共查询到20条相似文献,搜索用时 15 毫秒
1.
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
2.
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
3.
Barbara Kaltenbacher 《Numerische Mathematik》1998,79(4):501-528
This paper treats a class of Newton type methods for the approximate solution of nonlinear ill-posed operator equations,
that use so-called filter functions for regularizing the linearized equation in each Newton step. For noisy data we derive
an aposteriori stopping rule that yields convergence of the iterates to asolution, as the noise level goes to zero, under
certain smoothness conditions on the nonlinear operator. Appropriate closeness and smoothness assumptions on the starting
value and the solution are shown to lead to optimal convergence rates. Moreover we present an application of the Newton type
methods under consideration to a parameter identification problem, together with some numerical results.
Received November 29, 1996 / Revised version received April 25, 1997 相似文献
4.
Summary. In the study of the choice of the regularization parameter for Tikhonov regularization of nonlinear ill-posed problems, Scherzer, Engl and Kunisch proposed an a posteriori strategy in 1993. To prove the optimality of the strategy, they imposed many very restrictive conditions on the problem under consideration. Their results are difficult to apply to concrete problems since one can not make sure whether their assumptions are valid. In this paper we give a further study on this strategy, and show that Tikhonov regularization is order optimal for each with the regularization parameter chosen according to this strategy under some simple and easy-checking assumptions. This paper weakens the conditions needed in the existing results, and provides a theoretical guidance to numerical experiments. Received August 8, 1997 / Revised version received January 26, 1998 相似文献
5.
On convergence rates of inexact Newton regularizations 总被引:1,自引:0,他引:1
Andreas Rieder 《Numerische Mathematik》2001,88(2):347-365
Summary. REGINN is an algorithm of inexact Newton type for the regularization of nonlinear ill-posed problems [Inverse Problems 15 (1999), pp. 309–327]. In the present article convergence is shown under weak smoothness assumptions (source conditions). Moreover, convergence rates are established. Some computational illustrations support the theoretical results. Received March 12, 1999 / Published online October 16, 2000 相似文献
6.
Andreas Rieder 《Numerische Mathematik》1997,75(4):501-522
Summary. An additive Schwarz iteration is described for the fast resolution of linear ill-posed problems which are stabilized by Tikhonov
regularization. The algorithm and its analysis are presented in a general framework which applies to integral equations of
the first kind discretized either by spline functions or Daubechies wavelets. Numerical experiments are reported on to illustrate
the theoretical results and to compare both discretization schemes.
Received March 6, 1995 / Revised version received December 27, 1995 相似文献
7.
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
8.
K. Veselić 《Numerische Mathematik》1999,83(4):699-702
Summary. We prove that the diagonally pivoted symmetric LR algorithm on a positive definite matrix is globally convergent. Received December 23, 1997 / Revised version received August 3, 1998 / Published online August 19, 1999 相似文献
9.
Summary. We describe a new iterative method for the solution of large, very ill-conditioned linear systems of equations that arise
when discretizing linear ill-posed problems. The right-hand side vector represents the given data and is assumed to be contaminated
by measurement errors. Our method applies a filter function of the form with the purpose of reducing the influence of the errors in the right-hand side vector on the computed approximate solution
of the linear system. Here is a regularization parameter. The iterative method is derived by expanding in terms of Chebyshev polynomials. The method requires only little computer memory and is well suited for the solution of
large-scale problems. We also show how a value of and an associated approximate solution that satisfies the Morozov discrepancy principle can be computed efficiently. An application
to image restoration illustrates the performance of the method.
Received January 25, 1997 / Revised version received February 9, 1998 / Published online July 28, 1999 相似文献
10.
Stefano Serra 《Numerische Mathematik》1999,81(3):461-495
Summary. In previous works [21–23] we proposed the use of [5] and band Toeplitz based preconditioners for the solution of 1D and 2D boundary value problems (BVP) by means of the preconditioned
conjugate gradient (PCG) methods. As and band Toeplitz linear systems can be solved [4] by using fast sine transforms [8], these methods become especially attractive
in a parallel environment of computation. In this paper we extend this technique to the nonlinear, nonsymmetric case and,
in addition, we prove some clustering properties for the spectra of the preconditioned matrices showing why these methods
exhibit a convergence speed which results to be more than linear. Therefore these methods work much finer than those based on separable preconditioners [18,45], on incomplete LU factorizations
[36,13,27], and on circulant preconditioners [9,30,35] since the latter two techniques do not assure a linear rate of convergence.
On the other hand, the proposed technique has a wider range of application since it can be naturally used for nonlinear, nonsymmetric
problems and for BVP in which the coefficients of the differential operator are not strictly positive and only piecewise smooth.
