PN 5A-1

Analysis of Stochastic Partial Differential Equations and their Efficient Simulation

Project Description

To account for insufficient measurements in the description of natural phenomena via dynamical systems, discontinuous random fields are used. Such random fields are commonly used to model porosity or other physical coefficients in these dynamical systems. Often, measurement data for coefficient values is only available on certain points of the domain. In the mathematical formulation of a dynamical system the random field has to feature theses measured distributions when restricted to a point and the measured correlation structure throughout the domain. Thereby, the project contributes to FC3: Bridging data-poor & data-rich regimes. Focussing on theoretical advances and novel sampling techniques in the field of discontinuous random fields, the main contribution of the project is to FC4: Mastering stochastic models & total uncertainty. Through modelling tasks within the solution concept of the considered nonlinear hyperbolic stochastic partial differential equations, the project addresses PN5’s RQ3: Precision vs. accuracy. Numerical methods are developed to approximate the solution corresponding to this modelling and various novel methods are applied to speed-up simulation times. To give an example, we developed the “jump-adaptive wave-cell meshing” strategy for the discretization of conservation laws involving discontinuous flux functions. When such a conservation law is stochastic, applying this method significantly reduces the variance of the samplewise approximations of the solution. Thus, this novel method enables fast and precise Uncertainty Quantification of conservation laws, especially if combined with fast multilevel Monte Carlo methods. Thereby, the project also addresses RQ2: Precision vs. resources. The previous points strongly relate to Vision 1 Engineered Geosystems and a link to Vision 3 Next-Generation Material Design is established by a collaboration with project PN5-4. Furthermore, the project relates to the Uncertainty Quantification of the stochastic Gierer-Meinhardt system, the common demonstrator of PN5.

Project Information

Project Number PN 5A-1
Project Name Analysis of stochastic partial differential equations and their efficient simulation
Project Duration October 2018 - December 2021
Project Leader Andrea Barth
Project Members Lukas Brencher, PhD Researcher
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