This thesis presents a robust method for estimating the relaxations of a metallic object from its electromagnetic induction (EMI) response. The EMI response of a metallic object can be accurately modeled by a sum of real decaying exponentials. However, it is difficult to obtain the model parameters from measurements when the number of exponentials in the sum is unknown or the terms are strongly correlated. Traditionally, the relaxation constants are estimated by nonlinear iterative search that often leads to unsatisfactory results. An effective EMI modeling technique is developed by first linearizing the problem through enumeration and then solving the linearized model using a sparsity-regularized minimization. This approach overcomes several long-standing challenges in EMI signal modeling, including finding the unknown model order as well as handling the ill-posed nature of the problem. The resulting algorithm does not require a good initial guess to converge to a satisfactory solution. This new modeling technique is extended to incorporate multiple measurements in a single parameter estimation step. More accurate estimates are obtained by exploiting an invariance property of the EMI response, which states that the relaxation frequencies do not change for different locations and orientations of a metallic object. Using tests on synthetic data and laboratory measurement of known targets, the proposed multiple-measurement method is shown to provide accurate and stable estimates of the model parameters. The ability to estimate the relaxation constants of targets enables more robust subsurface target discrimination using the relaxations. A simple relaxation-based subsurface target detection algorithm is developed to demonstrate the potential of the estimated relaxations.

Ill-posed inverse problems with quadratic structure are introduced, studied and solved. As an example an inverse problem appearing in laser optics is solved numerically based on a new regularized inversion algorithm. In addition, the theory of sparsity promoting regularization is extended to situations in which sparsity cannot be expected and also to equations with non-injective operators.

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