Welcome to the new version of CaltechAUTHORS. Login is currently restricted to library staff. If you notice any issues, please email coda@library.caltech.edu
Published July 2017 | Published + Supplemental Material
Journal Article Open

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

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.

Additional Information

© 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. Received: 05 December 2015; Accepted: 05 May 2016; First Online: 02 June 2016. The authors gratefully acknowledge support from EPSRC Grant CRiSM EP/D002060/1, EPSRC Established Career Research Fellowship EP/J016934/2, EPSRC Programme Grant EQUIP EP/K034154/1, and Academy of Finland Research Fellowship 266940. Konstantinos Zygalakis was partially supported by a grant from the Simons Foundation. Part of this work was done during the author's stay at the Newton Institute for the program "Stochastic Dynamical Systems in Biology: Numerical Methods and Applications.;"

Attached Files

Published - art_3A10.1007_2Fs11222-016-9671-0.pdf

Supplemental Material - 11222_2016_9671_MOESM1_ESM.tex

Files

art_3A10.1007_2Fs11222-016-9671-0.pdf
Files (2.4 MB)
Name Size Download all
md5:186d4c79d09963a8d5404d29613d6fd7
2.4 MB Preview Download
md5:1173b9511bea5be8206b505c9341f61e
4.7 kB Download

Additional details

Created:
August 21, 2023
Modified:
March 5, 2024