Lanczos Vectors Research Articles

Broadband S-Parameter Simulation Based on the Mixed Finite-Element Method and Symmetric Lanczos Algorithm

In this paper, the symmetric Lanczos algorithm and mixed finite element method are combined to improve the efficiency and accuracy of S-parameter simulation in a broad frequency band. The mixed finite element method based on tree-cotree splitting is adopted to overcome the low-frequency (high-density) breakdown problem, thus ensuring the accuracy of low frequency (high integration density) results. The modified symmetric Lanczos algorithm for second order eigenvalue problem is employed for fast frequency sweep by one matrix inversion, then the field in the entire frequency band is obtained rapidly by solving the reduced order matrix equation. The accuracy of the high frequency results is improved by increasing the number of Lanczos vectors. Numerical simulations of two integrated passive filters (IPD filter) and an interconnect model are shown to verify the hybrid method along with commercial software HFSS.

A tale of two vectors: A Lanczos algorithm for calculating RPA mean excitation energies.

The experimental and theoretical determination of the mean excitation energy, I(0), and the stopping power, S(v), of a material is of great interest in particle and material physics and radiation therapy. For calculations of I(0), the complete set of electronic transitions in a given basis set is required, effectively limiting such calculations to systems with a small number of electrons, even at the random-phase approximation (RPA)/time-dependent Hartree-Fock (TDHF) or time-dependent density-functional theory level. To overcome such limitations, we present here the implementation of a Lanczos algorithm adapted for the paired RPA/TDHF eigenvalue problem in the Dalton program and show that it provides good approximation of the entire RPA eigenspectra in a reduced space. We observe rapid convergence of I(0) with the number of Lanczos vectors as the algorithm favors the transitions with large contributions. In most cases, the algorithm recovers RPA I(0) values of up to 0.5% accuracy at less than a quarter of the full space size. The algorithm not only exploits the RPA paired structure to save computational resources but also preserves certain sum-over-states properties, as first demonstrated by Johnson et al. [Comput. Phys. Commun. 120, 155 (1999)]. The block Lanczos RPA solver, as presented here, thus shows promise for computing mean excitation energies for systems larger than what was computationally feasible before.

When does the Lanczos algorithm compute exactly?

In theory, the Lanczos algorithm generates an orthogonal basis of the corresponding Krylov subspace. However, in finite precision arithmetic the orthogonality and linear independence of the computed Lanczos vectors is usually lost quickly. In this paper we study a class of matrices and starting vectors having a special nonzero structure that guarantees exact computations of the Lanczos algorithm whenever floating point arithmetic satisfying the IEEE 754 standard is used. Analogous results are formulated also for an implementation of the conjugate gradient method called cgLanczos. This implementation then computes approximations that agree with their exact counterparts to a relative accuracy given by the machine precision and the condition number of the system matrix. The results are extended to the Arnoldi algorithm, the nonsymmetric Lanczos algorithm, the Golub-Kahan bidiagonalization, the block-Lanczos algorithm, and their counterparts for solving linear systems.

Influence of structure and material on the vibration modal characteristics of novel combined flexible road wheel

Aiming at the independent development of tracked vehicles, it is urgent to improve its mobility, passability and ride comfort, a new type of flexible road wheel with a “wheel-hinge-hub” combined structure is proposed in this study. The vibration model characteristics of the flexible road wheel were studied by the combination of numerical simulation and experiments. The superelasticity of rubber is obtained through uniaxial tensile experiment of the material and a detail three-dimensional nolinear finite element model of the flexible road wheel is established through finite element software ABAQUS. The free vibration equation of the flexible road wheel is solved by Lanczos vector direct superposition method, and its predicted modes and natural frequencies are compared with experimental results, which verifies the accuracy and reliability of the established finite element model. On this basis, the effects of various key structural or material factors on the natural frequencies of the flexible road wheel are studied using orthogonal experimental design method. Besides, the vibration modal characteristics of the flexible road wheel are also compared with those of the rigid road wheel. The research results provide a theoretical basis for the vibration and noise reduction of flexible road wheel.

