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

Content Provider	Springer Nature Link
Author	Conrad, Patrick R. Girolami, Mark Särkkä, Simo Stuart, Andrew Zygalakis, Konstantis
Copyright Year	2016
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.
Starting Page	1065
Ending Page	1082
Page Count	18
File Format	PDF
ISSN	09603174
Journal	Statistics and Computing
Volume Number	27
Issue Number	4
e-ISSN	15731375
Language	English
Publisher	Springer US
Publisher Date	2016-06-02
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Numerical analysis Probabilistic numerics Inverse problems Uncertainty quantification Bayesian inference Probabilistic methods, particle methods, etc. Stability and convergence of numerical methods Statistics and Computing/Statistics Programs Artificial Intelligence (incl. Robotics) Statistical Theory and Methods Probability and Statistics in Computer Science
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Theoretical Computer Science Computational Theory and Mathematics Statistics, Probability and Uncertainty

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

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

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

Data free inference with processed data products

Exact Bayesian inference for the Bingham distribution

Cuts in Bayesian graphical models

Bayesian inference for Heston-STAR models

Bayesian inference via projections

Sequential Monte Carlo methods for Bayesian elliptic inverse problems

A statistical test for Nested Sampling algorithms

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

Similar Documents

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

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

Data free inference with processed data products

Exact Bayesian inference for the Bingham distribution

Cuts in Bayesian graphical models

Bayesian inference for Heston-STAR models

Bayesian inference via projections

Sequential Monte Carlo methods for Bayesian elliptic inverse problems

A statistical test for Nested Sampling algorithms

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