Volume preserving flows with cyclic winding numbers groups and without periodic orbits on 3-manifolds

We construct examples of volume preserving non-singular C ¹ vector fields on closed orientable 3-manifolds, which have cyclic winding numbers groups with respect to the preserved volume element, but have no periodic orbits. Received: 17 January 1998 / Revised version: 31 March 1998     

Minimal heteroclinic geodesics for the n-torus

This paper gives an extension of earlier work of Morse and of Hedlund on minimal heteroclinic geodesics for to the case of provided that an additional geometrical condition is satisfied. It also gives lower bounds on the number of such geodesics.     

Fixed mesh approximation of ordinary differential equations with impulses

When trains of impulse controls are present on the right-hand side of a system of ordinary differential equations, the solution is no longer smooth and contains jumps which can accumulate at several points in the time interval. In technological and physical systems the sum of the absolute value of all the impulses is finite and hence the total variation of the solution is finite. So the solution at best belongs to the space BV of vector functions with bounded variation. Unless variable node methods are used, the loss of smoothness of the solution would a priori make higher-order methods over a fixed mesh inactractive. Indeed in general the order of -convergence is and the nodal rate is . However in the linear case -convergence rate remains but the nodal rate can go up to by using one-step or multistep scheme with a nodal rate up to when the solution belongs to . Proofs are given of error estimates and several numerical experiments confirm the optimality of the estimates. Received March 15, 1996 / Revised version received January 3, 1997     

Rigorous computational shadowing of orbits of ordinary differential equations

Summary. The existence of a true orbit near a numerically computed approximate orbit -- shadowing -- of autonomous system of ordinary differential equations is investigated. A general shadowing theorem for finite time, which guarantees the existence of shadowing in ordinary differential equations and provides error bounds for the distance between the true and the approximate orbit in terms of computable quantities, is proved. The practical use and the effectiveness of this theorem is demonstrated in the numerical computations of chaotic orbits of the Lorenz equations. Received December 15, 1993     

Numerical Taylor expansions of invariant manifolds in large dynamical systems

Summary. In this paper we develop a numerical method for computing higher order local approximations of invariant manifolds, such as stable, unstable or center manifolds near steady states of a dynamical system. The underlying system is assumed to be large in the sense that a large sparse Jacobian at the equilibrium occurs, for which only a linear (black box) solver and a low dimensional invariant subspace is available, but for which methods like the QR–Algorithm are considered to be too expensive. Our method is based on an analysis of the multilinear Sylvester equations for the higher derivatives which can be solved under certain nonresonance conditions. These conditions are weaker than the standard gap conditions on the spectrum which guarantee the existence of the invariant manifold. The final algorithm requires the solution of several large linear systems with a bordered Jacobian. To these systems we apply a block elimination method recently developed by Govaerts and Pryce [12, 14]. Received March 12, 1996 / Revised version reveiced August 8, 1997     

Index estimates for strongly indefinite functionals, periodic orbits and homoclinic solutions of first order Hamiltonian systems

We improve Benci and Rabinowitz's Linking theorem for strongly indefinite functionals, giving estimates for a suitably defined relative Morse index of critical points. Such abstract result is applied to the existence problem of periodic orbits and homoclinic solutions of first order Hamiltonian systems in cases where the Palais-Smale condition does not hold. Received January 27, 1999 / Accepted January 14, 2000 / Published online July 20, 2000     

Shadowing orbits and Kantorovich's theorem

Summary. Some recent results on the existence of shadowing orbits are related to Kantorovich's theorems on convergence of Newton's method. This connection leads to short proofs and to better understanding of some dilatation measures and existence results. Received March 24, 1994 / Revised version received May 15, 1995     

Invariant manifolds and global error estimates of numerical integration schemes applied to stiff systems of singular perturbation type – Part II: Linear multistep methods

