No abstract

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 mandatory when large, sparse matrices are analyzed. We considered several preconditioners: Classical block-Jacobi, and IC(0), together with approximate inverse (AINV or FSAI) preconditioners. The rationale for using approximate inverse preconditioners is their high parallelization potential, combined with their efficiency in accelerating the iterative solution of the correction equation. Analysis was carried on the sequential performance of preconditioned JD for the spectral decomposition of large, sparse matrices, which originate in the numerical integration of partial differential equations arising in physical and engineering problems. It was found that JD is highly sensitive to preconditioning, and it can display an irregular convergence behavior. We parallelized JD by data-splitting techniques, combining them with techniques to reduce the amount of communication data. Our own parallel, preconditioned code was executed on a dedicated parallel machine, and we present the results of our experiments. Our JD code provides an appreciable parallel degree of computation. Its performance was also compared with those of PARPACK and parallel DACG.

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, DACG is more efficient than ARPACK. Effective convergence acceleration of DACG is shown to be performed by a suitable approximate inverse preconditioner (AINV). The performance of such a preconditioner is shown to be safe, i.e. not highly dependent on a drop tolerance parameter. On sequential machines, AINV preconditioning proves a practicable alternative to the effective incomplete Cholesky factorization, and is more efficient than Block Jacobi. Owing to its parallelizability, the AINV preconditioner is exploited for a parallel implementation of the DACG algorithm. Numerical tests account for the high degree of parallelization attainable on a Cray T3E machine and confirm the satisfactory scalability properties of the algorithm. A final comparison with PARPACK shows the (relative) higher efficiency of AINV‐DACG. Copyright © 2000 John Wiley & Sons, Ltd.

We investigated whether angle magnitude, similarly to numerical quantities (i.e., the spatial-numerical association of response codes effect), is associated to the side of response execution. In addition, we investigated whether this association has the properties of a spatially oriented mental line, since angles are taught in a right-to-left progression. We tested two groups of participants: civil engineering students (high familiarity with angles) and psychology students (low familiarity with angles). In Experiment 1, participants were asked to judge the continuity of the angles' arms (continuous vs. dashed). Magnitude of the angles was task-irrelevant. In Experiment 2, they were asked to judge whether the presented angles were smaller or larger than a right angle (90°). Therefore, the angle magnitude was relevant for performing the task. Overall, engineering students responded faster with their left hand to large angles and with their right hand to small angles. Conversely, psychology students did not show any reliable differences between left- and right-hand responses. In the case of engineering students, the spatial association has a right-to-left (counter clockwise) direction, suggesting the influence of education and practice on the mental representation of angle magnitude.

Numerical velocity fields arising from the solution of diffusion equations by the finite element (FE) and the mixed hybrid finite element (MHFE) schemes display different behaviors. In this paper we analyze the characteristics of the two different velocity fields in terms of both accuracy and mass balance properties. General theoretical findings are mostly concerned with the asymptotic behavior of the numerical schemes, i.e. they look at properties as the mesh size tends to zero. For practical applications, it is necessary to work with a fixed mesh of given size. Thus, we attempt to characterize the numerical flow field accuracy by analyzing the resulting mass balance characteristics on a fixed mesh. The comparison is carried out by using direct local mass balance evaluations and by calculating streamlines. We detail the important differences, advantages, and disadvantages of the two approaches. In particular, we show that both FE and MH are perfectly conservative (up to the residual of the linear system solution) if proper control volumes are used. MH streamlines are admissible, i.e. numerical normal fluxes across cell interfaces are continuous. Since continuity of the normal fluxes is not guaranteed by FE, the resulting streamlines are less accurate