Parameter Estimation of Partial Differential Equation Models

Xiaolei Xun, Jiguo Cao, Bani Mallick, Arnab Maity, Raymond J. Carroll

Research output: Contribution to journalArticlepeer-review

98 Scopus citations


Partial differential equation (PDE) models are commonly used to model complex dynamic systems in applied sciences such as biology and finance. The forms of these PDE models are usually proposed by experts based on their prior knowledge and understanding of the dynamic system. Parameters in PDE models often have interesting scientific interpretations, but their values are often unknown and need to be estimated from the measurements of the dynamic system in the presence of measurement errors. Most PDEs used in practice have no analytic solutions, and can only be solved with numerical methods. Currently, methods for estimating PDE parameters require repeatedly solving PDEs numerically under thousands of candidate parameter values, and thus the computational load is high. In this article, we propose two methods to estimate parameters in PDE models: a parameter cascading method and a Bayesian approach. In both methods, the underlying dynamic process modeled with the PDE model is represented via basis function expansion. For the parameter cascading method, we develop two nested levels of optimization to estimate the PDE parameters. For the Bayesian method, we develop a joint model for data and the PDE and develop a novel hierarchical model allowing us to employ Markov chain Monte Carlo (MCMC) techniques to make posterior inference. Simulation studies show that the Bayesian method and parameter cascading method are comparable, and both outperform other available methods in terms of estimation accuracy. The two methods are demonstrated by estimating parameters in a PDE model from long-range infrared light detection and ranging data. Supplementary materials for this article are available online. © 2013 American Statistical Association.
Original languageEnglish (US)
Pages (from-to)1009-1020
Number of pages12
JournalJournal of the American Statistical Association
Issue number503
StatePublished - Sep 2013
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUS-CI-016-04
Acknowledgements: The research of Mallick, Carroll, and Xun was supported by grants from the National Cancer Institute (R37-CA057030) and the National Science Foundation DMS (Division of Mathematical Sciences) grant 0914951. This publication is based in part on work supported by the Award Number KUS-CI-016-04, made by King Abdullah University of Science and Technology (KAUST). Can's research is supported by a discovery grant (PIN: 328256) from the Natural Science and Engineering Research Council of Canada (NSERC). Maity's research was performed while visiting the Department of Statistics, Texas A&M University, and was partially supported by the Award Number R00ES017744 from the National Institute of Environmental Health Sciences.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.


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