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1.
Martin Hanke 《Numerische Mathematik》1991,60(1):341-373
Summary In this paper, the potentials of so-calledlinear semiiterative methods are considered for the approximate solution of linear ill-posed problems and ill conditioned matrix equations. Several efficient two-step methods are presented, most of which have been introduced earlier in the literature. Stipulating certain conditions concerning the smoothness of the solution, a notion of optimal speed of convergence may be formulated. Various direct and converse results are derived to illustrate the properties of this concept.If the problem's right hand side data are contaminated by noise, semiiterative methods may be used asregularization methods. Assuming optimal rate of convergence of the iteration for the unperturbed problem, the regularized approximations will be of order optimal accuracy.To derive these results, specific properties of polynomials are used in connection with the basic theory of solving ill-posed problems. Rather recent results onfast decreasing polynomials are applied to answer an open question of Brakhage.Numerical examples are given including a comparison to the method of conjugate gradients.This research was sponsored by the Deutsche Forschungsgemeinschaft (DFG). 相似文献
2.
Robert Plato 《Numerische Mathematik》1996,75(1):99-120
Summary. For the numerical solution of (non-necessarily well-posed) linear equations in Banach spaces we consider a class of iterative
methods which contains well-known methods like the Richardson iteration, if the associated resolvent operator fulfils a condition
with respect to a sector. It is the purpose of this paper to show that for given noisy right-hand side the discrepancy principle
(being a stopping rule for the iteration methods belonging to the mentioned class) defines a regularization method, and convergence
rates are proved under additional smoothness conditions on the initial error. This extends similar results obtained for positive
semidefinite problems in Hilbert spaces. Then we consider a class of parametric methods which under the same resolvent condition
contains the method of the abstract Cauchy problem, and (under a weaker resolvent condition) the iterated method of Lavrentiev.
A modified discrepancy principle is formulated for them, and finally numerical illustrations are presented.
Received August 29, 1994 / Revised version received September 19, 1995 相似文献
3.
Andreas Neubauer 《Numerische Mathematik》2000,85(2):309-328
Summary. In this paper we derive convergence rates results for Landweber iteration in Hilbert scales in terms of the iteration index
for exact data and in terms of the noise level for perturbed data. These results improve the one obtained recently for Landweber iteration for nonlinear ill-posed problems
in Hilbert spaces. For numerical computations we have to approximate the nonlinear operator and the infinite-dimensional spaces
by finite-dimensional ones. We also give a convergence analysis for this finite-dimensional approximation. The conditions
needed to obtain the rates are illustrated for a nonlinear Hammerstein integral equation. Numerical results are presented
confirming the theoretical ones.
Received May 15, 1998 / Revised version received January 29, 1999 / Published online December 6, 1999 相似文献
4.
In this paper we investigate convergence of Landweber iteration in Hilbert scales for linear and nonlinear inverse problems.
As opposed to the usual application of Hilbert scales in the framework of regularization methods, we focus here on the case
s≤0, which (for Tikhonov regularization) corresponds to regularization in a weaker norm. In this case, the Hilbert scale operator
L−2s appearing in the iteration acts as a preconditioner, which significantly reduces the number of iterations needed to match
an appropriate stopping criterion. Additionally, we carry out our analysis under significantly relaxed conditions, i.e., we
only require instead of which is the usual condition for regularization in Hilbert scales. The assumptions needed for our analysis are verified for
several examples and numerical results are presented illustrating the theoretical ones.
supported by the Austrian Science Foundation (FWF) under grant SFB/F013 相似文献
5.
Summary In this paper we study a multi-grid method for the numerical solution of nonlinear systems of equations arising from the discretization
of ill-posed problems, where the special eigensystem structure of the underlying operator equation makes it necessary to use
special smoothers. We provide uniform contraction factor estimates and show that a nested multigrid iteration together with
an a priori or a posteriori chosen stopping index defines a regularization method for the ill-posed problem, i.e., a stable
solution method, that converges to an exact solution of the underlying infinite-dimensional problem as the data noise level
goes to zero, with optimal rates under additional regularity conditions.
Supported by the Fonds zur F?rderung der wissenschaftlichen Forschung under grant T 7-TEC and project F1308 within Spezialforschungsbereich
13 相似文献
6.
R. L. Voller 《Periodica Mathematica Hungarica》1984,15(4):267-280
In the present paper some Newton-like iteration methods are developed to enclose solutions of nonlinear operator equations of the kindF(x)=0. HereF maps a certain subset of a partially ordered vector space into another partially ordered vector space. The obtained results are proved without any special properties of the orderings by taking use of a new kind of a generalized divided difference operator, so that they even hold for nonconvex operators. Furthermore a method for constructing including starting points is presented and two examples are given. 相似文献
7.