On the block Lanczos and block Golub–Kahan reduction methods applied to discrete ill‐posed problems

AbstractThe reduction of a large‐scale symmetric linear discrete ill‐posed problem with multiple right‐hand sides to a smaller problem with a symmetric block tridiagonal matrix can easily be carried out by the application of a small number of steps of the symmetric block Lanczos method. We show that the subdiagonal blocks of the reduced problem converge to zero fairly rapidly with increasing block number. This quick convergence indicates that there is little advantage in expressing the solutions of discrete ill‐posed problems in terms of eigenvectors of the coefficient matrix when compared with using a basis of block Lanczos vectors, which are simpler and cheaper to compute. Similarly, for nonsymmetric linear discrete ill‐posed problems with multiple right‐hand sides, we show that the solution subspace defined by a few steps of the block Golub–Kahan bidiagonalization method usually can be applied instead of the solution subspace determined by the singular value decomposition of the coefficient matrix without significant, if any, reduction of the quality of the computed solution.

On the shift‐invert Lanczos method for the buckling eigenvalue problem

AbstractWe consider the problem of extracting a few desired eigenpairs of the buckling eigenvalue problem , where K is symmetric positive semi‐definite, KG is symmetric indefinite, and the pencil is singular, namely, K and KG share a nontrivial common nullspace. Moreover, in practical buckling analysis of structures, bases for the nullspace of K and the common nullspace of K and KG are available. There are two open issues for developing an industrial strength shift‐invert Lanczos method: (1) the shift‐invert operator does not exist or is extremely ill‐conditioned, and (2) the use of the semi‐inner product induced by K drives the Lanczos vectors rapidly toward the nullspace of K, which leads to a rapid growth of the Lanczos vectors in norms and causes permanent loss of information and the failure of the method. In this paper, we address these two issues by proposing a generalized buckling spectral transformation of the singular pencil and a regularization of the inner product via a low‐rank updating of the semi‐positive definiteness of K. The efficacy of our approach is demonstrated by numerical examples, including one from industrial buckling analysis.

The joint bidiagonalization process with partial reorthogonalization

The joint bidiagonalization(JBD) process is a useful algorithm for the computation of the generalized singular value decomposition(GSVD) of a matrix pair. However, it always suffers from rounding errors, which causes the Lanczos vectors to loss their mutual orthogonality. In order to maintain some level of orthongonality, we present a semiorthogonalization strategy. Our rounding error analysis shows that the JBD process with the semiorthogonalization strategy can ensure that the convergence of the computed quantities is not affected by rounding errors and the final accuracy is high enough. Based on the semiorthogonalization strategy, we develop the joint bidiagonalization process with partial reorthogonalization(JBDPRO). In the JBDPRO algorithm, reorthogonalizations occur only when necessary, which saves a big amount of reorthogonalization work compared with the full reorthogonalization strategy. Numerical experiments illustrate our theory and algorithm.

New fractional Lanczos vector polynomials and their application to system of Abel–Volterra integral equations and fractional differential equations

In this paper, the recursive approach of the τ-method is developed to construct new fractional order canonical polynomials for solving systems of Abel–Volterra integral equations. Due to the singular behavior of solutions of these equations, the existing spectral approaches suffer from low accuracy. To overcome this drawback we use Müntz polynomials as basis functions which give remarkable approximation to functions with singular behavior at origin and state Tau approximation to the exact solution based on a new sequence of basis vector canonical polynomials that is generated by a simple recursive formula in terms of fractional order Müntz polynomials. The efficiency and simplicity of the proposed method are illustrated by some examples. Convergence analysis of the method is also discussed. The paper is closed by providing application of this method to a linear multi-term fractional differential equations.

Conjugate Gradient Algorithm in the Four-Dimensional Variational Data Assimilation System in GRAPES

Minimization algorithms are singular components in four-dimensional variational data assimilation (4DVar). In this paper, the convergence and application of the conjugate gradient algorithm (CGA), which is based on the Lanczos iterative algorithm and the Hessian matrix derived from tangent linear and adjoint models using a non-hydrostatic framework, are investigated in the 4DVar minimization. First, the influence of the Gram-Schmidt orthogonalization of the Lanczos vector on the convergence of the Lanczos algorithm is studied. The results show that the Lanczos algorithm without orthogonalization fails to converge after the ninth iteration in the 4DVar minimization, while the orthogonalized Lanczos algorithm converges stably. Second, the convergence and computational efficiency of the CGA and quasi-Newton method in batch cycling assimilation experiments are compared on the 4DVar platform of the Global/Regional Assimilation and Prediction System (GRAPES). The CGA is 40% more computationally efficient than the quasi-Newton method, although the equivalent analysis results can be obtained by using either the CGA or the quasi-Newton method. Thus, the CGA based on Lanczos iterations is better for solving the optimization problems in the GRAPES 4DVar system.

