Davidson-type Methods Research Articles

Subspace Methods for the Simulation of Molecular Response Properties on a Quantum Computer.

We explore Davidson methods for obtaining excitation energies and other linear response properties within the recently developed quantum self-consistent linear response (q-sc-LR) method. Davidson-type methods allow for obtaining only a few selected excitation energies without explicitly constructing the electronic Hessian since they only require the ability to perform Hessian-vector multiplications. We apply the Davidson method to calculate the excitation energies of hydrogen chains (up to H10) and analyze aspects of statistical noise for computing excitation energies on quantum simulators. Additionally, we apply Davidson methods for computing linear response properties such as static polarizabilities for H2, LiH, H2O, OH-, and NH3, and show that unitary coupled cluster outperforms classical projected coupled cluster for molecular systems with strong correlation. Finally, we formulate the Davidson method for damped (complex) linear response, with application to the nitrogen K-edge X-ray absorption of ammonia, and the C6 coefficients of H2, LiH, H2O, OH-, and NH3.

A Cross-Product Free Jacobi–Davidson Type Method for Computing a Partial Generalized Singular Value Decomposition of a Large Matrix Pair

The FEAST eigensolver is extended to the computation of the singular triplets of a large matrix $A$ with the singular values in a given interval. The resulting FEAST SVDsolver is subspace iteration applied to an approximate spectral projector of $A^TA$ corresponding to the desired singular values in a given interval, and constructs approximate left and right singular subspaces corresponding to the desired singular values, onto which $A$ is projected to obtain Ritz approximations. Differently from a commonly used contour integral-based FEAST solver, we propose a robust alternative that constructs approximate spectral projectors by using the Chebyshev--Jackson polynomial series, which are symmetric positive semi-definite with the eigenvalues in $[0,1]$. We prove the pointwise convergence of this series and give compact estimates for pointwise errors of it and the step function that corresponds to the exact spectral projector. We present error bounds for the approximate spectral projector and reliable estimates for the number of desired singular triplets, establish numerous convergence results on the resulting FEAST SVDsolver, and propose practical selection strategies for determining the series degree and for reliably determining the subspace dimension. The solver and results on it are directly applicable or adaptable to the real symmetric and complex Hermitian eigenvalue problem. Numerical experiments illustrate that our FEAST SVDsolver is at least competitive with and is much more efficient than the contour integral-based FEAST SVDsolver when the desired singular values are extreme and interior ones, respectively, and it is also more robust than the latter.

On Inner Iterations of Jacobi--Davidson Type Methods for Large SVD Computations

We make a convergence analysis of the harmonic and refined harmonic extraction versions of Jacobi--Davidson SVD (JDSVD) type methods for computing one or more interior singular triplets of a large matrix $A$. At each outer iteration of these methods, a correction equation, i.e., inner linear system, is solved approximately by using iterative methods, which leads to two inexact JDSVD type methods, as opposed to the exact methods where correction equations are solved exactly. Accuracy of inner iterations critically affects the convergence and overall efficiency of the inexact JDSVD methods. A central problem is how accurately the correction equations should be solved to ensure that both of the inexact JDSVD methods can mimic their exact counterparts well, that is, they use almost the same outer iterations to achieve the convergence. In this paper, similar to the available results on the JD type methods for large matrix eigenvalue problems, we prove that each inexact JDSVD method behaves like its exact counterpart if all the correction equations are solved with low or modest accuracy during the outer iterations. Based on the theory, we propose practical stopping criteria for inner iterations. Numerical experiments confirm our theory and the effectiveness of the inexact algorithms.

A Jacobi–Davidson type method for computing real eigenvalues of the quadratic eigenvalue problem

This paper presents a new Jacobi---Davidson type method to compute several real eigenvalues of the Hermitian quadratic eigenvalue problem. This method uses a simple index to sort the eigenvalues of the projected quadratic eigenvalue problem and extracts the approximate eigenvectors for the quadratic eigenvalue problem with the eigenvectors of the projected quadratic eigenvalue problem corresponding to the eigenvalues with the smallest indices. Numerical examples show that our method is effective and efficient to compute real eigenvalues of the Hermitian quadratic eigenvalue problem.

