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1.
This paper is devoted to the convergence and stability analysis of a class of nonlinear subdivision schemes and associated
multiresolution transforms. As soon as a nonlinear scheme can be written as a specific perturbation of a linear and convergent
subdivision scheme, we show that if some contractivity properties are satisfied, then stability and convergence can be achieved.
This approach is applied to various schemes, which give different new results. More precisely, we study uncentered Lagrange
interpolatory linear schemes, WENO scheme (Liu et al., J Comput Phys 115:200–212, 1994), PPH and Power-P schemes (Amat and Liandrat, Appl Comput Harmon Anal 18(2):198–206, 2005; Serna and Marquina, J Comput Phys 194:632–658, 2004) and a nonlinear scheme using local spherical coordinates (Aspert et al., Comput Aided Geom Des 20:165–187, 2003). Finally, a stability proof is given for the multiresolution transform associated to a nonlinear scheme of Marinov et al.
(2005). 相似文献
2.
Roland and Varadhan (Appl. Numer. Math., 55:215–226, 2005) presented a new idea called “squaring” to improve the convergence of Lemaréchal’s scheme for solving nonlinear fixed-point
problems. Varadhan and Roland (Squared extrapolation methods: A new class of simple and efficient numerical schemes for accelerating
the convergence of the EM algorithm, Department of Biostatistics Working Paper. Johns Hopkins University, , 2004) noted that Lemaréchal’s scheme can be viewed as a member of the class of polynomial extrapolation methods with cycling that
uses two fixed-point iterations per cycle. Here we combine these two ideas, cycled extrapolation and squaring, and construct
a new class of methods, called squared polynomial methods (SQUAREM), for accelerating the convergence of fixed-point iterations.
Our main goal is to evaluate whether the squaring device is effective in improving the rate of convergence of cycled extrapolation
methods that use more than two fixed-point iterations per cycle. We study the behavior of the new schemes on an image reconstruction
problem for positron emission tomography (PET) using simulated data. Our numerical experiments show the effectiveness of first-
and higher-order squared polynomial extrapolation methods in accelerating image reconstruction, and also their relative superiority
compared to the classical, “unsquared” vector polynomial methods. 相似文献
3.
Fazlollah Soleymani 《Numerical Algorithms》2011,57(3):377-388
In this paper, some sixth-order modifications of Jarratt method for solving single variable nonlinear equations are proposed.
Per iteration, they consist of two function and two first derivative evaluations. The convergence analyses of the presented
iterative methods are provided theoretically and a comparison with other existing famous iterative methods of different orders
is given. Numerical examples include some of the newest and the most efficient optimal eighth-order schemes, such as Petkovic
(SIAM J Numer Anal 47:4402–4414, 2010), to put on show the accuracy of the novel methods. Finally, it is also observed that the convergence radii of our schemes
are better than the convergence radii of the optimal eighth-order methods. 相似文献
4.
Jon Eivind Vatne 《Annali dell'Universita di Ferrara》2012,58(1):199-215
In this paper we study Cohen–Macaulay monomial multiple structures (non-reduced schemes) on a linear subspace of codimension
two in projective space. We show that these structures determine smooth points in their respective Hilbert schemes, with (smooth)
neighbourhoods of two such points intersecting if their Hilbert functions are equal. We generalize a construction for multiple
structures on points in the plane to this setting, giving a kind of product of monomial multiple structures. For points, this
construction can be found in Nakajima’s book (Lectures on Hilbert schemes of points on surfaces, Univ Lecture Ser AMS, vol
18, 1999). The tools we use for studying multiple structures are developed in Vatne (Math Nachr 281(3):434–441, 2008; Comm Algebra 37(11):3861–3873, 2009) (see also Vatne in Towards a classification of multiple structures, PhD thesis, University of Bergen, 2001). 相似文献
5.
