Local convergence theorems for gradient methods in hilbert space

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.

In this paper, we consider the simplified Levenberg–Marquardt method for nonlinear ill-posed inverse problems in Hilbert spaces for obtaining stable approximations of solutions to the ill-posed nonlinear equations of the form F ⁢ ( u ) = y {F(u)=y} , where F : 𝒟 ⁢ ( F ) ⊂ 𝖴 → 𝖸 {F:\mathcal{D}(F)\subset\mathsf{U}\to\mathsf{Y}} is a nonlinear operator between Hilbert spaces 𝖴 {\mathsf{U}} and 𝖸 {\mathsf{Y}} . The method is defined as follows: u n + 1 δ = u n δ - ( T 0 ∗ ⁢ T 0 + α n ⁢ I ) - 1 ⁢ T 0 ∗ ⁢ ( F ⁢ ( u n δ ) - y δ ) , u_{n+1}^{\delta}=u_{n}^{\delta}-(T_{0}^{\ast}T_{0}+\alpha_{n}I)^{-1}T_{0}^{% \ast}(F(u_{n}^{\delta})-y^{\delta}), where T 0 = F ′ ⁢ ( u 0 ) {T_{0}=F^{\prime}(u_{0})} and T 0 ∗ = F ′ ⁢ ( u 0 ) ∗ {T_{0}^{\ast}=F^{\prime}(u_{0})^{\ast}} . Here F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} denotes the Frèchet derivative of F at an initial guess u 0 ∈ 𝒟 ⁢ ( F ) {u_{0}\in\mathcal{D}(F)} for the exact solution u † {u^{\dagger}} , F ′ ⁢ ( u 0 ) ∗ {F^{\prime}(u_{0})^{\ast}} is the adjoint of F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} and { α n } {\{\alpha_{n}\}} is an a priori chosen sequence of non-negative real numbers satisfying suitable properties. We use Morozov-type stopping rule to terminate the iterations. Under suitable non-linearity conditions on operator F, we show convergence of the method and also obtain a convergence rate result under a Hölder-type source condition on the element u 0 - u † {u_{0}-u^{\dagger}} . Furthermore, we derive convergence of the method for the case when no source conditions are used and the study concludes with numerical examples which validate the theoretical conclusions.

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.

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.

A Boundary Value Problem for 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.

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

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.

The Bayesian approach to ill-posed operator equations in Hilbert space recently gained attraction. In this context, and when the prior distribution is Gaussian, then two operators play a significant role, the one which governs the operator equation, and the one which describes the prior covariance. Typically it is assumed that these operators commute. Here we extend this analysis to non-commuting operators, replacing the commutativity assumption by a link condition. We discuss its relation to the commuting case, and we indicate that this allows to use interpolation type results to obtain tight bounds for the contraction of the posterior Gaussian distribution towards the data generating element.

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 a recent paper of the same title, L. A. Medeiros has considered the question of uniqueness for the Cauchy problem for ordinary differential equations in a complex Hilbert space. Section 1 contains a discussion of Medeiros' results, and provides a motivation for the improved uniqueness results in Section 2

The investigation has defined the characteristic criterion (and its modification) of solvability of the problem of differential realization of the bundle of controlled trajectory curves of determined chaotic dynamic processes in the class of bilinear non-autonomous ordinary second- and higher-order differential equations (with and without delay) in the separable Hilbert space. The problem statement under consideration belongs to the type of converse problems for the additive combination of nonstationary linear and bilinear operators of the evolution equation in the Hilbert space. The constructions of tensor products of the Hilbert spaces, structures of lattices with an orthocomplement, the theory of extension of M2 -operators and the functional apparatus of the entropy Relay Ritz operator represent the basis of this theory. It has been shown that in the case of the finite bundle of the controlled trajectory curves the existence of the property of sub-linearity of the given operator allows one to obtain sufficient conditions of existence of such realizations. Side by side with solving the main problems, grounded are topological-group conditions of continuity of projectivization of the Relay Ritz operator with computing the fundamental group (Poincare group) of its compact image. The results obtained give incentives for the development of the quantitative theory of converse problems of higher-order multilinear evolution equations with the operators of generalized delay describing, for example, differential modeling of nonlinear Van der Pol oscillators or Lorentz strange attractors.

In this paper, we consider a second-order abstract viscoelastic equation in Hilbert spaces with delay term in the nonlinear internal damping and a nonlinear source term. Under some suitable assumptions on the weight of the delayed feedback, the weight of the non-delayed feedback and the behavior of the relaxation function, we establish two explicit and general decay rate results of the energy by introducing a suitable Lyaponov functional and some properties of the convex functions. Moreover, we give some applications and examples. This work generalizes the previous results without time delay term to those with delay in the nonlinear damping.

In this work, we interest in the study of the wellposedness and the stability of fractional stochastic nonlinear heat equation in the Hilbert space $$ L^{2}(0,1) $$ ; perturbed by a trace-class noise and driven by the fractional Laplacian. Precisely, we use the fixed point theorem to prove the wellposedness of the problem. Moreover, we prove the $$ p^{th} $$ -moment exponential stability and the almost surely exponential stability by imposing an additional assumption. Some examples are considered in order to confirm and support the validity of our theoretical results.