Integrability on codimension one dominated splitting

We study the unique integrability of the center unstable subbundle of a codimension one dominated splitting. *Partially supported by CNPq-Brazil.     

Equivalent Conditions of Dominated Splitting for Volume-Preserving Diffeomorphisms

We discuss the equivalent conditions of dominated splitting for conservative diffeomorphisms in C^1 topology.     

Conditions for dominated splittings

In this paper, we solve the problem proposed by Lan Wen for the case of dimM = 3. Roughly speaking, we prove that for fixed i, f has C1 persistently no small angles of index i if and only if f has a dominated splitting of index i on the C1 i-preperiodic set P＊i（f）.     

线性对流占优扩散方程的后验误差估计

对于线性对流占优扩散方程,采用特征线有限元方法离散时间导数项和对流项,用分片线性有限元离散空间扩散项,并给出了一致的后验误差估计,其中估计常数不依赖与扩散项系数。     

Modulus‐based matrix splitting iteration methods for linear complementarity problems

For the large sparse linear complementarity problems, by reformulating them as implicit fixed‐point equations based on splittings of the system matrices, we establish a class of modulus‐based matrix splitting iteration methods and prove their convergence when the system matrices are positive‐definite matrices and H₊‐matrices. These results naturally present convergence conditions for the symmetric positive‐definite matrices and the M‐matrices. Numerical results show that the modulus‐based relaxation methods are superior to the projected relaxation methods as well as the modified modulus method in computing efficiency. Copyright © 2009 John Wiley & Sons, Ltd.     

A two-step matrix splitting iteration paradigm based on one single splitting for solving systems of linear equations

For solving large sparse systems of linear equations, we construct a paradigm of two-step matrix splitting iteration methods and analyze its convergence property for the nonsingular and the positive-definite matrix class. This two-step matrix splitting iteration paradigm adopts only one single splitting of the coefficient matrix, together with several arbitrary iteration parameters. Hence, it can be constructed easily in actual applications, and can also recover a number of representatives of the existing two-step matrix splitting iteration methods. This result provides systematic treatment for the two-step matrix splitting iteration methods, establishes rigorous theory for their asymptotic convergence, and enriches algorithmic family of the linear iteration solvers, for the iterative solutions of large sparse linear systems.     

Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming

A classical method for solving the variational inequality problem is the projection algorithm. We show that existing convergence results for this algorithm follow from one given by Gabay for a splitting algorithm for finding a zero of the sum of two maximal monotone operators. Moreover, we extend the projection algorithm to solveany monotone affine variational inequality problem. When applied to linear complementarity problems, we obtain a matrix splitting algorithm that is simple and, for linear/quadratic programs, massively parallelizable. Unlike existing matrix splitting algorithms, this algorithm converges under no additional assumption on the problem. When applied to generalized linear/quadratic programs, we obtain a decomposition method that, unlike existing decomposition methods, can simultaneously dualize the linear constraints and diagonalize the cost function. This method gives rise to highly parallelizable algorithms for solving a problem of deterministic control in discrete time and for computing the orthogonal projection onto the intersection of convex sets.This research is partially supported by the U.S. Army Research Office, contract DAAL03-86-K-0171 (Center for Intelligent Control Systems), and by the National Science Foundation under grant NSF-ECS-8519058.Thanks are due to Professor J.-S. Pang for his helpful comments.     

Stable weakly shadowable volume-preserving systems are volume-hyperbolic

We prove that any C1-stable weakly shadowable volume-preserving diffeomorphism defined on a compact manifold displays a dominated splitting E ⊕ F. Moreover, both E and F are volume-hyperbolic. Finally, we prove the version of this result for divergence-free vector fields. As a consequence, in low dimensions, we obtain global hyperbolicity.     

A two-step matrix splitting method for the mixed linear complementarity problem

In this paper, on the base of the methodology of the new modulus-based matrix splitting method in [Optim. Lett., (2022) 16:1427-1443], we establish a two-step matrix splitting (TMS) method for solving the mixed linear complementarity problem (MLCP). Two sufficient conditions to ensure the convergence of the proposed method are presented. Numerical examples are provided to illustrate the feasibility and efficiency of the proposed method.     

机器具有优势关系下的工件加工时间可控的流水作业排序问题

本文主要研究机器具有优势关系下的工件加工时间可控的流水作业排序问题.我们主要对以下两种情形进行了讨论:工件加工时间为线性恶化和线性学习.对于每一种加工模型,我们分别研究了几类不同的优势机器,并且对每种情况均给出了多项式时间算法.     

A modified iterated operator splitting method

In this paper we give a short overview of some traditional operator splitting methods. Furthermore, we introduce two recently developed methods, namely the additive splitting and the iterated splitting. We analyze the iterated splitting method in detail and give the suitable strategy for the choice of the initial elements in the iterations in order to get higher order discretization.     

