Error estimates for simplified Levenberg–Marquardt method for nonlinear ill-posed operator equations in Hilbert spaces

Iteratively regularized Gauss–Newton type methods for approximating quasi–solutions of irregular nonlinear operator equations in Hilbert space with an application to COVID–19 epidemic dynamics

A brief survey is given concerning iterative processes of Fejér type for basic statements of ill-posed problems, including constrained quadratic and convex minimization problems, variational inequalities, and linear and nonlinear operator equations in Hilbert spaces. By applying the method of successive approximations and its modification using correction factors, all these statements reduce to the problem of finding fixed points of nonexpansive Fejér operators. Material is also presented related to a two-stage method of constructing a regularizing algorithm for nonlinear ill-posed problems with monotone operators. An economic way is described by which the algorithm takes into account additional a priori information on the solution using Fejér maps.

We report on a new iterative method for regularizing a nonlinear operator equation in Hilbert spaces. The proposed IIGRA algorithm is combination of nonlinear implicit iterative method and a gradient method for minimizing a modified Tikhonov functional. Under some not very strict assumptions, we will give a convergence result. Finally we present two numerical results for integral equation and parameter identification.

Local convergence theorems for gradient methods in hilbert space

A general scheme for improving approximate solutions to irregular nonlinear operator equations in Hilbert spaces is proposed and analyzed in the presence of errors. A modification of this scheme designed for equations with quadratic operators is also examined. The technique of universal linear approximations of irregular equations is combined with the projection onto finite-dimensional subspaces of a special form. It is shown that, for finite-dimensional quadratic problems, the proposed scheme provides information about the global geometric properties of the intersections of quadrics.

In this paper, we deal with nonlinear ill-posed operator equations involving a monotone operator in the setting of Hilbert scales. Our convergence analysis of the proposed derivative-free method is based on the simple property of the norm of a self-adjoint operator. Using a general Hölder-type source condition, we obtain an optimal order error estimate. Also we consider the adaptive parameter choice strategy proposed by Pereverzev and Schock (2005) for choosing the regularization parameter. Finally, we applied the proposed method to the parameter identification problem in an elliptic PDE in the setting of Hilbert scales and compare the results with the corresponding method in Hilbert space.

A Boundary Value Problem for Operator Equations in Hilbert Spaces

Numerical solution of ill-posed operator equations requires regularization techniques. The convergence of regularized solutions to the exact solution can be usually guaranteed, but to also obtain estimates for the speed of convergence one has to exploit some kind of smoothness of the exact solution. We consider four such smoothness concepts in a Hilbert space setting: source conditions, approximate source conditions, variational inequalities, and approximate variational inequalities. Besides some new auxiliary results on variational inequalities the equivalence of the last three concepts is shown. In addition, it turns out that the classical concept of source conditions and the modern concept of variational inequalities are connected via Fenchel duality.

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.

- In this paper a convergence proof is given for the continuous analog of the Gauss-Newton method for nonlinear ill-posed operator equations and convergence rates are obtained. Convergence for exact data is proved for nonmonotone operators under weaker source conditions than before. Moreover, nonlinear ill-posed problems with noisy data are considered and a priori and a posteriori stopping rules are proposed. These rules yield convergence of the regularized approximations to the exact solution as the noise level tends to zero. The convergence rates are optimal rates under the source conditions considered.

Splittings and iterative methods for approximate solutions to singular operator equations in Hilbert spaces

As an application of the theory of reproducing kernel Hilbert spaces we prove a theorem on the existence of bounded solutions of a system of linear operator equations in Hilbert spaces. This result is obtained by using locally convex spaces and is a corollary to an extension of the well-known Parrott’s theorem. Our theorems extend the Strong Parrott Theorem and its extensions. Our tools for the proof are the generalized reproducing kernel Hilbert spaces due to Schwartz, and the complementary spaces due to de Branges and Rovnyak.

