Iowa State University

MiniSymposium-04: Inverse Problems for PDEs: Algorithms, Analysis and Applications

Abstract: The area of inverse problems for partial differential equations (PDEs) involves the problem of reconstructing PDEs and/or their domains of definition from external measurements of the solutions of these PDEs. Inverse problems for PDEs can arise from the questions of how to identify the source terms and how to inspect inaccessible regions. Therefore, these inverse problems directly involve applications such as non-destructive testing, medical imaging, photonics and optics, geophysical protesting and radar, etc. During the past two decades, this area has been one of the fastest growing ones in applied mathematics with significant developments in mathematical theory, computational algorithms, and applications. The goal of this minisymposium is to bring together researchers working on different aspects of the field to discuss recent and new results in the field and to promote idea exchange as well as potential future collaborations.


Saturday, October 19th, 2019 at 10:20 - 11:40 (232 Carver Hall)
10:20 - 10:40 Ngoc Do,
Missouri State University
Theoretically Exact Solution of the Inverse Source Problem for the Wave Equation with Spatially and Temporally Reduced Data

The inverse source problem for the wave equation arises in several promising emerging modalities of medical imaging. Of special interest here are theoretically exact inversion formulas, explicitly expressing solution of the problem in terms of the measured data. Almost all known formulas of this type require data to be measured on a closed surface completely surrounding the object. This, however, is too restrictive for practical applications. I will present an alternative approach that, under certain restriction on geometry, yields explicit, theoretically exact reconstruction from the data measured on a finite open surface. Numerical simulations illustrating the work of the method will be presented. This is joint work with Leonid Kunyansky.

10:40 - 11:00 Alexey Smirnov,
University of North Carolina at Charlotte
Convexification for an Inverse Problem with Single Measurement for a 1D Hyperbolic Equation

The progress made in the paper "On an inverse source problem for the full radiative transfer equation with incomplete data", recently accepted to SIAM Journal on Scientific Computing, will be discussed. While the major part of the talk is devoted to the study of the globally convergent numerical method for a coefficient inverse problem with a single measurement for a 1D hyperbolic equation. The questions of stability and global convergence for coefficient inverse problems are the most critical and challenging in numerical applications. To address these challenges, a novel numerical method, based on the concept of convexification was proposed. This approach constructs a weighted Tikhonov-like functional, which is strictly convex on an a priori chosen ball of an arbitrary radius in an appropriate Hilbert space. The strict convexity is ensured via the presence of the Carleman Weight Function in the functional. Then, the gradient projection method converges to the exact solution starting from an arbitrary point of that ball. The numerical simulations demonstrate the stability and accuracy of the proposed method.

11:00 - 11:20 Dinh-Liem Nguyen,
Kansas State University
Direct Sampling Method for Inverse Electromagnetic Scattering Problems

We consider the inverse electromagnetic scattering problem that is concerned with the determination of geometrical properties of the scattering medium from the electromagnetic scattering data. This is an important problem in scattering theory thanks its mathematical interests and applications in non-destructive testing, medical imaging, radar etc. In this talk, we will discuss our recently new results on the numerical solution to this nonlinear inverse problem. The so-called direct sampling method (DSM) is investigated to solve the inverse problem. Mathematical justifications and numerical simulations for the DSM are presented. The main advantages of the DSM are that it is very robust with noisy data, fast, and does not require a priori information of the solution.

11:20 - 11:40 Vikenty Mikheev,
Kansas State University
Christmas Jump in LIBOR

A short-term pattern in LIBOR dynamics was discovered. Namely, 2-month LIBOR experiences a jump after Xmas. The sign and size of the jump depend on the data trend on 21 days before Xmas.

Saturday, October 19th, 2019 at 14:40 - 16:00 (232 Carver Hall)
14:40 - 15:00 Isaac Harris,
Purdue University
Direct Sampling Algorithms for Inverse Scattering

In this talk, we will discuss a recent qualitative imaging method referred to as the Direct Sampling Method for inverse scattering. This method allows one to recover a scattering object by evaluating an imaging functional that is the inner-product of the far field data and a known function. It can be shown that the imaging functional is strictly positive and decay as the sampling point moves away from the scatterer. The analysis uses the factorization of the far field operator and the Funk-Hecke formula. This method can also be shown to be stable with respect to perturbations in the data. We will discuss the inverse scattering problem for both acoustic and electromagnetic waves.

