Operator Equations In Hilbert Spaces Research Articles

Non-resonance and existence for systems of non-linear operator equations

An existence theory for systems of two non-linear operator equations in Hilbert spaces is presented under non-resonance conditions with respect to two spectra and in terms of matrices convergent to zero. The theory is then applied to elliptic systems.

The exact solution of system of linear operator equations in reproducing kernel spaces

In this paper, we discuss the problem of how to solve the system of linear operator equations in Hilbert spaces and applied the result to reproducing kernel spaces. Under the assumption that the operator equations has a unique solution, its formal solution is given. As a result, when H , H 1 are both reproducing kernel spaces, the formal solution turned into analytic solution. Truncating the series (where formal solution and the analytic solution are denoted by a series), the approximate solution of the operator equations is obtained. When increasing the number of the nodes, the error of the approximate solution is monotone decreasing in the sense of the norm of the Hilbert space. The final numerical experiment shows the efficiency of our method.

Regularization by projection in variable Hilbert scales

We introduce and analyze projection schemes for solving ill-posed linear operator equations in Hilbert space. Control parameters are shown to yield convergence. Emphasis is on their approximation theoretic content. For smoothness given in terms of general source conditions we establish convergence rates, shown to be optimal in some cases. The study is accomplished with a discussion on condition numbers.

Newton methods for nonlinear inverse problems with random noise

AbstractWe study the convergence of regularized Newton methods applied to nonlinear operator equations in Hilbert spaces if the data are perturbed by random noise. We show that under certain conditions it is possible to achieve the minimax rates of the corresponding linearized problem if the smoothness of the solution is known. If the smoothness is unknown and the stopping index is determined by Lepskij's balancing principle, we show that the rates remain the same up to a logarithmic factor due to adaptation. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A modified Newton–Lavrentiev regularization for nonlinear ill-posed Hammerstein-type operator equations

Recently, a new iterative method, called Newton–Lavrentiev regularization (NLR) method, was considered by George (2006) for regularizing a nonlinear ill-posed Hammerstein-type operator equation in Hilbert spaces. In this paper we introduce a modified form of the NLR method and derive order optimal error bounds by choosing the regularization parameter according to the adaptive scheme considered by Pereverzev and Schock (2005).

Analysis of Profile Functions for General Linear Regularization Methods

The stable approximate solution of ill‐posed linear operator equations in Hilbert spaces requires regularization. Tight bounds for the noise‐free part of the regularization error are constitutive for bounding the overall error. Norm bounds of the noise‐free part which decrease to zero along with the regularization parameter are called profile functions and are the subject of our analysis. The interplay between properties of the regularization and certain smoothness properties of solution sets, which we shall describe in terms of sourcewise representations, is crucial for the decay of associated profile functions. On the one hand, we show that a given decay rate is possible only if the underlying true solution has appropriate smoothness. On the other hand, if smoothness fits the regularization, then decay rates are easily obtained. If smoothness does not fit, then we will measure this in terms of some distance function. Tight bounds for these allow us to obtain profile functions. Finally we study the most realistic case when smoothness is measured with respect to some operator which is related to the one governing the original equation only through a link condition. In many parts the analysis is done on geometric basis, extending classical concepts of linear regularization theory in Hilbert spaces. We emphasize intrinsic features of linear ill‐posed problems which are frequently hidden in the classical analysis of such problems.

Convergence of theL-regularization method in solving nonlinear operator equations

Necessary and sufficient conditions of convergence of the L-regularization methods for solution of nonlinear operator equations in Hilbert spaces are found.

Convergence of the <I>L</I>-regularization method in solving nonlinear operator equations

Necessary and sufficient conditions of convergence of the L-regularization methods for solution of nonlinear operator equations in Hilbert spaces are found.

Newton–Lavrentiev regularization of ill-posed Hammerstein type operator equation

In this paper we report on a new iterative method for regularizing a nonlinear Hammerstein type operator equation in Hilbert spaces. The proposed Newton–Lavrentiev method is a combination of Lavrentiev regularization and a Newton's iteration. Under the assumptions that the operator F is continuous Fréchet differentiable with a Lipschitz-continuous first derivative and that the solution of (1.1) fulfills a smoothness condition, we will give a convergence rate result.

Strong approximation of the solutions of a system of operator equations in Hilbert spaces

We give sharp conditions for the strong approximation of the solutions of a system of the form by the solutions of a system of difference equations of the form where F i are monotone operators in a Hilbert space and is the metric projection. Thus, the results complete (and correct) those by Chidume et al. [Chidume, C.E. and Zegeye, H., 2004, Approximation of solutions of nonlinear equations of Hammerstein type in Hilbert spaces, Proceedings of the American Mathematical Society, 133(3), 851–858]. Also, we are interested in what happens when small perturbations of F i and q ij occur. Meanwhile, we generalize a numerical difference inequality due to Alber (see, e.g. Alber, Ya. I., 1983, Recurrence relations and variational inequalities, Soviet Mathematics Doklady, 27, 511–517) which is used to obtain approximation results.

Newton–Tikhonov regularization of ill-posed Hammerstein operator equation

In this paper we report on a new iterative method for regularizing a nonlinear Hammerstein operator equation in Hilbert spaces. The proposed Newton-Tikhonov method is a combination of Tikhonov regularization and a Newton's iteration. Under the assumptions that the operator F is continuous Fréchet differentiable with a Lipschitz-continuous first derivative and that the solution of (1.1) fulfills a smoothness condition, we will give a convergence rate result.

