Time-Evolving Stochastic Manifolds

Project Information

StochMan

Grant agreement ID: 101088589

DOI

10.3030/101088589

EC signature date 3 April 2023

Start date 1 September 2023

End date 31 August 2028

Funded under

• European Research Council (ERC)

Total cost

€ 1 997 651,00

EU contribution

€ 1 997 651,00

Coordinated by

CHALMERS TEKNISKA HOGSKOLA AB
Sweden

Periodic Reporting for period 1 - StochMan (Time-Evolving Stochastic Manifolds)

Reporting period: 2023-09-01 to 2026-02-28

Summary of the context and overall objectives of the project

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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

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.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

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.

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