Hong Oh Kim Rae Young Kim Yeon Ju Lee Jungho Yoon 《Advances in Computational Mathematics》2010,33(3):255-283
We present a new family of compactly supported and symmetric biorthogonal wavelet systems. Each refinement mask in this family
has tension parameter ω. When ω = 0, it becomes the minimal length biorthogonal Coifman wavelet system (Wei et al., IEEE Trans Image Proc 7:1000–1013, 1998). Choosing ω away from zero, we can get better smoothness of the refinable functions at the expense of slightly larger support. Though
the construction of the new biorthogonal wavelet systems, in fact, starts from a new class of quasi-interpolatory subdivision
schemes, we find that the refinement masks accidently coincide with the ones by Cohen et al. (Comm Pure Appl Math 45:485–560,
1992, §6.C) (or Daubechies 1992, §8.3.5), which are designed for the purpose of generating biorthogonal wavelets close to orthonormal cases. However, the
corresponding mathematical analysis is yet to be provided. In this study, we highlight the connection between the quasi-interpolatory
subdivision schemes and the masks by Cohen, Daubechies and Feauveau, and then we study the fundamental properties of the new
biorthogonal wavelet systems such as regularity, stability, linear independence and accuracy. 相似文献
6.
Aurélien Deya 《Applied Mathematics and Optimization》2012,65(2):253-292
This paper is devoted to the study of numerical approximation schemes for a class of parabolic equations on (0,1) perturbed
by a non-linear rough signal. It is the continuation of Deya (Electron. J. Probab. 16:1489–1518, 2011) and Deya et al. (Probab. Theory Relat. Fields, to appear), where the existence and uniqueness of a solution has been established.
The approach combines rough paths methods with standard considerations on discretizing stochastic PDEs. The results apply
to a geometric 2-rough path, which covers the case of the multidimensional fractional Brownian motion with Hurst index H>1/3. 相似文献
7.
We determine shape-preserving regions and we describe a general setting to generate shape-preserving families for the 2-points
Hermite subdivision scheme introduced by Merrien (Numer. Algorithms 2:187–200, [1992]). This general construction includes the shape-preserving families presented in Merrien and Sablonníere (Constr. Approx.
19:279–298, [2003]) and Pelosi and Sablonníere (C
1 GP Hermite Interpolants Generated by a Subdivision Scheme, Prépublication IRMAR 06–23, Rennes, [2006]). New special families are presented as particular examples. Nonstationary and nonuniform versions of such schemes, which
produce smoother limits, are discussed.
相似文献
8.
In this paper, we study a variation of the equations of a chemotaxis kinetic model and investigate it in one dimension. In
fact, we use fractional diffusion for the chemoattractant in the Othmar–Dunbar–Alt system (Othmer in J Math Biol 26(3):263–298,
1988). This version was exhibited in Calvez in Amer Math Soc, pp 45–62, 2007 for the macroscopic well-known Keller–Segel model in all space dimensions. These two macroscopic and kinetic models are related
as mentioned in Bournaveas, Ann Inst H Poincaré Anal Non Linéaire, 26(5):1871–1895, 2009, Chalub, Math Models Methods Appl Sci, 16(7 suppl):1173–1197, 2006, Chalub, Monatsh Math, 142(1–2):123–141, 2004, Chalub, Port Math (NS), 63(2):227–250, 2006. The model we study here behaves in a similar way to the original model in two dimensions with the spherical symmetry assumption
on the initial data which is described in Bournaveas, Ann Inst H Poincaré Anal Non Linéaire, 26(5):1871–1895, 2009. We prove the existence and uniqueness of solutions for this model, as well as a convergence result for a family of numerical
schemes. The advantage of this model is that numerical simulations can be easily done especially to track the blow-up phenomenon. 相似文献
9.