Jacobi–Davidson methods for polynomial two-parameter eigenvalue problems

We propose Jacobi–Davidson type methods for polynomial two-parameter eigenvalue problems (PMEP). Such problems can be linearized as singular two-parameter eigenvalue problems, whose matrices are of dimension k(k+1)n/2, where k is the degree of the polynomial and n is the size of the matrix coefficients in the PMEP. When k2n is relatively small, the problem can be solved numerically by computing the common regular part of the related pair of singular pencils. For large k2n, computing all solutions is not feasible and iterative methods are required.When k is large, we propose to linearize the problem first and then apply Jacobi–Davidson to the obtained singular two-parameter eigenvalue problem. The resulting method may for instance be used for computing zeros of a system of scalar bivariate polynomials close to a given target. On the other hand, when k is small, we can apply a Jacobi–Davidson type approach directly to the original matrices. The original matrices are projected onto a low-dimensional subspace, and the projected polynomial two-parameter eigenvalue problems are solved by a linearization.

A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc

In the context of large-scale eigenvalue problems, methods of Davidson type such as Jacobi-Davidson can be competitive with respect to other types of algorithms, especially in some particularly difficult situations such as computing interior eigenvalues or when matrix factorization is prohibitive or highly inefficient. However, these types of methods are not generally available in the form of high-quality parallel implementations, especially for the case of non-Hermitian eigenproblems. We present our implementation of various Davidson-type methods in SLEPc, the Scalable Library for Eigenvalue Problem Computations. The solvers incorporate many algorithmic variants for subspace expansion and extraction, and cover a wide range of eigenproblems including standard and generalized, Hermitian and non-Hermitian, with either real or complex arithmetic. We provide performance results on a large battery of test problems.

Polynomial optimization and a Jacobi–Davidson type method for commuting matrices

In this paper we introduce a new Jacobi–Davidson type eigenvalue solver for a set of commuting matrices, called JDCOMM, used for the global optimization of so-called Minkowski-norm dominated polynomials in several variables. The Stetter–Möller matrix method yields such a set of real non-symmetric commuting matrices since it reformulates the optimization problem as an eigenvalue problem. A drawback of this approach is that the matrix most relevant for computing the global optimum of the polynomial under investigation is usually large and only moderately sparse. However, the other matrices are generally much sparser and have the same eigenvectors because of the commutativity. This fact is used to design the JDCOMM method for this problem: the most relevant matrix is used only in the outer loop and the sparser matrices are exploited in the solution of the correction equation in the inner loop to greatly improve the efficiency of the method. Some numerical examples demonstrate that the method proposed in this paper is more efficient than approaches that work on the main matrix (standard Jacobi–Davidson and implicitly restarted Arnoldi), as well as conventional solvers for computing the global optimum, i.e., SOSTOOLS, GloptiPoly, and PHCpack.

Robust Successive Computation of Eigenpairs for Nonlinear Eigenvalue Problems

Newton-based methods are well-established techniques for solving nonlinear eigenvalue problems. If a larger portion of the spectrum is sought, however, their tendency to reconverge to previously determined eigenpairs is a hindrance. To overcome this limitation, we propose and analyze a deflation strategy for nonlinear eigenvalue problems, based on the concept of minimal invariant pairs. We develop this strategy into a Jacobi--Davidson-type method and discuss its various algorithmic details. Finally, the efficiency of our approach is demonstrated by a sequence of numerical examples.

Laguerre collocation solutions to boundary layer type problems

In this paper a Laguerre collocation type method based on usual Laguerre functions is designed in order to solve high order nonlinear boundary value problems as well as eigenvalue problems, on semi-infinite domain. The method is first applied to Falkner---Skan boundary value problem. The solution along with its first two derivatives are computed inside the boundary layer on a fine grid which cluster towards the fixed boundary. Then the method is used to solve a generalized eigenvalue problem which arise in the study of the stability of the Ekman boundary layer. The method provides reliable numerical approximations, is robust and easy implementable. It introduces the boundary condition at infinity without any truncation of the domain. A particular attention is payed to the treatment of boundary conditions at origin. The dependence of the set of solutions to Falkner---Skan problem on the parameter embedded in the system is reproduced correctly. For Ekman eigenvalue problem, the critical Reynolds number which assure the linear stability is computed and compared with existing results. The leftmost part of the spectrum is validated using QZ as well as some Jacobi---Davidson type methods.

