The algebraic multigrid projection for eigenvalue problems; backrotations and multigrid fixed points

The periods of the theorem for the algebraic multigrid projection (MGP) for eigenvalue problems, and of the multigrid fixed point theorem for multigrid cycles combining MGP with backrotations, are presented. The MGP and the backrotations are central eigenvector separation techniques for multigrid eigenvalue algorithms. They allow computation on coarse levels of eigenvalues of a given eigenvalue problem, and are efficient tools in the computation of eigenvectors

A robust multilevel simultaneous eigenvalue solver

Multilevel (ML) algorithms for eigenvalue problems are often faced with several types of difficulties such as: the mixing of approximated eigenvectors by the solution process, the approximation of incomplete clusters of eigenvectors, the poor representation of solution on coarse levels, and the existence of close or equal eigenvalues. Algorithms that do not treat appropriately these difficulties usually fail, or their performance degrades when facing them. These issues motivated the development of a robust adaptive ML algorithm which treats these difficulties, for the calculation of a few eigenvectors and their corresponding eigenvalues. The main techniques used in the new algorithm include: the adaptive completion and separation of the relevant clusters on different levels, the simultaneous treatment of solutions within each cluster, and the robustness tests which monitor the algorithm's efficiency and convergence. The eigenvectors' separation efficiency is based on a new ML projection technique generalizing the Rayleigh Ritz projection, combined with a technique, the backrotations. These separation techniques, when combined with an FMG formulation, in many cases lead to algorithms of O(qN) complexity, for q eigenvectors of size N on the finest level. Previously developed ML algorithms are less focused on the mentioned difficulties. Moreover, algorithms which employ fine level separation techniques are of O(q(sub 2)N) complexity and usually do not overcome all these difficulties. Computational examples are presented where Schrodinger type eigenvalue problems in 2-D and 3-D, having equal and closely clustered eigenvalues, are solved with the efficiency of the Poisson multigrid solver. A second order approximation is obtained in O(qN) work, where the total computational work is equivalent to only a few fine level relaxations per eigenvector

Multigrid techniques for nonlinear eigenvalue probems: Solutions of a nonlinear Schroedinger eigenvalue problem in 2D and 3D

This paper presents multigrid (MG) techniques for nonlinear eigenvalue problems (EP) and emphasizes an MG algorithm for a nonlinear Schrodinger EP. The algorithm overcomes the mentioned difficulties combining the following techniques: an MG projection coupled with backrotations for separation of solutions and treatment of difficulties related to clusters of close and equal eigenvalues; MG subspace continuation techniques for treatment of the nonlinearity; an MG simultaneous treatment of the eigenvectors at the same time with the nonlinearity and with the global constraints. The simultaneous MG techniques reduce the large number of self consistent iterations to only a few or one MG simultaneous iteration and keep the solutions in a right neighborhood where the algorithm converges fast

MIKA: a multigrid-based program package for electronic structure calculations

A general real-space multigrid algorithm MIKA (Multigrid Instead of the K-spAce) for the self-consistent solution of the Kohn-Sham equations appearing in the state-of-the-art electronic-structure calculations is described. The most important part of the method is the multigrid solver for the Schr\"odinger equation. Our choice is the Rayleigh quotient multigrid method (RQMG), which applies directly to the minimization of the Rayleigh quotient on the finest level. Very coarse correction grids can be used, because there is in principle no need to represent the states on the coarse levels. The RQMG method is generalized for the simultaneous solution of all the states of the system using a penalty functional to keep the states orthogonal. Special care has been taken to optimize the iterations towards the self-consistency and to run the code in parallel computer architectures. The scheme has been implemented in multiple geometries. We show examples from electronic structure calculations employing nonlocal pseudopotentials and/or the jellium model. The RQMG solver is also applied for the calculation of positron states in solids.Comment: To appear in a special issue of Int J. Quant. Chem. devoted to the conference proceedings of 9th International Conference on the Applications of the Density Functional Theory in Chemistry and Physic

