Friday, June 15, 2018 - 14:15 in U2-135
Breaking the curse of dimensionality for approximating semilinear parabolic partial differential equations at single space-time points
A talk in the Oberseminar Analysis series by
Tuan Anh Nguyen from Universität Duisburg-Essen
| Abstract: |
We introduce an algorithm for solving high-dimensional PDEs of the form
$\partial_t u +\frac{1}{2}\Delta u+F(u)=0$ on $[0,T]\times \mathbb{R}^d$ and
$u(T,\cdot )=g$
where classical solutions usually suffer the curse of dimensionality. The algorithm is obtained by considering the PDE as solution to a fix-point problem via the Feynman-Kac formula
\begin{align*}
u(t,x)= \mathbb{E}\left[g(x+W_{T-t})+\int_t^T (F(u))(x+W_{s-t})\,ds\right],\quad t\in [0,T],\ x\in \mathbb{R}^d
\end{align*}
and a decomposition of the Picard iteration with Monte-Carlo approximation. Convergence of the iteration is shown by using a class of semi-norms and their recursive inequalities.
Joint work with Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse, and
Philippe von Wurstemberger.
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