Dimension reduction in stochastic modeling of coupled problems

Arnst, Maarten; Ghanem, Roger; Phipps, Eric Todd; Red-Horse, John

doi:10.1002/nme.4364

Cited by 42 publications

(59 citation statements)

References 27 publications

(67 reference statements)

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“…Previous work has explored the effect of sparsifying probabilistic graphical models where I(yi, yj yv\(i,j)) denotes the conditional mutual information for variables y, and yj [32,31]. When the variables are jointly Gaussian, the conditional mutual information is cheaply computed from the inverse covariance matrix, I'.…”

Section: B(a T)1|mentioning

confidence: 99%

Optimal Approximations of Coupling in Multidisciplinary Models

Baptista

Marzouk

Willcox

et al. 2017

58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference

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Design of complex engineering systems requires coupled analyses of the multiple disciplines affecting system performance. The coupling among disciplines typically contributes significantly to the computational cost of analyzing a system, and can become particularly burdensome when coupled analyses are embedded within a design or optimization loop. In many cases, disciplines may be weakly coupled, so that some of the coupling or interaction terms can be neglected without significantly impacting the accuracy of the system output. However, typical practice derives such approximations in an ad hoc manner using expert opinion and domain experience. In this thesis, we propose a new approach that formulates an optimization problem to find a model that optimally balances accuracy of the model outputs with the sparsity of the discipline couplings. An adaptive sequential Monte Carlo sampling-based technique is used to efficiently search the combinatorial model space of different discipline couplings. Finally, an algorithm for optimal model selection is presented and combined with three tractable approaches to quantify the accuracy of the system outputs with approximate couplings. These algorithms are applied to identify the important discipline couplings in three engineering problems: a fire detection satellite model, a turbine engine cycle analysis model, and a lifting surface aero-structural model.

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Section: B(a T)1|mentioning

confidence: 99%

Optimal Approximations of Coupling in Multidisciplinary Models

Baptista

Marzouk

Willcox

et al. 2017

58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference

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“…This approximation of the interface functions in a reduced basis is also used in the static condensation reduced basis method, where it is referred to as "port reduction" [18]. Another option for reducing the dimension of the coupling variables is to use the method of Arnst et al [6].…”

Section: Dimension Reduction For Interface Parametersmentioning

confidence: 99%

A Domain Decomposition Approach for Uncertainty Analysis

Liao¹,

Willcox²

2015

SIAM J. Sci. Comput.

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Abstract. This paper proposes a decomposition approach for uncertainty analysis of systems governed by partial differential equations (PDEs). The system is split into local components using domain decomposition. Our domain-decomposed uncertainty quantification (DDUQ) approach performs uncertainty analysis independently on each local component in an "offline" phase, and then assembles global uncertainty analysis results using precomputed local information in an "online" phase. At the heart of the DDUQ approach is importance sampling, which weights the precomputed local PDE solutions appropriately so as to satisfy the domain decomposition coupling conditions. To avoid global PDE solves in the online phase, a proper orthogonal decomposition reduced model provides an efficient approximate representation of the coupling functions. 1. Introduction. Many problems arising in computational science and engineering are described by mathematical models of high complexity-involving multiple disciplines, characterized by a large number of parameters, and impacted by multiple sources of uncertainty. Decomposition of a system into subsystems or component parts has been one strategy to manage this complexity. For example, modern engineered systems are typically designed by multiple groups, usually decomposed along disciplinary or subsystem lines, sometimes spanning different organizations and even different geographical locations. Mathematical strategies have been developed for decomposing a simulation task (e.g., domain decomposition methods [45]), for decomposing an optimization task (e.g., domain decomposition for optimal design or control [28,10,29,5]), and for decomposing a complex design task (e.g., decomposition approaches to multidisciplinary optimization [15,34,55]). In this paper we propose a decomposition approach for uncertainty quantification (UQ). In particular, we focus on the simulation of systems governed by stochastic partial differential equations (PDEs) and a domain decomposition approach to quantification of uncertainty in the corresponding quantities of interest.In the design setting, decomposition approaches are not typically aimed at improving computational efficiency, although in some cases decomposition can admit parallelism that would otherwise be impossible [57]. Rather, decomposition is typically employed in situations for which it is infeasible or impractical to achieve tight coupling among the system subcomponents. This inability to achieve tight coupling becomes a particular problem when the goal is to wrap an outer loop (e.g., optimization) around the simulation. In the field of multidisciplinary design optimization (MDO), decomposition is achieved through mathematical formulations such as col-

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“…We are also aware of several other directions of development in uncertainty analysis of fluid mechanics models, based on Kennedy and O'Hagan's work (for example, [12]). We point out a line of development that is perspective in general but is not effective on small training data, an approach currently called multifidelity kriging [13].…”

Section: Uncertainty Analysis In the Context Of Previous Workmentioning

confidence: 99%

Intrusive analysis for NEK5000: development of intrusive uncertainty quantification for high-dimensional, high-fidelity codes.

Roderick¹,

Anitescu²

2012

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We study the use of lower-fidelity training data for uncertainty quantification of complex simulation models. In our approach, computationally expensive full-model outputs are approximated applying proper orthogonal-decomposition-based dimensionality reduction to the full model.A Gaussian-processes-based machine learning approach is then used to model the difference (or, rather, the correspondence) between the higher-fidelity and the lowerfidelity data. This stochastic model can be constructed by using few additional full code evaluations; it is used to calibrate arbitrarily many lower-quality outputs in order to create additional training data for sampling-based analysis. In effect, an adequate approximation of the simulation model's response to uncertainty can be constructed at a modest computational cost. In fact, we aim at the number of full-model evaluations comparable to that required for a linear approximation (as opposed to numbers traditionally associated with sampling in high-dimensional spaces).In this report, we explain the basic algorithm, suggest some performance tuning options, and give a first characterization for the class of simulation models of nuclear engineering for which the suggested report is effective.The primarily goal of our work was to demonstrate the effectiveness of multifidelity analysis on high-performance fluid dynamics simulation code Nek5000. We have been successful at estimating statistics for an output of interest at low cost. While further demonstration exercises may be helpful, the information provided here clearly argues for the effectiveness of the approach. Logically, the next stage of work is to explore the full range of tasks related to Nek5000 code development and verification that could be made more computationally accessible with the use of calibrated lowerfidelity data.The suggested approach can be generalized to multiple levels of fidelity, enabling uncertainty analysis based on many different model approximations. The long-term goal of our research is to fit the developed method into a larger context of advanced uncertainty analysis tools for nuclear engineering applications. We foresee implementation as a suite of analysis tools that can become a component of SHARP / NEAMS codes.

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Dimension reduction in stochastic modeling of coupled problems

Cited by 42 publications

References 27 publications

Optimal Approximations of Coupling in Multidisciplinary Models

Optimal Approximations of Coupling in Multidisciplinary Models

A Domain Decomposition Approach for Uncertainty Analysis

Intrusive analysis for NEK5000: development of intrusive uncertainty quantification for high-dimensional, high-fidelity codes.

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