A Mathematical and Computational Framework for Multifidelity Design and Analysis With Computer Models

Allaire, Douglas; Willcox, Karen

doi:10.1615/int.j.uncertaintyquantification.2013004121

Cited by 46 publications

(57 citation statements)

References 36 publications

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“…For illustration, we set w 1 /w 2 = 0.1 and vary the correlation coefficients ρ 1,2 , ρ 1,3 and the costs ratio w 3 /w 1 . Figure 1 The plots show the contours of the variance reduction ratio (3.23) of an MFMC estimator with three models f (1) , f (2) , f (3) . The black region shows where condition (3.12) is violated and the gray region where the MFMC estimator leads to a higher variance than the Monte Carlo estimator.…”

Section: Illustrative Examples Let Us Consider An Example Withmentioning

confidence: 99%

“…Thus, if w 3 /w 1 is already low, the second term in γ dominates the variance reduction, which is independent of the costs w 3 . In Figure 1(b), the correlation coefficient of model f (2) is reduced to ρ 1,2 = 0.6. The gray region shows where γ evaluates to values larger than 1, and therefore where the MFMC estimator has a higher variance than the standard Monte Carlo estimator.…”

Section: Illustrative Examples Let Us Consider An Example Withmentioning

confidence: 99%

“…In the following, we have a high-fidelity model denoted as f (1) : D → Y and (lower-fidelity) surrogate models f (2) , . .…”

mentioning

confidence: 99%

“…We seek an estimator with a computational budget p ∈ R + that optimally exploits the surrogate models f (2) , . .…”

mentioning

confidence: 99%

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Optimal Model Management for Multifidelity Monte Carlo Estimation

Peherstorfer¹,

Willcox²,

Gunzburger³

2016

SIAM J. Sci. Comput.

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209

233

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Abstract. This work presents an optimal model management strategy that exploits multifidelity surrogate models to accelerate the estimation of statistics of outputs of computationally expensive high-fidelity models. Existing acceleration methods typically exploit a multilevel hierarchy of surrogate models that follow a known rate of error decay and computational costs; however, a general collection of surrogate models, which may include projection-based reduced models, data-fit models, support vector machines, and simplified-physics models, does not necessarily give rise to such a hierarchy. Our multifidelity approach provides a framework to combine an arbitrary number of surrogate models of any type. Instead of relying on error and cost rates, an optimization problem balances the number of model evaluations across the high-fidelity and surrogate models with respect to error and costs. We show that a unique analytic solution of the model management optimization problem exists under mild conditions on the models. Our multifidelity method makes occasional recourse to the high-fidelity model; in doing so it provides an unbiased estimator of the statistics of the high-fidelity model, even in the absence of error bounds and error estimators for the surrogate models. Numerical experiments with linear and nonlinear examples show that speedups by orders of magnitude are obtained compared to Monte Carlo estimation that invokes a single model only. 1. Introduction. Multilevel techniques have a long and successful history in computational science and engineering, e.g., multigrid for solving systems of equations [8,25,9], multilevel discretizations for representing functions [50,18,10], and multilevel Monte Carlo and multilevel stochastic collocation for estimating mean solutions of partial differential equations (PDEs) with stochastic parameters [27,22,45]. These multilevel techniques typically start with a fine-grid discretization-a high-fidelity model-of the underlying PDE or function. The fine-grid discretization is chosen to guarantee an approximation of the output of interest with the accuracy required by the current problem at hand. Additionally, a hierarchy of coarser discretizations-lowerfidelity surrogate models-is constructed, where a parameter (e.g., mesh width) controls the trade-off between error and computational costs. Changing this parameter gives rise to a multilevel hierarchy of discretizations with known error and cost rates. Multilevel techniques use these error and cost rates to distribute the computational work among the discretizations in the hierarchy, shifting most of the work onto the

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Section: Illustrative Examples Let Us Consider An Example Withmentioning

confidence: 99%

Section: Illustrative Examples Let Us Consider An Example Withmentioning

confidence: 99%

See 2 more Smart Citations

Optimal Model Management for Multifidelity Monte Carlo Estimation

Peherstorfer¹,

Willcox²,

Gunzburger³

2016

SIAM J. Sci. Comput.

Self Cite

209

233

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“…The approach that uses the tools of estimation theory to fuse together information from multifidelity analysis, resulting in a Bayesian-based approach to mitigating risk in complex design has been proposed [10]. …”

Section: Methodsmentioning

confidence: 99%

Novel Approach: Information Quantity for Calculating Uncertainty of Mathematical Model

Menin¹

2017

Proceedings of the IS4SI 2017 Summit DIGITALISATION FOR a SUSTAINABLE SOCIETY, Gothenburg, Sweden, 12–16 June 2017.

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Evaluation of the chosen mathematical model of a physical system (phenomena) is considered within the framework of thermodynamic theory and Mark Burgin's general theory of information. The finiteness of the model cased by the limited number of chosen variables and the need to use the International system of units cause the existing of the a-priori uncertainty of a physical-mathematical model. The proposed formula for calculation of this uncertainty provides its value comparison with the actual experimental measurement uncertainty. Examples of practical application in the framework of the considered concept are introduced regarding analysis of the Boltzmann constant and gravitational constant measurements Keywords: computer modelling; planning of experiment; system of primary variables, theory of information; theory of similarity; uncertainty of physical-mathematical model

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Sensitivity‐guided decision‐making for wind farm micro‐siting

Ulker

Allaire

Willcox

2016

Numerical Methods in Fluids

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Summary This paper presents a quantitative risk assessment for design and development of a renewable energy system to support decision‐making among design alternatives. Throughout the decision‐making phases, resources are allocated among exploration and exploitation tasks to manage the uncertainties in design parameters and to adapt designs to new information for enhanced performance. The resource allocation problem is formulated as a sequential decision feedback loop for a quantitative analysis of exploration and exploitation trade‐offs. We support decision‐making by tracking the evolution of uncertainties, the sensitivity of design alternatives to the uncertainties, and the performance, reliability, and robustness of each design. This is achieved by analyzing the uncertainties in the wind resource, the turbine performance and operation, and the models that define the power curve and wake deficiency. Comparison of the performance, reliability, and robustness of aligned and staggered turbine layouts before and after wind assessment experiments aids in improving micro‐siting decisions. The results demonstrate that design decisions can be supported by efficiently allocating resources towards improved estimates of achievable design objectives and by quantitatively assessing the risk in meeting those objectives. Copyright © 2016 John Wiley & Sons, Ltd.

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A Mathematical and Computational Framework for Multifidelity Design and Analysis With Computer Models

Cited by 46 publications

References 36 publications

Optimal Model Management for Multifidelity Monte Carlo Estimation

Optimal Model Management for Multifidelity Monte Carlo Estimation

Novel Approach: Information Quantity for Calculating Uncertainty of Mathematical Model

Sensitivity‐guided decision‐making for wind farm micro‐siting

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