Bayesian Solution Uncertainty Quantification for Differential Equations

Chkrebtii, Oksana A.; Campbell, David A.; Calderhead, Ben; Girolami, Mark

doi:10.1214/16-ba1017

Cited by 142 publications

(191 citation statements)

References 39 publications

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“…A recent Probabilistic Numerics methodology for ODEs (Chkrebtii et al 2013) [explored in parallel in Hennig and Hauberg (2014)] has two important shortcomings. First, it is impractical, only supporting first-order accurate schemes with a rapidly growing computational cost caused by the growing difference stencil [although Schober et al (2014) extends to Runge-Kutta methods].…”

Section: Review Of Existing Workmentioning

confidence: 99%

Statistical analysis of differential equations: introducing probability measures on numerical solutions

et al. 2016

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In this paper, we present a formal quantification of uncertainty induced by numerical solutions of ordinary and partial differential equation models. Numerical solutions of differential equations contain inherent uncertainties due to the finite-dimensional approximation of an unknown and implicitly defined function. When statistically analysing models based on differential equations describing physical, or other naturally occurring, phenomena, it can be important to explicitly account for the uncertainty introduced by the numerical method. Doing so enables objective determination of this source of uncertainty, relative to other uncertainties, such as those caused by data contaminated with noise or model error induced by missing physical or inadequate descriptors. As ever larger scale mathematical models are being used in the sciences, often sacrificing complete resolution of the differential equation on the grids used, formally accounting for the uncertainty in the numerical method is becoming increasingly more important. This paper provides the formal means to incorporate this uncer- School of Mathematics, University of Edinburgh, Edinburgh, Scotland tainty in a statistical model and its subsequent analysis. We show that a wide variety of existing solvers can be randomised, inducing a probability measure over the solutions of such differential equations. These measures exhibit contraction to a Dirac measure around the true unknown solution, where the rates of convergence are consistent with the underlying deterministic numerical method. Furthermore, we employ the method of modified equations to demonstrate enhanced rates of convergence to stochastic perturbations of the original deterministic problem. Ordinary differential equations and elliptic partial differential equations are used to illustrate the approach to quantify uncertainty in both the statistical analysis of the forward and inverse problems.

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Section: Review Of Existing Workmentioning

confidence: 99%

Statistical analysis of differential equations: introducing probability measures on numerical solutions

et al. 2016

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“…For algebraic as well as conceptual reasons [4], GPs are a preferable choice for the prior distribution. We first review GPs and then discuss their application in ODE solvers.…”

Section: Probabilistic Ode Solversmentioning

confidence: 99%

Probabilistic Shortest Path Tractography in DTI Using Gaussian Process ODE Solvers

Schober

Kasenburg

Feragen

et al. 2014

Medical Image Computing and Computer-Assisted Intervention – MICCAI 2014

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Abstract. Tractography in diffusion tensor imaging estimates connectivity in the brain through observations of local diffusivity. These observations are noisy and of low resolution and, as a consequence, connections cannot be found with high precision. We use probabilistic numerics to estimate connectivity between regions of interest and contribute a Gaussian Process tractography algorithm which allows for both quantification and visualization of its posterior uncertainty. We use the uncertainty both in visualization of individual tracts as well as in heat maps of tract locations. Finally, we provide a quantitative evaluation of different metrics and algorithms showing that the adjoint metric [8] combined with our algorithm produces paths which agree most often with experts.

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“…Therefore, as discussed in Dass (2016), the fact that C N → 0 alone is not enough to ensure convergence. In the Supplement (Chkrebtii et al, 2016b) we have shown that the posterior expectation …”

Section: Rate Of Convergencementioning

confidence: 90%

“…Our rejoinder is divided into six sections, which provide insight and clarification on the subjects that were raised by the discussants. The paper by Chkrebtii et al (2016a) and the proposed formalism will hereafter be referred to as UQDE (uncertainty quantification for differential equations).…”

Section: Introductionmentioning

confidence: 99%