A method for verifying the accuracy of numerical solutions of symmetric saddle point linear systems

A fast numerical verification method is proposed for evaluating the accuracy of numerical solutions for symmetric saddle point linear systems whose diagonal blocks of the coefficient matrix are semidefinite matrices. The method is based on results of an algebraic analysis of a block diagonal preconditioning. Some numerical experiments are present to illustrate the usefulness of the method.     

A comparison of overlapping Schwarz methods and block preconditioners for saddle point problems

Three domain decomposition methods for saddle point problems are introduced and compared. The first two are block‐diagonal and block‐triangular preconditioners with diagonal blocks approximated by an overlapping Schwarz technique with positive definite local and coarse problems. The third is an overlapping Schwarz preconditioner based on indefinite local and coarse problems. Numerical experiments show that while all three methods are numerically scalable, the last method is almost always the most efficient. Copyright © 2000 John Wiley & Sons, Ltd.     

Homoclinic orbits for a class of first-order nonperiodic asymptotically quadratic Hamiltonian systems with spectrum point zero

In this paper, we study the existence and multiplicity of homoclinic orbits for a class of first-order nonperiodic Hamiltonian systems. By applying two recent critical point theorems for strongly indefinite functionals, we give some new criteria to guarantee that Hamiltonian systems with asymptotically quadratic terms and spectrum point zero have at least one and a finite number of pairs of homoclinic orbits under some adequate conditions, respectively.     

A mixed finite element method for nonlinear elasticity: two-fold saddle point approach and a-posteriori error estimate

Summary. We extend the applicability of stable mixed finite elements for linear plane elasticity, such as PEERS, to a mixed variational formulation of hyperelasticity. The present approach is based on the introduction of the strain tensor as a further unknown, which yields a two-fold saddle point nonlinear operator equation for the corresponding weak formulation. We provide the uniqueness of solution for the continuous and discrete schemes, and derive the usual Cea estimate for the associated error. Finally, a reliable a-posteriori error estimate, based on the solution of local Dirichlet problems, and well suited for adaptive computations, is also given. Received August 5, 2000 / Published online August 17, 2001     

A symmetric design with parameters 2-(49, 16, 5)

A symmetric 2-design with parameters (v, k, λ) = (49, 16, 5) is constructed. Both this design and its residual, a design with parameters (v, b, r, k, λ) = (33, 48, 16, 11, 5), seem to be new. The derived designs do not have repeated blocks. The group of the design is cyclic of order 15. There is no polarity.     

Existence and multiplicity of nonzero periodic solution with saddle point character for some nonautonomous second order systems

The purpose of this paper is to study the existence and multiplicity of nonzero periodic solutions with saddle point character for the following nonautonomous second order systems:

     

A symmetric 2-(160, 54, 18) design

We give a construction for a symmetric 2-(160, 54, 18) design that effectively uses the incidence matrix of the design of the hyperplanes in PG(3, 3) as a tactical decomposition. This is the first design ever found with these parameters. © 1993 John Wiley & Sons, Inc.     

The existence of symmetric block designs with parameters (41, 16, 6) and (66, 26, 10)

In [1, 2] it is indicated that the existence of block designs with parameters (41, 16, 6) and (66, 26, 10) is still unknown. The purpose of this paper is to prove the existence of block designs with these parameters. The idea for the construction of these block designs is to assume that they admit certain automorphism group.     

A real-valued block conjugate gradient type method for solving complex symmetric linear systems with multiple right-hand sides

We consider solving complex symmetric linear systems with multiple right-hand sides. We assume that the coefficient matrix has indefinite real part and positive definite imaginary part. We propose a new block conjugate gradient type method based on the Schur complement of a certain 2-by-2 real block form. The algorithm of the proposed method consists of building blocks that involve only real arithmetic with real symmetric matrices of the original size. We also present the convergence property of the proposed method and an efficient algorithmic implementation. In numerical experiments, we compare our method to a complex-valued direct solver, and a preconditioned and nonpreconditioned block Krylov method that uses complex arithmetic.     

The exit problem for small random perturbations of dynamical systems with a hyperbolic fixed point

We consider the Markov diffusion process ξ^∈(t), transforming when ɛ=0 into the solution of an ordinary differential equation with a turning point ℴ of the hyperbolic type. The asymptotic behevior as ɛ→0 of the exit time, of its expectation of the probability distribution of exit points for the process ξ^∈(t) is studied. These indicate also the asymptotic behavior of solutions of the corresponding singularly perturbed elliptic boundary value problems.     

A new symmetric design with parameters (176, 50, 14)

A new symmetric non‐self‐dual design with parameters (176, 50, 14) is constructed, where its full automorphism group of order 1000 is a faithful extension of the non‐abelian group of order 125 and exponent 5 by the cyclic group of order 8. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 387–390, 2000     

Localization problem for one-dimensional disordered systems (a new approach)

The Molchanov theorem, proved as early as 1951, is applied to study the localization problem for one-electron states in one-dimensional disordered systems. The theorem permits one to approach the problem from a general standpoint and establishes a new localization criterion based on the asymptotic properties of a random potential. It is shown that in conditions of diagonal disorder, the theorem provides no new results, namely all states turn out to be localized. However, in the case of structural disorder, it implies that under some comparatively weak conditions, all states can be delocalized.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 3, pp. 425–437, March, 1996.Translated by V. M. Volosov.     

A new approach for finding a basis for the splitting preconditioner for linear systems from interior point methods

The class of splitting preconditioners for the iterative solution of linear systems arising from Mehrotra’s predictor-corrector method for large scale linear programming problems needs to find a basis through a sophisticated process based on the application of a rectangular LU factorization. This class of splitting preconditioners works better near a solution of the linear programming problem when the matrices are highly ill-conditioned. In this study, we develop and implement a new approach to find a basis for the splitting preconditioner, based on standard rectangular LU factorization with partial permutation of the scaled transpose linear programming constraint matrix. In most cases, this basis is better conditioned than the existing one. In addition, we include a penalty parameter in Mehrotra’s predictor-corrector method in order to reduce ill-conditioning of the normal equations matrix. Computational experiments show a reduction in the average number of iterations of the preconditioned conjugate gradient method. Also, the increased efficiency and robustness of the new approach become evident by the performance profile.