Summary. It is shown that appropriate linear multi-step methods (LMMs) applied to singularly perturbed systems of ODEs preserve the geometric properties of the underlying ODE. If the ODE admits an attractive invariant manifold so does the LMM. The continuous as well as the discrete dynamical system restricted to their invariant manifolds are no longer stiff and the dynamics of the full systems is essentially described by the dynamics of the systems reduced to the manifolds. These results may be used to transfer properties of the reduced system to the full system. As an example global error bounds of LMM-approximations to singularly perturbed ODEs are given. Received May 5, 1995 / Revised version received August 18, 1995     

Invariant manifolds and global error estimates of numerical integration schemes applied to stiff systems of singular perturbation type -- Part I: RK-methods

Summary. For implicit RK-methods applied to singularly perturbed systems of ODEs it is shown that the resulting discrete systems preserve the geometric properties of the underlying ODE. This invariant manifold result is used to derive sharp bounds on the global error of RK-solutions. Received August 26, 1993 / Revised version received May 10, 1994     

On the asymptotic stability properties of Runge-Kutta methods for delay differential equations

Summary. In this paper asymptotic stability properties of Runge-Kutta (R-K) methods for delay differential equations (DDEs) are considered with respect to the following test equation: where and is a continuous real-valued function. In the last few years, stability properties of R-K methods applied to DDEs have been studied by numerous authors who have considered regions of asymptotic stability for “any positive delay” (and thus independent of the specific value of ). In this work we direct attention at the dependence of stability regions on a fixed delay . In particular, natural Runge-Kutta methods for DDEs are extensively examined. Received April 15, 1996 / Revised version received August 8, 1996     

Tangential frequency filtering decompositions for symmetric matrices

Summary. The tangential frequency filtering decomposition (TFFD) is introduced. The convergence theory of an iterative scheme based on the TFFD for symmetric matrices is the focus of this paper. The existence of the TFFD and the convergence of the induced iterative algorithm is shown for symmetric and positive definite matrices. Convergence rates independent of the number of unknowns are proven for a smaller class of matrices. Using this framework, the convergence independent of the number of unknowns is shown for Wittum's frequency filtering decomposition. Some characteristic properties of the TFFD are illustrated and results of several numerical experiments are presented. Received April 1, 1996 / Revised version July 4, 1996     

Tangential frequency filtering decompositions for unsymmetric matrices

Summary. In this paper, tangential frequency filtering decompositions (TFFD) for unsymmetric matrices are introduced. Different algorithms for the construction of unsymmetric tangential frequency filtering decompositions are presented. These algorithms yield for a specified class of matrices equivalent decompositions. The convergence rates of an iterative scheme, which uses a sequence of TFFDs as preconditioners, are independent of the number of unknowns for this class of matrices. Several numerical experiments verify the efficiency of these methods for the solution of linear systems of equations which arise from the discretisation of convection-diffusion differential equations. Received April 1, 1996 / Revised version received July 4, 1996     

The convergence rate of the minimal residual method for the Stokes problem

Summary. Discretisation of the classical Stokes problem gives rise to symmetric indefinite matrices with eigenvalues which, in a precise way, are not symmetric about the origin, but which do depend on a mesh size parameter. Convergence estimates for the Conjugate Residual or Minimum Residual iterative solution of such systems are given by best minimax polynomial approximations on an inclusion set for the eigenvalues. In this paper, an analytic convergence estimate for such problems is given in terms of an asymptotically small mesh size parameter. Received November 16, 1993 / Revised version received August 2, 1994     

On hybrid iterative methods for nonsymmetric systems of linear equations

Summary. Hybrid methods for the solution of systems of linear equations consist of a first phase where some information about the associated coefficient matrix is acquired, and a second phase in which a polynomial iteration designed with respect to this information is used. Most of the hybrid algorithms proposed recently for the solution of nonsymmetric systems rely on the direct use of eigenvalue estimates constructed by the Arnoldi process in Phase I. We will show the limitations of this approach and propose an alternative, also based on the Arnoldi process, which approximates the field of values of the coefficient matrix and of its inverse in the Krylov subspace. We also report on numerical experiments comparing the resulting new method with other hybrid algorithms. Received May 27, 1993 / Revised version received November 14, 1994     

