<abstract xmlns="http://www.w3.org/1999/xhtml"><p> 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.</p></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.

A Dynamic BI–Orthogonal Field Equation Approach to Efficient Bayesian Inversion

International Journal of Applied Mathematics and Computer Science

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

This paper proposes a novel computationally efficient stochastic spectral projection based approach to Bayesian inversion of a computer simulator with high...

A Dynamic BI–Orthogonal Field Equation Approach to Efficient Bayesian Inversion

Published Online: Jul 08, 2017

Page range: 229 - 243

Received: Sep 14, 2016

Accepted: Mar 08, 2017

DOI: https://doi.org/10.1515/amcs-2017-0016

Keywords
Bayesian framework, stochastic partial differential equation, Karhunen-Loève expansion, generalized polynomial chaos, dynamically biorthogonal field equations

© by Han-Lim Choi

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

A Dynamic BI–Orthogonal Field Equation Approach to Efficient Bayesian Inversion

Published Online: Jul 08, 2017

Page range: 229 - 243

Received: Sep 14, 2016

Accepted: Mar 08, 2017

DOI: https://doi.org/10.1515/amcs-2017-0016

KeywordsBayesian framework, stochastic partial differential equation, Karhunen-Loève expansion, generalized polynomial chaos, dynamically biorthogonal field equations

© by Han-Lim Choi

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
Bayesian framework, stochastic partial differential equation, Karhunen-Loève expansion, generalized polynomial chaos, dynamically biorthogonal field equations