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

Abstract

A new method is proposed for efficient optimization under uncertainty that addresses the curse of dimensionality as it pertains to the evaluation of probabilistic objectives and constraints. A basis adaptation strategy previously introduced by the authors is integrated into a design optimization framework that construes the optimization cost function as the quantity of interest and computes stochastic adapted bases as functions of design space parameters. With these adapted bases, the stochastic integrations at each design point are evaluated as low-dimensional integrals (mostly one dimensional). The proposed approach is demonstrated on a well-placement problem where the uncertainty is in the form of a stochastic process describing the permeability of the subsurface. An analysis of the method is carried out to better understand the effect of design parameters on the smoothness of the adaptation isometry.

Optimal Well-Placement Using Probabilistic Learning

“…A polynomial chaos (PCE) surrogate [14] is also constructed using several eclipse runs. A basis adaptation procedure is implemented to accelerate the convergence while addressing the curse of dimensionality [26]. The purpose of the PCE surrogate is to enable a thorough convergence analysis of the optimization procedure presented in the present paper.…”

“…The purpose of the PCE surrogate is to enable a thorough convergence analysis of the optimization procedure presented in the present paper. Convergence of this surrogate and its properties are presented elsewhere [26].…”

“…It should be noted that the objective of this paper is not to present a stochastic analysis of the physical problem under consideration (such an analysis can be found in [26]), but to use the generated database for demonstrating that the novel method presented is efficient for solving the probabilistic optimization problem by significantly reducing the number of calls to the computational physical model. The reference solution, w …”

“…Some of these include a retrospective framework [27] where the intrinsic structure of genetic algorithms is leveraged to maximize the information gleaned from each expensive numerical calculation, a polynomial chaos methodology [3,26] which develops adapted surrogates to expedite the optimization task, specialized workflows to reuse expensive function evaluations in a sensible manner [19], and statistical proxies [2,25]. The present paper extends these statistical proxies, adapting them to intrinsic structure hidden within the data, thus allowing us to maximize the value of information being inferred form this data.…”

Gradient-Informed Basis Adaptation for Legendre Chaos Expansions

“…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.…”