This paper is inspired by recently proposed approach for interpreting data of Electrochemical Impedance Spectroscopy
(EIS) in terms of Distribution of Diffusion Times (DDT). Such an interpretation requires to solve a Fredholm integral
equation of the first kind, which may have a non-square-integrable kernel. We consider a class of equations with
above-mentioned peculiarity and propose to regularize them in weighted functional spaces. One more issue associated
with DDT-problem is that EIS data are available only for a finite number of frequencies. Therefore, a regularization
should unavoidably be combined with a collocation. In this paper we analyze a regularized collocation in weighted
spaces and propose a scheme for its numerical implementation. The performance of the proposed scheme is illustrated by
numerical experiments with synthetic data mimicking EIS measurements.
In this paper, we consider the problem of estimating the derivative of a function from its noisy version contaminated by a stochastic white noise and argue that in certain relevant cases the reconstruction of by the derivatives of the partial sums of Fourier–Legendre series of has advantage over some standard approaches. One of the interesting observations made in the paper is that in a Hilbert scale generated by the system of Legendre polynomials the stochastic white noise does not increase, as it might be expected, the loss of accuracy compared to the deterministic noise of the same intensity. We discuss the accuracy of the considered method in the spaces L2 and C and provide a guideline for an adaptive choice of the number of terms in differentiated partial sums (note that this number is playing the role of a regularization parameter). Moreover, we discuss the relation of the considered numerical differentiation scheme with the well-known Savitzky–Golay derivative filters, as well as possible applications in diabetes technology.
For solving linear ill-posed problems regularization methods are required when the right-hand side is with some noise. In the present paper regularized solutions are obtained by multi-parameter regularization and the regularization parameters are chosen by a multi-parameter discrepancy principle. Under certain smoothness assumptions we provide order optimal error bounds that characterize the accuracy of the regularized solutions. For the computation of the regularization parameters fast algorithms of Newton type are applied which are based on special transformations. These algorithms are globally and monotonically convergent. Some of our theoretical results are illustrated by numerical experiments. We also show how the proposed approach may be employed for multi-task approximation.
In this paper we continue our study of solving ill-posed problems with a noisy right-hand side and a noisy operator. Regularized approximations are obtained by Tikhonov regularization with differential operators and by dual regularized total least squares (dual RTLS) which can be characterized as a special multi-parameter regularization method where one of the two regularization parameters is negative. We report on order optimality results for both regularized approximations, discuss compu-tational aspects, provide special algorithms and show by experiments that dual RTLS is competitive to Tikhonov regularization with differential operators.