112 research outputs found
Application of the Jacobi Davidson method for spectral low-rank preconditioning in computational electromagnetics problems
[EN] We consider the numerical solution of linear systems arising from computational electromagnetics applications. For large scale problems the solution is usually obtained iteratively with a Krylov subspace method. It is well known that for ill conditioned problems the convergence of these methods can be very slow or even it may be impossible to obtain a satisfactory solution. To improve the convergence a preconditioner can be used, but in some cases additional strategies are needed. In this work we study the application of spectral lowrank updates (SLRU) to a previously computed sparse approximate inverse preconditioner.The updates are based on the computation of a small subset of the eigenpairs closest to the origin. Thus, the performance of the SLRU technique depends on the method available to compute the eigenpairs of interest. The SLRU method was first used using the IRA s method implemented in ARPACK. In this work we investigate the use of a Jacobi Davidson method,
in particular its JDQR variant. The results of the numerical experiments show that the application of the JDQR method to obtain the spectral low-rank updates can be quite competitive compared with the IRA s method.Mas MarĂ, J.; CerdĂĄn Soriano, JM.; Malla MartĂnez, N.; MarĂn Mateos-Aparicio, J. (2015). Application of the Jacobi Davidson method for spectral low-rank preconditioning in computational electromagnetics problems. Journal of the Spanish Society of Applied Mathematics. 67:39-50. doi:10.1007/s40324-014-0025-6S395067Bergamaschi, L., Pini, G., Sartoretto, F.: Computational experience with sequential, and parallel, preconditioned JacobiâDavidson for large sparse symmetric matrices. J. Comput. Phys. 188(1), 318â331 (2003)Carpentieri, B.: Sparse preconditioners for dense linear systems from electromagnetics applications. PhD thesis, Institut National Polytechnique de Toulouse, CERFACS (2002)Carpentieri, B., Duff, I.S., Giraud, L.: Sparse pattern selection strategies for robust Frobenius-norm minimization preconditioners in electromagnetism. Numer. Linear Algebr. Appl. 7(7â8), 667â685 (2000)Carpentieri, B., Duff, I.S., Giraud, L.: A class of spectral two-level preconditioners. SIAM J. Sci. Comput. 25(2), 749â765 (2003)Carpentieri, B., Duff, I.S., Giraud, L., Magolu monga Made, M.: Sparse symmetric preconditioners for dense linear systems in electromagnetism. Numer. Linear Algebr. Appl. 11(8â9), 753â771 (2004)Carpentieri, B., Duff, I.S., Giraud, L., Sylvand, G.: Combining fast multipole techniques and an approximate inverse preconditioner for large electromagnetism calculations. SIAM J. Sci. Comput. 27(3), 774â792 (2005)Darve, E.: The fast multipole method I: error analysis and asymptotic complexity. SIAM J. Numer. Anal. 38(1), 98â128 (2000)Fokkema, D.R., Sleijpen, G.L., Van der Vorst, H.A.: JacobiâDavidson style QR and QZ algorithms for the reduction of matrix pencils. SIAM J. Sci. Comput. 20(1), 94â125 (1998)Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(3), 325â348 (1987)Grote, M., Huckle, T.: Parallel preconditioning with sparse approximate inverses. SIAM J. Sci. Comput. 18(3), 838â853 (1997)Harrington, R.: Origin and development of the method of moments for field computation. IEEE Antenna Propag. Mag. (1990)Kunz, K.S., Luebbers, R.J.: The finite difference time domain method for electromagnetics. SIAM J. Sci. Comput. 18(3), 838â853 (1997)Maxwell, J.C.: A dynamical theory of the electromagnetic field. Roy. S. Trans. CLV, (1864). Reprinted in Tricker, R. A. R. The Contributions of Faraday and Maxwell to Electrial Science, Pergamon Press (1966)MarĂn, J., Malla M.: Some experiments preconditioning via spectral low rank updates for electromagnetism applications. In: Proceedings of the international conference on preconditioning techniques for large sparse matrix problems in scientific and industrial applications (Preconditioning 2007), Toulouse (2007)Meijerink, J.A., van der Vorst, H.A.: An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math. Comput. 31, 148â162 (1977)Sorensen, D.C., Lehoucq, R.B., Yang, C.: ARPACK usersâ guide: solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods. SIAM, Philadelphia (1998)Rao, S.M., Wilton, D.R., Glisson, A.W.: Electromagnetic scattering by surfaces of arbitrary shape. IEEE Trans. Antenna Propag. 30, 409â418 (1982)Saad, Y.: Iterative methods for sparse linear systems. PWS Publishing Company, Boston (1996)Silvester, P.P., Ferrari, R.L.: Finite elements for electrical engineers. Cambridge University Press, Cambridge (1990)Sleijpen, S.L., van der Vorst, H.A.: A JacobiâDavidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Appl. 17, 401â425 (1996)van der Vorst, H.A.: Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of non-symmetric linear systems. SIAM J. Sci. Stat. Comput. 12(6), 631â644 (1992
On acceleration of Krylov-subspace-based Newton and Arnoldi iterations for incompressible CFD: replacing time steppers and generation of initial guess
We propose two techniques aimed at improving the convergence rate of steady
state and eigenvalue solvers preconditioned by the inverse Stokes operator and
realized via time-stepping. First, we suggest a generalization of the Stokes
operator so that the resulting preconditioner operator depends on several
parameters and whose action preserves zero divergence and boundary conditions.