Previous article Next article The Generalized Overrelaxation Method for the Approximate Solution of Operator Equations in Hilbert SpaceW. V. PetryshynW. V. Petryshynhttps://doi.org/10.1137/0110052PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Julius Albrecht, Fehlerabschätzungen bei Relaxationsverfahren zur numerischen Auflösung linearer Gleichungssysteme, Numer. Math., 3 (1961), 188–201 10.1007/BF01386019 MR0146957 0111.12602 CrossrefGoogle Scholar[2] Hans Bückner, A special method of successive approximations for Fredholm integral equations, Duke Math. J., 15 (1948), 197–206 MR0025290 0030.39201 CrossrefISIGoogle Scholar[3] Yu. L. Daleckii˘, Integration and differentiation of functions of Hermitian operators depending on a parameter, Uspehi Mat. Nauk (N.S.), 12 (1957), 182–186 MR0086276 Google Scholar[4] W. Dück, Eine Fehlerabschätzung zum Einzelschrittverfahren bei linearen Gleichungssystemen, Numer. Math., 1 (1959), 73–77 MR0102910 0084.34504 CrossrefGoogle Scholar[5A] Gene H. Golub and , Richard S. Varga, Chebyshev semi-iterative methods, successive over-relaxation iterative methods, and second order Richardson iterative methods. I, Numer. Math., 3 (1961), 147–156 10.1007/BF01386013 MR0145678 0099.10903 CrossrefGoogle Scholar[5B] Gene H. Golub and , Richard S. Varga, Chebyshev semi-iterative methods, successive over-relaxation iterative methods, and second order Richardson iterative methods. II, Numer. Math., 3 (1961), 157–168 10.1007/BF01386014 MR0145679 0099.10903 CrossrefGoogle Scholar[6] G. J. Habetler and , E. L. Wachspress, Symmetric successive overrelaxation in solving diffusion difference equations, Math. Comp., 15 (1961), 356–362 MR0129139 0102.11403 CrossrefGoogle Scholar[7] P. R. Halmos, Introduction to Hilbert space, Chelsea, New York, 1957 0079.12404 Google Scholar[8] Einar Hille and , Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957xii+808 MR0089373 0078.10004 Google Scholar[9] Alston S. Householder, On the convergence of matrix iterations, Rep. ORNL 1883, Oak Ridge National Laboratory, Oak Ridge, Tenn., 1955, 47– MR0075677 Google Scholar[10] H. B. Keller, Numerical Methods, Lecture Notes, New York University, 1959, Ch. 2 Google Scholar[11] A. T. Lonseth, The propagation of error in linear problems, Trans. Amer. Math. Soc., 62 (1947), 193–212 MR0022315 0032.03203 CrossrefISIGoogle Scholar[12] W. V. Petryshyn, Masters Thesis, Direct and Iterative Methods for the Solution of Linear Operator Equations in Hilbert Space, Doctoral Thesis, Columbia University, 1961 Google Scholar[13] Angus E. Taylor, Introduction to functional analysis, John Wiley & Sons Inc., New York, 1958xvi+423 MR0098966 0081.10202 Google Scholar[14] Johannes Weissinger, Verallgemeinerungen des Seidelschen Iterationsverfahrens, Z. Angew. Math. Mech., 33 (1953), 155–163 MR0056375 0050.12401 CrossrefGoogle Scholar[15] G. Wiarda, Integralgleichungen unter besonderer Berücksichtung der Anwendungen, Leipzig, 1930 Google Scholar[16] David Young, Iterative methods for solving partial difference equations of elliptic type, Trans. Amer. Math. Soc., 76 (1954), 92–111 MR0059635 0055.35704 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Volume 10, Issue 4| 1962Journal of the Society for Industrial and Applied Mathematics History Submitted:18 January 1962Published online:13 July 2006 InformationCopyright © 1962 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0110052Article page range:pp. 675-690ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics

In the space H, a nonlinear operator equation of the first kind is studied, when the linear, nonlinear operator and the right-hand side of the equation are given approximately. Based on the method of Lavrent'ev M.M. an approximate solution of the equation in Hilbert space is constructed. The dependence of the regularization parameter on errors was selected. The rate of convergence of the approximate solution to the exact solution of the original equation is obtained.

It is demonstrated that the so-called “unavoidable quantum anomalies” can be avoided in the framework of a special nonlinear quantization scheme. In this scheme, the quantized Hamiltonians are represented by nonlinear but homogeneous operators in Hilbert space. The nonlinear terms are of the same order as quantum anomalies, and their role is to cancel anomalies. The quantization method proposed is applicable to integrable classical dynamical systems and the result of quantization is again an integrable (but, generally, nonlinear) “quantum” system. A simple example is discussed in detail. Irrespective of the existence of possible physical applications, the method provides a constructive way for extending the notion of quantum integrability to nonlinear spectral problems and gives a practical tool for building completely integrable nonlinear spectral equations in Hilbert space.