Multiscale Modelling and Inverse Problems

  • James Nolen
  • Grigorios A. Pavliotis
  • Andrew M. Stuart
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 83)


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.


Inverse Problem Linear Functional Multiscale Modelling Permeability Tensor Observation Noise 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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The authors thank A. Cliffe and Ch. Schwab for helpful discussions concerning the groundwater flow model.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • James Nolen
    • 1
  • Grigorios A. Pavliotis
    • 2
  • Andrew M. Stuart
    • 3
  1. 1.Department of MathematicsDuke University DurhamDurhamUSA
  2. 2.Department of Mathematics Imperial College LondonLondonUK
  3. 3.Mathematics InstituteWarwick UniversityCoventryUK

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