Nolen, James, Pavliotis, Grigorios A. and Stuart, A. M. (2012) Multiscale modelling and inverse problems. In: Graham, Ivan G. and Hou, Thomas Y. and Lakkis, Omar and Scheichl, Robert, (eds.) Numerical Analysis of Multiscale Problems. Lecture Notes in Computational Science and Engineering, Vol.83 . Berlin ; Heidelberg: Springer, pp. 1-34. ISBN 9783642220609

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Abstract

The need to blend observational data and mathematical models arises in many applications and leads naturally to inverse problems. Parameters appearing in the model, such as constitutive tensors, initial conditions, boundary conditions, and forcing can be estimated on the basis of observed data. The resulting inverse problems are usually ill-posed and some form of regularization is required. These notes discuss parameter estimation in situations where the unknown parameters vary across multiple scales. We illustrate the main ideas using a simple model for groundwater flow.
We will highlight various approaches to regularization for inverse problems, including Tikhonov and Bayesian methods. We illustrate three ideas that arise when considering inverse problems in the multiscale context. The first idea is that the choice of space or set in which to seek the solution to the inverse problem is intimately related to whether a homogenized or full multiscale solution is required. This is a choice of regularization. The second idea is that, if a homogenized solution to the inverse problem is what is desired, then this can be recovered from carefully designed observations of the full multiscale system. The third idea is that the theory of homogenization can be used to improve the estimation of homogenized coefficients from multiscale data.

Item Type:	Book Item
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Series Name:	Lecture Notes in Computational Science and Engineering
Publisher:	Springer
Place of Publication:	Berlin ; Heidelberg
ISBN:	9783642220609
ISSN:	1439-7358
Book Title:	Numerical Analysis of Multiscale Problems
Editor:	Graham, Ivan G. and Hou, Thomas Y. and Lakkis, Omar and Scheichl, Robert
Official Date:	2012
Dates:	Date Event 2012 Published
Volume:	Vol.83
Number of Pages:	371
Page Range:	pp. 1-34
DOI:	10.1007/978-3-642-22061-6_1
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access

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