Abstract:
Large (and moderate) deviations principle identifies the exponential rate of decay of probabilities for rare events in the context of small noise asymptotics. For a class of nonlinear stochastic partial differential equations that arise in fluid dynamics, I will present weak convergence approach to identify the exponential rate.
notice the special date. March 19th, 9-10am. Room Thackeray 427
Abstract:
The Navier-Stokes equations (NSE) model the motion of incompressible viscous fluids and are widely used in engineering and physics, etc. In many applications, the time-averaged information is very often the quantity of interest, so this talk focuses on finding the numerical solutions of the steady Navier-Stokes equations (SNSE). We introduce a few improvements to the commonly used Picard iteration for solving the SNSE from different aspects, like shortening the required number of iterations and reducing the computational cost for each iteration.
Abstract:
Irreversible, dissipative processes can be often naturally modeled as gradient flows; under certain assumptions, flow in deformable porous media is such a process. In this talk, we formulate the problem of linear poro-(visco-)elasticity as generalized gradient flow. By exploiting this structure, the analysis of well-posedness and construction of numerical solvers can be performed in a rather straight-forward and abstract manner. For this, we apply results from gradient flow theory and convex optimization.
Hopefully we know that $\displaystyle \sum_{i \geq 0} x^i $ $= \frac{1}{1-x}$. Similarly one computes that $ \displaystyle \sum_{i \leq 1} x^i $ $= \frac{x^2}{x-1}$. Interestingly, $ \frac{1}{1-x} + \frac{x^2}{x-1} = 1 + x$ which is the sum corresponding to the integers in the interval $[0, 1] = [0, \infty] \cap [-\infty, 1]$. We will explain generalization of this (called Brion's theorem) to integer points in convex polytopes of arbitrary dimension.