Partial Reorthogonalization Research Articles

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.

Residual Water Suppression Using the Squared Eigenfunctions of the Schrödinger Operator

Water suppression, in proton magnetic resonance spectroscopy (MRS) using post-processing techniques, is very challenging due to the large amplitude of the water line, which shadows the metabolic peaks with small amplitudes and complicates their quantification. In addition, the peak-shaped structure of these spectra and the relatively small number of data points representing them makes the suppression process more cumbersome. In this paper, a post-processing water suppression technique based on the Schrodinger operator is proposed. The method is based on the decomposition of the input MRS spectrum, using the squared eigenfunctions of a semi-classical Schrodinger operator. The proposed approach proceeds in three steps: first, the water peak is estimated using an optimal choice of the value of h to reconstruct the MRS spectrum with a minimum number of eigenfunctions. Second, these estimated eigenfunctions are further refined to ensure that they only represent the water line with no contribution from the metabolite peaks. Finally, the estimated water peak is subtracted from the input MRS spectrum. The proposed method is tested on simulated in vitro and real in vivo MRS data and compared with the Hankel-Lanczos singular value decomposition with partial reorthogonalization (HLSVD-PRO) method. The results obtained show that the semi-classical signal analysis (SCSA) performs comparably to the HLSVD-PRO in accurately suppressing the water peak.

Quantification method for MRSI based on Hankel matrix

Objective To study the quantification method based on Hankel matrix, the water suppression method and the metabolite imaging method for magnetic resonance spectroscopic imaging (MRSI) data. Methods Impact of Hankel data matrix on quantification MRSI methods were analyzed to obtain the most efficient Hankel matrix structure. The maximum amplitude method of the water signal peak was proposed for MRSI data water suppression. The interested metabolites information was extracted from MRSI data, and then metabolite image was obtained through bilinear interpolation algorithm. Results The minimum amplitude error and the minimum frequency error were acquired when columns number was 3/4 sampling points. The amplitude, frequency and the damping factor of the simulation data accuracy was 96.94%, 99.72% and 95.55% respectively. Hankel lanczos with partial reorthogonalization singular value decomposition (HLSVDPRO) method with 3/4 sampling points was used to form Hankel matrices. The speed of quantification decreased with the increase of sampling points. The error of quantification parameter reached minimum when the number of sampling points was 512. The water suppression degree of simulation data was 99.55% with the maximum amplitude water suppression method. Conclusions The accuracy and the speed of the quantification are promoted with an optimized Hankel matrix structure for the MRSI quantization method. The optimal length of sampling points is 512. The maximum amplitude method can suppress water perfectly. Doctors can detect the presence of tumor regions in human body at the (super) early stage by metabolite information images. Key words: Magnetic resonance spectrum imaging; Hankel matrix; Singular value decomposition; Water suppressed; Metabolite image

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.

A Filtered Lanczos Procedure for Extreme and Interior Eigenvalue Problems

When combined with Krylov projection methods, polynomial filtering can provide a powerful method for extracting extreme or interior eigenvalues of large sparse matrices. This general approach can be quite efficient in the situation when a large number of eigenvalues is sought. However, its competitiveness depends critically on a good implementation. This paper presents a technique based on such a combination to compute a group of extreme or interior eigenvalues of a real symmetric (or complex Hermitian) matrix. The technique harnesses the effectiveness of the Lanczos algorithm with partial reorthogonalization and the power of polynomial filtering. Numerical experiments indicate that the method can be far superior to competing algorithms when a large number of eigenvalues and eigenvectors is to be computed.

