Lanczos Eigensolver Research Articles

Fault Tolerant Lanczos Eigensolver via an Invariant Checking Method

An extensive survey of the literature shows that the Lanczos eigensolver is a popular iterative method for approximating a few maximal eigenvalues of a real symmetric matrix, particularly if the matrix is large and sparse. In recent years, graphics processing units (GPUs) have become a popular platform for scientific computing applications, many of which are based on linear algebra, and are increasingly being used as the main computational units in supercomputers. This trend is expected to continue as the number of computations required by scientific applications reach petascale and exascale range. In this paper, building on our earlier work [22], we investigate in detail the error checking mechanism for the Lanczos eigensolver. We identify a low cost invariant for efficient error checking, and through mathematical analysis determine the efficiency of our mechanism when used by the Lanczos eigensolver. We evaluate the proposed fault tolerant scheme using an open-source sparse eigensolver on a GPU platform, with and without the injection of faults. We use a large number of sparse matrices from real applications, to determine the efficiency and efficacy of our method and our implementation shows that the proposed fault tolerant method has good error coverage and low overhead. To the best of our knowledge, we are the first to introduce such a scheme for the Lanczos method.

Refined isogeometric analysis for generalized Hermitian eigenproblems

We use refined isogeometric analysis (rIGA) to solve generalized Hermitian eigenproblems (Ku=λMu). rIGA conserves the desirable properties of maximum-continuity isogeometric analysis (IGA) while it reduces the solution cost by adding zero-continuity basis functions, which decrease the matrix connectivity. As a result, rIGA enriches the approximation space and reduces the interconnection between degrees of freedom. We compare computational costs of rIGA versus those of IGA when employing a Lanczos eigensolver with a shift-and-invert spectral transformation. When all eigenpairs within a given interval [λs,λe] are of interest, we select several shifts σk∈[λs,λe] using a spectrum slicing technique. For each shift σk, the factorization cost of the spectral transformation matrix K−σkM controls the total computational cost of the eigensolution. Several multiplications of the operator matrix (K−σkM)−1M by vectors follow this factorization. Let p be the polynomial degree of the basis functions and assume that IGA has maximum continuity of p−1. When using rIGA, we introduce C0 separators at certain element interfaces to minimize the factorization cost. For this setup, our theoretical estimates predict computational savings to compute a fixed number of eigenpairs of up to O(p2) in the asymptotic regime, that is, large problem sizes. Yet, our numerical tests show that for moderate-size eigenproblems, the total observed computational cost reduction is O(p). In addition, rIGA improves the accuracy of every eigenpair of the first N0 eigenvalues and eigenfunctions, where N0 is the total number of modes of the original maximum-continuity IGA discretization.

A variational calculation of vibrational levels of vinyl radical.

We report the vibrational energy levels of vinyl radical (VR) that are computed with a Lanczos eigensolver and a contracted basis. Many of the levels of the two previous VR variational calculations differ significantly and differ also from those reported in this paper. We identify the source of and correct symmetry errors on the potential energy surfaces used in the previous calculations. VR has two equivalent equilibrium structures. By plotting wavefunction cuts, we show that two tunneling paths play an important role. Using the computed wavefunctions, it is possible to assign many states and thereby to determine tunneling splittings that are compared with their experimental counterparts. Our computed red shift of the hot band at 2897.23 cm-1, observed by Dong et al. [J Chem. Phys. 128, 044305 (2008)], is 4.47 cm-1, which is close to the experimental value of 4.63 cm-1.

Zero-site density matrix renormalization group and the optimal low-rank correction

A zero-site density matrix renormalization algorithm (DMRG0) is proposed to minimize the energy of matrix product states (MPS). Instead of the site tensors themselves, we propose to optimize sequentially the "message" tensors between neighbor sites, which contain the singular values of the bipartition. This leads to a local minimization step that is independent of the physical dimension of the site. Conceptually, it separates the optimization and decimation steps in DMRG. Furthermore, we introduce two new global perturbations based on the optimal low-rank correction to the current state, which are used to avoid local minima. They are determined variationally as the MPS closest to the one-step correction of the Lanczos or Jacobi-Davidson eigensolver, respectively. These perturbations mainly decrease the energy and are free of hand-tuned parameters. Compared to existing single-site enrichment proposals, our approach gives similar convergence ratios per sweep while the computations are cheaper by construction. Our methods may be useful in systems with many physical degrees of freedom per lattice site. We test our approach on the periodic Heisenberg spin chain for various spins, and on free electrons on the lattice.