15:00 - 15:20 Loc Nguyen,
University of North Carolina at Charlotte
A Numerical Method for an Inverse Source Problem for Parabolic Equations and its Application to a Coefficient Inverse Problem

Two main aims of this talk are to develop a numerical method to solve an inverse source problem for parabolic equations and apply it to solve a nonlinear coefficient inverse problem. The inverse source problem is the problem to reconstruct a source term from external observations. Our method to solve this inverse source problem consists of two stages. We first establish an equation of the derivative of the solution to the parabolic equation with respect to the time variable. Then, in the second stage, we solve this equation by the quasi-reversibility method. The inverse source problem above is the linearization of a nonlinear coefficient inverse problem. Hence, iteratively solving the inverse source problem provides the numerical solution to that coefficient inverse problem. Numerical results for the inverse source problem under consideration and the corresponding nonlinear coefficient inverse problem are presented.

15:20 - 15:40 Trung Truong,
Kansas State University
On the Inverse Scattering Problem for Anisotropic Periodic Structures

This talk is about a study of the Factorization method for shape reconstruction of anisotropic periodic structures from near field scattering data. This method provides a fast numerical algorithm and a unique determination for the shape reconstruction of the scatterer. We also present a rigorous justification for the Factorization method and several numerical examples to show how well it performs in different situations. This is joint work with I. Harris, D.-L. Nguyen, and J. Sands.

15:40 - 16:00 Nick Fisher,
Colorado School of Mines
An Analytically Divergence-free Quasi-interpolation Scheme

We propose and analyze an analytically divergence-free quasi-interpolation scheme. To this end, we first construct divergence-free quasi-interpolation and derive corresponding approximation orders to both the approximated function and its derivatives. The scheme does not require solving any linear system of equations and thus is time-efficient and easy to implement. To demonstrate the validity of the scheme, numerical simulations are presented. Both theoretical and numerical results show that our scheme is simple, time-efficient, and analytically divergence-free.

Saturday, October 19th, 2019 at 16:40 - 18:00 (232 Carver Hall)
16:40 - 17:00 Thi-Phong Nguyen,
University of Minnesota
Identifying Local Perturbation in Unknown Periodic Media

I will present and discuss in this talk a differential sampling method to recover the support of a local perturbation in an unknown periodic layer from measurements of scattered waves at a fixed frequency. This method relies on a careful analysis of so-called Generalized Linear Sampling Methods for a single Floquet-Bloch mode. The main novelty is the study of a new type of interior transmission problems that couples classical interior transmission problems for one Floquet-Bloch mode with scattering problems for the other modes. I will outline this analysis and the way it leads to the construction of an indicator function for the defect independently from the geometry of the background. Some related numerical results will be presented at the end.

17:00 - 17:20 Paul Sacks,
Iowa State University
Boundary Conditions and Inverse Problems for the Vocal Tract

There is a natural inverse problem associated with study of human speech production, namely determining the shape of the vocal tract, as various sounds are produced, from measurements of the corresponding acoustic waves which are generated. For one of the simplest models, in which we regard the vocal tract as cylindrical tube with circular cross sections of variable radius, we discuss the inverse problem with particular attention to the specification of boundary conditions for the acoustic waves.

17:20 - 17:40 Thuy (Jenny) Le,
University of North Carolina at Charlotte
A Numerical Method to Recover the Initial Condition for Nonlinear Parabolic Equations based on a the Truncation of a Fourier Series

We solve the problem of recovering the initial condition of nonlinear parabolic equations. An example of such a nonlinear problem is the important Fisher equation arising in ecology, physiology, combustion, crystallization, plasma physics, and in general phase transition problems. The widely used approach to solve nonlinear inverse problems is the optimal control method. The main drawback of this method is the lack of a good initial guess. Motivated by this, we suggest a numerical method that does not require any advanced knowledge of the true solution. Our method consists of two stages. In stage 1, we derive a system of elliptic equations whose solutions are the Fourier coefficients of the heat distribution. In stage 2, we propose an iterative numerical method to solve the system above. Numerical examples, obtained from our method with highly noisy data, are presented.

17:40 - 18:00 Manas Bhatnagar,
Iowa State University
Critical Thresholds in One Dimensional Damped Euler-Poisson Systems

The talk is about critical threshold phenomenon in one dimensional damped, pressureless Euler-Poisson equations with electric force induced by a constant background, originally studied in [S. Engelberg and H. Liu and E. Tadmor, Indiana Univ. Math. J., 50:109--157, 2001]. Using a simple transformation to linearize the characteristic system of equations allows us to study the geometrical structure of critical threshold curves for three damping cases: overdamped, underdamped and borderline damped through phase plane analysis. We will also see the explicit form of these critical curves. These sharp results state that if the initial data is within the threshold region, the solution will remain smooth for all time, otherwise it will have a finite time breakdown. At the end, we will see an application of these general results to identify critical thresholds for a non-local system subjected to initial data on the whole line.

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Iowa State University