The discretization of robust quadratic optimal control problems under uncertainty using the finite element method and the stochastic collocation method leads to large saddle-point systems, which are fully coupled across the random realizations. Despite its relevance for numerous engineering problems, the solution of such systems is notoriusly challenging. In this manuscript, we study efficient preconditioners for all-at-once approaches using both an algebraic and an operator preconditioning framework. We show in particular that for values of the regularization parameter not too small, the saddle-point system can be efficiently solved by preconditioning in parallel all the state and adjoint equations. For small values of the regularization parameter, robustness can be recovered by the additional solution of a small linear system, which however couples all realizations. A mean approximation and a Chebyshev semi-iterative method are investigated to solve this reduced system. Our analysis considers a random elliptic partial differential equation whose diffusion coefficient $\kappa(x,\omega)$ is modeled as an almost surely continuous and positive random field, though not necessarily uniformly bounded and coercive. We further provide estimates on the dependence of the preconditioned system on the variance of the random field. Such estimates involve either the first or second moment of the random variables $1/\min_{x\in \overline{D}} \kappa(x,\omega)$ and $\max_{x\in \overline{D}}\kappa(x,\omega)$, where $D$ is the spatial domain. The theoretical results are confirmed by numerical experiments, and implementation details are further addressed.
翻译:使用有限元素法和随机实现方法,在不确定情况下,强力二次最佳控制问题的离散性,使用有限元素法和随机同位法,导致大型马鞍点系统,这些系统在随机实现中完全结合。尽管这些系统对于许多工程问题具有相关性,但解决这些系统的方法具有明显的挑战性。在这份手稿中,我们研究使用代数框架和操作者先决条件框架的全方位方法的有效先决条件。我们特别表明,对于正规化参数值的值而言,马鞍点系统可以通过与所有状态方程和配套方程式平行的先决条件来有效解决。对于正规化参数的小型值来说,可以通过一个小线性系统的额外解决方案来恢复稳健性,而所有这些系统是同时实现的。一个平均近似和切比谢夫半化方法被调查,以解决这个被缩小的系统。我们的分析认为一个随机的椭脂部分差异方方程方程式,其传播系数$kppa(x, omegaga) 是一个几乎连续和积极的随机的字段,尽管不一定统一和胁迫性方程式。我们进一步提供了对一个随机性机序-x的域的数值的数值值的数值值值值值值值值值估算。