Surrogate Models for Uncertainty Propagation and Sensitivity Analysis

Abstract

For computationally intensive tasks such as design optimization, global sensitivity analysis, or parameter estimation, a model of interest needs to be evaluated multiple times exploring potential parameter ranges or design conditions. If a single simulation of the computational model is expensive, it is common to employ a precomputed surrogate approximation instead. The construction of an appropriate surrogate does still require a number of training evaluations of the original model. Typically, more function evaluations lead to more accurate surrogates, and therefore a careful accuracy-vs-efficiency tradeoff needs to take place for a given computational task. This chapter specifically focuses on polynomial chaos surrogates that are well suited for forward uncertainty propagation tasks, discusses a few construction mechanisms for such surrogates, and demonstrates the computational gain on select test functions.

Appendix

Table 19.1 shows six classes of functions employed in the numerical tests in this chapter. All functions, except the exponential one, are taken from the classical Genz family of test functions [18]. The weight parameters of the test functions are chosen according to a predefined “decay” rate w _i = C∕i and a normalization factor C = 1∕∑ _{i = 1} ^d i ⁻¹ to ensure ∑ _{i = 1} ^d w _i = 1. The exact moments are analytically available and are used for reference to compare against, with the exception of the corner-peak function, for which the variance estimator (19.30) with sampling size M = 10⁷ is used. The “true” reference values ‘true’ reference values for sensitivity indices S _i are also computed via Monte Carlo, using the estimates (19.34) with M = 10⁵. The functions are defined on $\boldsymbol{\lambda }\in [0,1]^{d}$; assuming the inputs are i.i.d uniform random variables, the underlying linear input PC expansions are simple linear transformations λ _i = 0. 5ξ _i + 0. 5, relating the “physical” model inputs λ _i ∈ [0, 1] to the PC surrogate inputs ξ _i ∈ [−1, 1], for i = 1, …, d.

Table 19.1 Test functions used in the studies of this section. The shift parameters are set to u _i = 0. 3 for all dimensions i = 1, …, d, while the weight parameters are selected as w _i = C∕i with normalization constant C = 1∕∑ _{i = 1} ^d i ⁻¹ to ensure ∑ _{i = 1} ^d w _i = 1. The variance formula for the product-peak function is $v(\boldsymbol{u},\boldsymbol{w}) =\prod _{ i=1}^{d}w_{i}^{4}\left ( \frac{1-u_{i}} {2(1+w_{i}^{2}(1-u_{i})^{2})} + \frac{u_{i}} {2(1+w_{i}^{2}u_{i}^{2})} + \frac{1} {2w_{i}}\left (\text{arctan}\left (w_{i}(1 - u_{i})\right ) + \text{arctan}\left (w_{i}u_{i}\right )\right )\right )-m(\boldsymbol{u},\boldsymbol{w})^{2}$