Inverse regression-based uncertainty quantification algorithms for high-dimensional models: Theory and practice

In decision‐making for groundwater management and contamination remediation, it is important to accurately evaluate the probability of the occurrence of a failure event. For small failure probability analysis, a large number of model evaluations are needed in the Monte Carlo (MC) simulation, which is impractical for CPU‐demanding models. One approach to alleviate the computational cost caused by the model evaluations is to construct a computationally inexpensive surrogate model instead. However, using a surrogate approximation can cause an extra error in the failure probability analysis. Moreover, constructing accurate surrogates is challenging for high‐dimensional models, i.e., models containing many uncertain input parameters. To address these issues, we propose an efficient two‐stage MC approach for small failure probability analysis in high‐dimensional groundwater contaminant transport modeling. In the first stage, a low‐dimensional representation of the original high‐dimensional model is sought with Karhunen‐Loève expansion and sliced inverse regression jointly, which allows for the easy construction of a surrogate with polynomial chaos expansion. Then a surrogate‐based MC simulation is implemented. In the second stage, the small number of samples that are close to the failure boundary are re‐evaluated with the original model, which corrects the bias introduced by the surrogate approximation. The proposed approach is tested with a numerical case study and is shown to be 100 times faster than the traditional MC approach in achieving the same level of estimation accuracy.

show abstract

“…The convergence of SIR‐based uncertainty quantification methods for high‐dimensional problems has been analyzed in Li et al . [].…”

Section: Methodsmentioning

confidence: 99%

“…A simple yet efficient technique to detect the central subspace is the sliced inverse regression (SIR) [ Li , ]. In a recent work, the SIR is used to enhance the computational efficiency of uncertainty quantification algorithms for high‐dimensional problems [ Li et al ., ].…”

Section: Introductionmentioning

confidence: 99%

Efficient evaluation of small failure probability in high‐dimensional groundwater contaminant transport modeling via a two‐stage Monte Carlo method

Zhang

Lin

et al. 2017

Water Resources Research

Self Cite

View full text Add to dashboard Cite

show abstract

“…To simplify the model u(ξ), an effective modeling strategy is to assume that only a few subspaces make major contributions to u. A formal definition tailored from [28] in [30] is as follows: Definition: Given the d-dimensional model u(ξ), a dimension reduction is a mapping from the d-dimensional input to ad-dimensional vector, η = Aξ, where A ∈ Rd ×d ,d < d, AA T = I is Algorithm 2 Sliced inverse regression algorithm. 1: Generate i.i.d.…”

Section: Sliced Inverse Regressionmentioning

confidence: 99%

Sliced-Inverse-Regression--Aided Rotated Compressive Sensing Method for Uncertainty Quantification

Yang

Tartakovsky

2018

SIAM/ASA J. Uncertainty Quantification

Self Cite

View full text Add to dashboard Cite

Compressive-sensing-based uncertainty quantification methods have become a powerful tool for problems with limited data. In this work, we use the sliced inverse regression (SIR) method to provide an initial guess for the alternating direction method, which is used to enhance sparsity of the Hermite polynomial expansion of stochastic quantity of interest. The sparsity improvement increases both the efficiency and accuracy of the compressive-sensingbased uncertainty quantification method. We demonstrate that the initial guess from SIR is suitable for cases when the available data are limited (Algorithm 4). We also propose another algorithm (Algorithm 5) that performs dimension reduction first with SIR. Then it constructs a Hermite polynomial expansion of the reduced model. This method affords the ability to approximate the statistics accurately with even less available data. Both methods are non-intrusive and require no a priori information of the sparsity of the system. The effectiveness of these two methods (Algorithms 4 and 5) are demonstrated using problems with up to 500 random dimensions.

show abstract

“…These techniques seek a low-dimensional subspace in the predictor space that is sufficient to statistically characterize the relationship between predictors and response. SDR methods are gaining interest for parameter reduction in computational science (Li et al, 2016;Zhang et al, 2017;Pan and Dias, 2017). The first use we know applies OLS, SIR, and pHd to a contaminant transport model from Los Alamos National Labs, where the results revealed simple, exploitable relationships between transformed inputs and the computational model's output (Cook, 1994b, Section 4).…”

Section: Introduction and Related Literaturementioning

confidence: 99%

Inverse regression for ridge recovery: a data-driven approach for parameter reduction in computer experiments

2019

View full text Add to dashboard Cite

Parameter reduction can enable otherwise infeasible design and uncertainty studies with modern computational science models that contain several input parameters. In statistical regression, techniques for sufficient dimension reduction (SDR) use data to reduce the predictor dimension of a regression problem. A computational scientist hoping to use SDR for parameter reduction encounters a problem: a computer prediction is best represented by a deterministic function of the inputs, so data comprised of computer simulation queries fail to satisfy the SDR assumptions. To address this problem, we interpret SDR methods sliced inverse regression (SIR) and sliced average variance estimation (SAVE) as estimating the directions of a ridge function, which is a composition of a low-dimensional linear transformation with a nonlinear function. of integrals whose column spaces are contained in the ridge directions' span; we analyze and numerically verify convergence of these column spaces as the number of computer model queries increases. Moreover, we show example functions that are not ridge functions but whose inverse conditional moment matrices are lowrank. Consequently, the computational scientist should beware when using SIR and SAVE for parameter reduction, since SIR and SAVE may mistakenly suggest that truly important directions are unimportant.

show abstract

Inverse regression-based uncertainty quantification algorithms for high-dimensional models: Theory and practice

Cited by 47 publications

References 36 publications

Efficient evaluation of small failure probability in high‐dimensional groundwater contaminant transport modeling via a two‐stage Monte Carlo method

Efficient evaluation of small failure probability in high‐dimensional groundwater contaminant transport modeling via a two‐stage Monte Carlo method

Sliced-Inverse-Regression--Aided Rotated Compressive Sensing Method for Uncertainty Quantification

Inverse regression for ridge recovery: a data-driven approach for parameter reduction in computer experiments

Contact Info

Product

Resources

About