An A Posteriori–A Priori Analysis of Multiscale Operator Splitting

Estep, D.; Ginting, Victor; Ropp, David L.; Shadid, John N.; Tavener, Simon

doi:10.1137/07068237x

Cited by 49 publications

(29 citation statements)

References 28 publications

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“…This case, called additive splitting, generally corresponds to problems in which the physics components act in the same spatial domain, such as reactive transport or radiation-hydrodynamics (Giraldo et al, 2003;Estep et al, 2008a;Giraldo and Restelli, 2009;Durran and Blossey, 2012;Giraldo et al, 2010;Ruuth, 1995). When a partitioned time integration method is used with physics components defined on the same spatial region, some work must be done to synchronize the timestepping in each physics component.…”

Section: Methods For Coupling Multiphysics Components In Timementioning

confidence: 99%

Multiphysics simulations: challenges and opportunities.

Keyes¹,

McInnes²,

Woodward³

et al. 2012

149

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We consider multiphysics applications from algorithmic and architectural perspectives, where "algorithmic" includes both mathematical analysis and computational complexity and "architectural" includes both software and hardware environments. Many diverse multiphysics applications can be reduced, en route to their computational simulation, to a common algebraic coupling paradigm. Mathematical analysis of multiphysics coupling in this form is not always practical for realistic applications, but model problems representative of applications discussed herein can provide insight. A variety of software frameworks for multiphysics applications have been constructed and refined within disciplinary communities and executed on leading-edge computer systems. We examine several of these, expose some commonalities among them, and attempt to extrapolate best practices to future systems. From our study, we summarize challenges and forecast opportunities.

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Section: Methods For Coupling Multiphysics Components In Timementioning

confidence: 99%

Multiphysics simulations: challenges and opportunities.

Keyes¹,

McInnes²,

Woodward³

et al. 2012

149

View full text Add to dashboard Cite

show abstract

“…In practice, we do not have access to g(u, U ) = 1 0 ∂ u g(x, t; su + (1 − s)U ) ds and must linearize around U , so even if the adjoint problem is solved exactly, (4.5) represents a computable approximation to the deterministic error. However, the effect of this linearization error on the estimate can be analyzed (see, e.g., [24]) and is generally not significant.…”

Section: Computable Estimate Ofmentioning

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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“…While many a posteriori techniques focus on estimating the error in a particular norm, such as the L 2 or the energy norm, the adjoint-based (dual-weighted residual) method, is motivated by the observation that oftentimes the goal of a simulation is to compute a small number of linear functionals of the solution, such as the average value in a region or the drag on an object, rather than controlling the error in a global norm. This method has been successfully utilized to estimate the error in quantities of interest for a class of transient nonlinear problems and has recently been extended to estimate numerical errors due to operator splittings [8] and operator decomposition for multiscale/multiphysics applications [3,10,11]. The a posteriori error estimation approach does not require as many simulations to infer the asymptotic values of the quantity of interest and the convergence rate, but it is also subject to accuracy concerns if the simulation is not near the asymptotic regime.…”

Section: Motivationmentioning

confidence: 99%

A comparison of adjoint and data-centric verification techniques.

Wildey¹,

Cyr²,

Shadid³

et al. 2013

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This document summarizes the results from a level 3 milestone study within the CASL VUQ effort. We compare the adjoint-based a posteriori error estimation approach with a recent variant of a data-centric verification technique. We provide a brief overview of each technique and then we discuss their relative advantages and disadvantages. We use Drekar::CFD to produce numerical results for steady-state Navier Stokes and SARANS approximations.

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An A Posteriori–A Priori Analysis of Multiscale Operator Splitting

Cited by 49 publications

References 28 publications

Multiphysics simulations: challenges and opportunities.

Multiphysics simulations: challenges and opportunities.

Propagation of Uncertainties Using Improved Surrogate Models

A comparison of adjoint and data-centric verification techniques.

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