| In this thesis, we apply Malliavin calculus to the sensitivity analysis of a stochastic partial differential equation of the Schrodinger type. The 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 all sorts of parameters. Because of the imperfectness of the time reversal mirror, the time-reversed signal is an integral of a cut-off function. This makes Monte Carlo numerical schemes ineffecient for sensitivity analysis. Here Malliavin calculus comes to the rescue since it emerged out of the stochastic calculus of variations. With its "Integration by Parts" formula, we avoid computing the derivative of the cut-off function. Instead, we obtain close form formulae for the sensitivities in terms of Skorohod integrals with respect to an infinite dimensional Wiener process. We also 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. Numerical implementation is done in both 2-D and 3-D. To the best of our knowledge, the numerical computation of the stochastic Schrodinger equation's solution was only carried out in 2-D, and even in that case our numerical algorithm seems better than those we found in the literature. |
