Active Learning-based Domain Adaptive Localized Polynomial Chaos Expansion

Abstract

The paper presents a novel methodology to build surrogate models of complicated functions by an active learning-based sequential decomposition of the input random space and construction of localized polynomial chaos expansions, referred to as domain adaptive localized polynomial chaos expansion (DAL-PCE). The approach utilizes sequential decomposition of the input random space into smaller sub-domains approximated by low-order polynomial expansions. This allows approximation of functions with strong nonlinearties, discontinuities, and/or singularities. Decomposition of the input random space and local approximations alleviates the Gibbs phenomenon for these types of problems and confines error to a very small vicinity near the non-linearity. The global behavior of the surrogate model is therefore significantly better than existing methods as shown in numerical examples. The whole process is driven by an active learning routine that uses the recently proposed $\Theta$ criterion to assess local variance contributions. The proposed approach balances both \emph{exploitation} of the surrogate model and \emph{exploration} of the input random space and thus leads to efficient and accurate approximation of the original mathematical model. The numerical results show the superiority of the DAL-PCE in comparison to (i) a single global polynomial chaos expansion and (ii) the recently proposed stochastic spectral embedding (SSE) method developed as an accurate surrogate model and which is based on a similar domain decomposition process. This method represents general framework upon which further extensions and refinements can be based, and which can be combined with any technique for non-intrusive polynomial chaos expansion construction.

References in corpus (20)

• Rejoinder to "Least angle regression" by Efron et al
• Least Angle Regression
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• Discussion of "Least angle regression" by Efron et al
• An active-learning algorithm that combines sparse polynomial chaos expansions and bootstrap for structural reliability analysis
• Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark
• Compressive Sampling of Polynomial Chaos Expansions: Convergence Analysis and Sampling Strategies
• Sparse Polynomial Chaos Expansions via Compressed Sensing and D-optimal Design
• A Christoffel function weighted least squares algorithm for collocation approximations
• Stochastic collocation approach with adaptive mesh refinement for parametric uncertainty analysis
• Reliability analysis of discrete-state performance functions via adaptive sequential sampling with detection of failure surfaces
• Rare event estimation using stochastic spectral embedding
• An $hp$-adaptive multi-element stochastic collocation method for surrogate modeling with information re-use
• Failure Probability Estimation and Detection of Failure Surfaces via Adaptive Sequential Decomposition of the Design Domain