A note on asymptotic splitting and its applications

Explicit forms of the remainder terms of the asymptotic splitting formulae associated with the first order splitting, Strange's splitting, and the parallel splitting are derived. Using the identities obtained, we establish the global error estimates for the asymptotic splitting formulae. Both the theoretical investigation and numerical experiments indicate that it is a more efficient and accurate way to use the asymptotic splitting than the conventional splitting formulae.     

Quasi‐Toeplitz splitting iteration methods for unsteady space‐fractional diffusion equations

We construct a class of quasi‐Toeplitz splitting iteration methods to solve the two‐sided unsteady space‐fractional diffusion equations with variable coefficients. By making full use of the structural characteristics of the coefficient matrix, the method only requires computational costs of O(n log n) with n denoting the number of degrees of freedom. We develop an appropriate circulant matrix to replace the Toeplitz matrix as a preconditioner. We discuss the spectral properties of the quasi‐circulant splitting preconditioned matrix. Numerical comparisons with existing approaches show that the present method is both effective and efficient when being used as matrix splitting preconditioners for Krylov subspace iteration methods.     

Generalized skew‐Hermitian triangular splitting iteration methods for saddle‐point linear systems

A generalized skew‐Hermitian triangular splitting iteration method is presented for solving non‐Hermitian linear systems with strong skew‐Hermitian parts. We study the convergence of the generalized skew‐Hermitian triangular splitting iteration methods for non‐Hermitian positive definite linear systems, as well as spectrum distribution of the preconditioned matrix with respect to the preconditioner induced from the generalized skew‐Hermitian triangular splitting. Then the generalized skew‐Hermitian triangular splitting iteration method is applied to non‐Hermitian positive semidefinite saddle‐point linear systems, and we prove its convergence under suitable restrictions on the iteration parameters. By specially choosing the values of the iteration parameters, we obtain a few of the existing iteration methods in the literature. Numerical results show that the generalized skew‐Hermitian triangular splitting iteration methods are effective for solving non‐Hermitian saddle‐point linear systems with strong skew‐Hermitian parts. Copyright © 2013 John Wiley & Sons, Ltd.     

Exponential splitting time integration for pseudospectral methods on moving meshes

Moving meshes are successfully used in many fields. Here we investigate how a recently proposed approach to combine the Strang splitting method for time integration with pseudospectral spatial discretization by orthogonal polynomials can be extended to include moving meshes. A double representation of a function (by coefficients of polynomial expansion and by values at the mesh nodes associated with a suitable quadrature formula) is an essential part of the numerical integration. Before numerical implementation the original PDE is transformed into a suitable form. The approach is illustrated on the linear heat transfer equation.     

On estimation of the optimal parameter of the modulus-based matrix splitting algorithm for linear complementarity problems on second-order cones

There are many studies on the well-known modulus-based matrix splitting (MMS) algorithm for solving complementarity problems, but very few studies on its optimal parameter, which is of theoretical and practical importance. Therefore and here, by introducing a novel mapping to explicitly cast the implicit fixed point equation and thus obtain the iteration matrix involved, we first present the estimation approach of the optimal parameter of each step of the MMS algorithm for solving linear complementarity problems on the direct product of second-order cones (SOCLCPs). It also works on single second-order cone and the non-negative orthant. On this basis, we further propose an iteration-independent optimal parameter selection strategy for practical usage. Finally, the practicability and effectiveness of the new proposal are verified by comparing with the experimental optimal parameter and the diagonal part of system matrix. In addition, with the optimal parameter, the effectiveness of the MMS algorithm can indeed be greatly improved, even better than the state-of-the-art solvers SCS and SuperSCS that solve the equivalent SOC programming.     

Diffeomorphisms with C 1-stably average shadowing

Let M be a closed smooth manifold M, and let f : M → M be a diffeomorphism. In this paper, we consider a nontrivial transitive set Λ of f . We show that if f has the C1-stably average shadowing property on Λ, then Λ admits a dominated splitting.     

Superrigidity for irreducible lattices and geometric splitting

We prove general superrigidity results for actions of irreducible lattices on CAT spaces, first in terms of the ideal boundary, and then for the intrinsic geometry (also for infinite-dimensional spaces). In particular, one obtains a new and self-contained proof of Margulis' superrigidity theorem for uniform irreducible lattices in non-simple groups. The proofs rely on simple geometric arguments, including a splitting theorem which can be viewed as an infinite-dimensional (and singular) generalization of the Lawson-Yau/Gromoll-Wolf theorem. Appendix A gives a very elementary proof of commensurator superrigidity; Appendix B proves that all our results also hold for certain non-uniform lattices.

     

A splitting theorem for degrees

We prove that, for any , and with _{T}A\oplus U$"> and r.e., in , there are pairs and such that ; ; and, for any and from and any set , if and , then . We then deduce that for any degrees , , and such that and are recursive in , , and is into , can be split over avoiding . This shows that the Main Theorem of Cooper (Bull. Amer. Math. Soc. 23 (1990), 151-158) is false.