The curse of dimensionality in inverse problems

Abstract

Nonlinear inverse problems in real problems in industry have typically a very underdetermined character due to the high number of parameters that are usually needed to achieve accurate forward predictions. The corresponding inverse problem is ill-posed, that is, there exist many solutions which are compatible with the prior information, fitting the observed data within the same error bounds. These solutions are located in (one or several) flat curvilinear and disconnected valleys of the cost function topography. The random sampling of these equivalent models is impossible due to the curse of dimensionality and to the high computational cost needed to provide the corresponding forward predictions. This paper generalizes the curse of dimensionality to linear and nonlinear inverse problems outlining the main differences between them. With a simple 2D example we show that nonlinearities allow for a reduction in size of the nonlinear equivalence region that could be embedded in a linear hyperquadric with smaller condition number than the corresponding linearized equivalence region. We also analyze the effect of the regularization in the posterior sampling, and that of the dimensionality reduction, which is needed to perform efficient sampling of the region of uncertainty equivalence in high dimensional problems. We hope that the additional theoretical knowledge provided by this research will help practitioners to design more efficient methods of sampling.