Abstract
The aim of this paper is to establish a support vector regression method for semi-discrete ill-posed problems. We consider the equation Af = g, where a linear integral operator A is known; discrete measurements of the right-hand side are observed and a solution f* is sought. For the reconstruction, instead of a standard square loss function, Vapnik's ϵ-intensive function is used to measure the distance between Af and g. This avoids an overfitting to disturbed data and guarantees additional stability given that the cut-off parameter ϵ is chosen appropriately. The resulting solution procedure can be formulated as a quadratic program. Besides the method, a Sobolev error analysis and a parameter strategy for the regularization parameters are provided. The results are substantiated with a numerical example.
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