Numerical Approximation of Parabolic SPDE’s
A talk in the BI.discrete Workshop series by
Noel Walkington from Pittsburgh
| Abstract: | onvergence theory for numerical schemes to approximate
solutions of stochastic parabolic equations of the form
form
$$
du + A(u) \, dt = f \, dt + g \, dW, \qquad u(0)=u^0,
$$
will be reviewed. Here $u$ is a random variable taking values in a
function space $U$, $A:U \rightarrow U'$ is partial differential
operator, $W = \{W_t\}_{t \geq 0}$ a Wiener process, and $f$, $g$, and
$u^0$ are data. This talk will illustrate how techniques from stochastic
analysis and numerical partial differential equations can be combined
to obtain a realization of the Lax--Richtmeyer meta--theorem:
A numerical scheme converges if (and only if) it is stable and consistent.
Structural properties of the partial differential operator(s) and
probabilistic methods will be developed to establish stability and a
version of Donsker's theorem for discrete processes taking values
in the dual space $U'$.
This is joint work with M. Ondrejat (Prague, CZ) and A. Prohl (Tuebingen, DE).
Within the CRC this talk is associated to the project(s): B7 |