Title: A Carleman-based numerical method for the 3D inverse scattering problem
Abstract: We study the inverse scattering problem for the three-dimensional Helmholtz equation with multi-frequency back scattering data. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasi-linear elliptic partial differential equations. We solve this coupled system by developing the Carleman convexification method. Using the Carleman weight function, we construct a globally strictly convex cost functional and prove the global convergence to the exact solution of the gradient projection method. The Lipschitz stability estimate of the Carleman convexification method is proved also via a Carleman estimate. Finally, our theoretical finding is verified via several numerical tests with computationally simulated data and experimental data. These tests demonstrate that we can accurately recover all three important components of targets of interest: locations, shapes, and dielectric constants.

Title: A second-order symplectic method for an advection-diffusion-reaction problem in Bioseparation
Abstract: An advection-diffusion-reaction problem with non-homogeneous boundary conditions is considered that modes the chromatography process, a vital stage in bioseparation. We prove stability and error estimates for both constant and affine adsorption, using the midpoint method for time discretization and finite elements for spatial discretization. In addition, we did the stability analysis for nonlinear, explicit adsorption in the continuous case. The numerical tests are performed that validate our theoretical results.

Title: Carleman Linearization of Nonlinear Systems and Its Finite-section Approximations
Abstract: The Carleman linearization is one of the mainstream approaches to lift a finite-dimensional nonlinear dynamical system into an infinite-dimensional linear system. Finite-section approximation of the lifted system has been widely used to study dynamical and control properties of the original nonlinear system. In this context, some of the outstanding problems are to determine under what conditions, as the finite-section order increases, the trajectory of the resulting approximate linear system from the finite-section scheme converges to that of the original nonlinear system, and whether the time interval over which the convergence happens can be quantified explicitly. In this talk, I will discuss explicit error bounds for the finite-section approximation and prove that the convergence is indeed exponential with respect to the finite-section order. For a class of nonlinear systems, it is shown that one can achieve exponential convergence over the entire time horizon up to infinity. Moreover, the proposed error bound estimates can be used to compute proper truncation lengths for a given application, e.g., determining the proper sampling period for model predictive control and reachability analysis for safety verifications. This talk is based on the joint paper, Carlemen linearization of nonlinear systems and its finite-section approximations, with Amini, Sun and Motee, arXiv 2207.07755.

Title: Narrow-Stencil Approximation Methods for Fully Nonlinear Elliptic Boundary Value Problems
Abstract: This talk will introduce a new convergent narrow-stencil finite difference method for approximating viscosity solutions of second order fully nonlinear elliptic partial differential equations including Hamilton-Jacobi-Bellman equations. The proposed finite difference method naturally extends the Lax-Friedrichs method for first order problems to second order problems by introducing a key stabilization term called a numerical moment. By abandoning the standard monotonicity assumption, the new methods do not require the use of wide-stencils. The new narrow-stencil methods are easy to formulate and implement, and they formally have higher-order truncation errors than monotone methods when first-order terms are present in the PDE. We will also discuss a new discontinuous Galerkin method that formally extends the narrow-stencil finite difference methods to achieve higher order accuracy.