Fioralba Cakoni, Transmission Eigenvalues and Nonscattering
Abstract: The transmission eigenvalue problem arises as a necessary condition for a compactly supported inhomogeneity to be invisible when probed by incident waves in the frequency-domain scattering regime. This problem, in which the wave number serves as the eigenvalue parameter, is inherently non-selfadjoint and exhibits a remarkably rich and subtle spectral structure. For wave numbers that are not transmission eigenvalues, every incident wave generates a nontrivial scattered field. However, the presence of a transmission eigenvalue is far from sufficient to guarantee non-scattering, even for a single incident wave. While the existence of real transmission eigenvalues can be established under suitable assumptions, the occurrence of non-scattering depends in a delicate way on the regularity of both the geometry and the coefficients of the underlying operator. As a result, the problem becomes closely connected to free-boundary regularity questions. In this presentation, we discuss the spectral properties of the transmission eigenvalue problem and their relationship to the non-scattering phenomenon. We show that non-scattering is deeply connected to the classical Schiffer and Pompeiu conjectures, revealing intriguing links between inverse scattering theory, spectral analysis, and geometric properties of domains.
Shixu Meng, Quantifying nonlinear information in the linear sampling method for inverse medium scattering
Abstract: We propose a new characterization of an imaging indicator in the class of linear sampling method for the inverse medium scattering problem. We demonstrate that the imaging indicator represents nonlinear information about the unknown contrast.
Thi Phong Nguyen, An imaging method for the detection of defects in a biperiodic layer
Abstract: In this talk, we will present an imaging method for the inverse scattering problem of detecting a local defect in an infinite biperiodic layer, using measurements of scattered electromagnetic waves taken at a distance. By exploiting properties of the Floquet–Bloch transform, we consider an approximate model that is periodic over a larger period consisting of multiple original cells. We then select a subset of the measured data to construct an imaging function for the defect that distinguishes sampling test points inside the defect from the background. The talk will focus on establishing the method and characterizing the defect and will include some numerical examples.
Jeffrey Ovall, Computational and theoretical tools for the magnetic Schrödinger eigenvalue problem
Abstract: The magnetic Schrödinger equation provides a probabilistic model of the motion of a charged particle in an electromagnetic field. The associated eigenvalue problem provides probability densities, via normalized eigenvectors, of the location of the charged particle at certain energies associated with the eigenvalues. Properties of the magnetic and electric potentials can cause eigenvectors to be strongly spatially localized. This phenomenon has been extensively studied in the case where the magnetic potential is absent, and we will briefly illustrate some known results about localization and its driving mechanisms in that context. Much less is known in the case where the behavior is dominated by the magnetic field. Our talk will focus on that case, providing computational tools that exploit the notion of gauge invariance to (dramatically) reduce the cost of eigenvector computations, and providing practical predictors of where eigenvectors lower in the spectrum are likely to localize.
Hongkai Zhao, Iterative Truncated SVD for Inverse Scattering Problems
Abstract: We propose an iterative truncated SVD (iTSVD) algorithms for Newton’s method for inverse scattering problems. The key ideas are: 1) frequency unbiased least square formulation; 2) effective rank estimation of the linearized operator and computing the truncated SVD solution using fast random SVD algorithm. A key point is that random SVD only needs the application of the matrix to a set of random vectors whose cardinality is comparable to the effective rank. Moreover, these matrix vector multiplications can be formulated as solving the same forward problem with different righthand sides, where fast solvers are available. In summary, the truncated SVD solution based on effective rank and the corresponding signal space, which are determined by the underlying forward model and the measurement data, provides the optimal (in the least squares sense) linearized reconstruction one can compute stably at the current iteration in one shot. We use extensive numerical tests on far-field and near-field data, limited aperture data, sparse random measurements, noisy data to demonstrate that iTSVD not only significantly reduces the number of iterations but also enhance the stability markedly.
Shuwang Li, A boundary integral scheme for computing the dynamics of rigid particles in unsteady stokes or linear viscoelastic fluids
Abstract: We present a numerical study of the flow due to oscillating three-dimensional rigid particles in either a linear viscoelastic fluid or a viscous Newtonian fluid. Using the viscous–viscoelastic correspondence between viscous flows and linear viscoelastic materials, we compute flow quantities in the frequency domain by solving linear partial differential equations via an accurate boundary integral method. We investigate the stresses distribution on the particle surfaces and the velocity fields near the particles. We discuss the flow characteristics under various conditions including different Weissenburg and frequency numbers, different particle separation distances, and particle sizes.
