Title: A variational quasi-reversibility method for a time-reversed nonlinear parabolic problem
Abstract: This talk is about a modified quasi-reversibility method for computing the exponentially unstable solution of a terminal-boundary value parabolic problem with noisy data. As a PDE-based approach, this variant relies on adding a suitable perturbing operator to the original PDE and, consequently, on gaining the corresponding fine stabilized operator. The designated approximate problem is a forward-like one. This new approximation is analyzed in a variational framework, where the finite element method can be applied. For each noise level, the Faedo-Galerkin method is exploited to study the weak solvability of the approximate problem. Relying on the energy-like analysis coupled with a suitable Carleman weight, convergence rates in $L^2$–$H^1$ of the proposed method are obtained when the true solution is sufficiently smooth.

Title: Application of Adomian Decomposition Method to certain Partial Differential Equations
Abstract: The Adomian decomposition method was introduced and developed by G. Adomian. A unique feature of this method is, that it deals directly with the nonlinear problem avoiding any linearization or discretization This is a semi-analytic method and assumes that the solution is decomposed into a rapidly convergent series and the nonlinear term as a series of Adomian Polynomials. This result in the reduction of any differential equation into a set of recursive relation for the Adomian solution series. We first present the methodology and show its application to a couple of PDEs that model fluid flow.

Title: Simulations of Photonic Crystal Ring Resonators using Domain Decomposition and Difference Potentials
Abstract: A photonic crystal ring resonator (PCRR) is a micro-scale optical device that combines a closed-loop waveguide with a light input and output. PCRRs are constructed with periodically placed scattering rods where the exclusion of rods is used to form the path of a waveguide. We simulate PCRRs numerically using a non-iterative domain decomposition approach that is insensitive to jumps in material properties, in particular, those between the scattering rods and surrounding medium. To approximate the governing Helmholtz equation, we use a compact fourth order accurate finite difference scheme combined with the method of difference potentials (MDP). The MDP renders exact coupling between the decomposition subdomains and maintains high order accuracy for non-conforming boundaries/interfaces on regular grids.

Title: Sticky particle Cucker-Smale dynamics and numerical simulations
Abstract: The 1D Euler-alignment system can be effectively described by a nonlocal scalar balance law. The entropy conditions for the scalar balance law serves as the selection principle that determines the unique weak solution of the 1D Euler-Alignment system, which corresponds to the sticky particle collision rule in the Cucker-Smale dynamics. In this talk, I will introduce a numerical algorithm to approximate the sticky particle Cucker-Smale dynamics based on established well-posedness and stability result. Under a reasonable assumption about the time step and collision states, we analyze the approximation error and demonstrate our method with plenty of numerical examples.