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

Patrick R. Conrad*, Mark Girolami, Simo Särkkä, Andrew Stuart, Konstantinos Zygalakis

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

28 Citations (Scopus)
140 Downloads (Pure)

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.

Original languageEnglish
Pages (from-to)1065-1082
Number of pages18
JournalSTATISTICS AND COMPUTING
Volume27
Issue number4
DOIs
Publication statusPublished - 1 Jul 2017
MoE publication typeA1 Journal article-refereed

Keywords

  • Inverse problems
  • Numerical analysis
  • Probabilistic numerics
  • Uncertainty quantification

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