Propagation of Uncertainties Using Improved Surrogate Models

Abstract

Abstract.We study the effect of various sources of error on the propagation of uncertain parameters and data through surrogate response surfaces approximating quantities of interest from stochastic differential equations. The main result centers on a novel approach for improving the pointwise accuracy of a surrogate with the use of an adjoint-based a posteriori estimate of its error. A general error analysis on propagated distribution functions for both forward and inverse problems is derived. To provide concrete examples focusing on the use of the improved surrogate, we consider standard polynomial spectral methods to approximate the surrogate. However, neither the definition of the improved surrogate nor the general error analysis for the computed distribution functions requires a specific surrogate formulation. Numerical results comparing pointwise errors in propagated distributions using a surrogate versus its improved counterpart demonstrate global decreases in both actual error and in error bounds for both forward and inverse problems. 1. Introduction. There is considerable interest in developing efficient and accurate methods to quantify the uncertainty in computational differential equation models [48,47,40]. Often, a two stage approach for solving this problem is formulated. First, a large number of samples of model parameters or input data are determined in terms of realizations of a stochastic process. Second, the probability distribution is approximately propagated through the computational model to the output or observable data.The first stage involves a priori knowledge and is often a modeling decision of the user, e.g., choosing to model a parameter as a random process with some specified distribution. The majority of the computational burden is in the second stage, involving the propagation of inputs to outputs. A simple approach is Monte Carlo simulation [43,57], where the model is solved for each randomly generated input sample resulting in samples of the output distribution. From these output samples, statistics or density estimates on specific quantities of interest may be computed. While the implementation is straightforward, this method can become computationally prohibitive for complex models where a large number of runs of the

Enhancing piecewise‐defined surrogate response surfaces with adjoints on sets of unstructured samples to solve stochastic inverse problems

“…The interested reader should refer to other works for more information on the theory and implementation of adjoint-based a posteriori error estimates in general [22][23][24][25][26][27] and on the application to certain classes of surrogate models. [14][15][16]20,28 For a more thorough introduction to the theory and application of adjoints in general, the reader is also referred to other recommended works. [35][36][37][38]…”

“…There is also interesting new research on using dimension reduction techniques and reduced‐order models for building surrogates . The surrogate modeling approach considered in this work most closely resembles techniques that exploit derivative information or error estimates using adjoints for building piecewise low‐order surrogate approximations to improve pointwise accuracy in propagations of uncertainties …”

“…The surrogate response surface is polluted by two sources of error affecting the accuracy of all samples: (1) the approximation error of the choice of surrogate and (2) the numerical error in evaluation of the samples used to construct the surrogate. We refer to both these types of error as discretization errors.…”

“…19 The surrogate modeling approach considered in this work most closely resembles techniques that exploit derivative information or error estimates using adjoints for building piecewise low-order surrogate approximations to improve pointwise accuracy in propagations of uncertainties. 13,20,21 The surrogate response surface is polluted by two sources of error affecting the accuracy of all samples 15,16,20 : (1) the approximation error of the choice of surrogate and (2) the numerical error in evaluation of the samples used to construct the surrogate. We refer to both these types of error as discretization errors.…”

Adjoint Error Estimation for a Pseudo‐Spectral Approach to Stochastic Field‐Circuit Coupled Problems

“…As error estimators relying on higher order solutions might be too expensive to obtain, we propose to use the adjoint DAE instead. This has been previously discussed in the context of parametrized linear systems [2] and partial differential equations [3].…”