Dual extragradient algorithms extended to equilibrium problems 总被引:1,自引:0,他引:1
In this paper we propose two iterative schemes for solving equilibrium problems which are called dual extragradient algorithms. In contrast with the primal extragradient methods in Quoc et al. (Optimization 57(6):749–776, 2008) which require to solve two general strongly convex programs at each iteration, the dual extragradient algorithms proposed
in this paper only need to solve, at each iteration, one general strongly convex program, one projection problem and one subgradient
calculation. Moreover, we provide the worst case complexity bounds of these algorithms, which have not been done in the primal
extragradient methods yet. An application to Nash-Cournot equilibrium models of electricity markets is presented and implemented
to examine the performance of the proposed algorithms. 相似文献
10.
John A. Ford Yasushi Narushima Hiroshi Yabe 《Computational Optimization and Applications》2008,40(2):191-216
Conjugate gradient methods are appealing for large scale nonlinear optimization problems, because they avoid the storage of
matrices. Recently, seeking fast convergence of these methods, Dai and Liao (Appl. Math. Optim. 43:87–101, 2001) proposed a conjugate gradient method based on the secant condition of quasi-Newton methods, and later Yabe and Takano (Comput.
Optim. Appl. 28:203–225, 2004) proposed another conjugate gradient method based on the modified secant condition. In this paper, we make use of a multi-step
secant condition given by Ford and Moghrabi (Optim. Methods Softw. 2:357–370, 1993; J. Comput. Appl. Math. 50:305–323, 1994) and propose two new conjugate gradient methods based on this condition. The methods are shown to be globally convergent
under certain assumptions. Numerical results are reported. 相似文献
11.
Alicia Cordero José L. Hueso Eulalia Martínez Juan R. Torregrosa 《Numerical Algorithms》2010,53(4):485-495
In this paper, we present two new three-step iterative methods for solving nonlinear equations with sixth convergence order.
The new methods are obtained by composing known methods of third order of convergence with Newton’s method and using an adequate
approximation for the derivative, that provides high order of convergence and reduces the required number of functional evaluations
per step. The first method is obtained from Potra-Pták’s method and the second one, from Homeier’s method, both reaching an
efficiency index of 1.5651. Our methods are comparable with the method of Parhi and Gupta (Appl Math Comput 203:50–55, 2008). Methods proposed by Kou and Li (Appl Math Comput 189:1816–1821, 2007), Wang et al. (Appl Math Comput 204:14–19, 2008) and Chun (Appl Math Comput 190:1432–1437, 2007) reach the same efficiency index, although they start from a fourth order method while we use third order methods and simpler
arithmetics. We prove the convergence results and check them with several numerical tests that allow us to compare the convergence
order, the computational cost and the efficiency order of our methods with those of the original methods. 相似文献
12.
Ioannis K. Argyros 《Numerical Algorithms》2010,54(4):485-501
We provide a semilocal convergence analysis for a certain class of secant-like methods considered also in Argyros (J Math
Anal Appl 298:374–397, 2004, 2007), Potra (Libertas Mathematica 5:71–84, 1985), in order to approximate a locally unique solution of an equation in a Banach space. Using a combination of Lipschitz and
center-Lipschitz conditions for the computation of the upper bounds on the inverses of the linear operators involved, instead
of only Lipschitz conditions (Potra, Libertas Mathematica 5:71–84, 1985), we provide an analysis with the following advantages over the work in Potra (Libertas Mathematica 5:71–84, 1985) which improved the works in Bosarge and Falb (J Optim Theory Appl 4:156–166, 1969, Numer Math 14:264–286, 1970), Dennis (SIAM J Numer Anal 6(3):493–507, 1969, 1971), Kornstaedt (1975), Larsonen (Ann Acad Sci Fenn, A 450:1–10, 1969), Potra (L’Analyse Numérique et la Théorie de l’Approximation 8(2):203–214, 1979, Aplikace Mathematiky 26:111–120, 1981, 1982, Libertas Mathematica 5:71–84, 1985), Potra and Pták (Math Scand 46:236–250, 1980, Numer Func Anal Optim 2(1):107–120, 1980), Schmidt (Period Math Hung 9(3):241–247, 1978), Schmidt and Schwetlick (Computing 3:215–226, 1968), Traub (1964), Wolfe (Numer Math 31:153–174, 1978): larger convergence domain; weaker sufficient convergence conditions, finer error bounds on the distances involved, and
a more precise information on the location of the solution. Numerical examples further validating the results are also provided. 相似文献
13.
