In this paper we clarify the relations between the existing sets of regularity conditions for convergence rates of nonparametric indirect regression (NPIR) and non-parametric instrumental variables (NPIV) regression models. We establish minimax risk lower bounds in mean integrated squared error loss for the NPIR and NPIV models under two basic regularity conditions: the approximation number and...

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A method for solving the eigen-problem of a general complex matrix can be constructed by first using a simple double-sided Lanczos algorithm to obtain the eigenvalues and next applying inverse iteration to find the eigenvectors. Even though (strict) convergence is not generally guaranteed, many well-posed physical and chemical problems can be transformed into eigen-problems of non-pathological ...

Recently, the metrics of Ky Fan and Prokhorov were introduced as a tool for studying convergence in stochastic ill-posed problems. In this work, we show that the Bayesian approach to linear inverse problems can be examined in the new framework as well. We consider the finitedimensional case where the measurements are disturbed by an additive normal noise and the prior distribution is normal. Co...

Motivated by Candes and Donoho′s work (Candés, E J, Donoho, D L, Recovering edges in ill-posed inverse problems: optimality of curvelet frames. Ann. Stat. 30, 784-842 (2002)), this paper is devoted to giving a lower bound of minimax mean square errors for Riesz fractional integration transforms and Bessel transforms.

To assess the quality of solutions in stochastic inverse problems, a proper measure for the distance of random variables is essential. The aim of this note is the comparison of the metrics of Ky Fan and Prokhorov with other concepts such as expected values, probability estimates and almost sure convergence. In ill-posed problems one aims to find an appropriate solution x† to an equation of the ...

The paper is concerned with the solution of nonlinear ill-posed problems by methods that utilise the second derivative. A general predictor{corrector approach is developed; one which avoids solving quadratic equations during the iteration process. Combining regularisation of each iteration step with an adequate stopping condition leads to a general regularisation scheme for nonlinear equations....

We consider a “local” Tikhonov regularization method for ill-posed Volterra problems. In addition to leading to efficient numerical schemes for inverse problems of this type, a feature of the method is that one may impose varying amounts of local smoothness on the solution, i.e., more regularization may be applied in some regions of the solution’s domain, and less in others. Here we present pro...

In this paper we consider the iteratively regularized Gauss–Newton method for solving nonlinear ill-posed inverse problems. Under merely the Lipschitz condition, we prove that this method together with an a posteriori stopping rule defines an order optimal regularization method if the solution is regular in some suitable sense.

In this paper we study some properties of the classical Arnoldi based methods for solving infinite dimensional linear equations involving compact operators. These problems are intrinsically ill-posed since a compact operator does not admit a bounded inverse. We study the convergence properties and the ability of these algorithms to estimate the dominant singular values of the operator.