Mathematics - Colloquium
Tuesday, April 1, 2014
11:00 AM-12:00 PM
ABSTRACT: Stochastic differential equations are usually used to model problems with extrinsic and intrinsic uncertainty. Although numerical methods for these equations have been developed for more than thirty years, there are great demands of efficient solvers for white noise driven equations with purposes varying in various applications. We have been focusing on two directions in this area: stochastic partial differential equations and nonlinear stochastic ordinary differential equations.
For stochastic partial differential equations, we have been developing efficient alternative methods to prohibitively expensive Monte Carlo methods using deterministic integration methods in random space, such as Wiener chaos methods and stochastic collocation methods. However, these deterministic integration methods are only efficient for short time integration of stochastic partial differential equations with white noise.
To achieve a longer-time numerical integration, we apply deterministic integration methods with a recursive strategy in time. For linear advection-reaction-diffusion equations, we show that Wiener chaos methods and stochastic collocation methods are comparable in computational performance and can be more efficient than Monte Carlo methods, when the recursive strategy in time is used.
We have also been investigating numerical schemes for nonlinear stochastic ordinary differential equations under some monotone conditions. We prove a fundamental theorem that is the relationship between local truncation error and global mean-square error. We propose several balanced explicit schemes to efficiently integrate the nonlinear equations and prove the convergence rate using our fundamental theorem.
Some future research directions are outlined--numerical methods for nonlinear stochastic differential equations and numerical methods for fractional partial differential equations, with applications to dynamical systems and stochastic volatility models.
Suggested Audiences: Adult, College
Last Modified: March 26, 2014 at 4:27 PM