A Recursive Polynomial Chaos Evolution Method for Stochastic Differential Equations
This work addresses the computational challenge of simulating stochastic differential equations over long time intervals, offering a method that avoids the curse of dimensionality in polynomial chaos expansions.
The paper proposes a recursive polynomial chaos evolution method for stochastic differential equations that maintains a fixed low-dimensional representation over long time intervals, achieving model reduction without sampling. Numerical results show high accuracy and low computational cost.
Numerical simulation of stochastic differential equations over long time intervals poses significant computational challenges. In this paper, we propose a novel recursive polynomial chaos evolution method that achieves model reduction without sampling by exploiting the Markov property to maintain a fixed low-dimensional representation throughout the time evolution. At each time step, we construct orthogonal polynomial bases adapted to the current probability measure, and project the one-step-ahead solution onto this new basis together with the new Brownian increments. This dynamic updating strategy effectively reduces the dimension of the random variables during long-time evolution. Under appropriate assumptions, we prove the convergence of the method, specifically that the distributions generated by the method preserve convergence in the Wasserstein-1 distance. We present numerical results demonstrating that the method can accurately capture complex dynamical behaviors with high accuracy and low computational cost.