共查询到20条相似文献,搜索用时 31 毫秒
1.
Summary. We construct and analyze combinations of rational implicit and explicit multistep methods for nonlinear evolution equations
and extend thus recent results concerning the discretization of nonlinear parabolic equations. The resulting schemes are linearly
implicit and include as particular cases implicit–explicit multistep schemes as well as the combination of implicit Runge–Kutta
schemes and extrapolation. We establish optimal order error estimates. The abstract results are applied to a third–order evolution
equation arising in the modelling of flow in a fluidized bed. We discretize this equation in space by a Petrov–Galerkin method.
The resulting fully discrete schemes require solving some linear systems to advance in time with coefficient matrices the
same for all time levels.
Received October 22, 2001 / Revised version received April 22, 2002 /
Published online December 13, 2002
Mathematics Subject Classification (1991): Primary 65M60, 65M12; Secondary 65L06
Correspondence to: G. Akrivis 相似文献
2.
Summary. When a system of linear equations is ill-conditioned, regularization techniques provide a quite useful tool for trying to
overcome the numerical inherent difficulties: the ill-conditioned system is replaced by another one whose solution depends
on a regularization term formed by a scalar and a matrix which are to be chosen. In this paper, we consider the case of several
regularizations terms added simultaneously, thus overcoming the problem of the best choice of the regularization matrix. The
error of this procedure is analyzed and numerical results prove its efficiency.
Received January 15, 2002 / Revised version received July 31, 2002 / Published online October 29, 2002
Mathematics Subject Classification (1991): 65F05 – 65F22 相似文献
3.
Summary. In this paper, a numerical scheme is presented by applying the finite element method to the viscosity equations of the Hamilton-Jacobi
equations on unstructured meshes. By improving the finite element scheme, another numerical scheme is constructed. Under certain
limitations, the numerical solutions of the two schemes converge to the viscosity solutions of the Hamilton-Jacobi equations.
The latter numerical scheme has weaker restrictions than the former scheme for convergence. Numerical examples are provided
to test the stability, convergence and sensitivity to different meshes.
Received November 5, 2001 / Revised version received March 5, 2002 / Published online October 29, 2002
RID="*"
ID="*" Current address: Department of Applied Mathematics, University of Petroleum, Dongying 257062, Shandong, P.R.China;
e-mail: xianggui_li@sina.com
Mathematics Subject Classification (1991): 65M60 相似文献
4.
This paper presents a renormalization and homogenization theory for fractional-in-space or in-time diffusion equations with
singular random initial conditions. The spectral representations for the solutions of these equations are provided. Gaussian
and non-Gaussian limiting distributions of the renormalized solutions of these equations are then described in terms of multiple
stochastic integral representations.
Received: 30 May 2000 / Revised version: 9 November 2001 / Published online: 10 September 2002
Mathematics Subject Classification (2000): Primary 62M40, 62M15; Secondary 60H05, 60G60
Key words or phrases: Fractional diffusion equation – Scaling laws – Renormalised solution – Long-range dependence – Non-Gaussian scenario – Mittag-Leffler
function – Stable distributions – Bessel potential – Riesz potential 相似文献
5.
Wensheng Wang 《Probability Theory and Related Fields》2003,126(2):203-220
Kesten and Spitzer have shown that certain random walks in random sceneries converge to stable processes in random sceneries.
In this paper, we consider certain random walks in sceneries defined using stationary Gaussian sequence, and show their convergence
towards a certain self-similar process that we call fractional Brownian motion in Brownian scenery.
Received: 17 April 2002 / Revised version: 11 October 2002 /
Published online: 15 April 2003
Research supported by NSFC (10131040).
Mathematics Subject Classification (2002): 60J55, 60J15, 60J65
Key words or phrases: Weak convergence – Random walk in random scenery – Local time – Fractional Brownian motion in Brownian scenery 相似文献
6.
In this article we investigate algebraic morphisms between toric varieties. Given presentations of toric varieties as quotients
we are interested in the question when a morphism admits a lifting to these quotient presentations. We show that this can
be completely answered in terms of invariant divisors. As an application we prove that two toric varieties, which are isomorphic
as abstract algebraic varieties, are even isomorphic as toric varieties. This generalizes a well-known result of Demushkin
on affine toric varieties.
