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

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Title: Statistical analysis of differential equations: introducing probability measures on numerical solutions
Authors: Conrad, PR
Girolami, M
Sarkka, S
Stuart, A
Zygalakis, K
Item Type: Journal Article
Abstract: 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 uncertainty 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.
Issue Date: 1-Jul-2017
Date of Acceptance: 5-May-2016
URI: http://hdl.handle.net/10044/1/66129
DOI: https://dx.doi.org/10.1007/s11222-016-9671-0
ISSN: 0960-3174
Publisher: Springer (part of Springer Nature)
Start Page: 1065
End Page: 1082
Journal / Book Title: Statistics and Computing
Volume: 27
Issue: 4
Copyright Statement: © 2016 The Author(s). This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Keywords: Science & Technology
Physical Sciences
Computer Science, Theory & Methods
Statistics & Probability
Computer Science
Numerical analysis
Probabilistic numerics
Inverse problems
Uncertainty quantification
0104 Statistics
0802 Computation Theory And Mathematics
Publication Status: Published
Open Access location: https://link.springer.com/content/pdf/10.1007%2Fs11222-016-9671-0.pdf
Online Publication Date: 2016-06-02
Appears in Collections:Mathematics