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We present exploratory data analysis methods to assess inversion estimates using examples based on 2- and 1-regularization. These methods can be used to reveal the presence of systematic errors such as bias and discretization effects, or to validate assumptions made on the statistical model used in the analysis. The methods include bounds on the performance of randomized estimators of a large matrix, confidence intervals and bounds for the bias, resampling methods for model validation and construction of training sets of functions with controlled local regularity. [ABSTRACT FROM AUTHOR]

Design of experiments for discrete ill-posed problems is a relatively new area of research. While there has been some limited work concerning the linear case, little has been done to study design criteria and numerical methods for ill-posed nonlinear problems. We present an algorithmic framework for nonlinear experimental design with an efficient numerical implementation. The data are modeled as indirect, noisy observations of the model collected via a set of plausible experiments. An inversion estimate based on these data is obtained by a weighted Tikhonov regularization whose weights control the contribution of the different experiments to the data misfit term. These weights are selected by minimization of an empirical estimate of the Bayes risk that is penalized to promote sparsity. This formulation entails a bilevel optimization problem that is solved using a simple descent method. We demonstrate the viability of our design with a problem in electromagnetic imaging based on direct current resistivity and magnetotelluric data. [ABSTRACT FROM AUTHOR]