Stochastic methods for solving high-dimensional partial differential equations
For researchers solving high-dimensional PDEs, this offers a novel hybrid method combining probabilistic and sparse interpolation techniques, though it appears incremental.
This work proposes algorithms for solving high-dimensional PDEs by combining probabilistic Feynman-Kac representation with sparse interpolation and sequential control variates. Numerical examples demonstrate the behavior of the algorithms, but no concrete performance numbers are provided.
We propose algorithms for solving high-dimensional Partial Differential Equations (PDEs) that combine a probabilistic interpretation of PDEs, through Feynman-Kac representation, with sparse interpolation. Monte-Carlo methods and time-integration schemes are used to estimate pointwise evaluations of the solution of a PDE. We use a sequential control variates algorithm, where control variates are constructed based on successive approximations of the solution of the PDE. Two different algorithms are proposed, combining in different ways the sequential control variates algorithm and adaptive sparse interpolation. Numerical examples will illustrate the behavior of these algorithms.