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1.
Summary In this paper, we develop a matrix framework to solve the problem of finding orthonormal rational function vectors with prescribed poles with respect to a certain discrete inner product that is defined by a set of data points and corresponding weight vectors wi,j. Our algorithm for solving the problem is recursive, and it is of complexity If all data points are real or lie on the unit circle, then the complexity is reduced by an order of magnitude. 相似文献
2.
Summary We introduce a class of n×n structured matrices which includes three well-known classes of generalized companion matrices: tridiagonal plus rank-one matrices (comrade matrices), diagonal plus rank-one matrices and arrowhead matrices. Relying on the structure properties of , we show that if A then A=RQ , where A=QR is the QR decomposition of A. This allows one to implement the QR iteration for computing the eigenvalues and the eigenvectors of any A with O(n) arithmetic operations per iteration and with O(n) memory storage. This iteration, applied to generalized companion matrices, provides new O(n2) flops algorithms for computing polynomial zeros and for solving the associated (rational) secular equations. Numerical experiments confirm the effectiveness and the robustness of our approach.The results of this paper were presented at the Workshop on Nonlinear Approximations in Numerical Analysis, June 22 – 25, 2003, Moscow, Russia, at the Workshop on Operator Theory and Applications (IWOTA), June 24 – 27, 2003, Cagliari, Italy, at the Workshop on Numerical Linear Algebra at Universidad Carlos III in Leganes, June 16 – 17, 2003, Leganes, Spain, at the SIAM Conference on Applied Linear Algebra, July 15 – 19, 2003, Williamsburg, VA and in the Technical Report [8]. This work was partially supported by MIUR, grant number 2002014121, and by GNCS-INDAM. This work was supported by NSF Grant CCR 9732206 and PSC CUNY Awards 66406-0033 and 65393-0034. 相似文献
3.
A multi-level spectral Galerkin method for the two-dimensional non-stationary Navier-Stokes equations is presented. The method
proposed here is a multiscale method in which the fully nonlinear Navier-Stokes equations are solved only on a low-dimensional
space subsequent approximations are generated on a succession of higher-dimensional spaces j=2, . . . ,J, by solving a linearized Navier-Stokes problem around the solution on the previous level. Error estimates depending on the
kinematic viscosity 0<ν<1 are also presented for the J-level spectral Galerkin method. The optimal accuracy is achieved when We demonstrate theoretically that the J-level spectral Galerkin method is much more efficient than the standard one-level spectral Galerkin method on the highest-dimensional
space .
The work of this author was supported in part by the NSF of China 10371095, City University of Hong Kong Research Project
7001093 Hong Kong and the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU
1084/02P) 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.
María López-Fernández Christian Lubich Cesar Palencia Achim Schädle 《Numerische Mathematik》2005,102(2):277-291
The result after N steps of an implicit Runge-Kutta time discretization of an inhomogeneous linear parabolic differential equation is computed,
up to accuracy ɛ, by solving only
linear systems of equations. We derive, analyse, and numerically illustrate this fast algorithm. 相似文献
6.
In this paper we study the approximation power, the existence of a normalized B-basis and the structure of a degree-raising process for spaces of the formrequiring suitable assumptions on the functions u and v. The results about degree raising are detailed for special spaces of this form which have been recently introduced in the area of CAGD. 相似文献
7.
Jemal Peradze 《Numerische Mathematik》2005,102(2):311-342
The initial boundary value problem is considered for the dynamic string equation . Its solution is found by means of an algorithm, the constituent parts of which are the Galerkin method, the modified Crank-Nicolson
difference scheme used to perform approximation with respect to spatial and time variables, and also a Picard type iteration
process for solving the system of nonlinear equations obtained by discretization. Errors of the three parts of the algorithm
are estimated and, as a result, its total error estimate is obtained. 相似文献
8.
