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1.
It is known that the unique thin plate spline interpolant to a function f ∈ C
3
IR sampled at the scaled integers h
Z converges at an optimal rate of h
3. In this paper we present results from a recent numerical investigation of the case where the function is sampled at equally
spaced points on the unit interval. In this setting the known theoretical error bounds predict a drop in the convergence rate
from h
3 to h. However, numerical experiments show that the usual rate of convergence is h
3/2 and that the deterioration occurs near the end points of the interval. We will examine the effect of the boundary on the
accuracy of the interpolant and also the effect of the smoothness of the target function. We will show that there exists functions
which enjoy an even faster order of convergence of h
5/2. 相似文献
2.
The construction of a multiresolution analysis starts with the specification of a scale function. The Fourier transform of
this function is defined by an infinite product. The convergence of this product is usually discussed in the context of L
2(R).Here, we treat the convergence problem by viewing the partial products as probabilities, converging weakly to a probability
defined on an appropriate sequence space. We obtain a sufficient condition for this convergence, which is also necessary in
the case where the scale function is continuous. These results extend and clarify those of Cohen [2] and Hernández et al.
[4]. The method also applies to more general dilation schemes that commute with translations by
Z
d
. 相似文献
3.
In this paper we study inexact inverse iteration for solving the generalised eigenvalue problem A
x=λM
x. We show that inexact inverse iteration is a modified Newton method and hence obtain convergence rates for various versions
of inexact inverse iteration for the calculation of an algebraically simple eigenvalue. In particular, if the inexact solves
are carried out with a tolerance chosen proportional to the eigenvalue residual then quadratic convergence is achieved. We
also show how modifying the right hand side in inverse iteration still provides a convergent method, but the rate of convergence
will be quadratic only under certain conditions on the right hand side. We discuss the implications of this for the preconditioned
iterative solution of the linear systems. Finally we introduce a new ILU preconditioner which is a simple modification to
the usual preconditioner, but which has advantages both for the standard form of inverse iteration and for the version with
a modified right hand side. Numerical examples are given to illustrate the theoretical results.
AMS subject classification (2000) 65F15, 65F10 相似文献
4.
Convergence of the ensemble Kalman filter in the limit for large ensembles to the Kalman filter is proved. In each step of
the filter, convergence of the ensemble sample covariance follows from a weak law of large numbers for exchangeable random
variables, the continuous mapping theorem gives convergence in probability of the ensemble members, and L
p
bounds on the ensemble then give L
p
convergence. 相似文献
5.
《Numerical Functional Analysis & Optimization》2013,34(3):385-405
ABSTRACT We analyze convergence domains of Newton's and the modified Newton methods for solving operator equations in Banach spaces assuming first that the operator in question is ω-smooth in a ball centered at the starting point. It is shown that the gap between convergence domains of these two methods cannot be closed under ω-smoothness. Its exact size for Hölder smooth operators is computed. Then we proceed to investigate their convergence domains under regular smoothness. As our analysis reveals, both domains are the same and wider than their counterparts in the previous case. 相似文献
6.
AbstractWe consider the minimization of a convex objective function subject to the set of minima of another convex function, under the assumption that both functions are twice continuously differentiable. We approach this optimization problem from a continuous perspective by means of a second-order dynamical system with Hessian-driven damping and a penalty term corresponding to the constrained function. By constructing appropriate energy functionals, we prove weak convergence of the trajectories generated by this differential equation to a minimizer of the optimization problem as well as convergence for the objective function values along the trajectories. The performed investigations rely on Lyapunov analysis in combination with the continuous version of the Opial Lemma. In case the objective function is strongly convex, we can even show strong convergence of the trajectories. 相似文献
7.
A. Brillard 《Monatshefte für Mathematik》1982,93(4):267-275
On the Interpolation of Maximal Monotone Operators. We study here one way to extend to the maximal monotone case the results of linear interpolation, exposed bybalakrishnan in [2]. We obtain a necessary and sufficient condition of convergence for sequences(A
n
)
n
of maximal monotone operators on a real Hilbert spaceH. 相似文献
8.