Computation of Large Invariant Subspaces Using Polynomial Filtered Lanczos Iterations with Applications in Density Functional Theory

The most expensive part of all electronic structure calculations based on density functional theory lies in the computation of an invariant subspace associated with some of the smallest eigenvalues of a discretized Hamiltonian operator. The dimension of this subspace typically depends on the total number of valence electrons in the system, and can easily reach hundreds or even thousands when large systems with many atoms are considered. At the same time, the discretization of Hamiltonians associated with large systems yields very large matrices, whether with planewave or real-space discretizations. The combination of these two factors results in one of the most significant bottlenecks in computational materials science. In this paper we show how to efficiently compute a large invariant subspace associated with the smallest eigenvalues of a symmetric/Hermitian matrix using polynomially filtered Lanczos iterations. The proposed method does not try to extract individual eigenvalues and eigenvectors. Instead, it constructs an orthogonal basis of the invariant subspace by combining two main ingredients. The first is a filtering technique to dampen the undesirable contribution of the largest eigenvalues at each matrix-vector product in the Lanczos algorithm. This technique employs a well-selected low pass filter polynomial, obtained via a conjugate residual-type algorithm in polynomial space. The second ingredient is the Lanczos algorithm with partial reorthogonalization. Experiments are reported to illustrate the efficiency of the proposed scheme compared to state-of-the-art implicitly restarted techniques.

Estimating the model resolution matrix for large seismic tomography problems based on Lanczos bidiagonalization with partial reorthogonalization

SUMMARY The PROPACK package developed by Larsen is able to efficiently and accurately estimate singular values and vectors for large matrices based on the Lanczos bidiagonalization with partial reorthogonalization. We incorporate the Pavtial Reorthogonalization Package (PROPACK) package into the double-difference seismic tomography code tomoDD and estimate the model resolution matrix for large seismic tomography problems. Compared to previous Least Squares QR (LSQR)-based methods for estimating the model resolution matrix the PROPACK-based method calculates the full resolution matrix and thus gives a complete description of how well the model is resolved. Several observations are drawn from the application to data from the 2001 eruption of Mt Etna: for this example, it is reasonable to use ray-sampling density information to characterize the model resolution qualitatively; the model resolution resulting from just velocity inversion bears a close linear relationship to that from simultaneous inversion but always overestimates resolution; and the inversion system using differential times has a greater ability to resolve the source region structure than the system using absolute times.

Frequency-selective quantitation of short-echo time 1H magnetic resonance spectra

Accurate and efficient filtering techniques are required to suppress large nuisance components present in short-echo time magnetic resonance (MR) spectra. This paper discusses two powerful filtering techniques used in long-echo time MR spectral quantitation, the maximum-phase FIR filter (MP-FIR) and the Hankel-Lanczos Singular Value Decomposition with Partial ReOrthogonalization (HLSVD-PRO), and shows that they can be applied to their more complex short-echo time spectral counterparts. Both filters are validated and compared through extensive simulations. Their properties are discussed. In particular, the capability of MP-FIR for dealing with macromolecular components is emphasized. Although this property does not make a large difference for long-echo time MR spectra, it can be important when quantifying short-echo time spectra.

Four-dimensional quantum study on exothermic complex-forming reactions: Cl−+CH3Br→ClCH3+Br−

The exothermic gas-phase bimolecular nucleophilic substitution (S(N)2) reaction Cl(-)+CH(3)Br (upsilon1',upsilon2',upsilon3')-->ClCH(3) (upsilon1,upsilon2,upsilon3)+Br- and the corresponding endothermic reverse reaction have been studied by time-independent quantum scattering calculations in hyperspherical coordinates on a coupled-cluster potential-energy surface. The dimensionality-reduced model takes four degrees of freedom into account [Cl-C and C-Br stretching modes (quantum numbers upsilon3' and upsilon3); totally symmetric modes of the methyl group, i.e., C-H stretching (upsilon1' and upsilon1) and umbrella bending vibrations (upsilon2' and upsilon2)]. Diagonalization of the Hamiltonian was performed employing the Lanczos algorithm with a variation of partial reorthogonalization. A narrow grid in the total energy was employed so that long-living resonance states could be resolved and extracted. While excitation of the reactant umbrella bending mode already leads to a considerable enhancement of the reaction probability, its combination with vibrational excitation of the broken C-Br bond, (0, 1, 1), results in a strong synergic effect that can be rationalized by the similarity with the classical transitional normal mode. Exciting the C-H stretch has a non-negligible effect on the reaction probability, while for larger translational energies this mode follows the expected spectatorlike behavior. Combination of C-Br stretch and symmetric C-H, (1,0,1), stretch does not show a cooperative effect. Contrary to the spectator mode concept, energy originally stored in the C-H stretching mode is by no means conserved, but almost completely released in other modes of the reaction products. Products are most likely formed in states with a high degree of excitation in the new C-Cl bond, while the internal modes of the methyl group are less important. Reactants with combined umbrella/C-Br stretch excitation, (0, 1, 1), may yield products with two quanta in the umbrella mode.

