A “parareal” time discretization for nonlinear PDE’s with application to the pricing of an American put. (English) Zbl 1022.65096
Pavarino, Luca F. (ed.) et al., Recent developments in domain decomposition methods. Some papers of the workshop on domain decomposition, ETH Zürich, Switzerland, June 7-8. 2001. Berlin: Springer. Lect. Notes Comput. Sci. Eng. 23, 189-202 (2002).
The authors consider the first-order time-dependent partial differential equation (PDE):
\[
F(\partial_t u, u)=0,
\]
where \(F\) is a functional of \(\partial_t u, u\) and, possibly derivatives of \(u\) with respect to the other variables than time. The objective is to devise a method that combines the use of the coarse solver on the whole interval \((0,T)\) and the fine solver in parallel on each small interval \((T^n, T^{n+1})\) and allows to provide an approximation of \(u\) over the whole interval \((0,T)\) with the same accuracy as a global use of fine, one-step solver.
The authors introduce a slightly modified implementation, in comparison with the method of J.-L. Lions, Y. Maday and G. Turinici [C. R. Acad. Sci. Paris, Sér. I, Math. 332, 661-668 (2001; Zbl 0984.65085)], which turns out to be equivalent to linear problems, but gives better answers for nonlinear problems and allows to tackle non-differentiable problems. An example of such problems is provided by the Black-Scholes equation for an American put in financial mathematics. Numerical advantages of the proposed “parareal” time discretization are presented.
For the entire collection see [Zbl 0989.00043].
The authors introduce a slightly modified implementation, in comparison with the method of J.-L. Lions, Y. Maday and G. Turinici [C. R. Acad. Sci. Paris, Sér. I, Math. 332, 661-668 (2001; Zbl 0984.65085)], which turns out to be equivalent to linear problems, but gives better answers for nonlinear problems and allows to tackle non-differentiable problems. An example of such problems is provided by the Black-Scholes equation for an American put in financial mathematics. Numerical advantages of the proposed “parareal” time discretization are presented.
For the entire collection see [Zbl 0989.00043].
Reviewer: Yuliya S.Mishura (Kyïv)
MSC:
65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |
35K90 | Abstract parabolic equations |
91G60 | Numerical methods (including Monte Carlo methods) |