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

高维 PDE 和空间离散 SPDE 背景下出现的大规模问题的低阶近似

• 批准号: 499366908
• 负责人: Professor Dr. Martin Redmann
• 金额: --
• 依托单位: Institut für Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/499366908?language=en
• 关键词: Low; order; approximations; large; scale

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.

在这个项目中，我们致力于发展和分析高维随机微分方程(SDE)的低阶近似，这种SDE与高维偏微分方程组(PDE)的解的随机表示相联系，或者以空间离散的随机PDE(SPDEs)的形式出现。对于空间离散的SPDEs，为了降低蒙特卡罗(MC)方法的复杂性或使稀疏网格表示或准MC可行，降维是必不可少的。例如，使用Feynman-Kac公式，高维线性PDE可以链接到大规模的SDE。将模型降阶(MOR)应用于这些SDE，得到与代表原始PDE的近似的低维PDE相关联的低阶随机系统。空间变量的这种减少具有使经典的PDE离散化方案可用的优点(与原始问题相反)。因此，MOR是计算非常复杂问题的有效数值解的一种很有前途的方法。系统理论MOR技术是确定性控制领域中公认的方法，它们在确定性控制系统中表现出了非常有前途的性能。这些方案是流行的和有益的，因为它们允许进行详细的理论分析，然而，它们基于频域和控制概念。这些想法不适用于本项目中考虑的不受控制的SDE。因此，需要替代的概念，以允许系统理论的MOR应用于随机情况。此外，我们的目标是研究不稳定和高度非线性的大规模随机微分方程的系统理论MOR方法，对于这些问题，已经在确定性环境中存在许多公开的理论问题，如误差和稳定性分析，本项目的目标是创建针对几个重要应用的定制的MOR方法(例如，平衡相关的和基于优化的MOR方法)以及这些算法的实现。一个很大的重点将是为应用于不稳定和非线性系统的这类方案建立误差和稳定性分析，这在随机框架中特别具有挑战性。我们在SDE的MOR方面的工作和最近在某些类型的非线性方程的MOR误差界方面的进展是这个项目的基础，该项目有可能导致具有分析基础的新算法。