From Bayes to Tarantola: New insights to understand uncertainty in inverse problems |
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| Institution: | 1. Departamento of Matemáticas, Universidad de Oviedo, Oviedo, Spain;2. ETSI en Topografía, Geodésia y Cartografía, Universidad Politécnica de Madrid, Madrid, Spain;1. National Research Institute of Astronomy and Geophysics (NRIAG), Helwan, Cairo, Egypt;2. Cairo University, Faculty of science, Geophysics department, Giza, Egypt;1. Deutsches Zentrum für Luft- und Raumfahrt (DLR), Institut für Methodik der Fernerkundung (IMF), Oberpfaffenhofen 82234, Germany;2. NASA Goddard Institute for Space Studies, 2880 Broadway, New York, NY 10025, USA;1. Deutsches Zentrum für Luft- und Raumfahrt (DLR), Institut für Methodik der Fernerkundung (IMF), Oberpfaffenhofen 82234, Germany;2. NASA Goddard Institute for Space Studies, 2880 Broadway, New York, NY 10025, USA;1. Deutsches Zentrum für Luft- und Raumfahrt (DLR), Institut für Methodik der Fernerkundung (IMF), Oberpfaffenhofen 82234, Germany;2. NASA Goddard Institute for Space Studies, 2880 Broadway, New York, NY 10025, USA |
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| Abstract: | Anyone working on inverse problems is aware of their ill-posed character. In the case of inverse problems, this concept (ill-posed) proposed by J. Hadamard in 1902, admits revision since it is somehow related to their ill-conditioning and the use of local optimization methods to find their solution. A more general and interesting approach regarding risk analysis and epistemological decision making would consist in analyzing the existence of families of equivalent model parameters that are compatible with the prior information and predict the observed data within the same error bounds. Otherwise said, the ill-posed character of discrete inverse problems (ill-conditioning) originates that their solution is uncertain. Traditionally nonlinear inverse problems in discrete form have been solved via local optimization methods with regularization, but linear analysis techniques failed to account for the uncertainty in the solution that it is adopted. As a result of this fact uncertainty analysis in nonlinear inverse problems has been approached in a probabilistic framework (Bayesian approach), but these methods are hindered by the curse of dimensionality and by the high computational cost needed to solve the corresponding forward problems. Global optimization techniques are very attractive, but most of the times are heuristic and have the same limitations than Monte Carlo methods. New research is needed to provide uncertainty estimates, especially in the case of high dimensional nonlinear inverse problems with very costly forward problems. After the discredit of deterministic methods and some initial years of Bayesian fever, now the pendulum seems to return back, because practitioners are aware that the uncertainty analysis in high dimensional nonlinear inverse problems cannot (and should not be) solved via random sampling methodologies. The main reason is that the uncertainty “space” of nonlinear inverse problems has a mathematical structure that is embedded in the forward physics and also in the observed data. Thus, problems with structure should be approached via linear algebra and optimization techniques. This paper provides new insights to understand uncertainty from a deterministic point of view, which is a necessary step to design more efficient methods to sample the uncertainty region(s) of equivalent solutions. |
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