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Uncertainty Quantification"],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>In multiscale studies of emergent phenomena, a common approach is to define a microscopic scale generative model and subsequently, by passing to a diffusion limit, to derive a mesoscopic or macroscopic model via a homogenization argument. Microscopic models are often inherently stochastic, while the diffusion limit model may represent distributions or mean field quantities. A canonical example of this type of limiting process is the classical diffusion equation as a limit of Brownian random motion. Being typically based on ordinary or partial differential equations formalism, large scale models are more amenable to inverse problems, where the model parameters are estimated from indirect observations. Due to the intrinsic ill-posed nature of inverse problems, the estimation of the parameters of large scale models may be itself a significant challenge. In some multiscale investigations, the quantities of primary interest are not those characterizing the large scale model, but rather the parameters of the associated microscopic model. The latter are sometimes related to our understanding of the processes that are described by the generative models. In many cases, there is no direct connection between the small scale parameters and the large scale parameters describing the aggregate effects of the underlying processes. Moreover, estimating the unknown parameters of a microscopic, stochastic model from the large scale observations poses a number of challenges, and requires the development of new methodological approaches. In the Bayesian statistical framework, stochastic forward models often lead to intractable likelihood densities, making standard statistical computations a challenge. Such models are addressed by approximate Bayesian computing (ABC). In this paper we propose an alternative approach combining Bayesian statistics and particle techniques for inference beyond the limits of model scales. The viability of the approach is demonstrated on three computed examples describing diffusion, chemical kinetics, and cell culture assay, and a connection with existing ABC methods is discussed.<\/jats:p>","DOI":"10.1137\/15m105505x","type":"journal-article","created":{"date-parts":[[2017,8,1]],"date-time":"2017-08-01T11:28:42Z","timestamp":1501586922000},"page":"665-693","source":"Crossref","is-referenced-by-count":0,"title":["Beyond the Model Limit: Parameter Inference Across Scales"],"prefix":"10.1137","volume":"5","author":[{"given":"Margaret","family":"Callahan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Daniela","family":"Calvetti","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Erkki","family":"Somersalo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,8,1]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/30\/10\/105008"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1007\/s11222-007-9043-x"},{"key":"atypb3","unstructured":"D. Calvetti and E. Somersalo (2007),\n                      Introduction to Bayesian Scientific Computing - Ten Lectures on Subjective Computing\n                      , Springer, New York."},{"key":"atypb4","doi-asserted-by":"crossref","unstructured":"D. Calvetti and E. Somersalo (2013),\n                      Computational Mathematical Modeling. An Integrated Approach through Scales\n                      , Math. Model. Comput., SIAM, Philadelphia.","DOI":"10.1137\/1.9781611972481"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1007\/s12210-015-0422-5"},{"key":"atypb6","doi-asserted-by":"crossref","unstructured":"J. A. Carrillo, M. Fornasier, G. Toscani, and F. Vecil (2010),\n                      Particle, kinetic, and hydrodynamic models of swarming\n                      , In Mathematical Modeling of Collective Behavior in Socio-economic and Life Sciences, Birkha\u0308user Boston, Boston, pp. 297-336.","DOI":"10.1007\/978-0-8176-4946-3_12"},{"key":"atypb7","doi-asserted-by":"crossref","unstructured":"T. Cui, C. Fox, and M. J. O'Sullivan (2011),\n                      Bayesian calibration of a large-scale geothermal reservoir model by a new adaptive delayed acceptance Metropolis Hastings algorithm\n                      , Water Resourc. Res 47, W10521,https:\/\/doi.org\/10.1029\/2010WR010352.","DOI":"10.1029\/2010WR010352"},{"key":"atypb8","doi-asserted-by":"crossref","unstructured":"J. E. Dennis and R. B. Schnabel (1996),\n                      Numerical Methods for Unconstrained Optimization and Nonlinear Equations\n                      , Classics Appl. 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Freedman (1971),\n                      Brownian Motion and Diffusion\n                      , Holder-Day, San Francisco."},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1198\/jasa.2009.0008"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(76)90041-3"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1021\/j100540a008"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1021\/jp806431b"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1111\/j.1541-0420.2005.00345.x"},{"key":"atypb20","doi-asserted-by":"crossref","unstructured":"P. C. Hansen (1998),\n                      Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion\n                      , SIAM Monogr. Math. Model. Comput. 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Fan (2011),\n                      Likelihood-free MCMC\n                      , in Handbook of Markov Chain Monte Carlo, CRC Press, Boca Raton, FL, pp. 313-335.","DOI":"10.1201\/b10905-13"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0607208104"},{"key":"atypb35","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1111\/j.2517-6161.1993.tb01911.x","volume":"55","author":"West M.","year":"1993","journal-title":"J. R. Stat. Soc. Ser. B Stat. Methodol."},{"key":"atypb36","unstructured":"M. West (1993),\n                      Mixture models, Monte Carlo, Bayesian updating and dynamic models\n                      , in Computing Science and Statistics: Proceedings of the 24th Symposium on the Interface, J. H. 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