Research on Laterally Restrained Built Up Steel Beam Under Dynamic Response

This paper presents the analytical investigation of laterally restrained built up steel beam under dynamic response using finite element software ABAQUS. The main objective of this study was to estimate the mode shapes and mode shape curvature of the laterally restrained built up steel beam. The three parameters such as modal frequencies, mode shapes and mode shape curvature are suggested for identifying the damages in built up steel beam. Damage assessment is done by linear perturbation free vibration study using finite element software ABAQUS. The frequency extraction methods for built up steel beam was done in ABAQUS to get the dynamic response. The Lanczos eigen solver was used for finding the mode shapes. Eigen value extraction, the natural frequencies and the corresponding mode shapes of a system were studied

Explicitly correlated MRCI-F12 potential energy surfaces for methane fit with several permutation invariant schemes and full-dimensional vibrational calculations

A data-set of nearly 100,000 symmetry unique multi-configurational ab initio points for methane were generated at the (AE)-MRCI-F12(Q)/CVQZ-F12 level, including energies beyond 30,000 cm−1 above the minimum and fit into potential energy surfaces (PESs) by several permutation invariant schemes. A multi-expansion interpolative fit combining interpolating moving least squares (IMLS) fitting and permutation invariant polynomials (PIP) was able to fit the complete data-set to a root-mean-square deviation of 1.0 cm−1 and thus was used to benchmark the other fitting methods. The other fitting methods include a single PIP expansion and two neural network (NN) based approaches, one of which combines NN with PIP. Full-dimensional variational vibrational calculations using a contracted-iterative method (and a Lanczos eigensolver) were used to assess the spectroscopic accuracy of the electronic structure method. The results show that the NN-based fitting approaches are able to fit the data-set remarkably accurately with the PIP–NN method producing levels in remarkably close agreement with the PIP–IMLS benchmark. The (AE)-MRCI-F12(Q)/CVQZ-F12 electronic structure method produces vibrational levels of near spectroscopic accuracy and a superb equilibrium geometry. The levels are systematically slightly too high, beginning at ∼ 1–2 cm−1 above the fundamentals and becoming correspondingly higher for overtones. The PES is therefore suitable for small ab initio or empirical corrections and since it is based on a multi-reference method, can be extended to represent dynamically relevant dissociation channels.

A hybrid variational–perturbational nuclear motion algorithm

A hybrid variational–perturbational nuclear motion algorithm based on the perturbative treatment of the Coriolis coupling terms of the Eckart–Watson kinetic energy operator following a variational treatment of the rest of the operator is described. The algorithm has been implemented in the quantum chemical code DEWE. Performance of the hybrid treatment is assessed by comparing selected numerically exact variational vibration-only and rovibrational energy levels of the C2H4, C2D4, and CH4 molecules with their perturbatively corrected counterparts. For many of the rotational–vibrational states examined, numerical tests reveal excellent agreement between the variational and even the first-order perturbative energy levels, whilst the perturbative approach is able to reduce the computational cost of the matrix-vector product evaluations, needed by the iterative Lanczos eigensolver, by almost an order of magnitude.

Theoretical study of the rovibrational spectrum of He2–OCS

We report calculated microwave and infrared rovibrational transitions of the van der Waals complex He2–OCS. The calculations were done using a product basis, a Lanczos eigensolver, and potentials built from He–OCS, and He–He potential functions taken from the literature. All five of the large amplitude coordinates are treated exactly and calculations are done for J values up to five. All rovibrational levels are converged to 0.001 cm–1 by using basis sets with as many as 87 million funcions. Good agreement is found with previously reported experimental results. Although we assume that the dipole moment is along the OCS axis, we find transitions with appreciable intensity between different torsion states.

Theoretical and Experimental Study of the Rovibrational Spectrum of He2−CO

We report calculated microwave and infrared ro-vibrational transitions of the van der Waals complex He(2)-CO. The calculations were done using a product basis and a Lanczos eigensolver, together with He-CO and He-He potential functions taken from the literature. The results are found to be in good agreement with previously reported experimental results, and they enable the experimental assignments to be clarified, augmented, and (in one case) corrected. Unlike some other van der Waals complexes with two He atoms such as He(2)-N(2)O, it is not possible to associate a set of energy levels with the "torsional" motion of the two He atoms on a ring encircling the dopant (in this case CO). Although we assume that the dipole moment is along the CO axis we find nonetheless that many transitions have appreciable intensity due to ro-vibrational coupling.

