Dr. Paul J. Atzberger (UCSB Mathematics)

Title: Spatially adaptive stochastic numerical methods for intrinsic fluctuations in reaction-diffusion systems

Abstract: Reaction diffusion systems are discussed which arise in the study of biological systems where concentration fluctuations play an important role. This includes the study of cell signaling, cell polarization, and directed cell motility through detection of shallow concentration gradients. To study these systems stochastic partial differential equations are formulated to model intrinsic fluctuations in the concentration fields associated with these processes. A number of challenges arise in approximating the solutions of such SPDE's. Further issues arise when considering non-periodic domains with complex geometries which arise naturally in modeling biological systems. To cope with these issues, methods are developed which allow for spatially adaptive numerical discretization of SPDE's. The methods specifically address how to discretize stochastic driving fields at coarse-refined interfaces.