This paper considers large-scale linear stochastic systems representing, e.g., spatially discretized stochastic partial differential equations. Since asymptotic stability can often not be ensured in such a stochastic setting (e.g. due to larger noise), the main focus is on establishing model order reduction (MOR) schemes applicable to unstable systems. MOR is vital to reduce the dimension of the problem in order to lower the enormous computational complexity of for instance sampling methods in high dimensions. In particular, a new type of Gramian-based MOR approach is proposed in this paper that can be used in very general settings. The considered Gramians are constructed to identify dominant subspaces of the stochastic system as pointed out in this work. Moreover, they can be computed via Lyapunov equations. However, covariance information of the underlying systems enters these equations which is not directly available. Therefore, efficient sampling based methods relying on variance reduction techniques are established to derive the required covariances and hence the Gramians. Alternatively, an ansatz to compute the Gramians by deterministic approximations of covariance functions is investigated. An error bound for the studied MOR methods is proved yielding an a-priori criterion for the choice of the reduced system dimension. This bound is new and beneficial even in the deterministic case. The paper is concluded by numerical experiments showing the efficiently of the proposed MOR schemes.
翻译:本文审议了代表空间离散的随机部分差异方程式的大规模线性线性随机系统。 由于在这种随机环境中通常无法确保无症状稳定性(例如,由于噪音较大),主要重点是建立适用于不稳定系统的减少订单(MOR)示范办法。 摩尔对于减少问题的规模至关重要,以便降低问题的复杂性,例如高维取样方法的巨大计算复杂性。 特别是,本文件提出了一种新的基于格莱米的摩尔方法,可以在非常普遍的环境下使用。 被认为的格拉米安方法的构建是为了确定本文所指出的随机系统的主要亚空间。 此外,它们可以通过Lyapunov方程式计算出适用于不稳定系统的减少订单(MOR)示范办法。 但是,基础系统的变量信息进入这些无法直接获得的方程式。 因此,基于差异减少技术的有效取样方法已经建立起来,以获得所需的 Coverity 和 Gramian 。 或者, 一种甚至用确定性误差的Gramian 构建的Gramian 的模型, 用于通过确定性误差的精确性实验方法来测量Gramians 。 这个确定性误差度的模型是测试结果性数值的新的测试。 的确定性精确度标准 。 。 测试的模型是测试的确定性误差值的测试 。 检验性 。 检验性 测试 的系统 的数值的精确性精确性精确性精确性 的精确性精确性 的精确性 的精确性 的精确性 的精确性 的精确性 。