Abstract
For non-linear inverse problems, the mathematical structure of the mapping from model parameters to data is usually unknown or partly unknown. Absence of information about the mathematical structure of this function prevents us from presenting an analytical solution, so our solution depends on our ability to produce efficient search algorithms. Such algorithms may be completely problem-independent (which is the case for the so-called ’meta-heuristics’ or ’blind-search’ algorithms), or they may be designed with the structure of the concrete problem in mind.
We show that pure meta-heuristics are inefficient for large-scale, non-linear inverse problems, and that the ’no-free-lunch’ theorem holds. We discuss typical objections to the relevance of this theorem.
A consequence of the no-free-lunch theorem is that algorithms adapted to the mathematical structure of the problem perform more efficiently than pure meta-heuristics. We study problem-adapted inversion algorithms that exploit the knowledge of the smoothness of the misfit function of the problem. Optimal sampling strategies exist for such problems, but many of these problems remain hard.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Nolet, G., van Trier, J., Huisman, R.: A formalism for nonlinear inversion of seismic surface waves. Geoph. Res. Lett. 13, 26–29 (1986)
Nolet, G.: Partitioned wave-form inversion and 2D structure under the NARS array. J. Geophys. Res. 95, 8499–8512 (1990)
Snieder, R.: The role of nonlinearity in inverse problems. Inverse Problems 14, 387–404 (1998)
Sambridge, M.: Exploring multi-dimensional landscapes without a map. Inverse Problems 14(3), 427–440 (1998)
Kirkpatrick, S.C., Gelatt, D., Vecchi, M.P.: Optimization by simulated annealing. Science 220, 671–680 (1983)
Holland, J.H.: Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor (1975)
Goldberg, D.E.: Genetic Algorithms in Search, Optimization and Machine Learning. Kluwer Academic Publishers, Boston (1989)
Glover, F.: Future Paths for Integer Programming and Links to Artificial Intelligence. Comput. & Ops. Res. 13(5), 533–549 (1986)
Sambridge, M.: Geophysical inversion with a Neighbourhood algorithm -I. Searching a parameter space. Geoph. Jour. Int. 138, 479–494 (1999a)
Sambridge, M.: Geophysical inversion with a Neighbourhood algorithm -II. Appraising the ensemble. Geoph. Jour. Int. 138, 727–746 (1999b)
Cauchy, A.L.: First Turin Memoir (1831)
Mosegaard, K., Tarantola, A.: Monte Carlo sampling of solutions to inverse problems. J. Geophys. Res. 100(B7), 12,431–12,447 (1995)
Khachiyan, L.G.: The problem of computing the volume of polytopes is np-hard. Uspekhi Mat. Nauk. 44, 199–200 (1989)
Wolpert, D.H., Macready, W.G.: No Free Lunch Theorems for Optimization. IEEE Transactions on Evolutionary Computation 1, 67–82 (1997)
Brown, A.L.: Uniform approximation by radial basis functions. In: Light, W.A. (ed.) Advances in Numerical Analysis, vol. 2, pp. 203–206. Oxford University Press, Oxford (1992)
Papadimitriou, C.H., Steiglitz: Combinatorial Optimization: Algorithms and Complexity. Dover Publications, Inc., Mineola (1998)
Hastings, W.K.: Monte Carlo sampling methods using Markov Chain and their applications. Biometrika 57, 97–109 (1970)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mosegaard, K. (2012). Limits to Nonlinear Inversion. In: Jónasson, K. (eds) Applied Parallel and Scientific Computing. PARA 2010. Lecture Notes in Computer Science, vol 7133. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28151-8_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-28151-8_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-28150-1
Online ISBN: 978-3-642-28151-8
eBook Packages: Computer ScienceComputer Science (R0)