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。这样的SDEs，例如，与高维偏微分方程（PDE）的解的随机表示相关联，或者作为空间离散随机PDE（SPDE）出现。为了降低蒙特-卡罗（MC）方法的复杂度，或者使稀疏网格表示或准MC方法成为可能，空间离散SPDE的降维是必不可少的。使用Feynman-Kac公式，高维线性偏微分方程可以与大规模的偏微分方程相联系。应用模型降阶（莫尔），这些SDES导致随机系统的低阶，与低维PDE表示原始PDE的近似。这种减少空间变量的优势，使经典的PDE离散计划（与原来的问题相比）。因此，莫尔是一个很有前途的方法，计算有效的数值解非常复杂的problem. System理论的莫尔技术是确定性控制系统领域的成熟的方法，在那里他们已经显示出非常有前途的性能。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。这些想法并不延伸到本项目中考虑的不受控制的SDEs。因此，需要替代的概念，以允许系统理论的莫尔被应用到随机的情况下。此外，我们的目标是研究系统理论的莫尔方法的不稳定和高度非线性的大规模SDES，有许多开放的理论问题，如误差和稳定性分析，已经在determinativesettings.The目标的这个项目是创建量身定制的莫尔方法（如平衡相关和优化为基础的莫尔）的几个重要应用程序，以及这些算法的实施。一个大的重点将是建立一个错误和稳定性分析，适用于不稳定和非线性系统，这是特别具有挑战性的随机框架等计划。我们在SDES的莫尔方面的工作以及最近的进展，例如，某些类型的非线性方程的莫尔误差界的基础上，这个项目有可能导致新的算法与分析的基础。

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.