Physics-based models for measurement correlations. Application to an inverse Sturm-Liouville problem

In many inverse problems, the measurement operator, which maps objects of interest to available measurements, is a smoothing (regularizing) operator. Its inverse is therefore unbounded and as a consequence, only the low frequency component of the object of interest is accessible from inevitably noisy measurements. In many inverse problems however, the neglected high frequency component may significantly affect the measured data. Using simple scaling arguments, we characterize the influence of the high frequency component. We then consider situations where the correlation function of such an influence may be estimated by asymptotic expansions, for instance as a random corrector in homogenization theory. This allows us to consistently eliminate the high frequency component and derive a closed-form, more accurate, inverse problem for the low frequency component of the object of interest. We present the asymptotic expression of the correlation matrix of the eigenvalues in a Sturm-Liouville problem with unknown potential. We propose an iterative algorithm for the reconstruction of the potential from knowledge of the eigenvalues and show that using the approximate correlation matrix significantly improves the reconstructions. keywords: inverse problem, measurement correlations, Sturm-Liouville, random fluctuations, central limit correction to homogenization.