Otmar Scherzer 《Numerische Mathematik》1998,80(4):579-600
Summary. The convergence analysis of Landweber's iteration for the solution of nonlinear ill–posed problem has been developed recently
by Hanke, Neubauer and Scherzer. In concrete applications, sufficient conditions for convergence of the Landweber iterates
developed there (although quite natural) turned out to be complicated to verify analytically. However, in numerical realizations,
when discretizations are considered, sufficient conditions for local convergence can usually easily be stated. This paper
is motivated by these observations: Initially a discretization is fixed and a discrete Landweber iteration is implemented
until an appropriate stopping criterion becomes active. The output is used as an initial guess for a finer discretization.
An advantage of this method is that the convergence analysis can be considered in a family of finite dimensional spaces. The
numerical performance of this multi level algorithm is compared with Landweber's iteration.
Received October 21, 1996 / Revised version received July 28, 1997 相似文献
8.
Charles E. Chidume 《Journal of Computational and Applied Mathematics》2010,234(3):861-882
This paper surveys some of the main convergence properties of the Mann-type iteration for the demicontractive mappings. Some variants of the Mann iteration that ensure the strong convergence, like the (CQ) algorithm and a variant for the asymptotically demicontractive mappings are also considered. The usual framework of our study is a (real) Hilbert space and only to a certain extent some particular Banach spaces. Historical aspects are pointed out and some applications for the convex feasibility problem are discussed. 相似文献
9.
J. A. Burnsf Z. Y. Liu R. E. Miller 《Numerical Functional Analysis & Optimization》2013,34(1-2):79-135
This paper deals with the development and analysis of well-posed models and computational algorithms for control of a class of partial differential equations that descrive the motions of thermo-viscoelastic structures. We first present an abstract “state space” framework and general well-posedness result that can be applied to a large class of thermo-elastic and thermo-viscoelastic models. This state space framework is used in the development of a computational scheme to be used in the solution of an LQR control problem. A detailed convergence proof is provided for the viscoelastic model, and several numerical results are presented to illustrate the theory and to analyze problems for which the theory is incomplete. 相似文献
10.
L. B. Rall 《BIT Numerical Mathematics》1982,22(2):233-251
In actual practice, iteration methods applied to the solution of finite systems of equations yield inconclusive results as to the existence or nonexistence of solutions and the accuracy of any approximate solutions obtained. On the other hand, construction of interval extensions of ordinary iteration operators permits one to carry out interval iteration computationally, with results which can give rigorous guarantees of existence or nonexistence of solutions, and error bounds for approximate solutions. Examples are given of the solution of a nonlinear system of equations and the calculation of eigenvalues and eigenvectors of a matrix by interval iteration. Several ways to obtain lower and upper bounds for eigenvalues are given.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041. 相似文献
11.
Maria Grazia Gasparo Alessandra Papini Aldo Pasquali 《Journal of Computational and Applied Mathematics》2007
In this work we are interested in the solution of nonlinear inverse problems of the form F(x)=y. We consider a two-stage method which is third order convergent for well-posed problems. Combining the method with Levenberg–Marquardt regularization of the linearized problems at each stage and using the discrepancy principle as a stopping criterion, we obtain a regularization method for ill-posed problems. Numerical experiments on some parameter identification and inverse acoustic scattering problems are presented to illustrate the performance of the method. 相似文献
12.
Xia CuiGuang-wei Yuan Jing-yan Yue 《Journal of Computational and Applied Mathematics》2011,236(2):253-264
A nonlinear iteration method named the Picard-Newton iteration is studied for a two-dimensional nonlinear coupled parabolic-hyperbolic system. It serves as an efficient method to solve a nonlinear discrete scheme with second spatial and temporal accuracy. The nonlinear iteration scheme is constructed with a linearization-discretization approach through discretizing the linearized systems of the original nonlinear partial differential equations. It can be viewed as an improved Picard iteration, and can accelerate convergence over the standard Picard iteration. Moreover, the discretization with second-order accuracy in both spatial and temporal variants is introduced to get the Picard-Newton iteration scheme. By using the energy estimate and inductive hypothesis reasoning, the difficulties arising from the nonlinearity and the coupling of different equation types are overcome. It follows that the rigorous theoretical analysis on the approximation of the solution of the Picard-Newton iteration scheme to the solution of the original continuous problem is obtained, which is different from the traditional error estimate that usually estimates the error between the solution of the nonlinear discrete scheme and the solution of the original problem. Moreover, such approximation is independent of the iteration number. Numerical experiments verify the theoretical result, and show that the Picard-Newton iteration scheme with second-order spatial and temporal accuracy is more accurate and efficient than that of first-order temporal accuracy. 相似文献
13.
