Polynomial-Chaos-based Kriging (1502.03939v1)

Abstract: Computer simulation has become the standard tool in many engineering fields for designing and optimizing systems, as well as for assessing their reliability. To cope with demanding analysis such as optimization and reliability, surrogate models (a.k.a meta-models) have been increasingly investigated in the last decade. Polynomial Chaos Expansions (PCE) and Kriging are two popular non-intrusive meta-modelling techniques. PCE surrogates the computational model with a series of orthonormal polynomials in the input variables where polynomials are chosen in coherency with the probability distributions of those input variables. On the other hand, Kriging assumes that the computer model behaves as a realization of a Gaussian random process whose parameters are estimated from the available computer runs, i.e. input vectors and response values. These two techniques have been developed more or less in parallel so far with little interaction between the researchers in the two fields. In this paper, PC-Kriging is derived as a new non-intrusive meta-modeling approach combining PCE and Kriging. A sparse set of orthonormal polynomials (PCE) approximates the global behavior of the computational model whereas Kriging manages the local variability of the model output. An adaptive algorithm similar to the least angle regression algorithm determines the optimal sparse set of polynomials. PC-Kriging is validated on various benchmark analytical functions which are easy to sample for reference results. From the numerical investigations it is concluded that PC-Kriging performs better than or at least as good as the two distinct meta-modeling techniques. A larger gain in accuracy is obtained when the experimental design has a limited size, which is an asset when dealing with demanding computational models.

• The paper introduces Polynomial-Chaos-Kriging (PC-Kriging), a novel surrogate modeling approach that integrates Polynomial Chaos Expansions (PCE) and Kriging to enhance accuracy and robustness in complex simulations.
• PC-Kriging is presented in two variants, Sequential (SPC-Kriging) and Optimal (OPC-Kriging), which strategically incorporate optimized polynomial sets into the Kriging framework.
• Numerical validation demonstrates that PC-Kriging, particularly OPC-Kriging, often outperforms traditional PCE and Kriging, especially with smaller sample sizes, offering enhanced efficiency for applications like reliability analysis.

An Overview of Polynomial-Chaos-based Kriging

The development of computational simulations has revolutionized engineering, improving the design and optimization of complex systems through high-fidelity models. However, these models often present significant computational burdens, necessitating the use of surrogate models or meta-models. This paper proposes a new meta-modelling approach, Polynomial-Chaos-Kriging (PC-Kriging), which synthesizes Polynomial Chaos Expansions (PCE) and Kriging to offer a more robust and accurate surrogate method. The combination aims to leverage PCE's ability to approximate global behaviors and Kriging's capability to capture local variations.

Background and Motivation

Polynomial Chaos Expansions and Kriging have independently evolved as powerful tools for managing the computational challenges associated with high-fidelity simulations. PCE uses orthonormal polynomials aligned with the input variables' distributions, exploiting least-square minimization to estimate expansion coefficients. Kriging, on the other hand, models the output as a Gaussian process, characterized by an auto-correlation function based on the spatial relationships in the input data.

Despite their respective strengths, the paper identifies that there has been little interaction between these two disciplines. By integrating these approaches, PC-Kriging is positioned to improve the fidelity and predictive power of meta-models, particularly when experimental designs are constrained in size.

Method: Polynomial-Chaos-Kriging

PC-Kriging is derived by embedding a sparse set of orthonormal polynomials, optimized through adaptive algorithms akin to least angle regression, into the universal Kriging framework. This combination allows for capturing both the broad functional representation achieved by PCE and the nuanced, localized variations modeled by Kriging.

Two variants of PC-Kriging are explored:

1. Sequential PC-Kriging (SPC-Kriging): Comprises using PCE to determine an optimal polynomial set which is then directly incorporated into a Kriging model as a trend.
2. Optimal PC-Kriging (OPC-Kriging): Iteratively adds polynomials based on their contribution to minimizing the leave-one-out cross-validation error, optimizing both polynomial selection and Kriging parameters.

Numerical Validation

The paper validates PC-Kriging using various benchmark functions such as Ishigami, Sobol', and Rastrigin. These functions are chosen for their diverse features, including input dimensionality and inherent complexity, which challenge the robustness of surrogate models.

The empirical comparisons showcase PC-Kriging, especially OPC-Kriging, often outperforming traditional PCE and Kriging approaches, particularly when constrained by smaller sample sizes. The integration of PCE with Kriging manages to preserve low empirical error with added flexibility, confirming its potential for broader application.

Implications and Future Directions

The implications of PC-Kriging are substantial for fields where frequent reevaluation of complex models is necessary, such as reliability analysis and design optimization. By effectively harnessing both global and local modeling strengths, PC-Kriging enhances efficiency without significant sacrifices in precision.

Future research endeavors could explore adaptive enrichment strategies tailored to focus computational resources efficiently, enhancing regions of interest in the input space guided by PC-Kriging's predictive accuracy. Additionally, further applications to real-world engineering problems would help refine these models and assess practical limits.

In summary, Polynomial-Chaos-Kriging represents a promising step forward in surrogate modeling by intelligently combining the individual merits of PCE and Kriging, addressing existing gaps in their standalone implementations. This paper provides strong numerical evidence that PC-Kriging can serve as a robust multipurpose meta-modeling technique capable of addressing complex simulation challenges across varied domains.