102,941 research outputs found
Hybrid preconditioning for iterative diagonalization of ill-conditioned generalized eigenvalue problems in electronic structure calculations
The iterative diagonalization of a sequence of large ill-conditioned
generalized eigenvalue problems is a computational bottleneck in quantum
mechanical methods employing a nonorthogonal basis for {\em ab initio}
electronic structure calculations. We propose a hybrid preconditioning scheme
to effectively combine global and locally accelerated preconditioners for rapid
iterative diagonalization of such eigenvalue problems. In partition-of-unity
finite-element (PUFE) pseudopotential density-functional calculations,
employing a nonorthogonal basis, we show that the hybrid preconditioned block
steepest descent method is a cost-effective eigensolver, outperforming current
state-of-the-art global preconditioning schemes, and comparably efficient for
the ill-conditioned generalized eigenvalue problems produced by PUFE as the
locally optimal block preconditioned conjugate-gradient method for the
well-conditioned standard eigenvalue problems produced by planewave methods
Block hybrid multilevel method to compute the dominant lambda-modes of the neutron diffusion equation
[EN] The dominant lambda-modes associated with a nuclear reactor configuration describe the neutron steady-state distribution and its criticality. Furthermore, they are useful to develop modal methods to study reactor instabilities. Different eigenvalues solvers have been successfully used to obtain such modes, most of them are implemented reducing the original generalized eigenvalue problem to an ordinary one. Thus, it is necessary to solve many linear systems making these methods not very efficient, especially for large problems. In this work, the original generalized eigenvalue problem is considered and two block iterative methods to solve it are studied: the block inverse-free preconditioned Arnoldi method and the modified block Newton method. All of these iterative solvers are initialized using a block multilevel technique. A hybrid multilevel method is also proposed based on the combination of the methods proposed. Two benchmark problems are studied illustrating the convergence and the competitiveness of the methods proposed. A comparison with the Krylov-Schur method and the Generalized Davidson is also included.This work has been partially supported by Spanish Ministerio de Economia y Competitividad under projects ENE2017-89029-P, MTM2017-85669-P and BES-2015-072901.Carreño, A.; Vidal-Ferràndiz, A.; Ginestar Peiro, D.; Verdú Martín, GJ. (2018). Block hybrid multilevel method to compute the dominant lambda-modes of the neutron diffusion equation. Annals of Nuclear Energy. 121:513-524. https://doi.org/10.1016/j.anucene.2018.08.010S51352412
Implicit solvers for unstructured meshes
Implicit methods were developed and tested for unstructured mesh computations. The approximate system which arises from the Newton linearization of the nonlinear evolution operator is solved by using the preconditioned GMRES (Generalized Minimum Residual) technique. Three different preconditioners were studied, namely, the incomplete LU factorization (ILU), block diagonal factorization, and the symmetric successive over relaxation (SSOR). The preconditioners were optimized to have good vectorization properties. SSOR and ILU were also studied as iterative schemes. The various methods are compared over a wide range of problems. Ordering of the unknowns, which affects the convergence of these sparse matrix iterative methods, is also studied. Results are presented for inviscid and turbulent viscous calculations on single and multielement airfoil configurations using globally and adaptively generated meshes
Block Iterative Eigensolvers for Sequences of Correlated Eigenvalue Problems
In Density Functional Theory simulations based on the LAPW method, each
self-consistent field cycle comprises dozens of large dense generalized
eigenproblems. In contrast to real-space methods, eigenpairs solving for
problems at distinct cycles have either been believed to be independent or at
most very loosely connected. In a recent study [7], it was demonstrated that,
contrary to belief, successive eigenproblems in a sequence are strongly
correlated with one another. In particular, by monitoring the subspace angles
between eigenvectors of successive eigenproblems, it was shown that these
angles decrease noticeably after the first few iterations and become close to
collinear. This last result suggests that we can manipulate the eigenvectors,
solving for a specific eigenproblem in a sequence, as an approximate solution
for the following eigenproblem. In this work we present results that are in
line with this intuition. We provide numerical examples where opportunely
selected block iterative eigensolvers benefit from the reuse of eigenvectors by
achieving a substantial speed-up. The results presented will eventually open
the way to a widespread use of block iterative eigensolvers in ab initio
electronic structure codes based on the LAPW approach.Comment: 12 Pages, 5 figures. Accepted for publication on Computer Physics
Communication
Preconditioned WR–LMF-based method for ODE systems
AbstractThe waveform relaxation (WR) method was developed as an iterative method for solving large systems of ordinary differential equations (ODEs). In each WR iteration, we are required to solve a system of ODEs. We then introduce the boundary value method (BVM) which is a relatively new method based on the linear multistep formulae to solve ODEs. In particular, we apply the generalized minimal residual method with the Strang-type block-circulant preconditioner for solving linear systems arising from the application of BVMs to each WR iteration. It is demonstrated that these techniques are very effective in speeding up the convergence rate of the resulting iterative processes. Numerical experiments are presented to illustrate the effectiveness of our methods
Inexact inner-outer Golub-Kahan bidiagonalization method: A relaxation strategy
We study an inexact inner-outer generalized Golub-Kahan algorithm for the
solution of saddle-point problems with a two-times-two block structure. In each
outer iteration, an inner system has to be solved which in theory has to be
done exactly. Whenever the system is getting large, an inner exact solver is,
however, no longer efficient or even feasible and iterative methods must be
used. We focus this article on a numerical study showing the influence of the
accuracy of an inner iterative solution on the accuracy of the solution of the
block system. Emphasis is further given on reducing the computational cost,
which is defined as the total number of inner iterations. We develop relaxation
techniques intended to dynamically change the inner tolerance for each outer
iteration to further minimize the total number of inner iterations. We
illustrate our findings on a Stokes problem and validate them on a mixed
formulation of the Poisson problem.Comment: 25 pages, 9 figure
Comparison of Iterative Back-Projection Inversion and Generalized Inversion Without Blocks: Case Studies In Attenuation Tomography
Iterative back-projection tomography and generalized inversion without blocks (‘no-block’) are two different inversion techniques developed recently for 3-D studies, and are commonly applied to the inversion of travel-time data. In this study, we compare the two methods and derive one from the other under certain assumptions. We then apply these two methods to the attenuation problem, inverting for the quality factor, Q, of the medium. Usually, travel-time inversion involves large data sets and fine resolution is not possible if generalized inversion is applied. A relatively small data set with little redundancy enables us to apply both techniques with similar resolution. We applied the methods to the data sets obtained for two areas in southern California, the Coso-Indian Wells region and Imperial Valley. The results obtained by the two methods are very similar. Back-projection tomography is a direct and fast method for this type of problem. However, it does not provide formal error estimates and resolution. The no-block inversion requires more computational time, but formal errors and resolution can be directly computed for the final model. Thus, application of the two methods to the same data set enhances the objectivity of the final result
- …