Abstract
Uncertainty is always present in inverse problems. The main reasons for that are noise in data and measurement error, solution non-uniqueness, data coverage and bandwidth limitations, physical assumptions and numerical approximations. In the context of nonlinear inversion, the uncertainty problem is that of quantifying the variability in the model space supported by prior information and the observed data. In this paper we outline a general nonlinear inverse uncertainty estimation method that allows for the comprehensive search of model posterior space while maintaining computational efficiencies similar to deterministic inversions. Integral to this method is the combination of model reduction techniques, a constrained mapping approach and a sparse sampling scheme. This approach allows for uncertainty quantification in inverse problems in high dimensional spaces and very costly forward evaluations. We show some results in non linear geophysical inversion (electromagnetic data).
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References
Avis, D., Fukuda, K.: A pivoting algorithm for convex bulls and vertex enumeration of arrangements and polyhedra. J. Discrete Comp. Geometry 8, 295–313 (1992)
Fernández-Martínez, J.L., Tompkins, M., Fernández-Muñiz, Z., Mukerji, T.: Inverse problems and model reduction techniques. In: Borgelt, C., González-Rodríguez, G., Trutschnig, W., Lubiano, M.A., Gil, M.A., Grzegorzewski, P., Hryniewicz, O. (eds.) Combining Soft Computing and Statistical Methods in Data Analysis. Advances in Soft Computing. Springer, Berlin (in this book, 2010)
Fernández-Martínez, J.L., Mukerji, T., García-Gonzalo, E.: Particle Swarm Optimization in high dimensional spaces. In: Proceedings of the Seventh International Conference on Swarm Intelligence, ANTS 2010, Bruxelles, Belgium (2010)
MacGregor, L.M., Sinha, M.C., Constable, S.: Electrical resistivity structure of the Valu Fa Ridge, Lau Basin, from marine controlled-source electromagnetic sounding. Geophys. J. Int. 146, 217–236 (2001)
Ganapathysubramanian, B., Zabaras, N.: Modeling diffusion in random heterogeneous media: Data-driven models, stochastic collocation and the variational multiscale method. J. Comput. Phys. 226, 326–353 (2007)
Smolyak, S.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Math. 4, 240–243 (1963)
Tompkins, M.J., Fernández-Martínez, J.L.: Scalable Solutions for Nonlinear Inverse Uncertainty Using Model Reduction, Constraint Mapping, and Sparse Sampling. In: Proceedings of the 72nd EAGE Conference & Exhibition, Barcelona, Spain (2010)
Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27, 1118–1139 (2005)
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Fernández-Martínez, J.L., Tompkins, M., Mukerji, T., Alumbaugh, D. (2010). Geometric Sampling: An Approach to Uncertainty in High Dimensional Spaces. In: Borgelt, C., et al. Combining Soft Computing and Statistical Methods in Data Analysis. Advances in Intelligent and Soft Computing, vol 77. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14746-3_31
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DOI: https://doi.org/10.1007/978-3-642-14746-3_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-14745-6
Online ISBN: 978-3-642-14746-3
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