Title: Exact asymptotics of the stochastic wave equation with time-independent noise

Abstract: In this talk, I will report a recent joint work with Raluca Balan and Xia Chen. In this work, we study the stochastic wave equation in dimensions $d\leq 3$, driven by a Gaussian noise $\dot{W}$ which does not depend on time. We assume that either the noise is white, or the covariance function of the noise satisfies a scaling property similar to the Riesz kernel. The solution is interpreted in the Skorohod sense using Malliavin calculus. We obtain the exact asymptotic behaviour of the p-th moment of the solution either when the time is large or when p is large. For the critical case, that is the case when d=3 and the noise is white, we obtain the exact transition time for the second moment to be finite. 

Title: Symplectic non-squeezing for the KdV flow on the line

Abstract: In this talk we prove that the KdV flow on the line cannot squeeze a ball in $\dot H^{-\frac 1 2}(\mathbb R)$ into a cylinder of lesser radius. This is a PDE analogue of Gromov’s famous symplectic non-squeezing theorem for an infinite dimensional PDE in infinite volume.

Title:Developing high order, efficient, and stable time-evolution methods using a time-filtering approach.

Abstract: Time stepping methods are critical to the stability, accuracy, and efficiency of the numerical solution of partial differential equations. In many legacy codes, well-tested low-order time-stepping modules are difficult to change; however, their accuracy and efficiency properties may form a bottleneck. Time filtering has been used to enhance the order of accuracy (as well as other properties) of time-stepping methods in legacy codes. In this talk I will describe our recent work on time filtering methods for the Navier Stokes equations as well as other applications. A rigorous development of such methods requires an understanding of the effect of the modification of inputs and outputs on the accuracy, efficiency, and stability of the time-evolution method. In this talk, we show that time-filtering a given method can be seen as equivalent to generating a new general linear method (GLM). We use this GLM approach to develop an optimization routine that enabled us to find new time-filtering methods with high order and efficient linear stability properties. In addition, understanding the dynamics of the errors allows us to combine the time-filteringGLM methods with the error inhibiting approach to produce a third order A-stable method based on alternating time-filtering of implicit Euler method. I will present our new methods and show their performance when tested on sample problems.

Title: Inverse problem for a spectral asymmetry function

Abstract: For the Schrödinger equation −u'' + q(x)u = λu on a finite x-interval, there is defined an “asymmetry function” a(λ;q), which is entire of order 1/2 and type 1 in λ.  The main result identifies the classes of square-integrable potentials q(x) that possess a common asymmetry function.  For any given a(λ), there is one potential for each Dirichlet spectral sequence.  This has applications to the spectral theory of multi-layer quantum graphs.  (Collaboration with Malcolm Brown, Karl Michael Schmidt, and Ian Wood)

Title: Recent advances in high order entropy stable discontinuous Galerkin schemes

Abstract: High order discontinuous Galerkin (DG) methods are known to be unstable when applied to nonlinear conservation laws with shocks and turbulence, and have traditionally required additional filtering, limiting, or artificial viscosity to avoid solution blow up.  Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi-discrete entropy inequality independently of numerical resolution. In this talk, we will review approaches for constructing entropy stable schemes and discuss recent developments, including positivity preserving strategies and the application of entropy stable methods to under-resolved compressible turbulent flows.

Title: A Posteriori Error Estimates in Finite Element Method by Preconditioning

Abstract:We present a framework that relates preconditioning with a posteriori error estimates in finite element methods. In particular, we use standard tools in subspace correction methods to obtain reliable and efficient error estimators. As a simple example, we recover the classical residual error estimators for the second order elliptic equations as well as present some new estimators for systems of PDEs. This is a joint work with Yuwen Li (Penn State).

Title: Rough coefficients in ordinary differential equations

Abstract: We investigate the spectral theory for the system $Ju’+qu=w(\lambda u+f)$ of ordinary differential equations where $J$ is constant invertible skew-hermitian matrix while $q$ is a hermitian and $w$ a non-negative matrix whose entries are distributions of order $0$. A major obstacle is the fact that, in general, the unique continuation of solutions of the differential equation is not possible. Basic information on distributions will be provided.

Title: Computational Approaches to Steklov Eigenvalue Problems and Free Boundary Minimal Surfaces 

Abstract: Recently Fraser and Schoen showed that the solution of a certain extremal Steklov eigenvalue problem on a compact surface with boundary can be used to generate a free boundary minimal surface, i.e., a surface contained in the ball that has (i) zero mean curvature and (ii) meets the boundary of the ball orthogonally (doi:10.1007/s00222-015-0604-x). In this talk, we discuss our new numerical methods that use this connection to realize free boundary minimal surfaces. Namely, on a compact surface, Σ, with genus γ and b boundary components, we maximize σj (Σ, g) L(∂Σ, g) over a class of smooth metrics, g, where σj (Σ, g) is the jth nonzero Steklov eigenvalue and L(∂Σ, g) is the length of ∂Σ. Our numerical method involves (i) using conformal uniformization of multiply connected domains to avoid explicit parameterization for the class of metrics, (ii) accurately solving a boundary-weighted Steklov eigenvalue problem in multi-connected domains, and (iii) developing gradient-based optimization methods for this non-smooth eigenvalue optimization problem. We numerically solve the extremal Steklov problem for the first eigenvalue and use corresponding eigenfunctions to generate a free boundary minimal surface, which we display in striking images. Many results are shown to demonstrate the accuracy and robustness of the proposed approaches.  

This is a joint work with Braxton Osting at Department of Mathematics, University of Utah, United States, and Edouard Oudet at LJK, Universit´e Grenoble Alpes, France. 

Title: Advances in fast high-order boundary integral equation solvers in complex geometries in three dimensions

Abstract: The fast multipole method (FMM) was invented over 30 years ago and has enabled the asymptotically optimal solution of dense linear systems arising from the discretization of integral equations appearing in acoustics, electromagnetics, fluid flow, etc. However, in order to build FMM-accelerated high-order solvers in three dimensions, additional advances in mesh generation and quadrature are also required; efficient coupling of these schemes to the FMM itself is also of importance. In this talk I will give an overview of the ingredients needed to build boundary integral equation solvers in three dimensions, and then detail some recent advances in high-order mesh generation and quadrature. Numerical examples for the Helmholtz and Laplace-Beltrami equations will be shown.