Received: 11 March 2002 / Revised: 14 June 2002
Mathematics Subject Classification (2000): 14M25, 14L30 相似文献
7.
Sándor Jenei 《Archive for Mathematical Logic》2003,42(5):489-514
We generalize the notions of Girard algebras and MV-algebras by introducing rotation-invariant semigroups. Based on a geometrical
characterization, we present five construction methods which result in rotation-invariant semigroups and in particular, Girard
algebras and MV-algebras. We characterize divisibility of MV-algebras, and point out that integrality of Girard algebras follows
from their other axioms.
Received: 7 January 2002 / Revised version: 4 April 2002 /
Published online: 19 December 2002
RID="*"
ID="*" Supported by the National Scientific Research Fund Hungary (OTKA F/032782).
Mathematics Subject Classification (2000): 20M14, 06F05
Key words or phrases: Residuated lattice – Conjunction for non-classical logics 相似文献
8.
Peter K. Moore 《Numerische Mathematik》2003,94(2):367-401
Summary. Hp-adaptive finite element codes require methods for estimating the error at several spatial orders and for interpolating solutions
between grids. Lobatto polynomial-based techniques are presented for both. An interpolation error-based error estimation strategy
for a posteriori error estimates is generalized to yield asymptotically exact error estimates one order higher than the computed solution.
The estimates involve high-order derivatives of the solution that must be approximated from the computed solution. Differentiating
a ``Taylor-like' series for error in the Lobatto interpolant and using the weak form of the equations yields the correct
derivative approximations. This leads to a more robust order selection strategy. Interpolation between grids is done over
each element using the Lobatto interpolating polynomial. Explicit formulas for the inverse of the resulting Lobatto interpolation
matrices are given. Computational results illustrate the theory.
Received June 25, 2001 / Revised version received February 12, 2002 / Published online October 29, 2002
Mathematics Subject Classification (1991): 65M15,65M20,65M60
This research was partially supported by NSF Grant #DMS-0196108. 相似文献
9.
Combining search directions using gradient flows 总被引:2,自引:0,他引:2
The efficient combination of directions is a significant problem in line search methods that either use negative curvature,
or wish to include additional information such as the gradient or different approximations to the Newton direction.
In this paper we describe a new procedure to combine several of these directions within an interior-point primal-dual algorithm.
Basically, we combine in an efficient manner a modified Newton direction with the gradient of a merit function and a direction
of negative curvature, if it exists. We also show that the procedure is well-defined, and it has reasonable theoretical properties
regarding the rate of convergence of the method.
We also present numerical results from an implementation of the proposed algorithm on a set of small test problems from the
CUTE collection.
Received: November 2000 / Accepted: October 2002 Published online: February 14, 2003
Key Words. negative curvature – primal-dual methods – interior-point methods – nonconvex optimization – line searches
Mathematics Subject Classification (1991): 49M37, 65K05, 90C30 相似文献
10.
Valery E. Grikurov Erkki Heikkola Pekka Neittaanmäki Boris A. Plamenevskii 《Numerische Mathematik》2003,94(2):269-288
Summary. A method for the numerical computation of the so-called augmented scattering matrices (ASM) is suggested for diffraction
gratings. To construct such (unitary) matrices one has to take into account not only the oscillating modes but also those
which exponentially grow (attenuate) in amplitude away from the grating. The method uses an optimization procedure to identify
the coefficients in the asymptotics of such modes. A justification of the approach is given and its numerical implementation
is discussed. Reliable numerical results allow us to study the occurrences of surface waves by means of a general existence
criterion based on the properties of ASM. To illustrate the method we give some examples of surface waves in gratings.
Received October 25, 2001 / Revised version received May 23, 2002 / Published online October 29, 2002
Mathematics Subject Classification (1991): 78M10, 78M50, 78A45 相似文献
11.
On polynomial collocation for second kind integral equations with fixed singularities of Mellin type
Summary. We consider a polynomial collocation for the numerical solution of a second kind integral equation with an integral kernel
of Mellin convolution type. Using a stability result by Junghanns and one of the authors, we prove that the error of the approximate
solution is less than a logarithmic factor times the best approximation and, using the asymptotics of the solution, we derive
the rates of convergence. Finally, we describe an algorithm to compute the stiffness matrix based on simple Gau? quadratures
and an alternative algorithm based on a recursion in the spirit of Monegato and Palamara Orsi. All together an almost best
approximation to the solution of the integral equation can be computed with 𝒪(n
2[log n]2) resp. 𝒪(n
2) operations, where n is the dimension of the polynomial trial space.