A refined a posteriori error analysis for symmetric eigenvalue problems and the convergence of the first-order adaptive finite
element method (AFEM) is presented. The H
1 stability of the L
2 projection provides reliability and efficiency of the edge-contribution of standard residual-based error estimators for P
1 finite element methods. In fact, the volume contributions and even oscillations can be omitted for Courant finite element
methods. This allows for a refined averaging scheme and so improves (Mao et al. in Adv Comput Math 25(1–3):135–160, 2006). The proposed AFEM monitors the edge-contributions in a bulk criterion and so enables a contraction property up to higher-order
terms and global convergence. Numerical experiments exploit the remaining L
2 error contributions and confirm our theoretical findings. The averaging schemes show a high accuracy and the AFEM leads to
optimal empirical convergence rates. 相似文献
14.
In this paper, we introduce two iterative schemes (one implicit and one explicit) for finding a common element of the set
of solutions of the generalized equilibrium problems and the set of all common fixed points of a nonexpansive semigroup in
the framework of a real Hilbert space. We prove that both approaches converge strongly to a common element of such two sets.
Such common element is the unique solution of a variational inequality, which is the optimality condition for a minimization
problem. Furthermore, we utilize the main results to obtain two mean ergodic theorems for nonexpansive mappings in a Hilbert
space. The results of this paper extend and improve the results of Li et al. (J Nonlinear Anal 70:3065–3071, 2009), Cianciaruso et al. (J Optim Theory Appl 146:491–509, 2010) and many others. 相似文献
15.
We present a rigorous analysis of the performance of some one-step discretization schemes for a class of PT-symmetric singular
boundary eigenvalue problem which encompasses a number of different problems whose investigation has been inspired by the
2003 article of Benilov et al. (J Fluid Mech 497:201–224, 2003). These discretization schemes are analyzed as initial value problems rather than as discrete boundary problems, since this
is the setting which ties in most naturally with the formulation of the problem which one is forced to adopt due to the presence
of an interior singularity. We also devise and analyze a variable step scheme for dealing with the singular points. Numerical
results show better agreement between our results and those obtained from small-ϵ asymptotics than has been shown in results presented hitherto. 相似文献
16.
We introduce the new idea of recurrent functions to provide a new semilocal convergence analysis for Newton-type methods,
under mild differentiability conditions. It turns out that our sufficient convergence conditions are weaker, and the error
bounds are tighter than in earlier studies in some interesting cases (Chen, Ann Inst Stat Math 42:387–401, 1990; Chen, Numer Funct Anal Optim 10:37–48, 1989; Cianciaruso, Numer Funct Anal Optim 24:713–723, 2003; Cianciaruso, Nonlinear Funct Anal Appl 2009; Dennis 1971; Deuflhard 2004; Deuflhard, SIAM J Numer Anal 16:1–10, 1979; Gutiérrez, J Comput Appl Math 79:131–145, 1997; Hernández, J Optim Theory Appl 109:631–648, 2001; Hernández, J Comput Appl Math 115:245–254, 2000; Huang, J Comput Appl Math 47:211–217, 1993; Kantorovich 1982; Miel, Numer Math 33:391–396, 1979; Miel, Math Comput 34:185–202, 1980; Moret, Computing 33:65–73, 1984; Potra, Libertas Mathematica 5:71–84, 1985; Rheinboldt, SIAM J Numer Anal 5:42–63, 1968; Yamamoto, Numer Math 51: 545–557, 1987; Zabrejko, Numer Funct Anal Optim 9:671–684, 1987; Zinc̆ko 1963). Applications and numerical examples, involving a nonlinear integral equation of Chandrasekhar-type, and a differential
equation are also provided in this study. 相似文献
17.
