This paper deals with an application of the Malliavin calculus to a stochastic partial differential equation of the Schrödinger type. This equation appears as the major building block in the analysis of the focusing properties of time-reversed waves in a random medium in the asymptotic regime where the parabolic approximation is valid. We consider the sensitivities of the solutions with respect to several parameters, and we provide closed form formulae in terms of Skorohod integrals with respect to an infinite dimensional Wiener process. We construct finite dimensional approximation schemes for these integrals. These schemes are based on a sieve of Wiener chaos expansions mixed with Galerkin approximations in a natural Fourier basis. In two space dimensions, our computational algorithm seems to perform better than those we found in the literature. Moreover, because it avoids finite difference methods, it can be implemented in three space dimensions without much ado. copy; 2009 Society for Industrial and Applied Mathematics.
All Science Journal Classification (ASJC) codes
- Applied Mathematics
- Malliavin calculus
- Parameter sensitivities
- Time reversal