Mori-Zwanzig reduced models for uncertainty quantification
Mori-Zwanzig reduced models for uncertainty quantification
复制标题
用于不确定性量化的 Mori-Zwanzig 简化模型
DOI：
10.3934/jcd.2019002
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
P. Stinis
中科院分区：
文献类型：
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作者：
Jing Li;P. Stinis
在许多与时间相关的实际问题中，进入描述各种量的演化的方程的参数和/或初始条件表现出不确定性。解决这种不确定性如何影响解决方案的问题的一种方法是使用多项式混沌展开来扩展解决方案，并获得用于展开系数演变的微分方程系统。我们提出了一个应用程序的Mori-Zwanzig（MZ）形式主义的问题，构建减少模型的微分方程系统。特别是，我们构造减少模型的多项式混沌展开系数的一个子集，需要一个完整的描述不确定性所造成的不确定性参数或初始条件。
即使MZ形式主义是准确的，它的直接应用的问题，构建简化模型估计不确定性涉及内存项的计算，其成本可能变得昂贵得令人望而却步。对于这些情况下，我们提出了一个马尔可夫重新制定的MZ形式主义，这可以导致近似，可以减轻一些计算费用，同时保持精度优势减少模型，完全放弃内存。我们的研究结果支持的结论，成功的简化模型需要包括记忆效应。
In many time-dependent problems of practical interest the parameters and/or initial conditions entering the equations describing the evolution of the various quantities exhibit uncertainty. One way to address the problem of how this uncertainty impacts the solution is to expand the solution using polynomial chaos expansions and obtain a system of differential equations for the evolution of the expansion coefficients. We present an application of the Mori-Zwanzig (MZ) formalism to the problem of constructing reduced models of such systems of differential equations. In particular, we construct reduced models for a subset of the polynomial chaos expansion coefficients that are needed for a full description of the uncertainty caused by uncertain parameters or initial conditions.
Even though the MZ formalism is exact, its straightforward application to the problem of constructing reduced models for estimating uncertainty involves the computation of memory terms whose cost can become prohibitively expensive. For those cases, we present a Markovian reformulation of the MZ formalism which can lead to approximations that can alleviate some of the computational expense while retaining an accuracy advantage over reduced models that discard the memory altogether. Our results support the conclusion that successful reduced models need to include memory effects.