Periodic Reporting for period 1 - StochMan (Time-Evolving Stochastic Manifolds)
Reporting period: 2023-09-01 to 2026-02-28
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 implicitly by stochastic partial differential equations. 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 ERC-funded StochMan 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 partial differential equations.
The main contribution of the ERC-funded StochMan 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 partial differential equations.
The project has developed efficient approximation methods for stochastic partial differential equations on surfaces that are a cornerstone for uncertainty quantification for spatio-temporal models with pre-described roughness or smoothness. The methods allow for random surfaces and their time evolution. Besides mathematically proven properties, the developed algorithms are ready to be used in applications.
Based on many small pieces of the puzzle, the main goal is to develop the mathematical foundations of stochastic manifolds and their approximation. The project bridges mathematical theory with algorithms that are ready for use in applications that suffer under uncertainty and require its simulation.