Certified Reduced Basis Model Characterization: a Frequentistic Uncertainty Framework

We introduce a frequentistic validation framework for assessment - acceptance or rejection - of the consistency of a proposed parametrized partial differential equation model with respect to (noisy) experimental data from a physical system. Our method builds upon the Hotelling T2 statistical hypothe...

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Bibliographic Details
Main Authors: Huynh, D B, Knezevic, D J, Patera, A T
Format: Report
Language:English
Subjects:
A POSTERIORI ERROR BOUND
BIAS
HYPOTHESES
HYPOTHESIS TESTING
MATHEMATICAL MODELS
Numerical Mathematics
PARTIAL DIFFERENTIAL EQUATIONS
REDUCED BASIS METHOD
UNCERTAINTY
UNCERTAINTY QUANTIFICATION
VALIDATION
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Summary:We introduce a frequentistic validation framework for assessment - acceptance or rejection - of the consistency of a proposed parametrized partial differential equation model with respect to (noisy) experimental data from a physical system. Our method builds upon the Hotelling T2 statistical hypothesis test for bias first introduced by Balci & Sargent in 1984 and subsequently extended by McFarland & Mahadevan 2008. Our approach introduces two new elements: a spectral representation of the misfit which reduces the dimensionality and variance of the underlying multivariate Gaussian but without introduction of the usual regression assumptions; a certified (verified) reduced basis approximation - reduced order model - which greatly accelerates computational performance but without any loss of rigor relative to the full (finite element) discretization. We illustrate our approach with examples from heat transfer and acoustics, both based on synthetic data. We demonstrate that we can efficiently identify possibility regions that characterize parameter uncertainty; furthermore, in the case that the possibility region is empty, we can deduce the presence of unmodeled physics such as cracks or heterogeneities. Computer Methods in Applied Mechanics and Engineering (submitted January 2011)