Towards a Linear-Scaling DFT Technique: The Density Matrix Approach

A recently proposed linear-scaling scheme for density-functional pseudopotential calculations is described in detail. The method is based on a formulation of density functional theory in which the ground state energy is determined by minimization with respect to the density matrix, subject to the condition that the eigenvalues of the latter lie in the range [0,1]. Linear-scaling behavior is achieved by requiring that the density matrix should vanish when the separation of its arguments exceeds a chosen cutoff. The limitation on the eigenvalue range is imposed by the method of Li, Nunes and Vanderbilt. The scheme is implemented by calculating all terms in the energy on a uniform real-space grid, and minimization is performed using the conjugate-gradient method. Tests on a 512-atom Si system show that the total energy converges rapidly as the range of the density matrix is increased. A discussion of the relation between the present method and other linear-scaling methods is given, and some problems that still require solution are indicated.Comment: REVTeX file, 27 pages with 4 uuencoded postscript figure

A novel multigrid method for electronic structure calculations

A general real-space multigrid algorithm for the self-consistent solution of the Kohn-Sham equations appearing in the state-of-the-art electronic-structure calculations is described. The most important part of the method is the multigrid solver for the Schroedinger equation. Our choice is the Rayleigh quotient multigrid method (RQMG), which applies directly to the minimization of the Rayleigh quotient on the finest level. Very coarse correction grids can be used, because there is no need to be able to represent the states on the coarse levels. The RQMG method is generalized for the simultaneous solution of all the states of the system using a penalty functional to keep the states orthogonal. The performance of the scheme is demonstrated by applying it in a few molecular and solid-state systems described by non-local norm-conserving pseudopotentials.Comment: 9 pages, 3 figure

A Real-Space Full Multigrid study of the fragmentation of Li11+ clusters

We have studied the fragmentation of Li11+ clusters into the two experimentally observed products (Li9+,Li2) and (Li10+,Li) The ground state structures for the two fragmentation channels are found by Molecular Dynamics Simulated Annealing in the framework of Local Density Functional theory. Energetics considerations suggest that the fragmentation process is dominated by non-equilibrium processes. We use a real-space approach to solve the Kohn-Sham problem, where the Laplacian operator is discretized according to the Mehrstellen scheme, and take advantage of a Full MultiGrid (FMG) strategy to accelerate convergence. When applied to isolated clusters we find our FMG method to be more efficient than state-of-the-art plane wave calculations.Comment: 9 pages + 6 Figures (in gzipped tar file

Three real-space discretization techniques in electronic structure calculations

A characteristic feature of the state-of-the-art of real-space methods in electronic structure calculations is the diversity of the techniques used in the discretization of the relevant partial differential equations. In this context, the main approaches include finite-difference methods, various types of finite-elements and wavelets. This paper reports on the results of several code development projects that approach problems related to the electronic structure using these three different discretization methods. We review the ideas behind these methods, give examples of their applications, and discuss their similarities and differences.Comment: 39 pages, 10 figures, accepted to a special issue of "physica status solidi (b) - basic solid state physics" devoted to the CECAM workshop "State of the art developments and perspectives of real-space electronic structure techniques in condensed matter and molecular physics". v2: Minor stylistic and typographical changes, partly inspired by referee comment

Real-Space Mesh Techniques in Density Functional Theory

This review discusses progress in efficient solvers which have as their foundation a representation in real space, either through finite-difference or finite-element formulations. The relationship of real-space approaches to linear-scaling electrostatics and electronic structure methods is first discussed. Then the basic aspects of real-space representations are presented. Multigrid techniques for solving the discretized problems are covered; these numerical schemes allow for highly efficient solution of the grid-based equations. Applications to problems in electrostatics are discussed, in particular numerical solutions of Poisson and Poisson-Boltzmann equations. Next, methods for solving self-consistent eigenvalue problems in real space are presented; these techniques have been extensively applied to solutions of the Hartree-Fock and Kohn-Sham equations of electronic structure, and to eigenvalue problems arising in semiconductor and polymer physics. Finally, real-space methods have found recent application in computations of optical response and excited states in time-dependent density functional theory, and these computational developments are summarized. Multiscale solvers are competitive with the most efficient available plane-wave techniques in terms of the number of self-consistency steps required to reach the ground state, and they require less work in each self-consistency update on a uniform grid. Besides excellent efficiencies, the decided advantages of the real-space multiscale approach are 1) the near-locality of each function update, 2) the ability to handle global eigenfunction constraints and potential updates on coarse levels, and 3) the ability to incorporate adaptive local mesh refinements without loss of optimal multigrid efficiencies.Comment: 70 pages, 11 figures. To be published in Reviews of Modern Physic