The Library
Statistical analysis of differential equations : introducing probability measures on numerical solutions
Tools
Conrad, Patrick R., Girolami, Mark, Särkkä, Simo, Stuart, A. M. and Zygalakis, Konstantinos (2017) Statistical analysis of differential equations : introducing probability measures on numerical solutions. Statistics and Computing, 27 (4). pp. 1065-1082. doi:10.1007/s11222-016-9671-0 ISSN 0960-3174.
|
PDF
WRAP_art%3A10.pdf - Published Version - Requires a PDF viewer. Available under License Creative Commons Attribution 4.0. Download (2803Kb) | Preview |
Official URL: http://dx.doi.org/10.1007/s11222-016-9671-0
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.
Item Type: | Journal Article | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Subjects: | Q Science > QA Mathematics | ||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics Faculty of Science, Engineering and Medicine > Science > Statistics |
||||||||||
Library of Congress Subject Headings (LCSH): | Numerical analysis, Probabilities, Inverse problems (Differential equations), Uncertainty -- Mathematical models | ||||||||||
Journal or Publication Title: | Statistics and Computing | ||||||||||
Publisher: | Springer | ||||||||||
ISSN: | 0960-3174 | ||||||||||
Official Date: | July 2017 | ||||||||||
Dates: |
|
||||||||||
Volume: | 27 | ||||||||||
Number: | 4 | ||||||||||
Page Range: | pp. 1065-1082 | ||||||||||
DOI: | 10.1007/s11222-016-9671-0 | ||||||||||
Status: | Peer Reviewed | ||||||||||
Publication Status: | Published | ||||||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||||||
Date of first compliant deposit: | 3 March 2017 | ||||||||||
Date of first compliant Open Access: | 3 March 2017 | ||||||||||
Funder: | Engineering and Physical Sciences Research Council (EPSRC), Suomen Akatemia [Academy of Finland], Simons Foundation (SF) | ||||||||||
Grant number: | EP/D002060/1, EP/J016934/2, EP/K034154/1 (EPSRC), 266940 (Suomen Akatemia) |
Request changes or add full text files to a record
Repository staff actions (login required)
View Item |
Downloads
Downloads per month over past year