Sparse approximate inverse preconditioners on high performance GPU platforms

Simulation with models based on partial differential equations often requires the solution of (sequences of) large and sparse algebraic linear systems. In multidimensional domains, preconditioned Krylov iterative solvers are often appropriate for these duties. Therefore, the search for efficient preconditioners for Krylov subspace methods is a crucial theme. Recent developments, especially in computing hardware, have renewed the interest in approximate inverse preconditioners in factorized form, because their application during the solution process can be more efficient. We present here some experiences focused on the approximate inverse preconditioners proposed by Benzi and Tůma from 1996 and the sparsification and inversion proposed by van Duin in 1999. Computational costs, reorderings and implementation issues are considered both on conventional and innovative computing architectures like Graphics Programming Units (GPUs)

Efficient approximation of functions of some large matrices by partial fraction expansions

Some important applicative problems require the evaluation of functions $\Psi$ of large and sparse and/or \emph{localized} matrices $A$. Popular and interesting techniques for computing $\Psi(A)$ and $\Psi(A)\mathbf{v}$, where $\mathbf{v}$ is a vector, are based on partial fraction expansions. However, some of these techniques require solving several linear systems whose matrices differ from $A$ by a complex multiple of the identity matrix $I$ for computing $\Psi(A)\mathbf{v}$ or require inverting sequences of matrices with the same characteristics for computing $\Psi(A)$. Here we study the use and the convergence of a recent technique for generating sequences of incomplete factorizations of matrices in order to face with both these issues. The solution of the sequences of linear systems and approximate matrix inversions above can be computed efficiently provided that $A^{-1}$ shows certain decay properties. These strategies have good parallel potentialities. Our claims are confirmed by numerical tests

Rational Krylov methods for functions of matrices with applications to fractional partial differential equations

In this paper, we propose a new choice of poles to define reliable rational Krylov methods. These methods are used for approximating function of positive definite matrices. In particular, the fractional power and the fractional resolvent are considered because of their importance in the numerical solution of fractional partial differential equations. The results of the numerical experiments we have carried out on some fractional models confirm that the proposed approach is promising

Why diffusion-based preconditioning of Richards equation works: spectral analysis and computational experiments at very large scale

We consider here a cell-centered finite difference approximation of the Richards equation in three dimensions, averaging for interface values the hydraulic conductivity $K=K(p)$, a highly nonlinear function, by arithmetic, upstream, and harmonic means. The nonlinearities in the equation can lead to changes in soil conductivity over several orders of magnitude and discretizations with respect to space variables often produce stiff systems of differential equations. A fully implicit time discretization is provided by \emph{backward Euler} one-step formula; the resulting nonlinear algebraic system is solved by an inexact Newton Armijo-Goldstein algorithm, requiring the solution of a sequence of linear systems involving Jacobian matrices. We prove some new results concerning the distribution of the Jacobians eigenvalues and the explicit expression of their entries. Moreover, we explore some connections between the saturation of the soil and the ill-conditioning of the Jacobians. The information on eigenvalues justifies the effectiveness of some preconditioner approaches which are widely used in the solution of Richards equation. We also propose a new software framework to experiment with scalable and robust preconditioners suitable for efficient parallel simulations at very large scales. Performance results on a literature test case show that our framework is very promising in the advance towards realistic simulations at extreme scale

Perceived Economic Uncertainty and Fertility Intentions in Couples: A Dyadic Extension of the Theory of Planned Behaviour

By adopting a dyadic extension of the Theory of Planned Behavior (Ajzen, 1991), this study examined whether perceived economic uncertainty afects fertility intentions. Three-hundred thirty one heterosexual couples living in Italy participated in a randomized between-group experimental study, in which we manipulated perceived economic uncertainty (low vs. high vs. control). The participants subsequently completed a questionnaire measuring their attitudes, subjective norms, perceived behavioral control, and fertility intentions. We employed Structural Equation Modelling in estimating the Actor–Partner Interdependence Model. The model showed a good ft to the data. Women’s attitudes, subjective norms, and perceived behavioral control were infuenced by the high economic uncertain scenario, whereas among men these variables were afected only by the positive economic scenario. Attitudes and perceived behavioral control were signifcant predictors of fertility intentions for both sexes. Signifcant partner efects were observed as well. These fndings suggest that fertility plans should be examined by adopting a dyadic perspective, as individuals’ intentions are afected not only by their own beliefs, but also by those of their partners.Perceived Economic Uncertainty and Fertility Intentions in Couples: A Dyadic Extension of the Theory of Planned BehaviourpublishedVersio

Perceived Economic Uncertainty and Fertility Intentions in Couples: A Dyadic Extension of The Theory of Planned Behavior

By adopting a dyadic extension of the Theory of Planned Behavior (Ajzen, 1991), this study examined whether perceived economic uncertainty affects fertility intentions. Three-hundred thirty one heterosexual couples living in Italy participated in a randomized between-group experimental study, in which we manipulated perceived economic uncertainty (low vs. high vs. control). The participants subsequently completed a questionnaire measuring their attitudes, subjective norms, perceived behavioral control, and fertility intentions. We employed Structural Equation Modelling in estimating the Actor–Partner Interdependence Model. The model showed a good fit to the data. Women’s attitudes, subjective norms, and perceived behavioral control were influenced by the high economic uncertain scenario, whereas among men these variables were affected only by the positive economic scenario. Attitudes and perceived behavioral control were significant predictors of fertility intentions for both sexes. Significant partner effects were observed as well. These findings suggest that fertility plans should be examined by adopting a dyadic perspective, as individuals’ intentions are affected not only by their own beliefs, but also by those of their partners