A Measure-Theoretic Computational Method for Inverse Sensitivity Problems I: Method and Analysis

Breidt, J.; Butler, Troy; Estep, Donald

doi:10.1137/100785946

Cited by 36 publications

(79 citation statements)

References 14 publications

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“…We follow [60,7] to formulate the forward sensitivity analysis problem for deterministic models with uncertain input parameters using the law of total probability. Let σ Λ (λ) denote a density on Λ and ρ D (q(λ)) = ρ D (q(u(λ))) denote a density on D. Both σ Λ (λ) and ρ D (q(λ)) define probability measures P Λ (λ) and P D (q), respectively.…”

Section: Forward Propagation Of Uncertainty and Errorsmentioning

confidence: 99%

Propagation of Uncertainties Using Improved Surrogate Models

Butler

Dawson

Wildey

2013

SIAM/ASA J. Uncertainty Quantification

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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

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Section: Forward Propagation Of Uncertainty and Errorsmentioning

confidence: 99%

Propagation of Uncertainties Using Improved Surrogate Models

Butler

Dawson

Wildey

2013

SIAM/ASA J. Uncertainty Quantification

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“…In Briedt et al (2011), we formulate this discussion in mathematical terms and then describe an approximate solution method for the stochastic inverse problem that provides the means to approximate the probability of an arbitrary event in the parameter domain. We also show that the approximate solution method converges.…”

Section: Introductionmentioning

confidence: 99%

“…In Briedt et al (2011), we develop a method for computing approximate probability densities that does not require random sampling. Our approach has two stages:…”

Section: Introductionmentioning

confidence: 99%

“… We review recent work (Briedt et al, 2011., 2012) on a new approach to the formulation and solution of the stochastic inverse parameter determination problem, i.e. determine the random variation of input parameters to a map that matches specified random variation in the output of the map, and then apply the various aspects of this method to the interesting Brusselator model.…”

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A numerical method for solving a stochastic inverse problem for parameters

Butler

Estep

2013

Annals of Nuclear Energy

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We review recent work (Briedt et al., 2011., 2012) on a new approach to the formulation and solution of the stochastic inverse parameter determination problem, i.e. determine the random variation of input parameters to a map that matches specified random variation in the output of the map, and then apply the various aspects of this method to the interesting Brusselator model. In this approach, the problem is formulated as an inverse problem for an integral equation using the Law of Total Probability. The solution method employs two steps: (1) we construct a systematic method for approximating set-valued inverse solutions and (2) we construct a computational approach to compute a measure-theoretic approximation of the probability measure on the input space imparted by the approximate set-valued inverse that solves the inverse problem. In addition to convergence analysis, we carry out an a posteriori error analysis on the computed probability distribution that takes into account all sources of stochastic and deterministic error.

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“…This method has been successfully extended to estimate numerical errors due to operator splittings [11] and operator decomposition for multiscale/multiphysics applications [6,15,16]. adaptive sampling algorithms [13,14], stochastic approximations [20,4], and inverse sensitivity analysis [3,5].…”

Section: Introductionmentioning

confidence: 99%

Adjoint based a posteriori error estimation in Drekar::CFD.

Wildey¹,

Cyr²,

Pawlowski³

et al. 2012

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This document summarizes the results from a level 3 milestone study within the CASL VUQ effort. It demonstrates the capability to perform adjoint-based a posteriori error estimation within Drekar::CFD. The a posteriori error estimates require the solution of a (linear) adjoint problem in a higher order approximation space with data chosen based on the quantity of interest. The a posteriori error estimates are verified using an analytical solution to a laminar flow problem. Finally, the framework is demonstrated on the TH-M Test Case #2 involving a 3-dimensional axisymmetric sudden expansion with a moderate Reynolds numbers ( 5000).3 Acknowledgment

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A Measure-Theoretic Computational Method for Inverse Sensitivity Problems I: Method and Analysis

Cited by 36 publications

References 14 publications

Propagation of Uncertainties Using Improved Surrogate Models

Propagation of Uncertainties Using Improved Surrogate Models

A numerical method for solving a stochastic inverse problem for parameters

Adjoint based a posteriori error estimation in Drekar::CFD.

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