Acceleration of Automatic Differentiation of Solutions to Parabolic Partial Differential Equations : A Higher Order Discretization

The paper proposes a new automatic/algorithmic differentiation for the solutions to partial differential equations of parabolic type. In particular, we provide a higher order discretization scheme which is a natural extension of the standard automatic differentiation. A Brownian polynomial approach is introduced to avoid the Lévy area simulation. The Lie brackets of vector fields associated with stochastic differential equation play an important role in the proposed scheme. The case that the test function is non-smooth but has Gateaux derivative is considered. Numerical examples are shown to confirm the effectiveness