Bayesian Uncertainty Quantification for Differential Equations

Duration: 1 hour 3 mins

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Description:	Girolami, M (University of Warwick) Tuesday 13 May 2014, 10:00-11:00


Created:	2014-05-16 11:58
Collection:	Advanced Monte Carlo Methods for Complex Inference Problems
Publisher:	Isaac Newton Institute
Copyright:	Girolami, M
Language:	eng (English)


Abstract:	This talk will make a case for expansion of the role of Bayesian statistical inference when formally quantifying uncertainty in computer models defined as systems of ordinary or partial differential equations by adopting the perspective that implicitly defined infinite dimensional functions representing model states are objects to be inferred probabilistically. I describe a general methodology for the probabilistic "integration" of differential equations via model based updating of a joint prior measure on the space of functions, their temporal and spatial derivatives. This results in a measure over functions reflecting how well they satisfy the system of differential equations and corresponding initial and boundary values. This measure can be naturally incorporated within the Kennedy and O'Hagan framework for uncertainty quantification and provides a fully Bayesian approach to model calibration and predictive analysis. By taking this probabilistic viewpoint, the full force of Bayesian inference can be exploited when seeking to coherently quantify and propagate epistemic uncertainty in computer models of complex natural and physical systems. A broad variety of examples are provided to illustrate the potential of this framework for characterising discretization uncertainty, including initial value, delay, and boundary value differential equations, as well as partial differential equations. I will also demonstrate the methodology on a large scale system, by modeling discretization uncertainty in the solution of the Navier-Stokes equations of fluid flow, reduced to over 16,000 coupled and stiff ordinary differential equations.

Abstract:

This talk will make a case for expansion of the role of Bayesian statistical inference when formally quantifying uncertainty in computer models defined as systems of ordinary or partial differential equations by adopting the perspective that implicitly defined infinite dimensional functions representing model states are objects to be inferred probabilistically. I describe a general methodology for the probabilistic "integration" of differential equations via model based updating of a joint prior measure on the space of functions, their temporal and spatial derivatives. This results in a measure over functions reflecting how well they satisfy the system of differential equations and corresponding initial and boundary values. This measure can be naturally incorporated within the Kennedy and O'Hagan framework for uncertainty quantification and provides a fully Bayesian approach to model calibration and predictive analysis. By taking this probabilistic viewpoint, the full force of Bayesian inference can be exploited when seeking to coherently quantify and propagate epistemic uncertainty in computer models of complex natural and physical systems. A broad variety of examples are provided to illustrate the potential of this framework for characterising discretization uncertainty, including initial value, delay, and boundary value differential equations, as well as partial differential equations. I will also demonstrate the methodology on a large scale system, by modeling discretization uncertainty in the solution of the Navier-Stokes equations of fluid flow, reduced to over 16,000 coupled and stiff ordinary differential equations.

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