Stochastic Partial Differential Equations and Their Numerical Solution

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Space-time phenomena as varied as turbulence, phase-field dynamics, surface growth, neuronal activity, population dynamics, and interest rate fluctuations are modelled by stochastic partial differential equations (SPDE). In comparison with partial differential equations (PDE), SPDEs are advantageous because they allow modelers to explicitly incorporate space-time randomness and capture the effect of random fluctuations inherent to systems with many degrees of freedom. However, they are trickier to numerically solve than PDEs and require new specialized techniques. Indeed, since SPDE solutions are very rough, straightforward adaptations of numerical PDE methods often lead to approximations that do not accurately represent the behaviors of the actual solutions. To address this issue, the project develops a new numerical method, called Spectrwm [spek-trum] (signifying spectral random walk method), that is tailored specifically to SPDEs. REU students are included in the work of the project.Spectrwm is a family of Markov jump process approximations for SPDEs. As such, these approximations can be viewed as infinite-dimensional analogues of random walk methods for Brownian motion. Aside from their mathematical interest, these methods address in a natural way several issues encountered when simulating SPDEs, including long-time simulation, accurately representing the effect of multiplicative noise, and approximation of their path-dependent expected values. by suitably constructing them, Spectrwm is faithful to the domain of their solution. The investigator and his collaborators aim to use probabilistic techniques to quantitatively understand the finite- and long-time behavior of Spectrwm methods for well-posed SPDEs with reflecting boundary conditions, and to develop accelerated variants of Spectrwm methods for a wider range of SPDE problems, including ones that are classically ill-posed or where the operators of the underlying processes are not well understood. As a major byproduct, the project lends insight into SPDEs themselves: the nature of the Kolmogorov equations associated to SPDEs, the influence of boundaries on SPDE solutions, metastability in SPDE solutions, and renormalizations for classically ill-posed SPDEs. REU students are included in the work of the project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.