耦合问题降维随机建模中随机系数的约化混沌展开
Reduced chaos expansions with random coefficients in reduced-dimensional stochastic modeling of coupled problems
摘要：
在科学和工程的许多领域都可以找到具有多种物理、尺度和领域的各种组合的耦合问题。在制定和实施相应的耦合模型的一个关键挑战是，以促进跨物理，规模和域接口，以及用于响应计算的求解器的迭代之间的信息通信。在概率上下文中，任何要在子问题或迭代之间传递的信息都应该由适当的概率表示来表征。在这项工作中，我们考虑随机耦合问题，其子问题只涉及不确定性，是统计上相互独立的，对于这些问题，我们提出了一个表征的交换信息，通过使用减少混沌扩展随机系数。这种扩展提供了一个降维表示的交换信息，同时保持隔离的不确定性来源，源于不同的子问题。此外，我们提出了一个措施的转换，允许随机扩展方法，利用这种降维，以获得一个有效解的子问题在降维空间。我们表明，由于不确定性源隔离，所需的正交多项式和求积规则可以很容易地得到张量化。最后，所提出的方法证明了它在核工程中的多物理场问题。
英文摘要：
Coupled problems with various combinations of multiple physics, scales, and domains can be found in numerous areas of science and engineering. A key challenge in the formulation and implementation of corresponding coupled models is to facilitate communication of information across physics, scale, and domain interfaces, as well as between iterations of solvers used for response computations. In a probabilistic context, any information that is to be communicated between subproblems or iterations should be characterized by an appropriate probabilistic representation. In this work, we consider stochastic coupled problems whose subproblems involve only uncertainties that are statistically independent of one another; for these problems, we present a characterization of the exchanged information by using a reduced chaos expansion with random coefficients. This expansion provides a reduced-dimensional representation of the exchanged information, while maintaining segregation between sources of uncertainty that stem from different subproblems. Further, we present a measure transformation that allows stochastic expansion methods to exploit this dimension reduction to obtain an efficient solution of subproblems in a reduced-dimensional space. We show that owing to the uncertainty source segregation, requisite orthonormal polynomials and quadrature rules can be readily obtained by tensorization. Finally, the proposed methodology is demonstrated by applying it to a multiphysics problem in nuclear engineering.
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