Estimation and model errors in Gaussian-process-based Sensitivity Analysis of functional outputs

2025-12-19

AI Summary

For global sensitivity analysis (GSA) of functional-output models, existing Gaussian process (GP) metamodeling approaches fail to decouple and quantify the intertwined effects of metamodeling error and Pick-Freeze sampling estimation error. Method: This paper introduces, for the first time within a GP framework, a joint quantification of both error sources. It proposes a low-dimensional coefficient GP model based on Karhunen–Loève expansion and a multi-condition vector-valued Pick-Freeze algorithm, integrated with conditional GP trajectory sampling. Contribution/Results: The approach significantly improves computational efficiency and interpretability in estimating Sobol’ indices and generalized sensitivity indices. In benchmark applications—including dam-break simulation of non-Newtonian fluids—it achieves a 15× speedup over Le Gratiet et al.’s baseline method while enabling precise separation and estimation of individual contributions from metamodeling and sampling errors.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Global sensitivity analysis (GSA) of functional-output models is usually performed by combining statistical techniques, such as basis expansions, metamodeling and sampling based estimation of sensitivity indices. By neglecting truncation error from basis expansion, two main sources of errors propagate to the final sensitivity indices: the metamodeling related error and the sampling-based, or pick-freeze (PF), estimation error. This work provides an efficient algorithm to estimate these errors in the frame of Gaussian processes (GP), based on the approach of Le Gratiet et al. [16]. The proposed algorithm takes advantage of the fact that the number of basis coefficients of expanded model outputs is significantly smaller than output dimensions. Basis coefficients are fitted by GP models and multiple conditional GP trajectories are sampled. Then, vector-valued PF estimation is used to speed-up the estimation of Sobol indices and generalized sensitivity indices (GSI). We illustrate the methodology on an analytical test case and on an application in non-Newtonian hydraulics, modelling an idealized dam-break flow. Numerical tests show an improvement of 15 times in the computational time when compared to the application of Le Gratiet et al. [16] algorithm separately over each output dimension.

Problem

• Estimates errors in Gaussian-process-based sensitivity analysis of functional outputs
• Improves computational efficiency for Sobol and generalized sensitivity indices
• Applies methodology to analytical and non-Newtonian hydraulics test cases

Innovation

• GP models fit basis coefficients for efficiency
• Multiple conditional GP trajectories sample error propagation
• Vector-valued pick-freeze estimation accelerates Sobol indices calculation