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Sparse Polynomial Chaos Expansion for Universal Stochastic Kriging

José Calos García-Marino, Carmen Calvo-Jurado, Enrique García-Macías

TL;DR

This paper tackles uncertainty quantification in surrogates for stochastic simulators by introducing a universal stochastic Kriging framework that uses an adaptive sparse polynomial chaos expansion as the trend. The trend is selected automatically via least-angle regression, balancing global and local predictive power, while a genetic algorithm calibrates the model hyper-parameters. The LAR-PCE SK method demonstrates superior accuracy over ordinary SK across benchmark problems (M/M/1 queue, egg-box surface, Ishigami function) under both known and unknown intrinsic noise, and it highlights the risk of overfitting when employing a full PCE trend. The work provides a practical, flexible surrogate modelling approach with improved uncertainty quantification and guidance for selecting sparse, expressive trend terms in stochastic kriging.

Abstract

Surrogate modelling techniques have opened up new possibilities to overcome the limitations of computationally intensive numerical models in various areas of engineering and science. However, while fundamental in many engineering applications and decision-making, the incorporation of uncertainty quantification into meta-models remains a challenging open area of research. To address this issue, this paper presents a novel stochastic simulation approach combining sparse polynomial chaos expansion (PCE) and Stochastic Kriging (SK). Specifically, the proposed approach adopts adaptive sparse PCE as the trend model in SK, achieving both global and local prediction capabilities and maximizing the role of the stochastic term to conduct uncertainty quantification. To maximize the generalization and computational efficiency of the meta-model, the Least Angle Regression (LAR) algorithm is adopted to automatically select the optimal polynomial basis in the PCE. The computational effectiveness and accuracy of the proposed approach are appraised through a comprehensive set of case studies and different quality metrics. The presented numerical results and discussion demonstrate the superior performance of the proposed approach compared to the classical ordinary SK model, offering high flexibility for the characterization of both extrinsic and intrinsic uncertainty for a wide variety of problems.

Sparse Polynomial Chaos Expansion for Universal Stochastic Kriging

TL;DR

This paper tackles uncertainty quantification in surrogates for stochastic simulators by introducing a universal stochastic Kriging framework that uses an adaptive sparse polynomial chaos expansion as the trend. The trend is selected automatically via least-angle regression, balancing global and local predictive power, while a genetic algorithm calibrates the model hyper-parameters. The LAR-PCE SK method demonstrates superior accuracy over ordinary SK across benchmark problems (M/M/1 queue, egg-box surface, Ishigami function) under both known and unknown intrinsic noise, and it highlights the risk of overfitting when employing a full PCE trend. The work provides a practical, flexible surrogate modelling approach with improved uncertainty quantification and guidance for selecting sparse, expressive trend terms in stochastic kriging.

Abstract

Surrogate modelling techniques have opened up new possibilities to overcome the limitations of computationally intensive numerical models in various areas of engineering and science. However, while fundamental in many engineering applications and decision-making, the incorporation of uncertainty quantification into meta-models remains a challenging open area of research. To address this issue, this paper presents a novel stochastic simulation approach combining sparse polynomial chaos expansion (PCE) and Stochastic Kriging (SK). Specifically, the proposed approach adopts adaptive sparse PCE as the trend model in SK, achieving both global and local prediction capabilities and maximizing the role of the stochastic term to conduct uncertainty quantification. To maximize the generalization and computational efficiency of the meta-model, the Least Angle Regression (LAR) algorithm is adopted to automatically select the optimal polynomial basis in the PCE. The computational effectiveness and accuracy of the proposed approach are appraised through a comprehensive set of case studies and different quality metrics. The presented numerical results and discussion demonstrate the superior performance of the proposed approach compared to the classical ordinary SK model, offering high flexibility for the characterization of both extrinsic and intrinsic uncertainty for a wide variety of problems.
Paper Structure (13 sections, 33 equations, 8 figures, 4 tables)

This paper contains 13 sections, 33 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Surrogate modelling: Deterministic and Stochastic Kriging
  3. LAR-PCE based Stochastic Kriging
  4. Polynomial-chaos expansion
  5. Least angle regression (LAR)
  6. LAR-PCE Stochastic Kriging
  7. Numerical results and discussion
  8. Case study I: M/M/1 Queue
  9. Known intrinsic noise
  10. Unknown intrinsic noise
  11. Case study II: two dimensional example.
  12. Case study III: Ishigami function.
  13. Conclusions

Figures (8)

  • Figure 1: Flowchart of the proposed PCE SK surrogate modelling approach for stochastic simulators.
  • Figure 2: Case Study I: M/M/1 example. Comparison between the true simulator (a,d) against the predictions by the proposed LAR-PCE SK (c,f) and Ordinary SK (b,e) under the assumption of known variance (one replication per ED sample) in Scenarios 1 (a,b,c) (10 ED samples and run-length=6000) and 2 (d,e,f) (30 ED samples and run-length=2000). Scatter points denote the ED samples, and light blue shaded areas represent the 95% confidence interval.
  • Figure 3: Performance analysis of Ordinary SK and LAR-PCE SK in Case Study I in terms of ERMS and NMAE error metrics using 100 known intrinsic noise M/M/1 experiments and sampling Scenarios 1 to 3.
  • Figure 4: Performance analysis of Ordinary SK and LAR-PCE SK in Case Study I in terms of ERMS and NMAE error metrics using 100 unknown intrinsic noise M/M/1 experiments and sampling Scenarios 1 to 3.
  • Figure 5: Response surface of the Case Study II (a), and predictions by SK (b), LAR-PCE SK (c) and full PCE SK (d) surrogate models with $k=64$ design points.
  • ...and 3 more figures