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Research Papers

Model-Form Calibration in Drift-Diffusion Simulation Using Fractional Derivatives

[+] Author and Article Information
Yan Wang

School of Mechanical Engineering, Georgia Institute of Technology,
Atlanta, GA 30332

Manuscript received July 13, 2015; final manuscript received November 19, 2015; published online July 1, 2016. Assoc. Editor: Ioannis Kougioumtzoglou.

ASME J. Risk Uncertainty Part B 2(3), 031006 (Jul 01, 2016) (9 pages) Paper No: RISK-15-1086; doi: 10.1115/1.4032312 History: Received July 13, 2015; Accepted November 20, 2015

In modeling and simulation, model-form uncertainty arises from the lack of knowledge and simplification during the modeling process and numerical treatment for ease of computation. Traditional uncertainty quantification (UQ) approaches are based on assumptions of stochasticity in real, reciprocal, or functional spaces to make them computationally tractable. This makes the prediction of important quantities of interest, such as rare events, difficult. In this paper, a new approach to capture model-form uncertainty is proposed. It is based on fractional calculus, and its flexibility allows us to model a family of non-Gaussian processes, which provides a more generic description of the physical world. A generalized fractional Fokker–Planck equation (fFPE) is used to describe the drift-diffusion processes under long-range correlations and memory effects. A new model-calibration approach based on the maximum mutual information is proposed to reduce model-form uncertainty, where an optimization procedure is taken.

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Figures

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Fig. 1

Effect of skewness θ on probability density functions (PDFs)

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Fig. 2

Effect of β on PDFs

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Fig. 3

Effect of α on PDFs

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Fig. 5

Search history of model-form parameters in the calibration process against random samples from a Gaussian distribution: (a) PDF evolution, (b) γ, and (c) θ

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Fig. 6

Search history of mutual information with different step sizes for Gaussian distribution: (a) step size λ=1.0 and (b) step size λ=2.0

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Fig. 7

PDF evolved in the calibration process against a Fréchet distribution

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Fig. 8

Search history of mutual information for Fréchet distribution

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Fig. 9

PDF evolved in the calibration process against random samples from a log-normal distribution

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Fig. 10

PDFs at three time steps are calibrated simultaneously against histograms of a diffusion process based on accumulative mutual information: (a) time step t=0.1, (b) time step t=0.2, and (c) time step t=0.3

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