Probabilistic models for stochastic elliptic partial differential equations
Probabilistic models for stochastic elliptic partial differential equations
复制标题
随机椭圆偏微分方程的概率模型
DOI：
10.1016/j.jcp.2010.07.023
复制
发表时间：
2010
期刊：
影响因子：
--
通讯作者：
M. Grigoriu
中科院分区：
文献类型：
--
作者：
M. Grigoriu
随机椭圆偏微分方程的随机系数必须满足才能具有唯一解的数学要求已被广泛研究。然而，这些系数必须满足才能提供物理量的实际表示的附加约束（称为物理要求）尚未得到系统的研究。结果表明，当前仅通过数学考虑构建的随机系数模型可能违反物理约束，因此实际用途有限。我们为随机微分方程的随机系数开发了满足数学和物理约束的替代模型。提出的理论论证表明了当前模型的潜在局限性并确定了本研究中开发的模型的属性。数值例子用于说明所提出模型的构造，评估这些模型的性能，并证明随机微分方程解对其随机系数的概率特征的敏感性。
Mathematical requirements that the random coefficients of stochastic elliptical partial differential equations must satisfy such that they have unique solutions have been studied extensively. Yet, additional constraints that these coefficients must satisfy to provide realistic representations for physical quantities, referred to as physical requirements, have not been examined systematically. It is shown that current models for random coefficients constructed solely by mathematical considerations can violate physical constraints and, consequently, be of limited practical use. We develop alternative models for the random coefficients of stochastic differential equations that satisfy both mathematical and physical constraints. Theoretical arguments are presented to show potential limitations of current models and establish properties of the models developed in this study. Numerical examples are used to illustrate the construction of the proposed models, assess the performance of these models, and demonstrate the sensitivity of the solutions of stochastic differential equations to probabilistic characteristics of their random coefficients.