Received February 18, 2002 / Revised version received May 15, 2002 / Published online October 29, 2002
RID="⋆"
ID="⋆" Correspondence to: A. Rathsfeld
Mathematics Subject Classification (1991): 65R20 相似文献
12.
Andrzej Rozkosz 《Probability Theory and Related Fields》2003,125(3):393-407
We extend the definition of solutions of backward stochastic differential equations to the case where the driving process
is a diffusion corresponding to symmetric uniformly elliptic divergence form operator. We show existence and uniqueness of
solutions of such equations under natural assumptions on the data and show its connections with solutions of semilinear parabolic
partial differential equations in Sobolev spaces.
Received: 22 January 2002 / Revised version: 10 September 2002 / Published online: 19 December 2002
Research supported by KBN Grant 0253 P03 2000 19.
Mathematics Subject Classification (2002): Primary 60H30; Secondary 35K55
Key words or phrases: Backward stochastic differential equation – Semilinear partial differential equation – Divergence form operator – Weak solution 相似文献
13.
Summary. Moving from the anisotropic interpolation error estimates derived in [12], we provide here both a-priori and a-posteriori
estimates for a generic elliptic problem. The a-priori result is deduced by following the standard finite element theory.
For the a-posteriori estimate, the analysis extends to anisotropic meshes the theory presented in [3–5]. Numerical test-cases
validate the derived results.
Received July 22, 2001 / Revised version received March 20, 2002 / Published online July 18, 2002
Mathematics Subject Classification (1991): 65N15, 65N50 相似文献
14.
We describe an efficient implementation of an interior-point algorithm for non-convex problems that uses directions of negative
curvature. These directions should ensure convergence to second-order KKT points and improve the computational efficiency
of the procedure. Some relevant aspects of the implementation are the strategy to combine a direction of negative curvature
and a modified Newton direction, and the conditions to ensure feasibility of the iterates with respect to the simple bounds.
The use of multivariate barrier and penalty parameters is also discussed, as well as the update rules for these parameters.
We analyze the convergence of the procedure; both the linesearch and the update rule for the barrier parameter behave appropriately.
As the main goal of the paper is the practical usage of negative curvature, a set of numerical results on small test problems
is presented. Based on these results, the relevance of using directions of negative curvature is discussed.
Received: July 2000 / Accepted: October 2002 Published online: December 19, 2002
Key words. Primal-dual methods – Nonconvex optimization – Linesearches
Research supported by Spanish MEC grant BEC2000-0167
Mathematics Subject Classification (1991): 49M37, 65K05, 90C30 相似文献
15.
Andreas Fischer 《Mathematical Programming》2002,94(1):91-124
An iterative framework for solving generalized equations with nonisolated solutions is presented. For generalized equations
with the structure , where is a multifunction and F is single-valued, the framework covers methods that, at each step, solve subproblems of the type . The multifunction approximates F around s. Besides a condition on the quality of this approximation, two other basic assumptions are employed to show Q-superlinear
or Q-quadratic convergence of the iterates to a solution. A key assumption is the upper Lipschitz-continuity of the solution
set map of the perturbed generalized equation . Moreover, the solvability of the subproblems is required. Conditions that ensure these assumptions are discussed in general
and by means of several applications. They include monotone mixed complementarity problems, Karush-Kuhn-Tucker systems arising
from nonlinear programs, and nonlinear equations. Particular results deal with error bounds and upper Lipschitz-continuity
properties for these problems.
Received: November 2001 / Accepted: November 2002 Published online: December 9, 2002
Key Words. generalized equation – nonisolated solutions – Newton's method – superlinear convergence – upper Lipschitz-continuity – mixed
complementarity problem – error bounds
Mathematics Subject Classification (1991): 90C30, 65K05, 90C31, 90C33 相似文献
16.
Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows 总被引:1,自引:0,他引:1
Summary. We propose and analyze a semi-discrete (in time) scheme and a fully discrete scheme for the Allen-Cahn equation u
t
−Δu+ɛ−2
f(u)=0 arising from phase transition in materials science, where ɛ is a small parameter known as an ``interaction length'. The
primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical methods, in particular,
by focusing on the dependence of the error bounds on ɛ. Optimal order and quasi-optimal order error bounds are shown for the
semi-discrete and fully discrete schemes under different constraints on the mesh size h and the time step size k and different regularity assumptions on the initial datum function u
0
. In particular, all our error bounds depend on only in some lower polynomial order for small ɛ. The cruxes of the analysis are to establish stability estimates for the
discrete solutions, to use a spectrum estimate result of de Mottoni and Schatzman [18, 19] and Chen [12] and to establish
a discrete counterpart of it for a linearized Allen-Cahn operator to handle the nonlinear term. Finally, as a nontrivial byproduct,
the error estimates are used to establish convergence and rate of convergence of the zero level set of the fully discrete
solution to the motion by mean curvature flow and to the generalized motion by mean curvature flow.
Received April 30, 2001 / Revised version received March 20, 2002 / Published online July 18, 2002
Mathematics Subject Classification (1991): 65M60, 65M12, 65M15, 35B25, 35K57, 35Q99, 53A10
Correspondence to: A. Prohl 相似文献
17.
Let Γ be a word hyperbolic group M. Gromov has constructed a compact space equipped with a flow which is defined up to orbit-equivalence and which is called the geodesic flow of Γ. In the special
case where Γ is the fundamental group of a Riemannian manifold of negative sectional curvature, is the unit tangent bundle of the manifold equipped with the usual geodesic flow. In this paper, we construct, for every
hyperbolic group Γ, a subshift of finite type and a continuous map from the suspension of this subshift onto , which is uniformly bounded-to-one and which sends each orbit of the suspension flow onto an orbit of the geodesic flow.
Received: 25 January 2002 / Revised version: 20 August 2002
Mathematics Subject Classification (2000): 20F67, 20F65, 20F69, 53C23, 53C21, 37D40, 37B10, 54H20 相似文献
18.
Summary. Impedance tomography seeks to recover the electrical conductivity distribution inside a body from measurements of current
flows and voltages on its surface. In its most general form impedance tomography is quite ill-posed, but when additional a-priori
information is admitted the situation changes dramatically. In this paper we consider the case where the goal is to find a
number of small objects (inhomogeneities) inside an otherwise known conductor. Taking advantage of the smallness of the inhomogeneities,
we can use asymptotic analysis to design a direct (i.e., non-iterative) reconstruction algorithm for the determination of
their locations. The viability of this direct approach is documented by numerical examples.
Received May 28, 2001 / Revised version received March 15, 2002 / Published online July 18, 2002
RID="⋆"
ID="⋆" Supported by the Deutsche Forschungsgemeinschaft (DFG) under grant HA 2121/2-3
RID="⋆⋆"
ID="⋆⋆" Supported by the National Science Foundation under grant DMS-0072556
Mathematics Subject Classification (2000): 65N21, 35R30, 35C20 相似文献
19.
In this paper, we describe how to reformulate a problem that has second-order cone and/or semidefiniteness constraints in
order to solve it using a general-purpose interior-point algorithm for nonlinear programming. The resulting problems are smooth
and convex, and numerical results from the DIMACS Implementation Challenge problems and SDPLib are provided.
Received: March 10, 2001 / Accepted: January 18, 2002 Published online: September 27, 2002
Key Words. semidefinite programming – second-order cone programming – interior-point methods – nonlinear programming
Mathematics Subject Classification (2000): 20E28, 20G40, 20C20 相似文献
20.
Tadahisa Funaki 《Probability Theory and Related Fields》2003,126(2):155-183
We consider random evolution of an interface on a hard wall under periodic boundary conditions. The dynamics are governed
by a system of stochastic differential equations of Skorohod type, which is Langevin equation associated with massless Hamiltonian
added a strong repelling force for the interface to stay over the wall. We study its macroscopic behavior under a suitable
large scale space-time limit and derive a nonlinear partial differential equation, which describes the mean curvature motion
except for some anisotropy effects, with reflection at the wall. Such equation is characterized by an evolutionary variational
inequality.
Received: 10 January 2002 / Revised version: 18 August 2002 /
Published online: 15 April 2003
Mathematics Subject Classification (2000): 60K35, 82C24, 35K55, 35K85
Key words or phrases: Hydrodynamic limit – Effective interfaces – Hard wall – Skorohod's stochastic differential equation – Evolutionary variational
inequality 相似文献