Isaac Harris, On the Clamped Transmission Eigenvalue Problem
Abstract: In this talk, we consider a transmission eigenvalue problem arising from the scattering of biharmonic waves by a clamped obstacle in a thin elastic plate. Using the Kirchhoff-Love model, the biharmonic scattering problem is decomposed into propagating and evanescent components, leading to a new transmission eigenvalue problem posed on the entire plane. We will discuss the existence and discreteness of these new eigenvalues as well as show that they can be identified from far-field measurements. In addition, we can relate the new eigenvalues to the Dirichlet and Neumann eigenvalues of the negative Laplacian on the obstacle. Numerical experiments will be discussed that suggest monotonicity and interlacing properties for future study.
Li Zhu, A Canonical Gauge for Efficient Computation of Magnetic Schrödinger Eigenpairs
Abstract: In this talk, we consider the eigenvalue problem for the magnetic Schrödinger operator. A major computational challenge in simulating these systems is that the eigenvectors can be highly oscillatory, demanding dense grids and high computational costs for accurate numerical approximation. To overcome this, we leverage gauge invariance to transform the original system into a mathematically equivalent problem that is significantly more amenable to numerical methods. Specifically, we introduce a canonical magnetic gauge computed by solving an auxiliary Poisson problem. This transformation yields a new operator that preserves the exact same spectrum but results in far less oscillatory eigenvectors. We will present extensive numerical tests demonstrating that this canonical magnetic gauge allows for the computation of eigenpairs with significantly higher efficiency, accuracy, and stability compared to standard approaches.
Zhimin Zhang, Natural superconvergence points for splines
Joseph Coyle, Series solutions to compartmental problems in epidemiology applications.
Abstracgt: Compartmental modeling is frequently used as a modelling technique when the variables of interest can be grouped into distinct categories, or compartments. This is typically the case when simulating the spread and behavior of infectious diseases. When the resulting differential system is coupled in a nonlinear way, numerical techniques are often employed to approximate the true solutions. Here, we employ a power series approximation, demonstrating a way to estimate the radius of convergence as part of an adaptive technique for long term approximations.
Yangwen Zhang, Sharp and unified $L^2$ error estimates for the nonsymmetric Nitsche method on convex polytopes
Abstract: Nitsche's method weakly imposes Dirichlet boundary conditions, but its nonsymmetric variant has long shown a gap between theory and computation: the classical $L^2$~analysis under $H^{k+1}$~regularity predicts a half-order convergence loss, whereas numerical experiments on smooth test problems consistently produce the optimal rate. Whether this discrepancy reflects a limitation of the analysis or an essential feature of the method has remained an open question.
On bounded convex polytopes in two and three dimensions, we prove a unified, regularity-dependent $L^2$~error estimate valid across the entire penalty scale $h^{-\alpha}$:
\begin{align*}
\|u-u_h\|_{L^2(\Omega)}\le C h^r |u|_{W^{k+1,p}(\Omega)},\qquad
r=\min\bigl\{k+1,\,k+\max\{1,\alpha\}-1/p\bigr\}.
\end{align*}
Numerical experiments in two and three dimensions, on a one-parameter family of manufactured solutions with tunable regularity, demonstrate the sharpness of the estimate and resolve the open question. First, under merely $H^{k+1}$~regularity the half-order loss is essential; second, the optimal convergence consistently observed on smooth test problems is therefore explained by their full $W^{k+1,\infty}$ regularity, not by a limitation of the standard analysis. The theory identifies \$\alpha\ge 1+1/p$ or $p=\infty$ as the sharp threshold for recovering the optimal rate $h^{k+1}$, and the experiments confirm this if-and-only-if condition in both dimensions.
Jichun Li, Analysis and simulation of the Cohen-Monk perfectly matched layer model
Abstract: In this talk, I'll first give a brief overview of the perfectly matched layer (PML) concept introduced by Berenger in 1994 to solve the time-domain Maxwell equations in unbounded domains. Then I will focus on a PML model developed in 1999 by Cohen and Monk for solving electromagnetic wave scattering problems in unbounded domains. In practical simulations, this PML model demonstrates excellent capability in absorbing the outgoing waves. Despite its empirical success, a direct theoretical proof of stability for the Cohen–Monk PML formulation has remained open. In this paper, by using the classic energy method with carefully constructed test functions, we successfully establish the stability for this PML model. Building upon this analysis, we develop a finite element scheme to solve this PML model and rigorously prove both discrete stability and optimal-order error estimates. Finally, we present numerical results to support our theoretical analysis and to demonstrate the effectiveness of this PML model in absorbing outgoing waves.