The parameters can be tuned for each problem to speed up the convergence of a
Krylov-subspace-based linear algebra solver. This operator can be inverted by
the Uzawa-like algorithm, and does not need a time-stepping. Second, we propose
to generate an initial guess of steady flow, leading eigenvalue and eigenvector
using orthogonal projection on a divergence-free basis satisfying all boundary
conditions. The approach, including the two proposed techniques, is illustrated
on the solution of the linear stability problem for laterally heated square and
cubic cavities
Refined saddle-point preconditioners for discretized Stokes problems
This paper is concerned with the implementation of efficient solution algorithms for elliptic problems with constraints. We establish theory which shows that including a simple scaling within well-established block diagonal preconditioners for Stokes problems can result in significantly faster convergence when applying the preconditioned MINRES method. The codes used in the numerical studies are available online
A global method for coupling transport with chemistry in heterogeneous porous media
Modeling reactive transport in porous media, using a local chemical
equilibrium assumption, leads to a system of advection-diffusion PDE's coupled
with algebraic equations. When solving this coupled system, the algebraic
equations have to be solved at each grid point for each chemical species and at
each time step. This leads to a coupled non-linear system. In this paper a
global solution approach that enables to keep the software codes for transport
and chemistry distinct is proposed. The method applies the Newton-Krylov
framework to the formulation for reactive transport used in operator splitting.
The method is formulated in terms of total mobile and total fixed
concentrations and uses the chemical solver as a black box, as it only requires
that on be able to solve chemical equilibrium problems (and compute
derivatives), without having to know the solution method. An additional
advantage of the Newton-Krylov method is that the Jacobian is only needed as an
operator in a Jacobian matrix times vector product. The proposed method is
tested on the MoMaS reactive transport benchmark.Comment: Computational Geosciences (2009)
http://www.springerlink.com/content/933p55085742m203/?p=db14bb8c399b49979ba8389a3cae1b0f&pi=1
On the Easy Use of Scientific Computing Services for Large Scale Linear Algebra and Parallel Decision Making with the P-Grade Portal
International audienceScientific research is becoming increasingly dependent on the large-scale analysis of data using distributed computing infrastructures (Grid, cloud, GPU, etc.). Scientific computing (Petitet et al. 1999) aims at constructing mathematical models and numerical solution techniques for solving problems arising in science and engineering. In this paper, we describe the services of an integrated portal based on the P-Grade (Parallel Grid Run-time and Application Development Environment) portal (http://www.p-grade.hu) that enables the solution of large-scale linear systems of equations using direct solvers, makes easier the use of parallel block iterative algorithm and provides an interface for parallel decision making algorithms. The ultimate goal is to develop a single sign on integrated multi-service environment providing an easy access to different kind of mathematical calculations and algorithms to be performed on hybrid distributed computing infrastructures combining the benefits of large clusters, Grid or cloud, when needed
Bidirectional Associations Between Sibling Relationships and Parental Support During Adolescence
Sibling relationships and parental support are important for adolescentsâ development and well-being, yet both are likely to change during adolescence. Since adolescents participate in both the sibling relationship and the parentâchild relationship, we can expect sibling relationships and parental support to be associated with each other. Theoretically, it can be expected that there is either a spillover from one relationship to another (congruence hypothesis) or that one relationship can compensate for the other (compensation hypothesis). However, research examining these associations in adolescence is limited. The present study longitudinally investigated the bidirectional associations between sibling relationships and parental support during adolescence. For five consecutive years, data were collected using self-reports of 428 families, consisting of a father, a mother, and two adolescent siblings. The mean ages of the first-born (52.8% males) and second-born (47.7% males) were 15 and 13Â years at T1, respectively. For the second-born siblings, prospective associations were found between sibling relationships and adolescent-reported parental support in early adolescence, with no differences between same-sex and mixed-sex dyads. These associations were not found for first-born siblings or for parentsâ reports of support. The findings suggest a spillover from the sibling relationship to adolescent-reported parental support only in early adolescence. Findings and implications are discussed in terms of the congruence/spillover and the compensation hypothesis
