Angeles Martinez

Bio: Angeles Martinez is an academic researcher from University of Padua. The author has contributed to research in topics: Preconditioner & Conjugate gradient method. The author has an hindex of 12, co-authored 33 publications receiving 367 citations. Previous affiliations of Angeles Martinez include Polytechnic University of Valencia.

Quasi-Newton preconditioners for the inexact Newton method.

Abstract: In this paper preconditioners for solving the linear systems of the Newton method in each nonlinear iteration are studied. In particular, we define a sequence of preconditioners built by means of Broyden-type rank-one updates. Optimality conditions are derived which guarantee that the preconditioned matrices are not far from the identity in a matrix norm. Some notes on the implementation of the corresponding inexact Newton method are given and some numerical results on two model problems illustrate the application of the proposed preconditioners.

Two-stage spectral preconditioners for iterative eigensolvers

Abstract: JEL Classification: 65F05 65F15 65H17 Summary In this paper, we present preconditioning techniques to accelerate the convergence of Krylov solvers at each step of an Inexact Newton’s method for the computation of the leftmost eigenpairs of large and sparse symmetric positive definite matrices arising in large-scale scientific computations. We propose a two-stage spectral preconditioning strategy: The first stage produces a very rough approximation of a number of the leftmost eigenvectors. The second stage uses these approximations as starting vectors and also to construct the tuned preconditioner from an initial inverse approximation of the coefficient matrix, as proposed by Martínez. In the framework of the Implicitly Restarted Lanczos method. The action of this spectral preconditioner results in clustering a number of the eigenvalues of the preconditioned matrices close to one. We also study the combination of this approach with a BFGS-style updating of the proposed spectral preconditioner as described by Bergamaschi and Martínez. Extensive numerical testing on a set of representative large SPD matrices gives evidence of the acceleration provided by these spectral preconditioners.

Comparing leja and krylov approximations of large scale matrix exponentials

Abstract: We have implemented a numerical code (ReLPM, Real Leja Points Method) for polynomial interpolation of the matrix exponential propagators exp (${\it \Delta}$tA) v and ϕ(${\it \Delta}$tA) v, ϕ(z) = (exp (z) – 1)/z. The ReLPM code is tested and compared with Krylov-based routines, on large scale sparse matrices arising from the spatial discretization of 2D and 3D advection-diffusion equations.

Physically based modeling in catchment hydrology at 50: Survey and outlook

Abstract: Integrated, process-based numerical models in hydrology are rapidly evolving, spurred by novel theories in mathematical physics, advances in computational methods, insights from laboratory and field experiments, and the need to better understand and predict the potential impacts of population, land use, and climate change on our water resources. At the catchment scale, these simulation models are commonly based on conservation principles for surface and subsurface water flow and solute transport (e.g., the Richards, shallow water, and advection-dispersion equations), and they require robust numerical techniques for their resolution. Traditional (and still open) challenges in developing reliable and efficient models are associated with heterogeneity and variability in parameters and state variables; nonlinearities and scale effects in process dynamics; and complex or poorly known boundary conditions and initial system states. As catchment modeling enters a highly interdisciplinary era, new challenges arise from the need to maintain physical and numerical consistency in the description of multiple processes that interact over a range of scales and across different compartments of an overall system. This paper first gives an historical overview (past 50 years) of some of the key developments in physically based hydrological modeling, emphasizing how the interplay between theory, experiments, and modeling has contributed to advancing the state of the art. The second part of the paper examines some outstanding problems in integrated catchment modeling from the perspective of recent developments in mathematical and computational science.

Nuclear Reactor Physics

Abstract: Neutron Physics By K H Beckurts and K Wirtz Pp x + 444 (Berlin: Springer-Verlag, 1964) 68 DM Introduction to Neutron Distribution Theory By L C Woods (Methuen's Monographs on Physical Subjects) Pp xii + 132 (London: Methuen and Co, Ltd; New York: John Wiley and Sons, Inc, 1964) 28s net

A class of spectral two-level preconditioners

Abstract: It is well known that the convergence of Krylov methods for solving the linear system often depends to a large extent on the eigenvalue distribution. In many cases, it is observed that ``removing'' the smallest eigenvalues can greatly improve the convergence. Several techniques have been proposed in the past few years that attempt to tackle this problem. The proposed approaches can be split into two main families depending on whether the scheme enlarges the generated Krylov space or adaptively updates the preconditioner. In this paper, we follow the second approach and propose a class of preconditioners both for unsymmetric and for symmetric linear systems that can also be adapted for symmetric positive definite problems. We effectively solve the preconditioned system exactly in the low dimensional space associated with the smallest eigenvalues and use this to update the preconditioned residual. This update results in shifting eigenvalues from close to the origin to near to one for the new preconditioner. This is ideal when there are only a few eigenvalues near the origin while all the others are close to one because the updated preconditioned system becomes close to the identity. We illustrate the performance of our method through extensive numerical experiments on a set of general linear systems. Finally, we show the advantages of the preconditioners for solving dense linear systems arising in electromagnetism applications, which were the main motivation for this work.