An application of the discrete-time Toda lattice to the progressive algorithm by Lanczos and related problems

The finite or the semi-infinite discrete-time Toda lattice has many applications to various areas in applied mathematics. The purpose of this paper is to review how the Toda lattice appears in the Lanczos algorithm through the quotient-difference algorithm and its progressive form (pqd). Then a multistep progressive algorithm (MPA) for solving linear systems is presented. The extended Lanczos parameters can be given not by computing inner products of the extended Lanczos vectors but by using the pqd algorithm with highly relative accuracy in a lower cost. The asymptotic behavior of the pqd algorithm brings us some applications of MPA related to eigenvectors.

Enhanced matrix function approximation

Matrix functions of the form $f(A)v$, where $A$ is a large symmetric matrix, $f$ is afunction, and $v\\ne 0$ is a vector, are commonly approximated by first applying a few,say $n$, steps of the symmetric Lanczos process to $A$ with the initial vector $v$ in order todetermine an orthogonal section of $A$. The latter is represented by a (small)$n\\times n$ tridiagonal matrix to which $f$ is applied. This approach uses the $n$ firstLanczos vectors provided by the Lanczos process. However, $n$ steps of the Lanczosprocess yield $n+1$ Lanczos vectors. This paper discusses how the $(n+1)$stLanczos vector can be used to improve the quality of the computed approximation of$f(A)v$. Also the approximation of expressions of the form $v^Tf(A)v$ is considered.

Extended Lanczos bidiagonalization algorithm for low rank approximation and its applications

We propose an extended Lanczos bidiagonalization algorithm for finding a low rank approximation of a given matrix. We show that this method can yield better low-rank approximations than standard Lanczos bidiagonalization algorithm, without increasing the cost too much. We also describe a partial reorthogonalization process that can be used to maintain an adequate level of orthogonality of the Lanczos vectors in order to produce accurate low-rank approximations. We demonstrate the effectiveness and applicability of our algorithm for a number of applications.

Development of numerical linear algebra algorithms in dynamic fixed‐point format: a case study of Lanczos tridiagonalization

SummaryProper range and precision analysis play an important role in the development of fixed‐point algorithms for embedded system applications. Numerical linear algebra algorithms used to find singular value decomposition of symmetric matrices are suitable for signal and image‐processing applications. These algorithms have not been attempted much in fixed‐point arithmetic. The reason is wide dynamic range of data and vulnerability of the algorithms to round‐off errors. For any real‐time application, the range of the input matrix may change frequently. This poses difficulty for constant and variable fixed‐point formats to decide on integer wordlengths during float‐to‐fixed conversion process because these formats involve determination of integer wordlengths before the compilation of the program. Thus, these formats may not guarantee to avoid overflow for all ranges of input matrices. To circumvent this problem, a novel dynamic fixed‐point format has been proposed to compute integer wordlengths adaptively during runtime. Lanczos algorithm with partial orthogonalization, which is a tridiagonalization step in computation of singular value decomposition of symmetric matrices, has been taken up as a case study. The fixed‐point Lanczos algorithm is tested for matrices with different dimensions and condition numbers along with image covariance matrix. The accuracy of fixed‐point Lanczos algorithm in three different formats has been compared on the basis of signal‐to‐quantization‐noise‐ratio, number of accurate fractional bits, orthogonality and factorization errors. Results show that dynamic fixed‐point format either outperforms or performs on par with constant and variable formats. Determination of fractional wordlengths requires minimization of hardware cost subject to accuracy constraint. In this context, we propose an analytical framework for deriving mean‐square‐error or quantization noise power among Lanczos vectors, which can serve as an accuracy constraint for wordlength optimization. Error is found to propagate through different arithmetic operations and finally accumulate in the last Lanczos vector. It is observed that variable and dynamic fixed‐point formats produce vectors with lesser round‐off error than constant format. All the three fixed‐point formats of Lanczos algorithm have been synthesized on Virtex 7 field‐programmable gate array using Vivado high‐level synthesis design tool. A comparative study of resource usage and power consumption is carried out. Copyright © 2015 John Wiley & Sons, Ltd.