Principals' reports of adults' alcohol use in Australian secondary schools
Background - Schools provide opportunities for parents and the wider community to connect and support the physical and emotional wellbeing of their children. Schools therefore have the potential to play a role in the socialisation of alcohol use through school policies and practices regarding consumption of alcohol by adults at school events in the presence of children. Methods - This survey was undertaken to a) compare the extent to which alcohol is used at secondary school events, when children are present, in the states of New South Wales (NSW) and Victoria (VIC), Australia; b) describe principalsâ level of agreement with these practices; c) their awareness of state policies on this issue; and d) the predictors of such events. A random sample of secondary schools, stratified to represent metropolitan and non-metropolitan schools were invited to participate. Bivariate and multivariate analysis were conducted with p valuesâ<â0.05 considered significant. Results - A total of 241 (43 %) schools consented to participate in the study. Fifteen percent of participating NSW schools and 57 % of VIC schools held at least one event in which alcohol was consumed by adults in the presence of children in the year before the survey. Of the 100 reported events, 78 % were Year 12 graduation dinners, and 18 % were debutante balls. Compared to NSW principals, VIC principals were significantly more likely to agree with the use of alcohol at these events; significantly less likely to be aware of their state education department policy on this issue; have a policy at their own school or support policy that prohibits alcohol use at such events; and less likely to report having enough Information to make decisions about this. Conclusions - There is a growing focus on adultsâ use of alcohol at school events when children are present. Schools can play an important role in educating and socialising children about alcohol via both the curriculum and policies regarding adultsâ alcohol use at school events. Findings from this study suggest education department and school-based policies that prohibit or restrict the use of alcohol, are significant predictors of adultsâ alcohol use at school events when children are present
Order-of-magnitude speedup for steady states and traveling waves via Stokes preconditioning in Channelflow and Openpipeflow
Steady states and traveling waves play a fundamental role in understanding
hydrodynamic problems. Even when unstable, these states provide the
bifurcation-theoretic explanation for the origin of the observed states. In
turbulent wall-bounded shear flows, these states have been hypothesized to be
saddle points organizing the trajectories within a chaotic attractor. These
states must be computed with Newton's method or one of its generalizations,
since time-integration cannot converge to unstable equilibria. The bottleneck
is the solution of linear systems involving the Jacobian of the Navier-Stokes
or Boussinesq equations. Originally such computations were carried out by
constructing and directly inverting the Jacobian, but this is unfeasible for
the matrices arising from three-dimensional hydrodynamic configurations in
large domains. A popular method is to seek states that are invariant under
numerical time integration. Surprisingly, equilibria may also be found by
seeking flows that are invariant under a single very large Backwards-Euler
Forwards-Euler timestep. We show that this method, called Stokes
preconditioning, is 10 to 50 times faster at computing steady states in plane
Couette flow and traveling waves in pipe flow. Moreover, it can be carried out
using Channelflow (by Gibson) and Openpipeflow (by Willis) without any changes
to these popular spectral codes. We explain the convergence rate as a function
of the integration period and Reynolds number by computing the full spectra of
the operators corresponding to the Jacobians of both methods.Comment: in Computational Modelling of Bifurcations and Instabilities in Fluid
Dynamics, ed. Alexander Gelfgat (Springer, 2018
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