Joint with the Special OxPDE Seminar (Oxford Centre for Nonlinear PDE)
Title: An Adjoint Method for Optimization of the Boltzmann Equation
Speaker: Professor Russel Caflisch
Abstract: We present an adjoint method for optimization of the spatially inhomogeneous Boltzmann equation for rarefied gas dynamics. The adjoint method is derived using a “discretize then optimize” approach. Discretization (in time and velocity) is via the Direct Simulation Monte Carlo (DSMC) method, and adjoint equations are derived from an augmented Lagrangian. The boundary conditions that are included in this analysis include spectral reflection, thermal reflection, and inflow boundary conditions. For thermal reflection, a “score function” is included as a statistical regularization. This is joint work with Yunan Yang (Cornell).
About the Speaker: Professor Russel Caflisch is Director of the Courant Institute of Mathematical Sciences at New York University (NYU), and a Professor in the Mathematics Department. He was a founding member of California NanoSystems Institute (CNSI). Caflisch’s expertise includes topics in the field of applied mathematics, including partial differential equations, fluid dynamics, plasma physics, materials science, Monte Carlo methods, and computational finance. More precisely, his research is on analysis and numerical methods for physical systems, in particular for special solutions and robust numerical methods in singular limits. For kinetics of fluids and plasmas, his results include analysis of the fluid limit and of shock wave solutions for the nonlinear Boltzmann equation, and accelerated simulation methods that are a hybrid of a continuum fluid solver and a Monte Carlo particle solver. For fluid dynamics, he analyzed the development of singularities for vortex sheets, extended this method to the Muskat problem and to a complexified generalization of the incompressible Euler equations, and analyzed the Prandtl equations for a viscous boundary layer. He developed a Brownian bridge method and other adapted and accelerated numerical techniques for quasi-Monte Carlo methods, which are now widely used in finance. He was a leader of a group of mathematicians and materials scientists that developed a level set method and an island dynamics model for epitaxial growth. As a generalization of compressed sensing, he applied techniques of sparsity and soft-thresholding to PDEs and physics. He was named a fellow of the Society for Industrial and Applied Mathematics in 2009, the American Mathematical Society in 2012, and the American Academy of Arts and Sciences in 2013. He was elected a member of the National Academy of Sciences in April 2019.
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