## Math/ICES Center of Numerical Analysis Seminar (Fall 2015)

Time and Location: Friday, 3:00-4:00PM, POB 6.304 (Previously known as ACE 6.304). Special time and locations are indicated in color.

 Dates Spekers and Hosts Title and Abstract 09/22/2015 Tuesday 3:30PM Diego del-Castillo-Negrete (ORNL) Host 1: Luis Caffarelli Host 2: Irene Gamba Nonlocal transport modeling: fundamentals, applications, and numerical methods: Lecture #1 09/24/2015 Thursday 3:30PM Diego del-Castillo-Negrete (ORNL) Host 1: Luis Caffarelli Host 2: Irene Gamba Nonlocal transport modeling: fundamentals, applications, and numerical methods: Lecture #2 09/25/2015 09/29/2015 Tuesday 1:30PM Diego del-Castillo-Negrete (ORNL) Host 1: Luis Caffarelli Host 2: Irene Gamba Nonlocal transport modeling: fundamentals, applications, and numerical methods: Lecture #3 10/01/2015 Thursday 1:30PM Diego del-Castillo-Negrete (ORNL) Host 1: Luis Caffarelli Host 2: Irene Gamba Nonlocal transport modeling: fundamentals, applications, and numerical methods: Lecture #4 10/02/2015 10/09/2015 10/16/2015 KI-Net Conference in honor of Profesor Bjorn Engquist 10/19/2015 Monday 1:00PM RLM 10.176 Wenjia Jing (University of Chicago) Host: Kui Ren Homogenization of Hamilton-Jacobi equations in dynamic random environments We consider stochastic homogenization of Hamilton-Jacobi equations in dynamic random environments, where the coefficients of the equations, namely the Hamiltonian and, for second order equations, the diffusion matrix, are highly oscillatory in space and time. I will discuss how to generalize the metric approach of stochastic homogenization developed for static random environment to the dynamic random setting, when uniform continuity (uniform with respect to the scale of oscillation and the random realization) of the minimal cost function is available. This talk is based on joint work with Takis Souganidis and Hung Tran. 10/23/2015 11/02/2015 3:00-4:00 Monday Alexander Bobylev (Keldysh Institute of Applied Mathematics, Russian Academy of Sciences and Karlstad University, Sweden) Host: Irene M. Gabma On some properties of the Landau kinetic equation We discuss some some general properties of the Landau kinetic equation. In particular, the difference between "true" Landau equation, which formally follows from classical mechanics, and "generalized" Landau equation, which is just an interesting mathematical object, is stressed. It is shown how to approximate the Landau equation by the Wild sum. It is the so-called quasi-Maxwellian approximation related to Monte Carlo methods. This approximation can be also useful for mathematical problems. A model equation which can be reduced to a "local" nonlinear parabolic equation is also constructed in connection with existence of the strong solution to the initial value problem.The self-similar asymptotic solution to the Landau equation for large v and t is discussed in detail. The solution, earlier confirmed by numerical experiments, describes a formation of Maxwellian tails for a wide class of initial data concentrated in the thermal domain. It is shown that the corresponding rate of relaxation ( fractional exponential ) is in exact agreement with recent mathematically rigorous estimates. The talk is based on a joint paper with Irene Gamba and Irina Potapenko. 11/06/2015 11/10/2015 11:00AM-12:00 Tuesday Alexander Mamonov (University of Houston) Host: Kui Ren Nonlinear seismic imaging via reduced order model backprojection We introduce a novel nonlinear seismic imaging method based on model order reduction. The reduced order model (ROM) is an orthogonal projection of the wave equation propagator on the subspace of snapshots of solutions of the wave equation. It can be computed entirely from the knowledge of the time domain seismic data. The image is a backprojection of the ROM using the subspace basis for a known smooth kinematic velocity model. The implicit orthogonalization of solution snapshots is a nonlinear procedure that differentiates our approach from the conventional linear methods (Kirchhoff, reverse time migration - RTM). It allows for automatic removal of multiple reflection artifacts. It also doubles the resolution (in depth) compared to conventional RTM. This is a joint work with V. Druskin and M. Zaslavsky. 11/13/2015 11:00AM-12:00 Irina F. Potapenko (Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow) Host: Irene Gamba Numerical solution for the nonlinear electron kinetic equation in self-similar variables We present numerical solution of fully nonlinear electron kinetic equation in self-similar variables, which has all features of a “standard” hydrodynamics (ratios of the electron mean free path to the scale length, $\gamma=\lambda_C/L \ll 1$) from one hand and, in the other hand, has no restriction on the smallness of the parameter $\gamma$. The self-similar variables approach reduces dimensionality of the space dependent kinetic equation thereby providing numerical analysis of the electron heat transport in the velocity space. The electron distribution structure and its super thermal power-law tail are examined. 11/20/2015 12/04/2015