The goal of this paper is to investigate the Tikhonov-Phillips method for semi-discrete linear ill-posed problems in order to determine tight error bounds and to obtain a good parameter choice. We consider the equation Af = g, where the operator A is known, noisy discrete data g δ 1 , . . . , g δ n with g(xi) ≈ g δ i can be observed and a solution f * is sought. Assuming that f * is an element of a Sobolev space, we use the well-known theory of optimal recovery in Hilbert spaces for the reconstruction process. We then provide L2-error estimates in terms of the data density and derive an a priori selection for the regularization parameter, which guarantees an optimal compromise between approximation and stability. Finally, we illustrate the parameter selection with a simple example.
In this paper, a size-structured model for cell division is examined and the question of determining the division (birth) rate from a measurable stable size distribution of the population is addressed. This inverse problem can be formulated as a differential-dilation equation. We propose a novel solution scheme based on mollification. The method of approximate inverse allows us to shift the derivative from the data to a precomputable reconstruction kernel. To comprise all available a priori information, a presmoothing step based on regression in reproducing kernel Hilbert spaces is introduced. We establish an error theory for the emerging algorithm, prove convergence and deduce a parameter strategy. The results are substantiated with extensive numerical tests both for artificial and real data based on proliferating tumor cells.
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 -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 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.
In this paper, a population balance equation, originating from applications in chemical engineering, is considered and novel solution techniques for a related inverse problem are presented. This problem consists in the determination of the breakage rate and the daughter drop distribution of an evolving drop size distribution from time-dependent measurements under the assumption of self-similarity. We analyze two established solution methods for this ill-posed problem and improve the two procedures by adapting suitable data fitting and inversion algorithms to the specific situation. In addition, we introduce a novel technique that, compared to the former, does not require certain a priori information. The improved stability properties of the resulting algorithms are substantiated with numerical examples.
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