On the variational computation of a large number of vibrational energy levels and wave functions for medium-sized molecules

In a recent publication [J. Chem. Phys. 127, 084102 (2007)], the nearly variational DEWE approach (DEWE denotes Discrete variable representation of the Watson Hamiltonian using the Eckart frame and an Exact inclusion of a potential energy surface expressed in arbitrarily chosen coordinates) was developed to compute a large number of (ro)vibrational eigenpairs for medium-sized semirigid molecules having a single well-defined minimum. In this publication, memory, CPU, and hard disk usage requirements of DEWE, and thus of any DEWE-type approach, are carefully considered, analyzed, and optimized. Particular attention is paid to the sparse matrix-vector multiplication, the most expensive part of the computation, and to rate-determining steps in the iterative Lanczos eigensolver, including spectral transformation, reorthogonalization, and restart of the iteration. Algorithmic improvements are discussed in considerable detail. Numerical results are presented for the vibrational band origins of the (12)CH(4) and (12)CH(2)D(2) isotopologues of the methane molecule. The largest matrix handled on a personal computer during these computations is of the size of (4x10(8))x(4x10(8)). The best strategy for determining vibrational eigenpairs depends largely on the actual details of the required computation. Nevertheless, for a usual scenario requiring a large number of the lowest eigenpairs of the Hamiltonian matrix the combination of the thick-restart Lanczos method, shift-fold filtering, and periodic reorthogonalization appears to result in the computationally most feasible approach.

Theoretical and experimental study of infrared spectra of He2-CO2This article is part of a Special Issue on Spectroscopy at the University of New Brunswick in honour of Colan Linton and Ron Lees.

Additional high-resolution lines in the ν3 CO2 fundamental band of the rovibrational spectrum of He2-CO2 have been observed using a tunable infrared laser to probe a pulsed supersonic jet expansion. The ro-vibrational spectrum was calculated using Euler angles and five vibrational coordinates that specify the positions of the He atoms with respect to the CO2 molecule, a product basis, and a Lanczos eigensolver. Rotational states with J = 0, 1, 2, and 3 associated with the vibrational ground state and different states (νt) of the torsional motion of the two He atoms about the CO2 axis are identified and assigned. The assignment is consistent with having different principle axis orientiations in νt = 0 and νt = 1 states. Δνt = 1 infrared transistions are forbidden due to the symmetry of CO2, but Δνt = 1 microwave transistions are possible. Theoretical line positions and intensities are predicted. Good agreement between experiment and theory was obtained. The calculated energy levels and intensities were crucial in assigning some of the weaker observed transitions.

Variational quantum approaches for computing vibrational energies of polyatomic molecules

In this article, we review state-of-the-art methods for computing vibrational energies of polyatomic molecules using quantum mechanical, variationally-based approaches. We illustrate the power of those methods by presenting applications to molecules with more than four atoms. This demonstrates the great progress that has been made in this field in the last decade in dealing with the exponential scaling with the number of vibrational degrees of freedom. In this review we present three methods that effectively obviate this bottleneck. The first important idea is the n-mode representation of the Hamiltonian and notably the potential. The potential (and other functions) is represented as a sum of terms that depend on a subset of the coordinates. This makes it possible to compute matrix elements, form a Hamiltonian matrix, and compute its eigenvalues and eigenfunctions. Another approach takes advantage of this multimode representation and represents the terms as a sum of products. It then exploits the powerful multiconfiguration Hartree time-dependent method to solve the time-dependent Schrödinger equation and extract the eigenvalue spectrum. The third approach we present uses contracted basis functions in conjunction with a Lanczos eigensolver. Matrix vector products are done without transforming to a direct-product grid. The usefulness of these methods is demonstrated for several example molecules, e.g. methane, methanol and the Zundel cation.

Calculating vibrational energies and wave functions of vinylidene using a contracted basis with a locally reorthogonalized coupled two-term Lanczos eigensolver

We use a contracted basis+Lanczos eigensolver approach to compute vinylidene-like vibrational states of the acetylene-vinylidene system. To overcome problems caused by loss of orthogonality of the Lanczos vectors we reorthogonalize Lanczos vector and use a coupled two-term approach. The calculations are done in CC-HH diatom-diatom Jacobi coordinates which make it easy to compute states one irreducible representation at a time. The most costly parts of the calculation are parallelized and scale well. We estimate that the vinylidene energies we compute are converged to approximately 1 cm(-1).

