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Paper details
Number 2 - June 2017
Volume 27 - 2017
A dynamic bi-orthogonal field equation approach to efficient Bayesian inversion
Piyush M. Tagade, Han-Lim Choi
Abstract
This paper proposes a novel computationally efficient stochastic spectral projection based approach to Bayesian inversion of
a computer simulator with high dimensional parametric and model structure uncertainty. The proposed method is based on
the decomposition of the solution into its mean and a random field using a generic Karhunen–Loève expansion. The random
field is represented as a convolution of separable Hilbert spaces in stochastic and spatial dimensions that are spectrally
represented using respective orthogonal bases. In particular, the present paper investigates generalized polynomial chaos
bases for the stochastic dimension and eigenfunction bases for the spatial dimension. Dynamic orthogonality is used to
derive closed-form equations for the time evolution of mean, spatial and the stochastic fields. The resultant system of
equations consists of a partial differential equation (PDE) that defines the dynamic evolution of the mean, a set of PDEs
to define the time evolution of eigenfunction bases, while a set of ordinary differential equations (ODEs) define dynamics
of the stochastic field. This system of dynamic evolution equations efficiently propagates the prior parametric uncertainty
to the system response. The resulting bi-orthogonal expansion of the system response is used to reformulate the Bayesian
inference for efficient exploration of the posterior distribution. The efficacy of the proposed method is investigated for
calibration of a 2D transient diffusion simulator with an uncertain source location and diffusivity. The computational
efficiency of the method is demonstrated against a Monte Carlo method and a generalized polynomial chaos approach.
Keywords
Bayesian framework, stochastic partial differential equation, Karhunen–Loève expansion, generalized polynomial chaos, dynamically biorthogonal field equations