Computing charge densities with partially reorthogonalized Lanczos

This paper considers the problem of computing charge densities in a density functional theory (DFT) framework. In contrast to traditional, diagonalization-based, methods, we utilize a technique which exploits a Lanczos basis, without explicit reference to individual eigenvectors. The key ingredient of this new approach is a partial reorthogonalization strategy whose goal is to ensure a good level of orthogonality of the basis vectors. The experiments reveal that the method can be a few times faster than ARPACK, the implicit restart Lanczos method. This is achievable by exploiting more memory and BLAS3 (dense) computations while avoiding the frequent updates of eigenvectors inherent to all restarted Lanczos methods.

Thick-Restart Lanczos Method for Large Symmetric Eigenvalue Problems

In this paper, we propose a restarted variant of the Lanczos method for symmetric eigenvalue problems named the thick-restart Lanczos method. This new variant is able to retain an arbitrary number of Ritz vectors from the previous iterations with a minimal restarting cost. Since it restarts with Ritz vectors, it is simpler than similar methods, such as the implicitly restarted Lanczos method. We carefully examine the effects of the floating-point round-off errors on stability of the new algorithm and present an implementation of the partial reorthogonalization scheme that guarantees accurate Ritz values with a minimal amount of reorthogonalization. We also show a number of heuristics on deciding which Ritz pairs to save during restart in order to maximize the overall performance of the thick-restart Lanczos method.

A parallel Lanczos method for symmetric generalized eigenvalue problems

The Lanczos algorithm is a very effective method for finding extreme eigenvalues of symmetric matrices. In this paper, we present our parallel version of the Lanczos method for symmetric generalized eigenvalue problem, PLANSO. PLANSO is based on a sequential package called LANSO which implements the Lanczos algorithm with partial reorthogonalization. It is portable to all parallel machines that support MPI and it is easy to interface with most parallel computing packages. Through numerical experiments, we demonstrate that it achieves similar parallel efficiency as PARPACK, but uses considerably less time.

Lanczos Bidiagonalization With Partial Reorthogonalization

A partial reorthogonalization procedure (BPRO) for maintaining semi-orthogonality among the left and right Lanczos vectors in the Lanczos bidiagonalization (LBD) is presented. The resulting algorithm is mathematically equivalent to the symmetric Lanczos algorithm with partial reorthogonalization (PRO) developed by Simon but works directly on the Lanczos bidiagonalization of A. For computing the singular values and vectors of a large sparse matrix with high accuracy, the BPRO algorithm uses only half the amount of storage and a factor of 3-4 less work compared to methods based on PRO applied to an equivalent symmetric system. Like PRO the algorithm presented here is based on simple recurrences which enable it to monitor the loss of orthogonality among the Lanczos vectors directly without forming inner products. These recurrences are used to develop a Lanczos bidiagonalization algorithm with partial reorthogonalization which has been implemented in a MATLAB package for sparse SVD and eigenvalue problems called PROPACK. Numerical experiments with the routines from PROPACK are conducted using a test problem from inverse helioseismology to illustrate the properties of the method. In addition a number of test matrices from the Harwell-Boeing collection are used to compare the accuracy and efficiency of the MATLAB implementations of BPRO and PRO with the svds routine in MATLAB 5.1, which uses an implicitly restarted Lanczos algorithm.

