Sparse Matrix Methods Research Articles

Localized and extended electronic eigenstates in proteins: A tight-binding approach

We present a numerical study of the electronic structure of the cytochrome subunit of the photoreaction center of the bacterium Rhodopseudomonas viridis. The electronic structure is described by a tight-binding Hamiltonian, the resulting large secular matrix is diagonalized applying sparse matrix methods. In the highly structured density of states, localized chromophore eigenstates can be observed around the HOMO-LUMO gap. In the low energy range, optical excitations originate from transitions between these states. Eigenfunctions that are extended on a length scale of up to ∼190 Å can be found in the bulk of the valence band density of states. Stretched along α-helices, conduction band eigenstates are characterized by a cylindrical shape. We discuss the role of these states within electron transport and possible extensions of the model presented here.

Introduction to the Special Section on Sparse and Structured Matrices and Their Applications

The field of sparse matrices is a broad and important area of the computational sciences that includes structured matrices and those with seemingly little or no structure. The relevance of the field is highlighted by the wide range of application areas that require the exploitation of matrix sparsity and structure in order to achieve a solution given real-world constraints on computing resources and/or time. Applications in which sparse matrices appear include structural analysis, computational fluid dynamics, economic modeling, financial analysis, numerical optimization, statistical modeling, power network analysis, electromagnetics, meteorology, medical imaging, data mining, and many more. A number of significant advancements in sparse matrix computations have been made in recent years. These advances have led to new challenges as multidisciplinary problems are now ambitiously posed, and they symbolize the growth in the area and demonstrate the dependence of the future of many fields on sparse matrix methods. Moreover, they give rise to a host of new problems that have yet to be addressed by the sparse matrix community. The Second SIAM Conference on Sparse Matrices, which was held in Coeur d'Alene, Idaho, October 9--11, 1996, was organized specifically to address some of the challenging issues in sparse matrix computations. This special issue of SIAM Journal on Matrix Analysis and Applications consists of some of the outstanding papers that were presented at the Sparse Matrix conference. Because of an error in scheduling, the paper titled ``Using Generalized Cayley Transformations within an Inexact Rational Krylov Sequence Method" by Richard Lehoucq, which should have appeared in this section, was published in SIAM Journal on Matrix Analysis and Applications, volume 20, number 1. We are grateful to Paul van Dooren, the Editor-in-Chief of SIAM Journal on Matrix Analysis and Applications, and the SIAM office for agreeing to publish a special section on sparse matrices. We would like to thank Roland Freund, Anne Greenbaum, Joseph Liu, and Zdenek Strakos for serving on the Guest Editorial Board; their effort and cooperation in handling the papers were much appreciated. Finally, we would like to express thanks to all the authors who have submitted papers to the special section; it would not have become a reality without their support.

Resolving seismic anisotropy: Sparse matrix methods for geophysical inverse problems

Two techniques for the singular value decomposition (SVD) of large sparse matrices are directly applicable to geophysical inverse problems: subspace iteration and the Lanczos method. Both methods require very little in‐core storage and efficiently compute a set of singular values and singular vectors. A comparison of the singular value and vector estimates of these iterative approaches with the results of a conventional in‐core SVD algorithm demonstrates their accuracy. Hence, it is possible to conduct an assessment of inversion results for larger sparse inverse problems such as those arising in seismic tomography. As an example, we examine the resolution matrix associated with a crosswell seismic inversion of first arrival times for lateral variations in anisotropy. The application to a set of first arrival times from a crosswell survey at the Grimsel Laboratory emphasizes the utility of including anisotropy in a traveltime inversion. The isotropic component of the estimated velocity structure appears to be well constrained even when anisotropy is included in the inversion. In the case of the Grimsel experiment, we are able to resolve a fracture zone, in agreement with borehole fracture intersections. Elements of the resolution matrix reveal moderate averaging among anisotropy coefficients as well as between anisotropy coefficients and source‐receiver static terms. The information on anisotropy, such as the directions of maximum velocity, appears sensitive to lateral variations in velocity and must be interpreted with some caution.