Theoretical and experimental studies of the infrared rovibrational spectrum of He2–N2O

Rovibrational spectra of the He(2)-N(2)O complex in the nu(1) fundamental band of N(2)O (2224 cm(-1)) have been observed using a tunable infrared laser to probe a pulsed supersonic jet expansion, and calculated using five coordinates that specify the positions of the He atoms with respect to the NNO molecule, a product basis, and a Lanczos eigensolver. Vibrational dynamics of the complex are dominated by the torsional motion of the two He atoms on a ring encircling the N(2)O molecule. The resulting torsional states could be readily identified, and they are relatively uncoupled to other He motions up to at least upsilon(t) = 7. Good agreement between experiment and theory was obtained with only one adjustable parameter, the band origin. The calculated results were crucial in assigning many weaker observed transitions because the effective rotational constants depend strongly on the torsional state. The observed spectra had effective temperatures around 0.7 K and involved transitions with J < or =3, with upsilon(t) = 0 and 1, and (with one possible exception) with Deltaupsilon(t)=0. Mixing of the torsion-rotation states is small but significant: some transitions with Deltaupsilon(t) not equal 0 were predicted to have appreciable intensity even assuming that the dipole transition moment coincides perfectly with the NNO axis. One such transition was tentatively assigned in the observed spectra, but confirmation will require further work.

USING LEBEDEV GRIDS, SINE SPHERICAL HARMONICS, AND MONOMER CONTRACTED BASIS FUNCTIONS TO CALCULATE BENDING ENERGY LEVELS OF HF TRIMER

We calculate energy levels of a six-dimensional bending Hamiltonian for HF trimer using a finite basis representation (FBR) in conjunction with the Lanczos eigensolver. We improve on our previous method [J. Chem. Phys.115, 9781 (2001)] using three techniques: (1) Lebedev's quadrature scheme is used to reduce the size of quadrature grid by a factor of 3.4. (2) Since the barrier separating the two equivalent versions of HF trimer is high and wide, it is a good approximation to confine the bending motion to one well by using sine spherical harmonics basis functions (this reduces the size of the basis by a factor of 8). (3) The sine spherical harmonic basis is contracted for each monomer to generate a very efficient basis. It is shown that the best approach is to combine all the three techniques.

New ideas for using contracted basis functions with a Lanczos eigensolver for computing vibrational spectra of molecules with four or more atoms

We propose new methods for using contracted basis functions in conjunction with the Lanczos algorithm to calculate vibrational (or rovibrational) spectra. As basis functions we use products of eigenfunctions of reduced-dimension Hamiltonians obtained by freezing coordinates at equilibrium. The basis functions represent the desired wave functions well, yet are simple enough that matrix-vector products may be evaluated efficiently. The methods we suggest obviate the need to transform from the contracted to an original product basis each time a matrix-vector product is evaluated. For HOOH the most efficient of the methods we present is about an order of magnitude faster than a product basis Lanczos calculation.

Using a Lanczos Eigensolver in the Computation of Empirical Orthogonal Functions

Abstract In the framework of physical field studies, EOF analysis allows the scientist to determine the modes that govern the variability of a phenomenon. The analysis requires the resolution of a linear algebra problem. This paper focuses on this part of the EOF analysis, the computation of some singular values, and the associated vectors of the data matrix D. After recalling some fundamentals of this type of problem, the authors compare the usually employed singular value decomposition strategy with a Lanczos eigensolver technique. The latter consists of computing some eigenvalues of a small symmetric matrix. The authors demonstrate its mathematical and numerical stability and discuss its main features. A comparison of the two strategies shows the advantages of the Lanczos technique. Finally, the approach is illustrated with an example based on the study of oceanographic datasets.

A parallel implementation of tight-binding molecular dynamics

We introduce an efficient and scalable parallel implementation of tight-binding molecular dynamics which employs reordering of the atoms. This implementation also exploits the sparse character of the Hamiltonian matrix. Reordering allows our algorithm to scale well on parallel machines since it asssures data locality. A sparse storage format and a stabilized parallel Lanczos eigensolver allow consideration of large problem sizes relevant to materials science. We present a showcase application to a 1000-atom silicon sample.

A Parallel Implementation of Tight-Binding Molecular Dynamics Based on Reordering of Atoms and the Lanczos Eigen-Solver

AbstractWe introduce an efficient and scalable parallel implementation of tight-binding molecular dynamics (TBMD) which employs reordering of the atoms in order to maximize datalocality of the distributed tight-binding (TB) Hamiltonian matrix. Reordering of the atom labels allows our new algorithm to scale well on parallel machines since most of the TB hopping integrals for a given atom are local to the processing element (PE) therefore minimizing communication. The sparse storage format and the distribution of the required eigenvectors reduces memory requirements per PE. The sparse storage format and a stabilized parallel Lanczos eigen-solver allow consideration of large problem sizes relevant to materials science. In addition, the implementation allows the calculation of the full spectrum of individual eigen-values/-vectors of the TB matrix at each time-step. This feature is a key issue when the dielectric and optical response must be computed during a TBMD simulation. We present a benchmark of our code and an analysis of the overall efficiency.