IGA-based multi-index stochastic collocation for random PDEs on arbitrary domains

Abstract

This paper proposes an extension of the Multi-Index Stochastic Collocation (MISC) method for forward uncertainty quantification (UQ) problems in computational domains of shape other than a square or cube, by exploiting isogeometric analysis (IGA) techniques. Introducing IGA solvers to the MISC algorithm is very natural since they are tensor-based PDE solvers, which are precisely what is required by the MISC machinery. Moreover, the combination-technique formulation of MISC allows the straight-forward reuse of existing implementations of IGA solvers. We present numerical results to showcase the effectiveness of the proposed approach. Highlights• Isogeometric solvers used in a MISC framework for forward UQ problems.• The combination-technique formulation of the method allows straight-forward reuse of legacy IGA solvers.

Multilevel double loop Monte Carlo and stochastic collocation methods with importance sampling for Bayesian optimal experimental design

“…The difference between the MLDLSC estimator (55) and a standard MISC method is that we use a full-tensor approximation of the inner expectation, which expands the multi-index set Λ to include a multi-index set for the full-tensor approximation as described in Section 4.2. The method uses a greedy algorithm, based on a priori estimates of the error and work contribution of the multi-indices to solve a knapsack problem in which the most profitable multi-indices are sequentially included in the multi-index set Λ; see Algorithm 1 of [8]. For example, the a priori estimates of multi-index profits have been widely used for sparse-grid stochastic collocation, e.g.…”

“…The function Z depends on f , (8), and, in turn, f depends on the approximate evidence p (16). Therefore, to evaluate the approximate evidence, we resort to another approximation by combining Monte Carlo (MC) sampling with the Laplace-based importance sampling described in Section 2.3, to obtain a sample average approximate evidence,…”

“…; (28) p ,p are defined in (10), (24), respectively. Here, E [·] denotes the expectation operator under the probability measure p (Y |θ) as defined in (8). The approximate expected information gain at level when using the sample average approximate evidence,p , (24), iŝ…”

“…Then, we combine these approximations using the combination-technique formula (51) to create a more accurate approximation. Various approaches have been proposed for selecting the multi-index set Λ, such as using the classical sets given in [4] or selecting the set adaptively as discussed in [8,15,31,36].…”

Comparing multi-index stochastic collocation and multi-fidelity stochastic radial basis functions for forward uncertainty quantification of ship resistance

“…A different option is to resort to methods that build a polynomial approximation over the parameter space: methods such as Multi-Level Stochastic Collocation [14], Multi-Index Stochastic Collocation [15][16][17][18], Multi-Level Least-Squares polynomial approximation [19], etc. fall in this category.…”