A spectral transform Krylov subspace iteration approach to quantum scattering

A spectral transform Lanczos iteration method is suggested to find the eigenvalues and eigenvectors of the coupling matrix appearing in quantum scattering calculations. Three main attributes which improve previous implementations are (a) partial reorthogonalization, (b) exponential filter, and (c) discrete variable representation basis. The method is applied to the 3D rotating line umbrella calculations of the reaction Cl+CH4⇌ HCl+CH3 which shows that it is substantially faster than the direct diagonalization method. Calculated state-to-state reaction probabilities as well as adiabats are in good agreement with previous results.

Bi-Lanczos with partial orthogonalization

In theory the bi-Lanczos algorithm is an attractive possibility for solving the unsymmetric eigenproblem. In practice, however, it turns out to be unstable due to loss of orthogonality of the iteration vectors. In this paper we discuss the possibilities of reorthogonalizing the bi-Lanczos iteration vectors. To save part of the advantage of the bi-Lanczos method over the other most commonly used Krylov subspace based method of Arnoldi (bi-Lanczos lacks the growth of the number of vector operations per iteration), we propose partial reorthogonalization. In that case reorthogonalization takes place if and only if too much orthogonality is lost, so that the results are accurate enough, whereas the algorithm is still less time consuming than Arnoldi. The theory presented here is an extension of the theory available for partial reorthogonalization of the symmetric Lanczos algorithm.

Analysis of the symmetric Lanczos algorithm with reorthogonalization methods

We present an error analysis of the symmetric Lanczos algorithm in finite precision arithmetic. The loss of orthogonality among the computed Lanczos vectors is explained with the help of a recurrence formula. A backward error analysis shows that semiorthogonality among the Lanczos vectors is enough to guarantee the accuracy of the computed quantities up to machine precision. The results of this analysis are then extended to the more general case of the Lanczos algorithm with a semiorthogonalization strategy. Based on the recurrence formula, a new reorthogonalization method called partial reorthogonalization is introduced. We show that both partial reorthogonalization and selective orthogonalization as introduced by Parlett and Scott [15] are semiorthogonalization strategies. Finally we discuss the application of our results to the solution of linear systems of equations and to the eigenvalue problem.

The Lanczos algorithm with partial reorthogonalization

The Lanczos algorithm is becoming accepted as a powerful tool for finding the eigenvalues and for solving linear systems of equations. Any practical implementation of the algorithm suffers however from roundoff errors, which usually cause the Lanczos vectors to lose their mutual orthogonality. In order to maintain some level of orthogonality, full reorthogonalization (FRO) and selective orthogonalization (SO) have been used in the past as a remedy. Here partial reorthogonalization (PRO) is proposed as a new method for maintaining semiorthogonality among the Lanczos vectors. PRO is based on a simple recurrence, which allows us to monitor the loss of orthogonality among the Lanczos vectors directly without computing the inner products. Based on the information from the recurrence, reorthogonalizations occur only when necessary. Thus substantial savings are made as compared to FRO. In some numerical examples we apply the Lanczos algorithm with PRO to the solution of large symmetric systems of linear equations and show that it is a robust and efficient algorithm for maintaining semiorthogonality among the Lanczos vectors. The results obtained compare favorably with the conjugate gradient method.

The Lanczos Algorithm With Partial Reorthogonalization

The Lanczos algorithm is becoming accepted as a powerful tool for finding the eigenvalues and for solving linear systems of equations. Any practical implementation of the algorithm suffers however from roundoff errors, which usually cause the Lanczos vectors to lose their mutual orthogonality. In order to maintain some level of orthogonality, full reorthogonalization (FRO) and selective orthogonalization (SO) have been used in the past as a remedy. Here partial reorthogonalization (PRO) is proposed as a new method for maintaining semiorthogonality among the Lanczos vectors. PRO is based on a simple recurrence, which allows us to monitor the loss of orthogonality among the Lanczos vectors directly without computing the inner products. Based on the information from the recurrence, reorthogonalizations occur only when necessary. Thus substantial savings are made as compared to FRO. In some numerical examples we apply the Lanczos algorithm with PRO to the solution of large symmetric systems of linear equations and show that it is a robust and efficient algorithm for maintaining semiorthogonality among the Lanczos vectors. The results obtained compare favorably with the conjugate gradient method.