Optimization of Projection Methods for Solving Ill-Posed Problems

Optimization of Projection Methods for Solving ill-posed Problems. In this paper we propose a modification of the projection scheme for solving ill-posed problems. We show that this modification allows to obtain the best possible order of accuracy of Tikhonov Regularization using an amount of information which is far less than for the standard projection technique.Optimisierung von Projektionsverfahren für die Lösung von inkorrekt gestellten Problemen. In dieser Arbeit wird eine Modifizierung des Projektionsschemas zur Lösung inkorrekt gestellter Probleme vorgeschlagen. Wir zeigen, daß diese Modifizierung es ermöglicht, eine Genauigkeit der Tikhonov-Regularisierung von bestmöglicher Ordnung zu erhalten, wobei man eine wesentlich kleinere Menge von Informationen benutzt als beim Standard-Projektionsschema

Optimal discretization and Degrees of ill-posedness for inverse estimation in Hilbert scales in the presence of random noise

The problem of minimizing the difficulty of the inverse estimation of some unknown element x0 from noisy observations yδ = Ax0 + δξ in dependence of the nature of the random noise ξ is considered. It is shown that a combination of a Tikhonov regularization estimator with a certain projection scheme is order optimal in the sense of difficulty for a wide class of operators A acting along Hilbert scales

The discretized discrepancy principle under general source conditions

AbstractWe discuss adaptive strategies for choosing regularization parameters in Tikhonov–Phillips regularization of discretized linear operator equations. Two rules turn out to be based entirely on data from the underlying regularization scheme. Among them, only the discrepancy principle allows us to search for the optimal regularization parameter from the easiest problem. This potential advantage cannot be achieved by the standard projection scheme. We present a modified scheme, in which the discretization level varies with the successive regularization parameters, which has the advantage, mentioned before

A discretization of Volterra integrals equations of the third kind with weakly singular kernels

In this paper we propose a method of piecewise constant approximation for the solution of ill-posed third kind Volterra equations $$p(t)z(t) + int_0^t frac{h(t,tau)}{(t-tau)^{1-alpha}}z(tau) dtau = f(t),quad tin[0,1],enspace 0<alpha< 1.$$ Here $p(t)$ vanishes on some subset of $[t_1,t_2]subset[0,1]$ and mbox{$p(t) <delta$} for {mbox{$tin[t_1,t_2]$}}, where $delta$ is a sufficiently small positive number. The proposed method gives the accuracy $O(delta^{2nu/(2nu+1)})$ with respect to the mbox{$L_2$-norm,} where $nu$ is the parameter of sourcewise representation of the exact solution on $[t_1,t_2]$, and uses no more than $O(delta^{-(2-lambda)/alpha} log^{2+1/alpha}frac{1}{delta})$ values of Galer-kin functionals, where $lambdain(0,1/2)$ is determined in the act of choosing the regularization parameter within the framework of Morozov's discrepancy principle

Optimal discretization of inverse problems in Hilbert scales. Regularization and self-regularization of projection methods

We study the efficiency of the approximate solution of ill-posed problems, based on discretized observations, which we assume to be given afore-hand. We restrict ourselves to problems which can be formulated in Hilbert scales. Within this framework we shall quantify the degree of ill-posedness, provide general conditions on projection schemes to achieve the best possible order of accuracy. We pay particular attention on the problem of self-regularization vs. Tikhonov regularization. Moreover, we study the information complexity. Asymptotically, any method, which achieves the best possible order of accuracy must use at least such amount of noisy observations. We accomplish our study with two specific problems, Abel's integral equation and the recovery of continuous functions from noisy coefficients with respect to a given orthonormal system, both classical ill-posed problems

Self-regularization of projection methods with a posteriori discretization level choice for severely ill-posed problems

It is well known that projection schemes for certain linear ill-posed problems A퓍 = y can be regularized by a proper choice of the discretization level only, where no additional regularization is needed. The previous study of this self-regularization phenomenon was restricted to the case of so-called moderately ill-posed problems, i.e., when the singular values σ푘(A), 푘 = 1,2,..., of the operator A tend to zero with polynomial rate. The main accomplishment of the present paper is a new strategy for a discretization level choice that provides optimal order accuracy also for severely ill-posed problems, i.e., when σ푘(A) tend to zero exponentially. The proposed strategy does not require a priori information regarding the solution smoothness and the exact rate of σ푘(A)

On adaptive inverse estimating linear functionals of unknown smoothness in Hilbert scales

We address the problem of estimating the value of a linear functional 〈 ƒ,푥 〉 from random noisy observations of 푦 = A 푥 in Hilbert scales. Both the white noise and density observation models are considered. We develop an inverse estimator of 〈 ƒ,푥 〉 which automatically adapts to unknown smoothness of 푥 and ƒ. It is shown that accuracy of this adaptive estimator is only by a logarithmic factor worse than one could achieve in the case of known smoothness. As an illustrative example, the problem of deconvolving a bivariate density with singular support is considered

A discretization of Volterra integrals equations of the third kind with weakly singular kernels

In this paper we propose a method of piecewise constant approximation for the solution of ill-posed third kind Volterra equations $$p(t)z(t) + int_0^t frac{h(t,tau)}{(t-tau)^{1-alpha}}z(tau) dtau = f(t),quad tin[0,1],enspace 0<alpha< 1.$$ Here $p(t)$ vanishes on some subset of $[t_1,t_2]subset[0,1]$ and mbox{$p(t) <delta$} for {mbox{$tin[t_1,t_2]$}}, where $delta$ is a sufficiently small positive number. The proposed method gives the accuracy $O(delta^{2nu/(2nu+1)})$ with respect to the mbox{$L_2$-norm,} where $nu$ is the parameter of sourcewise representation of the exact solution on $[t_1,t_2]$, and uses no more than $O(delta^{-(2-lambda)/alpha} log^{2+1/alpha}frac{1}{delta})$ values of Galer-kin functionals, where $lambdain(0,1/2)$ is determined in the act of choosing the regularization parameter within the framework of Morozov's discrepancy principle

A discretization of Volterra integrals equations of the third kind with weakly singular kernels

In this paper we propose a method of piecewise constant approximation for the solution of ill-posed third kind Volterra equations $$p(t)z(t) + int_0^t frac{h(t,tau)}{(t-tau)^{1-alpha}}z(tau) dtau = f(t),quad tin[0,1],enspace 0<alpha< 1.$$ Here $p(t)$ vanishes on some subset of $[t_1,t_2]subset[0,1]$ and mbox{$p(t) <delta$} for {mbox{$tin[t_1,t_2]$}}, where $delta$ is a sufficiently small positive number. The proposed method gives the accuracy $O(delta^{2nu/(2nu+1)})$ with respect to the mbox{$L_2$-norm,} where $nu$ is the parameter of sourcewise representation of the exact solution on $[t_1,t_2]$, and uses no more than $O(delta^{-(2-lambda)/alpha} log^{2+1/alpha}frac{1}{delta})$ values of Galer-kin functionals, where $lambdain(0,1/2)$ is determined in the act of choosing the regularization parameter within the framework of Morozov's discrepancy principle