共查询到20条相似文献,搜索用时 875 毫秒
1.
Summary. Conformal maps from the exterior of the closed unit disk onto the exterior of ‘bratwurst’ shape sets in the complex plane
are constructed. Using these maps, coefficients for the computation of the corresponding Faber polynomials are derived. A
‘bratwurst’ shape set is the result of deforming an ellipse with foci on the real axis, by conformally mapping the real axis
onto the unit circle. Such sets are well suited to serve as inclusion sets for sets associated with a matrix, for example
the spectrum, field of values or a pseudospectrum. Hence, the sets can be applied in the construction and analysis of a broad
range of iterative methods for the solution of linear systems. The main advantage of the approach is that the conformal maps
are derived from elementary transformations, allowing an easy computation of the associated transfinite diameter, asymptotic
convergence factor and Faber polynomials. Numerical examples are given.
Received October 7, 1998 / Revised version received March 15, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000 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.
Summary A number of numerical solutions are presented as examples of a new iterative method for variational inequalities. The iterative method is based on the reduction of variational inequalities to the Wiener-Hopf equations. For obstacle problems the convergence is guaranteed inW
1,p
spaces forp2. The examples presented are one and two dimensional obstacle problems in cases when the Greens function is known, but the method is very general. 相似文献
4.
For solving nonsymmetric linear systems, the well-known GMRES method is considered to be a stable method; however, the work
per iteration increases as the number of iterations increases. We consider two new iterative methods GGMRES and MGMRES, which
are a generalization and a modification of the GMRES method, respectively. Instead of using a minimization condition as in
the derivation of GGMRES, we use a Galerkin condition to derive the MGMRES method. We also introduce another new iterative
method, LAN/MGMRES, which is designed to combine the reliability of GMRES with the reduced work of a Lanczos-type method.
A computer program has been written based on the use of the LAN/MGMRES algorithm for solving nonsymmetric linear systems arising
from certain elliptic problems. Numerical tests are presented comparing this algorithm with some other commonly used iterative
algorithms. These preliminary tests of the LAN/MGMRES algorithm show that it is comparable in terms of both the approximate
number of iterations and the overall convergence behavior.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
5.
Summary A parallel projection algorithm is proposed to solve the generalized linear least-squares problem: find a vector to minimize the 2-norm distance from its image under an affine mapping to a closed convex cone. In each iteration of the algorithm the problem is decomposed into several independent small problems of finding projections onto subspaces, which are simple and can be tackled parallelly. The algorithm can be viewed as a dual version of the algorithm proposed by Han and Lou [8]. For the special problem under consideration, stronger convergence results are established. The algorithm is also related to the block iterative methods of Elfving [6], Dennis and Steihaug [5], and the primal-dual method of Springarn [14].This material is based on work supported in part by the National Science foundation under Grant DMS-8602416 and by the Center for Supercomputing Research and Development, University of Illinois at Urbana-Champaign 相似文献
6.
Summary.
We analyze the convergence of a substructuring iterative method
with Lagrange multipliers, proposed recently by Farhat and Roux.
The method decomposes finite element
discretization of an elliptic boundary value problem into
Neumann problems on the subdomains plus a coarse problem for the
subdomain nullspace components. For linear conforming elements and
preconditioning by the Dirichlet problems on the subdomains,
we prove the asymptotic bound on the condition number
,
or ,where
is the characteristic element size and
subdomain size.
Received January 3, 1995 相似文献
7.
Summary. We construct a quadrature formula for integration on the unit disc which is based on line integrals over distinct chords in the disc and integrates exactly all polynomials in two variables of total degree .
Received August 8, 1996 / Revised version received July 2, 1997 相似文献
8.
In this paper, some semismooth methods are considered to solve a nonsmooth equation which can arise from a discrete version of the well-known Hamilton-Jacobi-Bellman equation. By using the slant differentiability introduced by Chen, Nashed and Qi in 2000, a semismooth Newton method is proposed. The method is proved to have monotone convergence by suitably choosing the initial iterative point and local superlinear convergence rate. Moreover, an inexact version of the proposed method is introduced, which reduces the cost of computations and still preserves nice convergence properties. Some numerical results are also reported. 相似文献
9.
Summary.
In recent years, it has been shown that many modern iterative algorithms
(multigrid schemes, multilevel preconditioners, domain decomposition
methods etc.)
for solving problems resulting from the discretization
of PDEs can be
interpreted as additive (Jacobi-like) or multiplicative
(Gauss-Seidel-like) subspace correction methods. The key to their
analysis is the study of certain metric properties of the underlying
splitting of the discretization space into a sum of subspaces
and the splitting of the variational problem on into auxiliary problems on
these subspaces.
In this paper, we propose a modification of the abstract convergence
theory of the additive and multiplicative Schwarz methods, that
makes the relation to traditional iteration methods more explicit.
The analysis of the additive and multiplicative Schwarz iterations
can be carried out in almost the same spirit as in the
traditional block-matrix
situation, making convergence proofs of multilevel and domain decomposition
methods clearer, or, at least, more classical.