In this paper, the second order boundary value problem −∇·((x,y)∇u)=f is discretized by the Finite Element Method using piecewise polynomial functions of degree p on a triangular mesh. On the reference element, we define integrated Jacobi polynomials as interior ansatz functions. If
is a constant function on each triangle and each triangle has straight edges, we prove that the element stiffness matrix
has not more than nonzero matrix entries. An application for preconditioning is given. Numerical examples show the advantages of the proposed
basis. 相似文献
9.
Paul Houston Ilaria Perugia Anna Schneebeli Dominik Schötzau 《Numerische Mathematik》2005,100(3):485-518
Summary In this paper, we introduce and analyze the interior penalty discontinuous Galerkin method for the numerical discretization of the indefinite time-harmonic Maxwell equations in the high-frequency regime. Based on suitable duality arguments, we derive a-priori error bounds in the energy norm and the L2-norm. In particular, the error in the energy norm is shown to converge with the optimal order (hmin{s,}) with respect to the mesh size h, the polynomial degree , and the regularity exponent s of the analytical solution. Under additional regularity assumptions, the L2-error is shown to converge with the optimal order (h+1). The theoretical results are confirmed in a series of numerical experiments.Supported by the EPSRC (Grant GR/R76615).Supported by the Swiss National Science Foundation under project 21-068126.02.Supported in part by the Natural Sciences and Engineering Council of Canada. 相似文献
10.
Marcus Stiemer 《Numerische Mathematik》2005,99(3):533-555
Summary. Let be an analytic Jordan curve in the unit disk We regard the hyperbolic minimal energy problem where () denotes the set of all probability measures on . There exist several extremal point discretizations of *, among others introduced by M. Tsuji (Tsuji points) or by K. Menke (hyperbolic Menke points). In the present article, it is proven that hyperbolic Menke points approach the images of roots of unity under a conformal map from onto geometrically fast if the number of points tends to infinity. This establishes a conjecture of K. Menke. In particular, explicit bounds for the approximation error are given. Finally, an effective method for the numerical determination of * providing a geometrically shrinking error bound is presented.Mathematics Subject Classification (1991): 30C85, 30E10, 31C20The notation Menke points has been introduced by D. Gaier. 相似文献
11.
For a random closed set obtained by exponential transformation of the closed range of a subordinator, a regenerative composition of generic positive integer n is defined by recording the sizes of clusters of n uniform random points as they are separated by the points of . We focus on the number of parts Kn of the composition when is derived from a gamma subordinator. We prove logarithmic asymptotics of the moments and central limit theorems for Kn and other functionals of the composition such as the number of singletons, doubletons, etc. This study complements our previous
work on asymptotics of these functionals when the tail of the Lévy measure is regularly varying at 0+.
Research supported in part by N.S.F. Grant DMS-0071448 相似文献
12.
Summary We derive rates of convergence for regularization procedures (characterized by a parameter ) and finite element approximations of the total variation flow, which arises from image processing, geometric analysis and materials sciences. Practically useful error estimates, which depend on only in low polynomial orders, are established for the proposed fully discrete finite element approximations. As a result, scaling laws which relate mesh parameters to the regularization parameter are also obtained. Numerical experiments are provided to validate the theoretical results and show efficiency of the proposed numerical methods. 相似文献
13.
We introduce in this paper two original Mixed methods for the numerical resolution of the (stationary) Primitive Equations
(PE) of the Ocean. The PE govern the behavior of oceanic flows in shallow domains for large time scales. We use a reduced
formulation (Lions et al. [28]) involving horizontal velocities and surface pressures. By using bubble functions constructed
ad-hoc, we are able to define two stable Mixed Methods requiring a low number of degrees of freedom. The first one is based on the
addition of bubbles of reduced support to velocities elementwise. The second one makes use of conic bubbles of extended support along the vertical coordinate. The latter constitutes a genuine mini-element for the PE, e.g., it requires the least number of extra degrees of freedom to stabilize piecewise linear hydrostatic pressures.