Qinghui Zhang 《BIT Numerical Mathematics》2011,51(2):459-480
In this paper, we study effects of numerical integration on Galerkin meshless methods for solving elliptic partial differential
equations with Neumann boundary conditions. The shape functions used in the meshless methods reproduce linear polynomials.
The numerical integration rules are required to satisfy the so-called zero row sum condition of stiffness matrix, which is
also used by Babuška et al. (Int. J. Numer. Methods Eng. 76:1434–1470, 2008). But the analysis presented there relies on a certain property of the approximation space, which is difficult to verify.
The analysis in this paper does not require this property. Moreover, the Lagrange multiplier technique was used to handle
the pure Neumann condition. We have also identified specific numerical schemes, diagonal elements correction and background
mesh integration, that satisfy the zero row sum condition. The numerical experiments are carried out to verify the theoretical
results and test the accuracy of the algorithms. 相似文献
18.
The existence and construction of symplectic 2s-stage variable coefficients Runge-Kutta (RK) methods that integrate exactly IVPs whose solution is a trigonometrical polynomial
of order s with a given frequency ω is considered. The resulting methods, that can be considered as trigonometrical collocation methods, are fully implicit,
symmetric and symplectic RK methods with variable nodes and coefficients that are even functions of ν=ω
h (h is the step size), and for ω→0 they tend to the conventional RK Gauss methods. The present analysis extends previous results on two-stage symplectic exponentially
fitted integrators of Van de Vyver (Comput. Phys. Commun. 174: 255–262, 2006) and Calvo et al. (J. Comput. Appl. Math. 218: 421–434, 2008) to symmetric and symplectic trigonometrically fitted methods of high order. The algebraic order of the trigonometrically
fitted symmetric and symplectic 2s-stage methods is shown to be 4s like in conventional RK Gauss methods. Finally, some numerical experiments with oscillatory Hamiltonian systems are presented. 相似文献
19.
Recently, O’Hara, Pillay and Xu (Nonlinear Anal. 54, 1417–1426, 2003) considered an iterative approach to finding a nearest common fixed point of infinitely many nonexpansive mappings in a Hilbert
space. Very recently, Takahashi and Takahashi (J. Math. Anal. Appl. 331, 506–515, 2007) introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions
of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. In this paper, motivated
by these authors’ iterative schemes, we introduce a new iterative approach to finding a common element of the set of solutions
of an equilibrium problem and the set of common fixed points of infinitely many nonexpansive mappings in a Hilbert space.
The main result of this work is a strong convergence theorem which improves and extends results from the above mentioned papers. 相似文献
20.
This paper systematically studies numerical solution of fourth order problems in any dimensions by use of the Morley–Wang–Xu
(MWX) element discretization combined with two-grid methods (Xu and Zhou (Math Comp 69:881–909, 1999)). Since the coarse and fine finite element spaces are nonnested, two intergrid transfer operators are first constructed
in any dimensions technically, based on which two classes of local and parallel algorithms are then devised for solving such
problems. Following some ideas in (Xu and Zhou (Math Comp 69:881–909, 1999)), the intrinsic derivation of error analysis for nonconforming finite element methods of fourth order problems (Huang et al.
(Appl Numer Math 37:519–533, 2001); Huang et al. (Sci China Ser A 49:109–120, 2006)), and the error estimates for the intergrid transfer operators, we prove that the discrete energy errors of the two classes
of methods are of the sizes O(h + H
2) and O(h + H
2(H/h)(d−1)/2), respectively. Here, H and h denote respectively the mesh sizes of the coarse and fine finite element triangulations, and d indicates the space dimension of the solution region. Numerical results are performed to support the theory obtained and
to compare the numerical performance of several local and parallel algorithms using different intergrid transfer operators. 相似文献