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

An Optimized and Scalable Eigensolver for Sequences of Eigenvalue Problems

In many scientific applications the solution of non-linear differential equations are obtained through the set-up and solution of a number of successive eigenproblems. These eigenproblems can be regarded as a sequence whenever the solution of one problem fosters the initialization of the next. In addition, in some eigenproblem sequences there is a connection between the solutions of adjacent eigenproblems. Whenever it is possible to unravel the existence of such a connection, the eigenproblem sequence is said to be correlated. When facing with a sequence of correlated eigenproblems the current strategy amounts to solving each eigenproblem in isolation. We propose a alternative approach which exploits such correlation through the use of an eigensolver based on subspace iteration and accelerated with Chebyshev polynomials (ChFSI). The resulting eigensolver is optimized by minimizing the number of matrix-vector multiplications and parallelized using the Elemental library framework. Numerical results show that ChFSI achieves excellent scalability and is competitive with current dense linear algebra parallel eigensolvers.Comment: 23 Pages, 6 figures. First revision of an invited submission to special issue of Concurrency and Computation: Practice and Experienc

Regularni politopi

U ovom članku promatramo dvodimenzionalne, trodimenzionalne i višedimenzionalne regularne politope. Pozabavili smo se ponajprije njihovom egzistencijom, konstrukcijom te svojstvima koja zadovoljavaju njihovi elementi. Najvažniji teorem članka, poznat pod nazivom Euler–Poincaréova formula, nalazi se u šestom poglavlju te povezuje broj svih elemenata n–dimenzionalnog politopa. http://e.math.hr/math_e_article/br18/berljafa_susnjar

Rational Krylov Decompositions: Theory and Applications

Numerical methods based on rational Krylov spaces have become an indispensable tool of scientific computing. In this thesis we study rational Krylov spaces by considering rational Krylov decompositions; matrix relations which, under certain conditions, are associated with these spaces. We investigate the algebraic properties of such decompositions and present an implicit Q theorem for rational Krylov spaces. We derive standard and harmonic Ritz extraction strategies for approximating the eigenpairs of a matrix and for approximating the action of a matrix function onto a vector. While these topics have been considered previously, our approach does not require the last pole to be infinite, which makes the extraction procedure computationally more efficient. Typically, the computationally most expensive component of the rational Arnoldi algorithm for computing a rational Krylov basis is the solution of a large linear system of equations at each iteration. We explore the option of solving several linear systems simultaneously, thus constructing the rational Krylov basis in parallel. If this is not done carefully, the basis being orthogonalized may become poorly conditioned, leading to numerical instabilities in the orthogonalization process. We introduce the new concept of continuation pairs which gives rise to a near-optimal parallelization strategy that allows to control the growth of the condition number of this nonorthogonal basis. As a consequence we obtain a more accurate and reliable parallel rational Arnoldi algorithm. The computational benefits are illustrated using our high performance C++ implementation. We develop an iterative algorithm for solving nonlinear rational least squares problems. The difficulty is in finding the poles of a rational function. For this purpose, at each iteration a rational Krylov decomposition is constructed and a modified linear problem is solved in order to relocate the poles to new ones. Our numerical results indicate that the algorithm, called RKFIT, is well suited for model order reduction of linear time invariant dynamical systems and for optimisation problems related to exponential integration. Furthermore, we derive a strategy for the degree reduction of the approximant obtained by RKFIT. The rational function obtained by RKFIT is represented with the aid of a scalar rational Krylov decomposition and an additional coefficient vector. A function represented in this form is called an RKFUN. We develop efficient methods for the evaluation, pole and root finding, and for performing basic arithmetic operations with RKFUNs. Lastly, we discuss RKToolbox, a rational Krylov toolbox for MATLAB, which implements all our algorithms and is freely available from http://rktoolbox.org. RKToolbox also features an extensive guide and a growing number of examples. In particular, most of our numerical experiments are easily reproducible by downloading the toolbox and running the corresponding example files in MATLAB

The RKFIT algorithm for nonlinear rational approximation

The RKFIT algorithm outlined in [M. Berljafa and S. Guettel, Generalized rational Krylov decompositions with an application to rational approximation, SIAM J. Matrix Anal. Appl., 2015] is a Krylov-based approach for solving nonlinear rational least squares problems. This paper puts RKFIT into a general framework, allowing for its extension to nondiagonal rational approximants and a family of approximants sharing a common denominator. Furthermore, we derive a strategy for the degree reduction of the approximants, as well as methods for their conversion to partial fraction form, for the efficient evaluation, and root-finding. We also discuss commons and differences of RKFIT and the popular vector fitting algorithm. A MATLAB implementation of RKFIT is provided and numerical experiments, including the fitting of a MIMO dynamical system and an optimization problem related to exponential integration, demonstrate its applicability

A Rational Krylov Toolbox for MATLAB

The Rational Krylov Toolbox contains MATLAB implementations of Ruhe's rational Krylov sequence method, algorithms for the implicit and explicit relocation of the poles of a rational Krylov space, an implementation of RKFIT, a robust algorithm for rational least squares fitting, and the RKFUN class for numerical computations with rational functions

An Optimized and Scalable Iterative Eigensolver for Sequences of Dense Eigenvalue Problems

Sequences of eigenvalue problems consistently appear in a large class of applications based on the iterative solution of a non-linear eigenvalue problem. A typical example is given by the chemistry and materials science ab initio simulations relying on computational methods developed within the framework of Density Functional Theory (DFT). DFT provides the means to solve a high-dimensional quantum mechanical problem by representing it as a non-linear generalized eigenvalue problem which is solved self-consistently through a series of successive outer-iteration cycles. As a consequence each self-consistent simulation is made of several sequences of generalized eigenproblems P : Ax

Parallelization of the rational Arnoldi algorithm

Rational Krylov methods are applicable to a wide range of scientific computing problems, and the rational Arnoldi algorithm is a commonly used procedure for computing an orthonormal basis of a rational Krylov space. Typically, the computationally most expensive component of this algorithm is the solution of a large linear system of equations at each iteration. We explore the option of solving several linear systems simultaneously, thus constructing the rational Krylov basis in parallel. If this is not done carefully, the basis being orthogonalized may become badly conditioned, leading to numerical instabilities in the orthogonalization process. We introduce the new concept of continuation pairs which gives rise to a near-optimal parallelization strategy that allows to control the growth of the condition number of this nonorthogonal basis. As a consequence we obtain a significantly more accurate and reliable parallel rational Arnoldi algorithm. The computational benefits are illustrated using several numerical examples from different application areas

Generalized rational Krylov decompositions with an application to rational approximation

Generalized rational Krylov decompositions are matrix relations which, under certain conditions, are associated with rational Krylov spaces. We study the algebraic properties of such decompositions and present an implicit Q theorem for rational Krylov spaces. Transformations on rational Krylov decompositions allow for changing the poles of a rational Krylov space without recomputation, and two algorithms are presented for this task. Using such transformations we develop a rational Krylov method for rational least squares fitting. Numerical experiments indicate that the proposed method converges fast and robustly. A MATLAB toolbox with implementations of the presented algorithms and experiments is provided