A Jacobi–Davidson type method with a correction equation tailored for integral operators

We propose two iterative numerical methods for eigenvalue computations of large dimensional problems arising from finite approximations of integral operators, and describe their parallel implementation. A matrix representation of the problem on a space of moderate dimension, defined from an infinite dimensional one, is computed along with its eigenpairs. These are taken as initial approximations and iteratively refined, by means of a correction equation based on the reduced resolvent operator and performed on the moderate size space, to enhance their quality. Each refinement step requires the prolongation of the correction equation solution back to a higher dimensional space, defined from the infinite dimensional one. This approach is particularly adapted for the computation of eigenpair approximations of integral operators, where prolongation and restriction matrices can be easily built making a bridge between coarser and finer discretizations. We propose two methods that apply a Jacobi---Davidson like correction: Multipower Defect-Correction (MPDC), which uses a single-vector scheme, if the eigenvalues to refine are simple, and Rayleigh---Ritz Defect-Correction (RRDC), which is based on a projection onto an expanding subspace. Their main advantage lies in the fact that the correction equation is performed on a smaller space while for general solvers it is done on the higher dimensional one. We discuss implementation and parallelization details, using the PETSc and SLEPc packages. Also, numerical results on an astrophysics application, whose mathematical model involves a weakly singular integral operator, are presented.

New eigensolvers for large-scale nanoscience simulations

We present results for applications to nanosystems of state-of-the-art iterative eigensolvers based on conjugate gradients and variants of Davidson in the context of semi-empirical plane wave electronic structure calculations. We are concerned with the computation of electronic and optical properties of nanosystems using the Energy SCAN method to compute interior eigenstates around the band gap that determine their properties. Numerically, this interior Hermitian eigenvalue problem poses several challenges, with respect to both accuracy and efficiency. All the iterative eigensolvers are seeking the minimal eigenvalues of the folded operator with reference shift in the band-gap. The tested methods include standard conjugate-gradient (CG)-based Rayleigh quotient minimization, Locally optimal block-preconditioned CG (LOBPCG) and two variants of the (Jacobi-)Davidson method: JDQMR and GD+1. Our experimental results indicated that Davidson-type methods are often the fastest.

Nearly Optimal Preconditioned Methods for Hermitian Eigenproblems Under Limited Memory. Part II: Seeking Many Eigenvalues

In a recent companion paper, we proposed two methods, GD+k and JDQMR, as nearly optimal methods for finding one eigenpair of a real symmetric matrix. In this paper, we seek nearly optimal methods for a large number, $nev$, of eigenpairs that work with a search space whose size is $O(1)$, independent from $nev$. The motivation is twofold: avoid the additional $O(nev N)$ storage and the $O(nev^2N)$ iteration costs. First, we provide an analysis of the oblique projectors required in the Jacobi–Davidson method and identify ways to avoid them during the inner iterations, either completely or partially. Second, we develop a comprehensive set of performance models for GD+k, Jacobi–Davidson type methods, and ARPACK. Based both on theoretical arguments and on our models we argue that any eigenmethod with $O(1)$ basis size, preconditioned or not, will be superseded asymptotically by Lanczos-type methods that use $O(nev)$ vectors in the basis. However, this may not happen until $nev > O(1000)$. Third, we perform an extensive set of experiments with our methods and against other state-of-the-art software that validates our models and confirms our GD+k and JDQMR methods as nearly optimal within the class of O(1) basis size methods.

A Chebyshev–Davidson Algorithm for Large Symmetric Eigenproblems

A polynomial filtered Davidson-type algorithm is proposed for symmetric eigenproblems, in which the correction-equation of the Davidson approach is replaced by a polynomial filtering step. The new approach has better global convergence and robustness properties when compared with standard Davidson-type methods. The typical filter used in this paper is based on Chebyshev polynomials. The goal of the polynomial filter is to amplify components of the desired eigenvectors in the subspace, which has the effect of reducing both the number of steps required for convergence and the cost in orthogonalizations and restarts. Numerical results are presented to show the effectiveness of the proposed approach.

