Parameter Estimation of Partial Differential Equation Models

Handle URI:
http://hdl.handle.net/10754/599138
Title:
Parameter Estimation of Partial Differential Equation Models
Authors:
Xun, Xiaolei; Cao, Jiguo; Mallick, Bani; Maity, Arnab; Carroll, Raymond J.
Abstract:
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.
Citation:
Xun X, Cao J, Mallick B, Maity A, Carroll RJ (2013) Parameter Estimation of Partial Differential Equation Models. Journal of the American Statistical Association 108: 1009–1020. Available: http://dx.doi.org/10.1080/01621459.2013.794730.
Publisher:
Informa UK Limited
Journal:
Journal of the American Statistical Association
KAUST Grant Number:
KUS-CI-016-04
Issue Date:
Sep-2013
DOI:
10.1080/01621459.2013.794730
PubMed ID:
24363476
PubMed Central ID:
PMC3867159
Type:
Article
ISSN:
0162-1459; 1537-274X
Sponsors:
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.
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Full metadata record

DC FieldValue Language
dc.contributor.authorXun, Xiaoleien
dc.contributor.authorCao, Jiguoen
dc.contributor.authorMallick, Banien
dc.contributor.authorMaity, Arnaben
dc.contributor.authorCarroll, Raymond J.en
dc.date.accessioned2016-02-25T13:53:34Zen
dc.date.available2016-02-25T13:53:34Zen
dc.date.issued2013-09en
dc.identifier.citationXun X, Cao J, Mallick B, Maity A, Carroll RJ (2013) Parameter Estimation of Partial Differential Equation Models. Journal of the American Statistical Association 108: 1009–1020. Available: http://dx.doi.org/10.1080/01621459.2013.794730.en
dc.identifier.issn0162-1459en
dc.identifier.issn1537-274Xen
dc.identifier.pmid24363476en
dc.identifier.doi10.1080/01621459.2013.794730en
dc.identifier.urihttp://hdl.handle.net/10754/599138en
dc.description.abstractPartial 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.en
dc.description.sponsorshipThe 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.en
dc.publisherInforma UK Limiteden
dc.subjectAsymptotic theoryen
dc.subjectBasis function expansionen
dc.subjectBayesian methoden
dc.subjectDifferential equationsen
dc.subjectMeasurement erroren
dc.subjectParameter cascadingen
dc.titleParameter Estimation of Partial Differential Equation Modelsen
dc.typeArticleen
dc.identifier.journalJournal of the American Statistical Associationen
dc.identifier.pmcidPMC3867159en
dc.contributor.institutionNovartis Pharma, Rueil-Malmaison, Franceen
dc.contributor.institutionSimon Fraser University, Burnaby, Canadaen
dc.contributor.institutionTexas A and M University, College Station, United Statesen
dc.contributor.institutionNorth Carolina State University, Raleigh, United Statesen
kaust.grant.numberKUS-CI-016-04en
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