In addition, we present a
new bound for the convergence rate of the appropriately scaled
multiplicative Schwarz method directly in terms
of the condition number of the corresponding additive
Schwarz operator.
These results may be viewed as an appendix to the
recent surveys [X], [Ys].
Received February 1, 1994 / Revised version received August
1, 1994 相似文献
10.
Xue-Cheng Tai 《Numerical Algorithms》1992,3(1):427-440
Extrapolation with a parallel splitting method is discussed. The parallel splitting method reduces a multidimensional problem into independent one-dimensional problems and can improve the convergence order of space variables to an order as high as the regularity of the solution permits. Therefore, in order to match the convergence order of the space variables, a high order method should also be used for the time integration. Second and third order extrapolation methods are used to improve the time convergence and it was found that the higher order extrapolation method can produce a more accurate solution than the lower order extrapolation method, but the convergence order of high order extrapolation may be less than the actual order of the extrapolation. We also try to show a fact that has not been studied in the literature, i.e. when the extrapolation is used, it may decrease the convergence of the space variables. The higher the order of the extrapolation method, the more it decreases the convergence of the space variables. The global extrapolation method also improves the parallel degree of the parallel splitting method. Numerical tests in the paper are done in a domain of a unit circle and a unit square.Supported by the Academy of Finland. 相似文献
11.
For the augmented system of linear equations, Golub, Wu and Yuan recently studied an SOR-like method (BIT 41(2001)71–85).
By further accelerating it with another parameter, in this paper we present a generalized SOR (GSOR) method for the augmented
linear system. We prove its convergence under suitable restrictions on the iteration parameters, and determine its optimal
iteration parameters and the corresponding optimal convergence factor. Theoretical analyses show that the GSOR method has
faster asymptotic convergence rate than the SOR-like method. Also numerical results show that the GSOR method is more effective
than the SOR-like method when they are applied to solve the augmented linear system. This GSOR method is further generalized
to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear
systems by using the techniques of vector extrapolation, matrix relaxation and inexact iteration. Besides, we also demonstrate
a complete version about the convergence theory of the SOR-like method.
Subsidized by The Special Funds For Major State Basic Research Projects (No. G1999032803) and The National Natural Science
Foundation (No. 10471146), P.R. China 相似文献
12.
C.V. Pao 《Numerische Mathematik》1995,72(2):239-262
Summary.
Two block monotone iterative schemes for a nonlinear
algebraic system, which is a finite difference approximation of a
nonlinear elliptic boundary-value problem, are presented and are
shown to converge monotonically either from above or from below to
a solution of the system. This monotone convergence result yields
a computational algorithm for numerical solutions as well as an
existence-comparison theorem of the system, including a sufficient
condition for the uniqueness of the solution. An advantage of the
block iterative schemes is that the Thomas algorithm can be used to
compute numerical solutions of the sequence of iterations in the
same fashion as for one-dimensional problems. The block iterative
schemes are compared with the point monotone iterative schemes of
Picard, Jacobi and Gauss-Seidel, and various theoretical comparison
results among these monotone iterative schemes are given. These
comparison results demonstrate that the sequence of iterations from
the block iterative schemes converges faster than the corresponding
sequence given by the point iterative schemes. Application of the
iterative schemes is given to a logistic model problem in ecology
and numerical ressults for a test problem with known analytical
solution are given.
Received
August 1, 1993 / Revised version received November 7, 1994 相似文献
13.
This paper deals with a semi-discrete version of the Galerkin method for the single-layer equation in a plane, in which the
outer integral is approximated by a quadrature rule. A feature of the analysis is that it does not require high precision
quadrature or exceptional smoothness of the data. Instead, the assumptions on the quadrature rule are that constant functions
are integrated exactly, that the rule is based on sufficiently many quadrature points to resolve the approximation space,
and that the Peano constant of the rule is sufficiently small. It is then shown that the semi-discrete Galerkin approximation is well posed. For the
trial and test spaces we consider quite general piecewise polynomials on quasi-uniform meshes, ranging from discontinuous
piecewise polynomials to smoothest splines. Since it is not assumed that the quadrature rule integrates products of basis
functions exactly, one might expect degradation in the rate of convergence. To the contrary, it is shown that the semi-discrete
Galerkin approximation will converge at the same rate as the corresponding Galerkin approximation in the and norms.
Received March 15, 1996 / Revised version received June 2, 1997 相似文献
14.
Summary.
Large, sparse nonsymmetric systems of linear equations with a
matrix whose eigenvalues lie in the right half plane may be solved by an
iterative method based on Chebyshev polynomials for an interval in the
complex plane. Knowledge of the convex hull of the spectrum of the
matrix is required in order to choose parameters upon which the
iteration depends. Adaptive Chebyshev algorithms, in which these
parameters are determined by using eigenvalue estimates computed by the
power method or modifications thereof, have been described by Manteuffel
[18]. This paper presents an adaptive Chebyshev iterative method, in
which eigenvalue estimates are computed from modified moments determined
during the iterations. The computation of eigenvalue estimates from
modified moments requires less computer storage than when eigenvalue
estimates are computed by a power method and yields faster convergence
for many problems.