Using Ergodic Theory and Epiconvergence notion, we study the rate of convergence of solutions relative to random Dirichlet problems in domains ofR
d
with random holes whose size tends to 0. This stochastic analysis allows to extend the results already obtained in the corresponding periodic case. 相似文献
9.
Lev Kapitanski 《偏微分方程通讯》2013,38(7-8):1303-1323
Abstract The asymptotic behavior of eigenvalues of an elliptic operator with a divergence form is discussed. The coefficients of the operator are discontinuous through a boundary of a subdomain and degenerate to zero on the subdomain when a parameter tends to zero. We will prove that the eigenvalues approach eigenvalues of the Laplacian on the subdomain or on the complement. We will obtain precise asymptotic behavior of their convergence. 相似文献
10.
Jun Shan Xie 《数学学报(英文版)》2013,29(3):477-488
The paper focuses on the largest eigenvalues of the β-Hermite ensemble and the β-Laguerre ensemble. In particular, we obtain the precise moment convergence rates of their largest eigenvalues. The results are motivated by the complete convergence for partial sums of i.i.d. random variables, and the proofs depend on the small deviations for largest eigenvalues of the β ensembles and tail inequalities of the general β Tracy-Widom law. 相似文献
11.
Pointwise convergence of double trigonometric Fourier series of functions in the Lebesgue space L
p[0,2]2was studied by M. I. Dyachenko. In this paper, we also consider the problems of the convergence of double Fourier series in Pringsheim"s sense with respect to the trigonometric as well as the Walsh systems of functions in the Lebesgue space L
p[0,1]2, p=(p1,p2), endowed with a mixed norm, in the particular case when the coefficients of the series in question are monotone with respect to each of the indices. We shall obtain theorems which generalize those of M. I. Dyachenko to the case when p is a vector. We shall also show that our theorems in the case of trigonometric Fourier series are best possible. 相似文献
12.
Summary. We consider spline collocation methods for a class of parabolic pseudodifferential operators. We show optimal order convergence
results in a large scale of anisotropic Sobolev spaces. The results cover for example the case of the single layer heat operator
equation when the spatial domain is a disc.
Received December 15, 1997 / Revised version received November 16, 1998 / Published online September 24, 1999 相似文献
13.
Summary. We present a convergence analysis of an algorithm for the numerical computation of the rank-one convex envelope of a function
. A rate of convergence for the scheme is established, and numerical experiments are presented to illustrate the analytical
results and applications of the algorithm.
Received February 1, 1999 / Published online April 20, 2000 相似文献
14.
AbstractIn this article, we discuss Jacobi spectral Galerkin and iterated Jacobi spectral Galerkin methods for Volterra-Urysohn integral equations with weakly singular kernels and obtain the convergence results in both the infinity and weighted L2-norm. We show that the order of convergence in iterated Jacobi spectral Galerkin method improves over Jacobi spectral Galerkin method. We obtain the convergence results in two cases when the exact solution is sufficiently smooth and non-smooth. For finding the improved convergence results, we also discuss Jacobi spectral multi-Galerkin and iterated Jacobi spectral multi-Galerkin method and obtain the convergence results in weighted L2-norm. In fact, we prove that the iterated Jacobi spectral multi-Galerkin method improves over iterated Jacobi spectral Galerkin method. We provide numerical results to verify the theoretical results. 相似文献
15.
《Numerical Functional Analysis & Optimization》2012,33(2):192-208
AbstractIn this article, we introduce a new class of contractive mappings and study analytical and computational aspects of a special case of Jungck-Khan iterative algorithm generated by this class of mappings. In particular, we improve upon strong convergence, rate of convergence and data dependence results existing in the current literature. Analytical as well numerical illustrative examples are given to support the new results. 相似文献
16.