A Jacobi–Davidson type method for the product eigenvalue problem

We propose a Jacobi–Davidson type method to compute selected eigenpairs of the product eigenvalue problem A m ⋯ A 1 x = λ x , where the matrices may be large and sparse. To avoid difficulties caused by a high condition number of the product matrix, we split up the action of the product matrix and work with several search spaces. We generalize the Jacobi–Davidson correction equation and the harmonic and refined extraction for the product eigenvalue problem. Numerical experiments indicate that the method can be used to compute eigenvalues of product matrices with extremely high condition numbers.

Harmonic and Refined Extraction Methods for the Singular Value Problem, with Applications in Least Squares Problems

For the accurate approximation of the minimal singular triple (singular value and left and right singular vector) of a large sparse matrix, we may use two separate search spaces, one for the left, and one for the right singular vector. In Lanczos bidiagonalization, for example, such search spaces are constructed. In SIAM J. Sci. Comput., 23(2) (2002), pp. 606–628, the author proposes a Jacobi–Davidson type method for the singular value problem, where solutions to certain correction equations are used to expand the search spaces. As noted in the mentioned paper, the standard Galerkin subspace extraction works well for the computation of large singular triples, but may lead to unsatisfactory approximations to small and interior triples. To overcome this problem for the smallest triples, we propose three harmonic and a refined approach. All methods are derived in a number of different ways. Some of these methods can also be applied when we are interested in the largest or interior singular triples. Theoretical results as well as numerical experiments indicate that the results of the alternative extraction processes are often better than the standard approach. We show that when Lanczos bidiagonalization is used for the subspace expansion, the standard, harmonic, and refined extraction methods are all essentially equivalent. This gives more insight in the success of Lanczos bidiagonalization to find the smallest singular triples. Finally, we show that the extraction processes for the smallest singular values may give an approximation to a least squares problem at low additional costs. The truncated SVD is also discussed in this context.

Jacobi–Davidson methods for cubic eigenvalue problems

AbstractSeveral Jacobi–Davidson type methods are proposed for computing interior eigenpairs of large‐scale cubic eigenvalue problems. To successively compute the eigenpairs, a novel explicit non‐equivalence deflation method with low‐rank updates is developed and analysed. Various techniques such as locking, search direction transformation, restarting, and preconditioning are incorporated into the methods to improve stability and efficiency. A semiconductor quantum dot model is given as an example to illustrate the cubic nature of the eigenvalue system resulting from the finite difference approximation. Numerical results of this model are given to demonstrate the convergence and effectiveness of the methods. Comparison results are also provided to indicate advantages and disadvantages among the various methods. Copyright © 2004 John Wiley & Sons, Ltd.

A Jacobi--Davidson Type Method for the Two-Parameter Eigenvalue Problem

We present a new numerical method for computing selected eigenvalues and eigenvectors of the two-parameter eigenvalue problem. The method does not require good initial approximations and is able to tackle large problems that are too expensive for methods that compute all eigenvalues. The new method uses a two-sided approach and is a generalization of the Jacobi--Davidson type method for right definite two-parameter eigenvalue problems [M. E. Hochstenbach and B. Plestenjak, SIAM J. Matrix Anal. Appl., 24 (2002), pp. 392--410]. Here we consider the much wider class of nonsingular problems. In each step we first compute Petrov triples of a small projected two-parameter eigenvalue problem and then expand the left and right search spaces using approximate solutions to appropriate correction equations. Using a selection technique, it is possible to compute more than one eigenpair. Some numerical examples are presented.

Parallel computation of spectral portrait of large matrices by Davidson type methods

The eigenanalysis of matrices or operators based only on the knowledge of the spectrum may be misleading in the non-normal case. Instead of the spectrum, one may fully characterize the spectral behavior of a non-normal matrix by analyzing its spectral portrait, i.e., the set of its resolvent norm. In this paper, we propose a parallel version of the generalized Davidson method for analyzing and plotting the spectral portrait of large non-normal matrices. We report the performance results obtained on the machine Paragon.