn relies on the existence of very weak solutions to linear nonlocal wave equations with sources in $L^2(0,T;H^{-s}(\Omega))$.