On Convergence of Convex Minorant Algorithms for Distribution Estimation with Interval-Censored Data
Abstract Local convergence results of the convex minorant (CM) algorithm to obtain the nonparametric maximum-likelihood estimator of a distribution under interval-censored observations are given. We also provide a variation of the CM algorithm, which yields global convergence. The algorithm is illustrated with data on AIDS survival time in 92 members of the U.S. Air Force. 相似文献
17.
18.
Deconvolution: a wavelet frame approach 总被引:1,自引:0,他引:1
This paper devotes to analyzing deconvolution algorithms based on wavelet frame approaches, which has already appeared in
Chan et al. (SIAM J. Sci. Comput. 24(4), 1408–1432, 2003; Appl. Comput. Hormon. Anal. 17, 91–115, 2004a; Int. J. Imaging Syst. Technol. 14, 91–104, 2004b) as wavelet frame based high resolution image reconstruction methods. We first give a complete formulation
of deconvolution in terms of multiresolution analysis and its approximation, which completes the formulation given in Chan
et al. (SIAM J. Sci. Comput. 24(4), 1408–1432, 2003; Appl. Comput. Hormon. Anal. 17, 91–115, 2004a; Int. J. Imaging Syst. Technol. 14, 91–104, 2004b). This formulation converts deconvolution to a problem of filling the missing coefficients of wavelet frames
which satisfy certain minimization properties. These missing coefficients are recovered iteratively together with a built-in
denoising scheme that removes noise in the data set such that noise in the data will not blow up while iterating. This approach
has already been proven to be efficient in solving various problems in high resolution image reconstructions as shown by the
simulation results given in Chan et al. (SIAM J. Sci. Comput. 24(4), 1408–1432, 2003; Appl. Comput. Hormon. Anal. 17, 91–115, 2004a; Int. J. Imaging Syst. Technol. 14, 91–104, 2004b). However, an analysis of convergence as well as the stability of algorithms and the minimization properties
of solutions were absent in those papers. This paper is to establish the theoretical foundation of this wavelet frame approach.
In particular, a proof of convergence, an analysis of the stability of algorithms and a study of the minimization property
of solutions are given. 相似文献
19.
Ioannis K. Argyros 《Journal of Mathematical Analysis and Applications》2004,298(2):374-397
We provide a local as well as a semilocal convergence analysis for two-point Newton-like methods in a Banach space setting under very general Lipschitz type conditions. Our equation contains a Fréchet differentiable operator F and another operator G whose differentiability is not assumed. Using more precise majorizing sequences than before we provide sufficient convergence conditions for Newton-like methods to a locally unique solution of equation F(x)+G(x)=0. In the semilocal case we show under weaker conditions that our error estimates on the distances involved are finer and the information on the location of the solution at least as precise as in earlier results. In the local case a larger radius of convergence is obtained. Several numerical examples are provided to show that our results compare favorably with earlier ones. As a special case we show that the famous Newton-Kantorovich hypothesis is weakened under the same hypotheses as the ones contained in the Newton-Kantorovich theorem. 相似文献
20.
We consider implicit integration methods for the solution of stiff initial value problems for second-order differential equations
of the special form y' = f(y). In implicit methods, we are faced with the problem of solving systems of implicit relations. This paper focuses on the construction
and analysis of iterative solution methods which are effective in cases where the Jacobian of the right‐hand side of the differential
equation can be split into a sum of matrices with a simple structure. These iterative methods consist of the modified Newton
method and an iterative linear solver to deal with the linear Newton systems. The linear solver is based on the approximate
factorization of the system matrix associated with the linear Newton systems. A number of convergence results are derived
for the linear solver in the case where the Jacobian matrix can be split into commuting matrices. Such problems often arise
in the spatial discretization of time‐dependent partial differential equations. Furthermore, the stability matrix and the
order of accuracy of the integration process are derived in the case of a finite number of iterations.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献