Reduced Wiener Chaos representation of random fields via basis adaptation and projection

Abstract

A new characterization of random fields appearing in physical models is presented that is based on their well-known Homogeneous Chaos expansions. We take advantage of the adaptation capabilities of these expansions where the core idea is to rotate the basis of the underlying Gaussian Hilbert space, in order to achieve reduced functional representations that concentrate the induced probability measure in a lower dimensional subspace. For a smooth family of rotations along the domain of interest, the uncorrelated Gaussian inputs are transformed into a Gaussian process, thus introducing a mesoscale that captures intermediate characteristics of the quantity of interest.

Homogeneous chaos basis adaptation for design optimization under uncertainty: Application to the oil well placement problem

“…Then, we have that almost surely and the coefficients of the new expansion can be written in terms of as We have for |α| ≠ |β|, while for |α| = |β| the inner products can be computed explicitly (Tsilifis & Ghansem, 2016). The motivation behind applying a change of basis as described above is that, by choosing an appropriate matrix , the new expansion can concentrate the dominant effects of the original expansion to fewer coefficients of and thus provide sparse representations depending on only a few components of , say, with d 0 < d , and therefore, we can discard terms of the adapted expansion via projection (see Tipieddy & Ghanem, 2014; Tsifilis & Ghanem, 2016, for details as well as several popular choices of ). If we let be the set of d -dimensional multi-indices of order up to P and take any for some p 0 < p , d 0 < d , which implies that , then for given , the reduced decomposition approaches as p 0 → p and d 0 → d .…”

“…(11), is often a smooth function of , and therefore, we should be able to estimate it; for instance, we could evaluate 1 at several points and then apply some interpolation scheme. A brief theoretical discussion on the smoothness of the entries of can also be found in (Tsifiis & Ghanem, 2016), where it is shown that if are square integrable functions of , then the covariance kernels of η i s define Hilbert–Schmidt operators. Here, we only demonstrate that in our example this claim is true and the entries of 1 or some properties of them could be estimated.…”

Bayesian adaptation of chaos representations using variational inference and sampling on geodesics

“…The rotation in question is adapted to specific quantities of interest and provides a reduction of full polynomial expansions through a rotation of the independent variables. This formulation enables the concentration of the probabilistic content into a very low-dimensional structure that can be easily explored using low-level quadrature rules, active subspaces [25,26], 1 -minimization [27] and even analytical mappings [28], allowing the segregation and reordering of uncertainties. In this work, we attempt a Bayesian approach to the basis adaptation framework for Homogeneous (Hermite) Chaos that views both the chaos coefficients and the input rotation matrix as random variables defined on probability spaces and we employ inference techniques for characterization of their probability measures.…”

Gradient-Informed Basis Adaptation for Legendre Chaos Expansions

“…Computational schemes for efficiently computing the coefficients of the adapted expansion were also developed and the numerical results were impressive. The basis adaptation concept was further extended from scalar QoI's to random fields [46], where explicit formulas for computing the coefficients with respect to any rotation were derived and the case of a parametric family of rotations was discussed that gives rise to expansions with a Gaussian process input. Even though the idea of rotating the basis has been further applied to design optimization problems [47] and has been used to develop efficient schemes coupled with compressing sensing methods [48,49], it is still quite restricted to the Homogeneous Chaos expansions, where the Gaussian distribution remains invariant under rotations, a property that is not valid in other cases.…”