Both methods verify a specific inf-sup condition and provide stability and convergence. Finally, we compare several numerical
features of the proposed pairs in the context of other FE methods found in the literature. 相似文献
14.
If E and F are real Banach lattices and there is an algebra and order isomorphism Φ:(E) →(F) between their respective ordered Banach algebras of regular operators then there is a linear order isomorphism U:E→F such that Φ(T) =UTU−1 for all T ∈ (E). 相似文献
15.
Let R be a commutative noetherian henselian non-Gorenstein local ring of depth zero. Denote by modR the category of all finitely generated R-modules, and by the full subcategory of modR consisting of all R-modules of Gorenstein dimension zero. We prove in this paper that if contains a non-free module, then it is not precovering in modR, in particular, there exist infinitely many isomorphism classes of indecomposable R-modules of Gorenstein dimension zero. 相似文献
16.
Valentin Savin 《manuscripta mathematica》2006,119(3):287-303
Let = [X/G] be the quotient stack of a scheme X by an affine group scheme G over a field k. Assume that there is a line bundle on whose underlying line bundle on X is very ample. Let VB() be the category of vector bundles on .We show that is canonically isomorphic to the stack of fiber functors on VB(). This is an analogue of the Tannaka duality for affine groups.
Partially supported by CNCSIS contract no. 33079/2004 相似文献
17.
Summary. We introduce the Jacobi-weighted Besov and Sobolev spaces in the one-dimensional setting. In the framework of these spaces, we analyze lower and upper bounds for approximation errors in the p-version of the boundary element method for hypersingular and weakly singular integral operators on polygons. We prove the optimal rate of convergence for the p-version in the energy norms of and respectively.Mathematics Subject Classification (2000): 65N38This author is supported by NSERC of Canada under Grant OGP0046726 and partially supported by the FONDAP Program (Chile) on Numerical Analysis during his visit of the Universidad de Concepción in 2001.This author is supported by Fondecyt project no. 1010220 and by the FONDAP Program (Chile) on Numerical Analysis.Revised version received January 28, 2004 相似文献
18.
Summary. We employ a data-sparse, recursive matrix representation, so-called -matrices, for the efficient treatment of discretized integral operators. We obtain this format using local tensor product interpolants of the kernel function and replacing high-order approximations with piecewise lower-order ones. The scheme has optimal, i.e., linear, complexity in the memory requirement and time for the matrix-vector multiplication. We present an error analysis for integral operators of order zero. In particular, we show that the optimal convergence (h) is retained for the classical double layer potential discretized with piecewise constant functions.Corrigendum This revised version was published online in February 2005 due to typesetting mistakes in the author correction process. 相似文献
19.
Christoph Scheven 《Mathematische Zeitschrift》2006,253(1):135-157
Let and be smooth Riemannian manifolds, of the dimension n≥2 with nonempty boundary, and compact without boundary. We consider stationary harmonic maps u ∈ H1(, ) with a free boundary condition of the type u(∂) ⊂ Γ, given a submanifold Γ⊂. We prove partial boundary regularity, namely (sing(u))=0, a result that was until now only known in the interior of the domain (see [B]). The key of the proof is a new lemma
that allows an extension of u by a reflection construction. Once the partial regularity theorem is known, it is possible to reduce the dimension of the
singular set further under additional assumptions on the target manifold and the submanifold Γ. 相似文献
20.
Summary. We develop a new algorithm for the fast evaluation of linear combinations of radial functions based on the recently developed fast Fourier transform at nonequispaced knots. For smooth kernels, e.g. the Gaussian, our algorithm requires arithmetic operations. In case of singular kernels an additional regularization procedure must be incorporated and the algorithm has the arithmetic complexity if either the points yj or the points xk are reasonably uniformly distributed. We prove error estimates to obtain clues about the choice of the involved parameters and present numerical examples for various singular and smooth kernels in two dimensions.Mathematics Subject Classification (2000): 65T40, 65T50, 65F30Revised version received December 3, 2003 相似文献