On the Lanczos and Golub–Kahan reduction methods applied to discrete ill‐posed problems

SummaryThe symmetric Lanczos method is commonly applied to reduce large‐scale symmetric linear discrete ill‐posed problems to small ones with a symmetric tridiagonal matrix. We investigate how quickly the nonnegative subdiagonal entries of this matrix decay to zero. Their fast decay to zero suggests that there is little benefit in expressing the solution of the discrete ill‐posed problems in terms of the eigenvectors of the matrix compared with using a basis of Lanczos vectors, which are cheaper to compute. Similarly, we show that the solution subspace determined by the LSQR method when applied to the solution of linear discrete ill‐posed problems with a nonsymmetric matrix often can be used instead of the solution subspace determined by the singular value decomposition without significant, if any, reduction of the quality of the computed solution. Copyright © 2015 John Wiley & Sons, Ltd.

Estimates of ocean forecast error covariance derived from Hessian Singular Vectors

Experience in numerical weather prediction suggests that singular value decomposition (SVD) of a forecast can yield useful a priori information about the growth of forecast errors. It has been shown formally that SVD using the inverse of the expected analysis error covariance matrix to define the norm at initial time yields the Empirical Orthogonal Functions (EOFs) of the forecast error covariance matrix at the final time. Because of their connection to the 2nd derivative of the cost function in 4-dimensional variational (4D-Var) data assimilation, the initial time singular vectors defined in this way are often referred to as the Hessian Singular Vectors (HSVs). In the present study, estimates of ocean forecast errors and forecast error covariance were computed using SVD applied to a baroclinically unstable temperature front in a re-entrant channel using the Regional Ocean Modeling System (ROMS). An identical twin approach was used in which a truth run of the model was sampled to generate synthetic hydrographic observations that were then assimilated into the same model started from an incorrect initial condition using 4D-Var. The 4D-Var system was run sequentially, and forecasts were initialized from each ocean analysis. SVD was performed on the resulting forecasts to compute the HSVs and corresponding EOFs of the expected forecast error covariance matrix. In this study, a reduced rank approximation of the inverse expected analysis error covariance matrix was used to compute the HSVs and EOFs based on the Lanczos vectors computed during the 4D-Var minimization of the cost function. This has the advantage that the entire spectrum of HSVs and EOFs in the reduced space can be computed. The associated singular value spectrum is found to yield consistent and reliable estimates of forecast error variance in the space spanned by the EOFs. In addition, at long forecast lead times the resulting HSVs and companion EOFs are able to capture many features of the actual realized forecast error at the largest scales. Forecast error growth via the HSVs was found to be significantly influenced by the non-normal character of the underlying forecast circulation, and is accompanied by a forward energy cascade, suggesting that forecast errors could be effectively controlled by reducing the error at the largest scales in the forecast initial conditions. A predictive relation for the amplitude of the basin integrated forecast error in terms of the mean aspect ratio of the forecast error hyperellipse (quantified in terms of the mean eccentricity) was also identified which could prove useful for predicting the level of forecast error a priori. All of these findings were found to be insensitive to the configuration of the 4D-Var data assimilation system and the resolution of the observing network.

Accuracy of the $s$-Step Lanczos Method for the Symmetric Eigenproblem in Finite Precision

The $s$-step Lanczos method can achieve an $O(s)$ reduction in data movement over the classical Lanczos method for a fixed number of iterations, allowing the potential for significant speedups on modern computers. However, although the $s$-step Lanczos method is equivalent to the classical Lanczos method in exact arithmetic, it can behave quite differently in finite precision. Increased roundoff errors can manifest as a loss of accuracy or deterioration of convergence relative to the classical method, reducing the potential performance benefits of the $s$-step approach. In this paper, we present for the first time a complete rounding error analysis of the $s$-step Lanczos method. Our methodology is analogous to Paige's rounding error analysis for classical Lanczos [IMA J. Appl. Math., 18 (1976), pp. 341--349]. Our analysis gives upper bounds on the loss of normality of and orthogonality between the computed Lanczos vectors, as well as a recurrence for the loss of orthogonality. We further demonstrate that bounds on accuracy for the finite precision Lanczos method given by Paige [Linear Algebra Appl., 34 (1980), pp. 235--258] can be extended to the $s$-step Lanczos case assuming a bound on the maximum condition number of the precomputed $s$-step Krylov bases. Our results confirm that the conditioning of the precomputed Krylov bases plays a large role in determining finite precision behavior. In particular, if one can enforce that the condition numbers of the precomputed $s$-step Krylov bases are not too large in any iteration, then the finite precision behavior of the $s$-step Lanczos method will be similar to that of classical Lanczos.

