Sensitivity analysis techniques applied to a system of hyperbolic conservation laws

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Abstract

Sensitivity analysis is comprised of techniques to quantify the effects of the input variables on a set of outputs. In particular, sensitivity indices can be used to infer which input parameters most significantly affect the results of a computational model. With continually increasing computing power, sensitivity analysis has become an important technique by which to understand the behavior of large-scale computer simulations. Many sensitivity analysis methods rely on sampling from distributions of the inputs. Such sampling-based methods can be computationally expensive, requiring many evaluations of the simulation; in this case, the Sobol' method provides an easy and accurate way to compute variance-based measures, provided a sufficient number of model evaluations are available. As an alternative, meta-modeling approaches have been devised to approximate the response surface and estimate various measures of sensitivity. In this work, we consider a variety of sensitivity analysis methods, including different sampling strategies, different meta-models, and different ways of evaluating variance-based sensitivity indices. The problem we consider is the 1-D Riemann problem. By a careful choice of inputs, discontinuous solutions are obtained, leading to discontinuous response surfaces; such surfaces can be particularly problematic for meta-modeling approaches. The goal of this study is to compare the estimated sensitivity indices with exact values and to evaluate the convergence of these estimates with increasing samples sizes and under an increasing number of meta-model evaluations.

Highlights

► Sensitivity analysis techniques for a model shock physics problem are compared. ► The model problem and the sensitivity analysis problem have exact solutions. ► Subtle details of the method for computing sensitivity indices can affect the results.

Introduction

Sensitivity analysis is a broad field, with many methods used for many different applications. Here we consider variance-based sensitivity indices developed by Sobol' [1], [2], which express relative sensitivities as the fraction of the variance of a model output that can be attributed to each uncertain input. These indices are used, for example, to identify the most influential inputs; with this information, limited resources can be focused on reducing the uncertainty of the most influential inputs so the variance of the outputs can be reduced by the greatest amount. Another use is to identify unimportant inputs to fix (hold constant) in a subsequent uncertainty quantification effort; fixing unimportant inputs reduces the size and complexity of the uncertainty quantification problem.

In this paper we examine several sensitivity analysis approaches. We consider Latin hypercube sampling (LHS) and Sobol' sequences (a type of quasi-Monte Carlo (QMC) sequence) for sampling the input hypercube. We consider two approaches to meta-modeling, which aims to reduce the number of computationally expensive function evaluations, including several state-of-the-art regression based meta-models and non-intrusive polynomial chaos expansion (PCE), a stochastic expansion method. Section 2 describes these techniques and the computation of variance-based sensitivity indices.

In Section 3 we apply these techniques to a canonical shock physics problem. The key features of such problems are that the solutions to the governing equations are often discontinuous, even when the initial conditions are smooth; the character of the governing nonlinear partial differential equations is hyperbolic, with waves propagating in various directions at finite speeds and interacting with other waves; and most solutions are time-dependent. Numerical simulations provide approximate solutions to the governing equations. The numerical solutions, in turn, provide the outputs (responses) for the sensitivity analysis, so our simulation code is referred to as the simulation model. Particular care was taken so that some outputs are discontinuous functions of the inputs, reflecting the properties of the underlying physics. An exact solution to the governing equations is also known for this shock physics problem, and provides the exact model, which can be used to generate alternative output values for the sensitivity analysis.

This shock physics problem is very familiar to those in the field, and a working knowledge of the sensitivities of the problem is generally known. Consequently, this problem was chosen as a test problem on which the various sensitivity analysis techniques could be tested. Since simulations of our test problem are not costly, an exact solution to the sensitivity analysis itself can be obtained by full factorial sampling. This provides an unambiguous metric against which the sensitivity analysis techniques can be compared. Our initial results are presented in Section 4. Our ultimate goal, in the context of discontinuous responses, is to examine the performance of the sensitivity analysis techniques in a rigorous fashion. We hope to determine, for example, the accuracy of the Sobol' indices as a function of the sample size; the accuracy as a function of the PCE order; the accuracy as a function of the number of samples to build a particular meta-model; and what, if anything, can be learned about the response function when various meta-models yield different results.

Section snippets

Sensitivity analysis

For a given model with a number of inputs and outputs, sensitivity analysis identifies the inputs that have the greatest influence on each output. In global SA methods including those considered here, this is achieved by repeatedly sampling input values from their distributions, evaluating the model with these values, and measuring the outputs.

The Riemann problem

The shock physics problem we consider is the Riemann problem for ideal gases. The governing equations are the one-dimensional inviscid Euler equations, which express the conservation of mass, momentum, and energy ρt+(ρu)x=0,(ρu)t+(ρu2+p)x=0,(ρE)t+[(ρE+p)u]x=0,where ρ is the density, u is the x-component of velocity, E=e+u2/2 is the total energy per unit mass, p is the pressure, and e is the internal energy per unit mass, also called the specific internal energy (SIE). The Euler

Results and discussion

We performed a number of sensitivity analyses using the methods described in Section 2. The sensitivity analyses were conducted with the software packages DAKOTA [34] and SimLab [39] and other MATLAB routines developed at the Joint Research Center by some of the authors. The analyses are summarized in Table 2. Each “Name” labels a sensitivity analysis with a label that includes the number of function evaluations for that analysis. The “Method Parameter” has different information depending on

Conclusions

We have compared several different sensitivity analysis approaches for a canonical shock physics problem. We examine variance-based Sobol' sensitivity indices produced by these approaches to learn how well they perform as a function of the sample size and how accurate they are for discontinuous response surfaces. Our simulation model provides approximate numerical solutions to this problem, and can be executed quickly enough to generate as many function evaluations as needed. This allows us to

Acknowledgments

Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy's National Nuclear Security Administration under Contract DE-AC04-94AL85000.

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