A Trust Region Method for Nonlinear Programming Based on Primal Interior-Point Techniques

This paper describes a new trust region method for solving large-scale optimization problems with nonlinear equality and inequality constraints. The new algorithm employs interior-point techniques from linear programming, adapting them for more general nonlinear problems. A software implementation based entirely on sparse matrix methods is described. The software handles infeasible start points, identifies the active set of constraints at a solution, and can use second derivative information to solve problems. Numerical results are reported for large and small problems, and a comparison is made with other large-scale optimization codes.

Efficient ordering algorithms for sparse matrix/vector methods

This paper introduces four node ordering algorithms that can be used to improve the efficiency of the sparse matrix and spare vector methods. The algorithms are based on the diagonalization of the A matrix via the use of the C transformation matrix while minimizing the fill-in elements in C. The node ordering sequence and the C matrix are generated in a single stage via Gaussian elimination. Unlike the conventional approach, the LDU decomposition is not needed in the pivot selection of the proposed ordering algorithms. The performance of the algorithms is compared with that of seven other commonly used ordering algorithms using actual power systems data. Test results show that the proposed ordering algorithms can reduce the computation burden of the sparse matrix/vector method effectively and can be used for parallel computation of large sparse systems.

Parallel Computation for Electronic Waves in Quantum Corrals

Recent scanning tunneling microscopy (STM) studies on the (111) faces of noble metals have directly imaged electronic surface‐confined states and dramatic standing‐wave patterns have been observed 1,2]. We solve for the local density of electronic states in these “leaky” quantum corral confinement structures using a coherent elastic scattering theory. We seek solutions of the two‐dimensional Schrödinger equation compatible with non‐reflecting boundary conditions which asymptotically satisfy the Sommerfeld radiation condition [11,14]. The large matrices generated by the discretization of realistic quantum corral structures require the use of sparse matrix methods. In addition, a parallel finite element solution was undertaken using the message passing interface standard (MPI) and the Portable, Extensible, Toolkit for Scientific Computation (PETSc) [5] for an efficient computational solution on both distributed and shared memory architectures. Our calculations reveal excellent agreement with the reported experimental dl/dV STM data.

Sparse matrix tools for Gaussian models on lattices

Sparse matrix methods are considered for Gaussian models on lattices. The approach depends on the applicability of mixed model methodology, which is feasible when observational errors are present. Moreover, lattice models with nearest-neighbor interactions lead to a spatial variance-covariance matrix having a sparse inverse, and this lends itself to sparse-matrix manipulation. Modern technology involving the Cholesky decomposition and backward differentiation are described for prediction, parameter estimation, cross validation, and conditional simulation of spatial processes.

Fast cars

This paper develops methods for quickly computing maximum likelihood conditional autoregressions(CARs).By using sparse matrix methods,reorganizing the sum-of-squared errors function to avoid unnecessary calculations,and precomputing a set of determinants,simulations of large CARs become possible. As an illustration of the power of these approaches,a simulation of 250 CARs of 2,905 observations can take fewer than three minutes on a personal computer,despite the necessity of evaluating 100 determinants of 2,905 by 2,905 matrices.The computation of each estimate via examining the profile likelihood sampled at 100 points avoids problems of local optima.Simulating estimates avoids other problems associated with the traditional information matrix approach to inference.

A sparse formulation and implementation of the transient energy function method for dynamic security analysis

A new solution for the Transient Energy Function (TEF) method of transient stability analysis is presented. Its speed, robustness, unrestricted problem size and adaptability to modeling fulfill requirements for production use. Major features are preservation of network structure, sparse matrix methods, TEF analysis based on the controlling Unstable Equilibrium Point (UEP), automatic variable step/order integration and a new UEP solution. Extensive tests have been performed on large practical problems.

