Effect of using the 1-norm on the data and regularization terms of inverse problems, and implementation with Primal Dual Interior Point Methods

Andrea Borsic

Thayer School, Dartmouth College.

Parameter estimation in inverse problems is traditionally formulated as a least squares model fitting problem, with quadratic regularization. In the presence of outliers on the data, an in the context of ill-posed problems, the use of least squares approaches is particularly sensitive to data errors. Use of least absolute values fitting (1-norm on the data term) approaches is robust to presence of outliers and can produce meaningful results where least squares approaches fail. The use of the 1-norm is also interesting as an approach to regularization, leading to Total Variation regularization. This approach allows reconstructing parameters with much faster spatial variations compared to traditional quadratic regularization approaches. A Primal Dual Interior Point framework for the estimation of parameters in inverse problems using the 1-norm on the data and regularization terms will be discussed and practical application examples will be shown.

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