Received May 13, 1992/Revised version received May 13,
1993 相似文献
15.
Summary. This analysis of convergence of a coupled FEM-IEM is based on our previous work on the FEM and the IEM for exterior Helmholtz
problems. The key idea is to represent both the exact and the numerical solution by the Dirichlet-to-Neumann operators that
they induce on the coupling hypersurface in the exterior of an obstacle. The investigation of convergence can then be related
to a spectral analysis of these DtN operators. We give a general outline of our method and then proceed to a detailed investigation
of the case that the coupling surface is a sphere. Our main goal is to explore the convergence mechanism. In this context,
we show well-posedness of both the continuous and the discrete models. We further show that the discrete inf-sup constants
have a positive lower bound that does not depend on the number of DOF of the IEM. The proofs are based on lemmas on the spectra
of the continuous and the discrete DtN operators, where the spectral characterization of the discrete DtN operator is given
as a conjecture from numerical experiments. In our convergence analysis, we show algebraic (in terms of N) convergence of arbitrary order and generalize this result to exponential convergence.
Received April 10, 1999 / Revised version received November 10, 1999 / Published online October 16, 2000 相似文献
16.
Mikko Lyly 《Numerische Mathematik》2000,85(1):77-107
Summary. We consider three triangular plate bending elements for the Reissner-Mindlin model. The elements are the MIN3 element of
Tessler and Hughes [19], the stabilized MITC3 element of Brezzi, Fortin and Stenberg [5] and the T3BL element of Xu, Auricchio
and Taylor [2, 17, 20]. We show that the bilinear forms of the stabilized MITC3 and MIN3 elements are equivalent and that
their implementation may be simplified by using numerical integration of reduced order. The T3BL element is shown to be essentially
the same as the MIN3 and stabilized MITC3 elements with reduced integration. We finally introduce a general stabilized finite
element formulation which covers all three methods. For this class of methods we prove the stability and optimal convergence
properties.
Received November 4, 1996 / Revised version received May 29, 1997 / Published online January 27, 2000 相似文献
17.
Summary. The Schur complement of a model problem is considered as a preconditioner for the Uzawa type schemes for the generalized
Stokes problem (the Stokes problem with the additional term in the motion equation). The implementation of the preconditioned method requires for each iteration only one extra solution
of the Poisson equation with Neumann boundary conditions. For a wide class of 2D and 3D domains a theorem on its convergence
is proved. In particular, it is established that the method converges with a rate that is bounded by some constant independent
of . Some finite difference and finite element methods are discussed. Numerical results for finite difference MAC scheme are
provided.
Received May 2, 1997 / Revised version received May 10, 1999 / Published online May 8, 2000 相似文献
18.
Kristian Witsch 《Numerische Mathematik》1978,31(2):209-230
Summary In this paper we consider the following Newton-like methods for the solution of nonlinear equations. In each step of the Newton method the linear equations are solved approximatively by a projection method. We call this a Projective Newton method. For a fixed projection method the approximations often are the same as those of the Newton method applied to a nonlinear projection method. But the efficiency can be increased by adapting the accuracy of the projection method to the convergence of the approximations. We investigate the convergence and the order of convergence for these methods. The results are applied to some Projective Newton methods for nonlinear two point boundary value problems. Some numerical results indicate the efficiency of these methods. 相似文献
19.
A preconditioned minimal residual method for nonsymmetric saddle point problems is analyzed. The proposed preconditioner
is of block triangular form. The aim of this article is to show that a rigorous convergence analysis can be performed by using
the field of values of the preconditioned linear system. As an example, a saddle point problem obtained from a mixed finite
element discretization of the Oseen equations is considered. The convergence estimates obtained by using a field–of–values
analysis are independent of the discretization parameter h. Several computational experiments supplement the theoretical results and illustrate the performance of the method.
Received March 20, 1997 / Revised version received January 14, 1998 相似文献
20.
Alfredo N. Iusem 《Acta Appl Math》1995,38(2):163-184
We analyze an algorithm for the problem minf(x) s.t.x 0 suggested, without convergence proof, by Eggermont. The iterative step is given by x
j
k+1
=x
j
k
(1-kf(x
k)j) with k > 0 determined through a line search. This method can be seen as a natural extension of the steepest descent method for unconstrained optimization, and we establish convergence properties similar to those known for steepest descent, namely weak convergence to a KKT point for a generalf, weak convergence to a solution for convexf and full convergence to the solution for strictly convexf. Applying this method to a maximum likelihood estimation problem, we obtain an additively overrelaxed version of the EM Algorithm. We extend the full convergence results known for EM to this overrelaxed version by establishing local Fejér monotonicity to the solution set.Research for this paper was partially supported by CNPq grant No 301280/86. 相似文献