We will consider various models in hydrodynamic, including the 2d Navier-Stokes, Boussinesq equations, and a Brinkman-Forchheimer-Navier-Stokes equations in 3d. These models are driven by an external stochastic Brownian perturbation. We will implement space-time numerical schemes and prove their convergence. We will show some rates of convergence as well. Furthermore, we will show the difference between…

Recent developments in the practice of numerical programming require optimization problems not only to be solved numerically, but also to be differentiated. This allows to integrate the computational operation of evaluating a solution in larger models, which are themselves trained or optimized using gradient methods. Most well known applications include bilevel optimization and implicit input-output…

We will discuss a special family of 2D incompressible inviscid fluid flows in the form of logarithmic spiral vortex sheets. Such flows are determined by a vorticity distribution of a curve R^2, and they are notoriously hard to study analytically. In the talk we will discuss several results regarding logarithmic spiral vortex sheets: well-posedness of the spirals as…

I first give a quick introduction to front propagations, Hamilton-Jacobi equations, level-set forced mean curvature flows, and homogenization theory. I will then show the optimal rates of convergence for homogenization of both first-order and second-order Hamilton-Jacobi equations. Based on joint works with J. Qian, T. Sprekeler, and Y. Yu. Zoom meeting: Link