Time-Evolving Stochastic Manifolds

Uncertainty is all around us and caused, for example, by the nature of a problem as in quantum mechanics, the lack of our precise knowledge as in porous media, or inaccuracies in measurements as in experiments with imperfect equipment. While traditionally and due to the lack of computing power, science and technology relied on deterministic models, recent developments allow to include randomness. This trend requires efficient simulation methods for models with uncertainty. In space-time problems such as moving biological cells and the surface of the ocean, the randomness could be modeled by a stochastic process given explicitly or described by stochastic PDEs. Fast and accurate methods for sampling the stochastic processes are the key when computing statistical quantities of the advanced models.The main contribution of the project is the development of a theoretical framework for evolving stochastic manifolds and their efficient simulation with analyzed algorithms. Special emphasis is paid to the situation when the evolving stochastic manifold is a moving surface disturbed by external forces and described by stochastic PDEs. The main steps of the project are divided into three objectives: Obj. (A) From random fields on manifolds to stochastic manifolds. Obj. (B) From stochastic processes to evolving stochastic manifolds. Obj. (C) Solving PDEs on stochastic manifolds.The challenges are tackled based on recent advances in the simulation of Gaussian random fields on manifolds and their analysis obtained by the research team of the PI. This new breakthrough paves the way for the development of sampling methods for stochastic processes on manifolds and ultimately to evolving stochastic manifolds.To reach these goals, the PI's research group is complemented by specialists in geometric numerical integration, numerical methods for (stochastic) PDEs, and spatial statistics.