Abstract. Every Newton step in an interior-point method for optimization requires a solution of a symmetric indefinite system of linear equations. Most of today's codes apply direct solution methods to perform this task. The use of logarithmic barriers in interior point methods causes unavoidable ill-conditioning of linear systems and, hence, iterative methods fail to provide sufficient accuracy unless appropriately preconditioned. Two types of preconditioners which use some form of incomplete Cholesky factorization for indefinite systems are proposed in this paper. Although they involve significantly sparser factorizations than those used in direct approaches they still capture most of the numerical properties of the preconditioned system. The spectral analysis of the preconditioned matrix is performed: for convex optimization problems all the eigenvalues of this matrix are strictly positive. Numerical results are given for a set of public domain large linearly constrained convex quadratic programming problems with sizes reaching tens of thousands of variables. The analysis of these results reveals that the solution times for such problems on a modern PC are measured in minutes when direct methods are used and drop to seconds when iterative methods with appropriate preconditioners are used.

We propose and analyze the ReLPM (Real Leja Points Method) for evaluating the propagator '(tB)v via matrix interpolation polynomials at spectral Leja sequences. Here B is the large, sparse, nonsymmetric matrix arising from stable 2D or 3D nite-dierence discretization of linear advection-diusion equations, and '(z) is the entire function '(z) = (e z − 1)=z. The corresponding sti dierential systemẏ(t) = By(t) + g; y(0) = y 0 , is solved by the exact time marching scheme y i+1 = y i + t i '(t i B)(By i + g), i = 0; 1; : : : ; where the time-step is controlled simply via the variation percentage of the solution, and can be large. Numerical tests show substantial speed-ups (up to one order of magnitude) with respect to a classical variable step-size Crank-Nicolson solver.

SUMMARYWe present the development of a two-dimensional Mixed-Hybrid Finite Element (MHFE) model for the solution of the non-linear equation of variably saturated ow in groundwater on unstructured triangular meshes. By this approach the Darcy velocity is approximated using lowest-order Raviart-Thomas (RT0) elements and is 'exactly' mass conserving. Hybridization is used to overcome the ill-conditioning of the mixed system. The scheme is globally ÿrst-order in space. Nevertheless, numerical results employing non-uniform meshes show second-order accuracy of the pressure head and normal uxes on speciÿc grid points. The non-linear systems of algebraic equations resulting from the MHFE discretization are solved using Picard or Newton iterations. Realistic sample tests show that the MHFE-Newton approach achieves fast convergence in many situations, in particular, when a good initial guess is provided by either the Picard scheme or relaxation techniques.

Abstract. Issues of indefinite preconditioning of reduced Newton systems arising in optimization with interior point methods are addressed in this paper. Constraint preconditioners have shown much promise in this context. However, there are situations in which an unfavorable sparsity pattern of Jacobian matrix may adversely affect the preconditioner and make its inverse representation unacceptably dense hence too expensive to be used in practice. A remedy to such situations is proposed in this paper. An approximate constraint preconditioner is considered in which sparse approximation of the Jacobian is used instead of the complete matrix. Spectral analysis of the preconditioned matrix is performed and bounds on its non-unit eigenvalues are provided. Preliminary computational results are encouraging.

In this paper we compare Krylov subspace methods with Chebyshev series expansion for approximating the matrix exponential operator on large, sparse, symmetric matrices. Experimental results upon negative-definite matrices with very large size, arising from (2D and 3D) FE and FD spatial discretization of linear parabolic PDEs, demonstrate that the Chebyshev method can be an effective alternative to Krylov techniques, especially when memory bounds do not allow the storage of all Ritz vectors. We discuss also sensitivity of Chebyshev convergence to extreme eigenvalue approximation, as well as reliability of various a priori and a posteriori error estimates for both methods.