Nonlinear inverse problems in industry typically have an underdetermined character because of the high number of model parameters needed to achieve accurate data predictions. The fact that inverse problems are highly underdetermined implies that many solutions are compatible with the prior information and fit the observed data within the same error bounds. These solutions are called equivalent and are located in one or several flat disconnected curvilinear valleys of the cost-function topography. Random sampling of these equivalent models is impossible because of the curse of dimensionality and the high computational cost needed for providing the corresponding forward predictions because in most of the currently used sampling methods, sampling and forward evaluation are coupled. In addition, in most cases, the sampling methodologies have a random character. Analyzing the importance of model reduction in the uncertainty analysis of nonlinear inverse and optimization problems shows that model reduction alleviates the ill-posed character of the corresponding inverse problem and allows efficient deterministic sampling of the equivalent solutions and/or optimization in the reduced basis set. Such problems are common in geophysical inversion and in reservoir production optimization and history matching.

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