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

在这个项目中，我们的目标是开发和分析高维随机微分方程（SDEs）的低阶近似。例如，这种随机偏微分方程与高维偏微分方程（PDEs）解的随机表示有关，或者以空间离散随机偏微分方程（SPDEs）的形式出现。对于空间离散化的spde，降维是降低蒙特卡罗（MC）方法复杂度或使稀疏网格表示或拟MC可行的必要条件。例如，使用费曼-卡茨公式，高维线性偏微分方程可以与大规模偏微分方程联系起来。将模型降阶（MOR）应用于这些sde，可以得到与低维PDE相关的低阶随机系统，这些系统代表了原始PDE的近似值。这种空间变量的减少具有使经典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技术是确定性控制系统领域中成熟的方法，在该领域中表现出非常有前途的性能。这些方案是流行和有益的，因为它们允许详细的理论分析，然而，它们是基于频域和控制概念。这些想法并没有扩展到本项目中考虑的不受控制的sde。因此，需要其他概念来允许系统理论的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SDEs的系统理论MOR方法，这些方法已经在确定性设置中存在许多开放的理论问题，例如误差和稳定性分析。该项目的目标是为几个重要的应用程序以及这些算法的实现创建量身定制的MOR方法（例如，平衡相关的和基于优化的MOR）。一个很大的焦点将是建立一个误差和稳定性分析，这些方案应用于不稳定和非线性系统，这在随机框架中特别具有挑战性。我们在SDEs的MOR方面的工作和最近的进展，例如，某些类型的非线性方程的MOR误差界限，是这个项目的基础，它有可能导致具有分析基础的新算法。

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.