Low-order approximations for large-scale problems arising in the context of high-dimensional PDEs and spatially discretized SPDEs

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In this project, we aim to develop and analyze low-order approximations for high-dimensional stochastic differential equations (SDEs).Such SDEs are, e.g., associated with stochastic representations of solutions to high-dimensional partial differential equations (PDEs) or occur as spatially discretized stochastic PDEs (SPDEs). For spatially discretized SPDEs, dimension reduction is essential in order to lower the complexity of Monte-Carlo (MC) methods or make sparse grid representation or quasi-MC feasible. Using the Feynman-Kac formula, high-dimensional linear PDEs can for instance be linked to large-scale SDEs. Applying model order reduction (MOR) to these SDEs leads to stochastic systems of low-order that are associated with low-dimensional PDEs representing an approximation of the original PDE. This reduction in the spatial variable has the advantage of making classical PDE discretization schemes available (in contrast to the original problem). Consequently, MOR is a promising approach for computing efficient numerical solutions to very complex problems.System-theoretic MOR techniques are well-established methods in the field of deterministic control systems, where they have shown a very promising performance. These schemes are popular and beneficial since they allow a detailed theoretical analysis, however, they are based on frequency domain and control concepts. These ideas do not extend to uncontrolled SDEs that are considered in this project. Therefore, alternative concepts are required to allow for system-theoretic MOR to be applied to the stochastic case. In addition, we aim to investigate system-theoretic MOR methods for unstable and highly nonlinear large-scale SDEs for which there are many open theoretical questions, such as error and stability analysis, already in deterministic settings.The goal of this project is to create tailor-made MOR methods (e.g. balancing related and optimization based MOR) for several important applications as well as the implementation of these algorithms. A large focus will be on establishing an error and stability analysis for such schemes applied to unstable and nonlinear systems which is particularly challenging in a stochastic framework. Our work on MOR for SDEs and recent progress on, e.g., MOR error bounds for certain types of nonlinear equations are the basis for this project which has the potential to lead to new algorithms with analytic foundations.