Probabilistic Interpretation of Conjugate Gradient Iterations in Spectral Stochastic Finite Element Method

摘要

Some of the properties of the CG in a deterministic setting are found to have their stochastic counterparts. For instance, conjugacy of the search directions and monotonic nonincreasing property of the error also hold in the product space, with the usual inner product and norm, whereas the orthogonality of the residual does not hold in the usual inner product in the product space. However, here the orthogonality holds with a different norm that does not involve the expectation operator. These observations imply that, although there is a strong connection between deterministic CG iterations in SSFEM and the iterates in the product space, all the properties in the deterministic setting are not directly extendible to the stochastic case. These observations also open up the possibility to develop OR methods for solving SSFEM problems.