A block MINRES algorithm based on the band Lanczos method

We develop a block minimum residual (MINRES) algorithm for symmetric indefinite matrices. This version is built upon the band Lanczos method that generates one basis vector of the block Krylov subspace per iteration rather than a whole block as in the block Lanczos process. However, we modify the method such that the most expensive operations are still performed in a block fashion. The benefit of using the band Lanczos method is that one can detect breakdowns from scalar values arising in the computation, allowing for a handling of breakdown which is straightforward to implement. We derive a progressive formulation of the MINRES method based on the band Lanczos process and give some implementation details. Specifically, a simple reordering of the steps allows us to perform many of the operations at the block level in order to take advantage of communication efficiencies offered by the block Lanczos process. This is an important concern in the context of next-generation super computing applications. We also present a technique allowing us to maintain the block size by replacing dependent Lanczos vectors with pregenerated random vectors whose orthogonality against all Lanczos vectors is maintained. Numerical results illustrate the performance on some sample problems. We present experiments that show how the relationship between right-hand sides can effect the performance of the method.

Inexact and incremental bilinear Lanczos components algorithms for high dimensionality reduction and image reconstruction

Recently, matrix-based methods have gained wide attentions in pattern recognition and machine learning communities. The generalized low rank approximations of matrices (GLRAM) and the bilinear Lanczos components (BLC) algorithm are two popular algorithms that treat data as the native two-dimensional matrix patterns. However, these two algorithms often require heavy computation time and memory space in practice, especially for large scale problems. In this paper, we propose inexact and incremental bilinear Lanczos components algorithms for high dimensionality reduction and image reconstruction. We first introduce the thick-restarting strategy to the BLC algorithm, and present a thick-restarted Lanczos components (TRBLC) algorithm. In this algorithm, we use the Ritz vectors as approximations to dominant eigenvectors instead of the Lanczos vectors. In our implementation, the iterative matrices are neither formed nor stored explicitly, thanks to the characteristic of the Lanczos procedure. Then, we explore the relationship between the reconstruction error and the accuracy of the Ritz vectors, so that the computational complexities of eigenpairs can be reduced significantly. As a result, we propose an inexact thick-restarted Lanczos components (Inex-TRBLC) algorithm. Moreover, we investigate the problem of incremental generalized low rank approximations of matrices, and propose an incremental and inexact TRBLC (Incr-TRBLC) algorithm. Numerical experiments illustrate the superiority of the new algorithms over the GLRAM algorithm and its variations, as well as the BLC algorithm for some real-world image reconstruction and face recognition problems.

Prewhitening High-Dimensional fMRI Data Sets Without Eigendecomposition

This letter proposes an algorithm for linear whitening that minimizes the mean squared error between the original and whitened data without using the truncated eigendecomposition (ED) of the covariance matrix of the original data. This algorithm uses Lanczos vectors to accurately approximate the major eigenvectors and eigenvalues of the covariance matrix of the original data. The major advantage of the proposed whitening approach is its low computational cost when compared with that of the truncated ED. This gain comes without sacrificing accuracy, as illustrated with an experiment of whitening a high-dimensional fMRI data set.

An efficient method to set up a Lanczos based preconditioner for discrete ill-posed problems

Rezghi and Hosseini [M. Rezghi, S.M. Hosseini, Lanczos based preconditioner for discrete ill-posed problems, Computing 88 (2010) 79–96] presented a Lanczos based preconditioner for discrete ill-posed problems. Their preconditioner is constructed by using few steps (e.g., k) of the Lanczos bidiagonalization and corresponding computed singular values and right Lanczos vectors. In this article, we propose an efficient method to set up such preconditioner. Some numerical examples are given to show the effectiveness of the method.