In this paper, we investigate the convergence behavior of a Runge–Kutta type modified Landweber method for nonlinear ill-posed operator equations. In order to improve the stability and convergence of the Landweber iteration, a 2-stage Gauss-type Runge–Kutta method is applied to the continuous analogy of the modified Landweber method, to give a new modified Landweber method, called R–K type modified Landweber method. Under some appropriate conditions, we prove the convergence of the proposed method. We conclude with a numerical example confirming the theoretical results, including comparisons to the modified Landweber iteration. 相似文献
14.
We design, analyse and test a class of incomplete orthogonal factorization preconditioners constructed from Givens rotations, incorporating some dropping strategies and updating tricks, for the solution of large sparse systems of linear equations. Comprehensive accounts about how the preconditioners are coded, what storage is required and how the computation is executed for a given accuracy are presented. A number of numerical experiments show that these preconditioners are competitive with standard incomplete triangular factorization preconditioners when they are applied to accelerate Krylov subspace iteration methods such as GMRES and BiCGSTAB. 相似文献
15.
The present paper is concerned with the convergence problem of the variants of the Chebyshev–Halley iteration family with parameters for solving nonlinear operator equations in Banach spaces. Under the assumption that the first derivative of the operator satisfies the Hölder condition of order p, a convergence criterion of order 1+p for the iteration family is established. An application to a nonlinear Hammerstein integral equation of the second kind is provided. 相似文献
16.
Finite element approximation of multi-scale elliptic problems using patches of elements 总被引:1,自引:0,他引:1
Roland Glowinski Jiwen He Alexei Lozinski Jacques Rappaz Joël Wagner 《Numerische Mathematik》2005,101(4):663-687
In this paper we present a method for the numerical solution of elliptic problems with multi-scale data using multiple levels
of not necessarily nested grids. The method consists in calculating successive corrections to the solution in patches whose
discretizations are not necessarily conforming. This paper provides proofs of the results published earlier (see C. R. Acad.
Sci. Paris, Ser. I 337 (2003) 679–684), gives a generalization of the latter to more than two domains and contains extensive
numerical illustrations. New results including the spectral analysis of the iteration operator and a numerical method to evaluate
the constant of the strengthened Cauchy-Buniakowski-Schwarz inequality are presented.
Supported by CTI Project 6437.1 IWS-IW. 相似文献
17.
Huidong Yang Walter Zulehner 《Journal of Computational and Applied Mathematics》2011,235(18):5367-5379
Fluid-structure interaction problems arise in many fields of application such as flows around elastic structures and blood flow in arteries. The method presented in this paper for solving such a problem is based on a reduction to an equation at the interface, involving the so-called Steklov-Poincaré operators. This interface equation is solved by a Newton iteration, for which directional derivatives involving shape derivatives with respect to the interface perturbation have to be evaluated appropriately. One step of the Newton iteration requires the solution of several decoupled linear sub-problems in the structure and the fluid domains. These sub-problems are spatially discretized by a finite element method on hybrid meshes. For the time discretization, implicit first-order methods are used for both sub-problems. The discretized equations are solved by algebraic multigrid methods. 相似文献
18.
Strong convergence theorems are obtained for the CQ method for an Ishikawa iteration process, a contractive-type iteration process for nonexpansive mappings, and the proximal point algorithm for maximal monotone operators in Hilbert spaces. 相似文献
19.
In this paper, we proposed a modified extragradient method for solving variational inequalities. The method can be viewed as an extension of the method proposed by He and Liao [Improvement of some projection methods for monotone variational inequalities, J. Optim. Theory Appl. 112 (2002) 111–128], by performing an additional projection step at each iteration and another optimal step length is employed to reach substantial progress in each iteration. We used a self-adaptive technique to adjust parameter ρ at each iteration. Under certain conditions, the global convergence of the proposed method is proved. Preliminary numerical experiments are included to compare our method with some known methods. 相似文献
20.
Summary.
An adaptive Richardson iteration method is described for the solution of
large sparse symmetric positive definite linear systems of equations with
multiple right-hand side vectors. This scheme ``learns' about the linear
system to be solved by computing inner products of residual matrices during
the iterations. These inner products are interpreted as block modified moments.
A block version of the modified Chebyshev algorithm is presented which yields
a block tridiagonal matrix from the block modified moments and the recursion
coefficients of the residual polynomials. The eigenvalues of this block
tridiagonal matrix define an interval, which determines the choice of relaxation
parameters for Richardson iteration. Only minor modifications are necessary
in order to obtain a scheme for the solution of symmetric indefinite linear
systems with multiple right-hand side vectors. We outline the changes required.
Received April 22, 1993 相似文献