Teaching and Examining Sparse Matrix Methods

A method is presented for concise teaching and examining of the principles and advantages of sparse matrix methods. The method uses only mental arithmetic and is illustrated using Gauss elimination for the solution of simultaneous equations. Indications are given of the ways in which the ideas can be extended to methods other than Gauss elimination and to types of sparse matrix method other than those considered in detail. Indications are also given of how the material can be taught so as to integrate with related matters, such as the evaluation of determinants and the way that the savings obtained by using the most sophisticated sparse matrix methods increase rapidly as the order of the matrix increases.

A general approach to circuit equations

AbstractComputer‐aided circuit analysis is a complex procedure which may be roughly divided into three steps. the first stage is device modelling, where quantitative properties of the participating electronic devices are determined. In the second step the circuit's topology in conjunction with the device equations is exploited in order to formulate a linearized nonsingular equation set. Eventually the equation set is solved iteratively in the third phase, yielding a solution vector of circuit variables.This paper deals with the second and third steps, where diverse sparse matrix methods are employed. Its aim is to provide a general mathematical form by which most matrix manipulation techniques can be described, thus enabling their classification on a theoretical level. Usually these methods are presented descriptively rather than strictly mathematically.The paper introduces a matrix reduction operator from which a general matrix operator equation is deduced. This operator equation can be used to describe entire analysis approaches. After some necessary definitions, the purpose of this paper is illustrated by studying several classical examples rather than giving many mathematical proofs. the selected examples involve some well‐known linear equation set solution methods as well as several typical equation set transformations.

High-order multivariable transfer function curve fitting: Algorithms, sparse matrix methods and experimental results

This paper develops a computational approach to multivariable frequency domain curve fitting, based on two-norm minimization. The algorithm is specifically tailored to the identification of complex systems having a large number of parameters, and includes a sparse matrix method for reducing computation and memory requirements on large problems. The algorithm is also well-suited for identification of lightly damped systems such as flexible structures. The overall approach is successfully demonstrated on a high-order multivariable flexible structure experiment requiring the estimation of 780 parameters over a 100 Hz band width.

The control of difficult refinements

Each year, many single-crystal structure analyses are reported that show evidence of over- or under-refinement. Often, the refinement strategies have been naive or over-complex and alternative strategies might have been more effective. The many descriptions of crystallographic and numerical techniques suitable for assisting with the control of difficult refinements are distributed widely in the literature and so are not always easily accessible. Without being a review of these procedures (which would require a substantial book), this article attempts to list readily available procedures, together with a brief outline of their backgrounds and examples of their applications to organic and organometallic compounds. The analysis of extended-lattice materials (usually inorganic materials) often raises problems in addition to those covered here. The particular aim of this article is to remind the reader that X-ray structure analysis is a modelling process and that, while standard models may be adequate for most analyses, more care and imagination must be applied to the treatment of difficult cases. Principles of methods are described without detailed mathematical derivations, although sufficient references to the literature are provided to permit careful study. Future requirements for refinement processes are outlined, including the use of new machine architectures, applications of sparse-matrix methods and the development of expert systems.

Efficient solution for equilibrium points in transient energy function analysis

A new method for the determination of stable and unstable equilibrium points in the transient energy function method of power system stability analysis is presented. Computation of the unstable equilibrium point is one of the fundamental steps in determination of the energy margin. The basis of the new method is its formulation as a power flow problem and the use of well known power flow algorithms for its solution. Sparse matrix methods are used to achieve the indicated speedups. The new method has shown robust behavior in all systems tested and performance improvements over previous methods. Results using test systems of up to 161 generators, 901 buses, and 2000 lines are given.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Exactly solvable models of material breakdown.

We present the solutions to two simple models for the brittle failure of materials containing random flaws. These solutions provide support for simple scaling theories we had previously developed for more complex models, and refute recent claims that models with random dilution scale in a manner similar to a disorderless material. In particular, we find that for these models, the asymptotic size effect in the average strength is logarithmic, and the failure distribution is of an exponential of an exponential form (often with an algebraic prefactor). The method of solution is also interesting. The failure probability of the quasi-one-dimensional models we solve can be written in terms of a transition matrix introduced by Harlow. For large sample sizes, the largest eigenvalue of this transition matrix approaches one, and our solution rests on a perturbative expansion of the largest eigenvalue about one. The small and intermediate lattice behavior of the model is analyzed by using sparse matrix methods to find the largest eigenvalue of the transition matrix, and the trace of powers of the transition matrix.

