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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… (More)

A preconditioned scheme for solving sparse symmetric eigenproblems is proposed. The solution strategy relies upon the DACG algorithm, which is a Preconditioned Conjugate Gradient algorithm for minimizing the Rayleigh Quotient. A comparison with the well established ARPACK code, shows that when a small number of the leftmost eigenpairs is to be computed,… (More)

The Jacobi–Davidson (JD) algorithm was recently proposed for evaluating a number of the eigenvalues of a matrix. JD goes beyond pure Krylov-space techniques; it cleverly expands its search space, by solving the so-called correction equation, thus in principle providing a more powerful method. Preconditioning the Jacobi–Davidson correction equation is… (More)

- Marco Caliari, Marco Vianello, Luca Bergamaschi
- 1991

We implement a second-order exponential integrator for semidiscretized advection-diffusion-reaction equations, obtained by coupling exponential-like Euler and Mid-point integrators, and computing the relevant matrix exponentials by polynomial interpolation at Leja points. Numerical tests on 2D models discretized in space by Finite Differences or Finite… (More)

Recently an efficient method (DACG) for the partial solution of the symmetric generalized eigenproblem Ax = Bx has been developed, based on the conjugate gradient (CG) minimization of the Rayleigh quotient over successive deflated subspaces of decreasing size. The present paper provides a numerical analysis of the asymptotic convergence rate j of DACG in… (More)

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… (More)

Issues of indefinite preconditioning of reduced Newton systems arising in optimization with interior point methods are addressed in this paper. Constraint pre-conditioners 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… (More)

for the exponential integration of large-scale sparse systems of ODEs, generated by Finite Element or Finite Difference discretizations of 3-D advection-diffusion models. We present an efficient parallel implementation of ReLPM for polynomial interpolation of the matrix exponential propagators exp (∆tA) v and ϕ(∆tA) v, ϕ(z) = (exp (z) − 1)/z. A scalability… (More)