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Multiscale Modelling and Inverse Problems

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

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.

Keywords

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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Notes

Acknowledgements

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

References

  1. 1.
    G. Allaire. Homogenization and two-scale convergence. SIAM J. Math. Anal., 23(6):1482–1518, 1992.CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    M. Avellaneda and F.-H. Lin. Compactness methods in the theory of homogenization. Comm. Pure Appl. Math., 40(6):803–847, 1987.CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    R. Azencott, A. Beri, and I. Timofeyev. Adaptive sub-sampling for parametric estimation of Gaussian diffusions. J. Stat. Phys., 139(6):1066–1089, 2010.CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    G. Bal, J. Garnier, S. Motsch, and V. Perrier. Random integrals and correctors in homogenization. Asymptotic Analysis, 59:1–26, 2008.zbMATHMathSciNetGoogle Scholar
  5. 5.
    G. Bal and K. Ren. Physics-based models for measurement correlations: application to an inverse Sturm-Liouville problem. Inverse Problems, 25:055006, 2009.CrossRefMathSciNetGoogle Scholar
  6. 6.
    H.T. Banks and K. Kunisch. Estimation Techniqiues for Distributed Parameter Systems. Birkhäuser, 1989.Google Scholar
  7. 7.
    A. Bensoussan, J.-L. Lions, and G. Papanicolaou. Asymptotic Analysis for Periodic Structures, volume 5 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam, 1978.Google Scholar
  8. 8.
    J.O. Berger. Statistical Decision Theory and Bayesian Analysis. Springer, 1980.Google Scholar
  9. 9.
    P.J. Bickel and K.A. Doksum. Mathematical Statistics. Prentice-Hall, 2001.Google Scholar
  10. 10.
    V.I. Bogachev. Gaussian Meausures. American Mathematical Society, 1998.Google Scholar
  11. 11.
    D. Cioranescu and P. Donato. An Introduction to Homogenization. Oxford University Press, New York, 1999.zbMATHGoogle Scholar
  12. 12.
    S.L. Cotter, M. Dashti, J.C. Robinson, and A.M. Stuart. Bayesian inverse problems for functions and applications to fluid mechanics. Inverse Problems, 25:115008, 2009.CrossRefMathSciNetGoogle Scholar
  13. 13.
    B. Dacarogna. Direct Methods in the Calculus of Variations. Springer, New York, 1989.Google Scholar
  14. 14.
    G. DaPrato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. Cambridge University Press, 1992.Google Scholar
  15. 15.
    M. Dashti and A.M. Stuart. Uncertainty quantification and weak approximation of an elliptic inverse problem. Preprint, 2011.Google Scholar
  16. 16.
    Y. R. Efendiev, Th. Y. Hou, and X.-H. Wu. Convergence of a nonconforming multiscale finite element method. SIAM J. Numer. Anal., 37(3):888–910 (electronic), 2000.Google Scholar
  17. 17.
    H.K. Engl, M. Hanke, and A. Neubauer. Regularization of Inverse Problems. Kluwer, 1996.Google Scholar
  18. 18.
    L.C. Evans. Partial Differential Equations. AMS, Providence, Rhode Island, 1998.zbMATHGoogle Scholar
  19. 19.
    B. Fitzpatrick. Bayesian analysis in inverse problems. Inverse Problems, 7:675–702, 1991.CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbirch. Optimization with PDE Constraints, volume 23 of Mathematics Modelling: Theory and Applications. Springer, 2009.Google Scholar
  21. 21.
    H. König. Eigenvalue Distribution of Compact Operators. Birkhäuser Verlag, Basel, 1986.Google Scholar
  22. 23.
    M.A. Lifshits. Gaussian Random Functions, volume 322 of Mathematics and its Applications. Kluwer, Dordrecht, 1995.Google Scholar
  23. 24.
    J. Nolen and G. Papanicolaou. Inverse problems. Fine scale unertainty in parameter estimation for elliptic equations, 25:115021, 2009.MathSciNetGoogle Scholar
  24. 25.
    J. Nolen, G.A. Pavliotis, and A.M. Stuart. In preparation, 2010.Google Scholar
  25. 26.
    A. Papavasiliou, G. A. Pavliotis, and A. M. Stuart. Maximum likelihood drift estimation for multiscale diffusions. Stochastic Process. Appl., 119(10):3173–3210, 2009.CrossRefzbMATHMathSciNetGoogle Scholar
  26. 27.
    G. A. Pavliotis and A. M. Stuart. Multiscale Methods, volume 53 of Texts in Applied Mathematics. Springer, 2008. Averaging and Homogenization.Google Scholar
  27. 28.
    G.A. Pavliotis and A.M Stuart. Parameter estimation for multiscale diffusions. J. Stat. Phys., 127(4):741–781, 2007.Google Scholar
  28. 29.
    A.M. Stuart. Inverse problems: a Bayesian perspective. Acta Numerica, 19, 2010.Google Scholar

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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