Regions of exact convergence of the USSOR method applied on generalized consistently ordered matrices

Summary. Using the theory of nonnegative matrices and regular splittings, exact convergence and divergence domains of the Unsymmetric Successive Overrelaxation (USSOR) method, as it pertains to the class of Generalized Consistently Ordered (GCO) matrices, are determined. Our recently derived upper bounds, for the convergence of the USSOR method, re also used as effective tools. Received October 17, 1993 / Revised version received December 19, 1994     

Semiconvergence of nonnegative splittings for singular matrices

Summary. In this paper, we discuss semiconvergence of the matrix splitting methods for solving singular linear systems. The concepts that a splitting of a matrix is regular or nonnegative are generalized and we introduce the terminologies that a splitting is quasi-regular or quasi-nonnegative. The equivalent conditions for the semiconvergence are proved. Comparison theorem on convergence factors for two different quasi-nonnegative splittings is presented. As an application, the semiconvergence of the power method for solving the Markov chain is derived. The monotone convergence of the quasi-nonnegative splittings is proved. That is, for some initial guess, the iterative sequence generated by the iterative method introduced by a quasi-nonnegative splitting converges towards a solution of the system from below or from above. Received August 19, 1997 / Revised version received August 20, 1998 / Published online January 27, 2000     

On the convergence of line iterative methods for cyclically reduced non-symmetrizable linear systems

Summary. We derive analytic bounds on the convergence factors associated with block relaxation methods for solving the discrete two-dimensional convection-diffusion equation. The analysis applies to the reduced systems derived when one step of block Gaussian elimination is performed on red-black ordered two-cyclic discretizations. We consider the case where centered finite difference discretization is used and one cell Reynolds number is less than one in absolute value and the other is greater than one. It is shown that line ordered relaxation exhibits very fast rates of convergence. Received March 3, 1992/Revised version received July 2, 1993     

Numerical validation of solutions of complementarity The nonlinear case

Summary. This paper proposes a validation method for solutions of nonlinear complementarity problems. The validation procedure performs a computational test. If the result of the test is positive, then it is guaranteed that a given multi-dimensional interval either includes a solution or excludes all solutions of the nonlinear complementarity problem. Received September 22, 2000 / Revised version received April 11, 2001 / Published online October 17, 2001     

The rate of convergence of Toeplitz based PCG methods for second order nonlinear boundary value problems

Summary. In previous works [21–23] we proposed the use of [5] and band Toeplitz based preconditioners for the solution of 1D and 2D boundary value problems (BVP) by means of the preconditioned conjugate gradient (PCG) methods. As and band Toeplitz linear systems can be solved [4] by using fast sine transforms [8], these methods become especially attractive in a parallel environment of computation. In this paper we extend this technique to the nonlinear, nonsymmetric case and, in addition, we prove some clustering properties for the spectra of the preconditioned matrices showing why these methods exhibit a convergence speed which results to be more than linear. Therefore these methods work much finer than those based on separable preconditioners [18,45], on incomplete LU factorizations [36,13,27], and on circulant preconditioners [9,30,35] since the latter two techniques do not assure a linear rate of convergence. On the other hand, the proposed technique has a wider range of application since it can be naturally used for nonlinear, nonsymmetric problems and for BVP in which the coefficients of the differential operator are not strictly positive and only piecewise smooth. Finally the several numerical experiments performed here and in [22,23] confirm the effectiveness of the theoretical analysis. Received December 19, 1995 / Revised version received September 15, 1997     

Necessary and sufficient conditions for the convergence of Orthomin(k) on singular and inconsistent linear systems

Summary. We consider the convergence of Orthomin(k) on singular and inconsistent linear systems. Criteria for the breakdown of Orthomin(k) are discussed and analyzed. Moreover, necessary and sufficient conditions for the convergence of Orthomin(k) for any right hand side are given, and a rate of convergence is provided as well. Finally, numerical experiments are shown to confirm the convergence theorem. Received April 1, 1995 / Revised version received October 1, 1997 / Published online July 12, 2000