Linear and non‐linear iterative methods for the incompressible Navier‐Stokes equations

AbstractIn this study, the discretized finite volume form of the two‐dimensional, incompressible Navier‐Stokes equations is solved using both a frozen coefficient and a full Newton non‐linear iteration. The optimal method is a combination of these two techniques. The linearized equations are solved using a conjugate‐gradient‐like method (CGSTAB). Various types of preconditioning are developed. Completely general sparse matrix methods are used. Investigations are carried out to determine the effect of finite volume cell anisotropy on the preconditioner. Numerical results are given for several test problems.

Calculation of prediction error variances using sparse matrix methods

The use of exact and approximate algorithms to calculate prediction error variances using sparse matrix methods are demonstrated for an individual animal effect including maternal effects. One exact algorithm is substantially faster than two others. An approximation of the best exact method gave an acceptable level of reliabilities and reduced the computation by a factor of approximately fifty compared with the exact computation and is routine in national beef evaluation in Britain.

The use of sparse matrix methods in finite-element codes for structural mechanics applications

In this study, sparse matrix methods are considered as linear solvers in finite-element programs for structural mechanics applications. The factorization schemes used, both for the standard global solution technique and for the partial factorization in the superelement technique, are based on sparse matrix technology [George and Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, 1981; Duff and Reid, ACM Trans. Math Software 9, 302–325 (1983); Ashcraft et al., Int. J. Supercomputer Appl. 1, 10–30 (1987); Ashcraft, “A vector implementation of the multifrontal method for large sparse symmetric positive definite linear systems,” ETA-TR-51, Boeing Computer Services, Seattle, 1987], and tailored for use with the finite-element method for structural mechanics problems as described in [Damhaug, “Sparse solution of linear finite-element equations,” Dr Ing. Thesis 1992: 76, The Norwegian Institute of Technology, 1992].

Estimation of nonadditive genetic variances for a total-merit model including maternal effects

Henderson described a method to reduce the number of mixed-model equations when estimating additive and nonadditive genetic variances or predicting additive and nonadditive genetic merits. The extension to a maternal effects model is straight-forward. When maternal genetic effects are strictly additive, an algebraic identity was found that reduces by a factor of two the order of a matrix that must be inverted each round to account for the genetic covariances among direct and maternal genetic effects. An algorithm for derivative-free restricted maximum likelihood was developed based on Henderson's total-merit model that is the basis for a reduced number of equations. The same values for the logarithm of the likelihood can be calculated from components of the equations for the total-merit model and from components of the equations for the individual effects model. The computational properties of the equations for the total-merit model, however, do not lend themselves to sparse-matrix methods. Both memory and time requirements were much greater for the total-merit model than for the individual-effects model for a data set of 871 animals and a model with additive, dominance, and additive x additive direct and additive maternal genetic effects. Approximately 14 times more memory was required, although the number of equations decreased from 3,773 to 2,031. Computing time per round increased by a factor of 50.

Fast waveguide mode computation using wavelet-like basis functions

The use of wavelet-like basis functions for solving electromagnetics problems is demonstrated. In particular, the modes of an arbitrarily shaped hollow metallic waveguide are found, using a surface-integral-equation/method-of-moments (MOM) formulation. A class of wavelet-like basis functions is used to produce a sparse MOM impedance matrix, allowing the use of sparse matrix methods for fast solution of the problem. The same method applies directly to the external scattering problem. For the examples considered, the wavelet-domain impedance matrix has about 20% nonzero elements, and the time required to compute its LU factorization is reduced by approximately a factor of 10 compared to that for the original full matrix. >