Finally the several numerical experiments performed here and in [22,23] confirm the effectiveness of the theoretical analysis.
Received December 19, 1995 / Revised version received September 15, 1997 相似文献
11.
Summary. Graeffe iteration was the choice algorithm for solving univariate polynomials in the XIX-th and early XX-th century. In this paper, a new variation of Graeffe iteration is given, suitable to IEEE floating-point arithmetics of modern digital computers. We prove that under a certain generic assumption the proposed algorithm converges. We also estimate the error after N iterations and the running cost. The main ideas from which this algorithm is built are: classical Graeffe iteration and Newton Diagrams, changes of scale (renormalization), and replacement of a difference technique by a differentiation one. The algorithm was implemented successfully and a number of numerical experiments are displayed. Received May 29, 1998 / Revised version received September 13, 1999 / Published online April 5, 2001 相似文献
12.
K. Segeth 《Numerische Mathematik》1999,83(3):455-475
Summary. Convergence of a posteriori error estimates to the true error for the semidiscrete finite element method of lines is shown
for a nonlinear parabolic initial-boundary value problem.
Received June 15, 1997 / Revised version received May 15, 1998 / Published online: June 29, 1999 相似文献
13.
G.W. Stewart 《Numerische Mathematik》1994,68(1):143-147
Summary.
This note gives a new convergence proof for iterations based on
multipoint formulas. It rests on the very general assumption that if
the desired fixed point appears as an argument in the formula then
the formula returns the fixed point.
Received March 24, 1993 / Revised version received
January 1994 相似文献
14.
M.L. Seoane 《Numerische Mathematik》1995,70(3):353-377
Summary. In this paper we study a general theory for the numerical approximation of functional nonlinear two-parameter problems in
a neighbourhood of an isola center. The results are also valid for a certain class of perturbed bifurcation points. The abstract
theory is applied to the Galerkin approximation of nonlinear variational posed problems. In this case, as a consequence of
the error being orthogonal to the approximating space, we prove the superconvergence of the perturbation parameter, whereas
for the bifurcation parameter and the solution we obtain the same order as in the linear problem. Numerical results are given
for the one-dimensional Brussellator model.
Received June 10, 1992 / Revised version received May 16, 1994 相似文献
15.
Summary.
The ``L--curve' is a plot (in ordinary or doubly--logarithmic scale) of the
norm of (Tikhonov--) regularized solutions of an ill--posed
problem versus the norm of the residuals. We show that the popular criterion
of choosing the parameter corresponding to the point with maximal
curvature of the L--curve does not yield
a convergent regularization strategy to solve the ill--posed problem.
Nevertheless, the L--curve can be used to compute the
regularization
parameters produced by Morozov's discrepancy principle and by an
order--optimal variant of the
discrepancy principle proposed by Engl and Gfrerer in an alternate way.
Received June 29, 1993 /
Revised version received February 2, 1994 相似文献
16.
Qingchuan Yao 《Numerische Mathematik》1999,81(4):647-677
This paper proposes some modified Halley iterations for finding the zeros of polynomials. We investigate the non-overshoot
properties of the modified Halley iterations and other important properties that play key roles in solving symmetric eigenproblems.
We also extend Halley iteration to systems of polynomial equations in several variables.
Received March 20, 1996 / Revised version received December 5, 1997 相似文献
17.
In this paper, we build up a modification of the Midpoint method, reducing its operational cost without losing its cubical
convergence. Then we obtain a semilocal convergence result for this new iterative process and by means of several examples
we compare it with other iterative processes.
(Received 11 April 2000; in final form 27 March 2001) 相似文献
18.
Christian Kanzow 《Numerische Mathematik》1998,80(4):557-577
Summary. We consider a quadratic programming-based method for nonlinear complementarity problems which allows inexact solutions of
the quadratic subproblems. The main features of this method are that all iterates stay in the feasible set and that the method
has some strong global and local convergence properties. Numerical results for all complementarity problems from the MCPLIB
test problem collection are also reported.
Received February 24, 1997 / Revised version received September 5, 1997 相似文献
19.
20.
Summary. In this paper, we derive a posteriori error estimates for the finite element approximation of quadratic optimal control problem
governed by linear parabolic equation. We obtain a posteriori error estimates for both the state and the control approximation.
Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive
finite element approximation schemes for the control problem.
Received July 7, 2000 / Revised version received January 22, 2001 / Published online January 30, 2002
RID="*"
ID="*" Supported by EPSRC research grant GR/R31980 相似文献