Michael Roeckner
A Break of the Complexity of the Numerical Approximation of Nonlinear SPDEs with Multiplicative Noise.
Mon, 01/18/2010 - 16:01 — SomNambulAuthors: Michael Roeckner, Arnulf Jentzen
Subjects: Numerical Analysis
AbstractA new numerical method for stochastic partial differential equations (SPDEs)
of evolutionary type, which is in some sense the infinite dimensional analog of
Milstein's scheme for finite dimensional stochastic ordinary differential
equations (SODEs), is introduced and analyzed in this article. The Milstein
scheme is known to be impressively efficient for scalar one-dimensional SODEs
but only for some special multidimensional SODEs due to difficult simulations
of iterated stochastic integrals in the general multidimensional SODE case.Probabilistic representation for solutions of an irregular porous media type equation: the degenerate case.
Thu, 08/20/2009 - 23:06 — SomNambulAuthors: Viorel Barbu, Michael Roeckner, Francesco Russo
Subjects: Probability
AbstractWe consider a possibly degenerate porous media type equation over all of
$\R^d$ with $d = 1$, with monotone discontinuous coefficients with linear
growth and prove a probabilistic representation of its solution in terms of an
associated microscopic diffusion. This equation is motivated by some singular
behaviour arising in complex self-organized critical systems. The main idea
consists in approximating the equation by equations with monotone
non-degenerate coefficients and deriving some new analytical properties of the
solution.
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