On the use of fixed point iterations for the regularization of nonlinear ill-posed problems

We report on a new iterative method for regularizing a nonlinear operator equation in Hilbert spaces. The proposed algorithm is a combination of Tikhonov regularization and a fixed point algorithm for the minimization of the Tikhonov functional. Under the assumptions that the operator F is twice continuous Fréchet-differentiable with Lipschitz-continuous first derivative and that the solution of the equation F (x) = y fulfills a smoothness condition we will give a convergence rate result. Numerical results with data from Single Photon Emission Computed Tomography (SPECT) show the rapid convergence of the proposed algorithms.

Complete second order linear differential operator equations in Hilbert space and applications in hydrodynamics

We study the Cauchy problem for a complete second order linear differential operator equation in a Hilbert space H {\mathcal H} of the form \[ d 2 u d t 2 + ( F + i K ) d u d t + B u = f , u ( 0 ) = u 0 , u ′ ( 0 ) = u 1 . \frac {d^2u}{dt^2}+(F+\textrm {i}K)\frac {du}{dt}+Bu=f,\quad u(0)=u^0, \quad u’(0)=u^1. \] Problems of this kind arise, e.g., in hydrodynamics where the coefficients F F , K K , and B B are unbounded selfadjoint operators. It is assumed that F F is the dominating operator in the Cauchy problem above, i.e., \[ D ( F ) ⊂ D ( B ) , D ( F ) ⊂ D ( K ) . {\mathcal D}(F)\subset {\mathcal D}(B),\quad {\mathcal D}(F)\subset {\mathcal D}(K). \] We also suppose that F F and B B are bounded from below, but the operator coefficients are not assumed to commute. The main results concern the existence of strong solutions to the stated Cauchy problem and applications of these results to the Cauchy problem associated with small motions of some hydrodynamical systems.

How to solve nonlinear operator equation A( v2)+ Cv= f

In this paper, we discuss the problems of how to solve linear operator equation in Hilbert spaces and a kind of nonlinear operator equation A( v 2)+ Cv= f in reproducing kernel space W 2 1[ a, b]. For linear operator equation, we obtain a criterion about the existence of solutions. If it has solutions, we get the analytic representation of its minimum norm solution and the structure of its solution space; For nonlinear operator equation A( v 2)+ Cv= f, by using the method of solving linear operator equation, one exact solution is given. Besides, we obtain the representation of its solution set. Final examples show our methods are effective.

TIGRA—an iterative algorithm for regularizing nonlinear ill-posed problems

We report on a new iterative method for regularizing a nonlinear operator equation inHilbert spaces. The proposed TIGRA algorithm is a combination of Tikhonov regularization and a gradientmethod for minimizing the Tikhonov functional. Under the assumptions that the operatorF is twice continuousFréchet differentiable with a Lipschitz-continuous first derivative and that the solution of theequation F (x) = yfulfils a smoothness condition, we will give a convergence rate result. Finally wepresent some applications and a numerical result for the reconstruction of theactivity function in single-photon-emission computed tomography.

Variable Preconditioning via Quasi-Newton Methods for Nonlinear Problems in Hilbert Space

The aim of this paper is to develop stepwise variable preconditioning for the iterative solution of monotone operator equations in Hilbert space and apply it to nonlinear elliptic problems. The paper is built up to reflect the common character of preconditioned simple iterations and quasi-Newton methods. The main feature of the results is that the preconditioners are chosen via spectral equivalence. The latter can be executed in the corresponding Sobolev space in the case of elliptic problems, which helps both the construction and convergence analysis of preconditioners. This is illustrated by an example of a preconditioner using suitable domain decomposition.

A new iterative algorithm for reconstructing a signal from its dyadic wavelet transform modulus maxima

A new algorithm for reconstructing a signal from its wavelet transform modulus maxima is presented based on an iterative method for solutions to monotone operator equations in Hilbert spaces. The algorithm’s convergence is proved. Numerical simulations for different types of signals are given. The results indicate that compared with Mallat’s alternate projection method, the proposed algorithm is simpler, faster and more effective.

Bounds for perturbed solutions of linear operator equations in Hilbert space

Some perturbation results on relative errors of solutions to singular linear operator equations in Hilbert space are given for consistent systems and for weighted least-squares problems with the help of weighted generalized inverses.

The Fast Solution of Periodic Integral and Pseudodifferential Equations by Gmres

Abstract In this paper we consider GMRES to solve finite-dimensional approxi- mations of a class of well-posed linear operator equations in Hilbert spaces. It is shown that the speed of convergence is superlinear. As a consequence we have that GMRES can be used as a fast solver of a fully discrete variant of the trigonometric Galerkin equations associated with periodic integral equations.

Identification of the diffusion coefficient in aone-dimensional parabolic equation

This paper deals with the problem of the identification of the diffusioncoefficient in a parabolic equation. This inverse problem is formulated asa nonlinear operator equation in Hilbert spaces. Continuity and differentiability ofthe corresponding operator is shown. In the one-dimensional case uniqueness and conditionalstability results are obtained by using the heat equation transform. Finally, the problem issolved numerically by iteratively regularized Gauss-Newton method.