Hao-Min Zhou
(Department of Applied and Computational Mathematics,
California Institute of Technology)
Efficient Numerical Computations of
Stochastic Partial Differential Equations
Abstract
Stochastic PDE's with solutions depending on multiple scales play fundamental
and important roles in many problems such as composite materials, flows
and transports in porous media, and turbulence. Numerical simulations become
an important strategy in gaining understandings to the phenomena and exploring
their applications. However, direct numerical simulations are often very
difficult due to the problem's multiscale nature and randomness. In this
talk, I will report two different approaches that we have been explored
recently. One is using Wiener Chaos expansions which separate randomness
from the problems, to convert random problems into deterministic ones.
Therefore, by solving these deterministic equations, all statistical properties,
such as mean and variance of the original problems can be fully recovered.
We have demonstrated that this approach can be applied to a wide range
of problems. The other approach is to use a dynamic nonlinear transformation
and to characterize the probability density functions (PDF) of the transformed
random variable by using Fokker-Planck equations. This enables us to compute
the desired statistical properties efficiently and accurately using quadratures.
Both approaches need not involve any randomness in the computations, thus
avoiding, e.g., random number generating. Therefore we can use well developed
deterministic techniques to solve nonlinear stochastic differential equations.
In many applications, they can drastically reduce the computation load
and provide reliable control over the computational errors.
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