A comparative study of iterative solutions to linear systems arising in quantum mechanics

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Abstract

This study is mainly focused on iterative solutions with simple diagonal preconditioning to two complex-valued nonsymmetric systems of linear equations arising from a computational chemistry model problem proposed by Sherry Li of NERSC. Numerical experiments show the feasibility of iterative methods to some extent when applied to the problems and reveal the competitiveness of our recently proposed Lanczos biconjugate A-orthonormalization methods to other classic and popular iterative methods. By the way, experiment results also indicate that application specific preconditioners may be mandatory and required for accelerating convergence.

Introduction

We consider the numerical solution of the coupled equations for the two-dimensional radial functions on a two-dimensional radial grid for solving a three-body problem in quantum mechanics [1]. Given a total energy E, a Hamiltonian H describing the interaction of two electrons with each other and with the nucleus, an initial state Ψki0 of an electron with momentum ki incident on a ground state hydrogen atom, find two-dimensional radial functions ψl1l2L which solveE-Hl1(r1)-Hl2(r2)ψl1l2L(r1,r2)-l1,l2l1l2l1l2Lψl1l2L(r1,r2)=χl1l2L(r1,r2),where r1 and r2 are the underlying coordinates, L denotes the total angular momentum quantum number while l1 and l2 denote the single-electron angular momentum quantum numbers, χl1l2L are the radial functions from the expansion of (H-E)Ψki0(r1,r2), l1l2l1l2L are two-dimensional coupling potentials arising from the electron–electron interaction, and Hl are the one-dimensional, Coulomb radial Hamiltonians. Refer to the excellent paper [1] by Baertschy and Sherry Li for a complete and detailed description of the original problem as well as its significance in scientific applications.

Discretization of Eq. (1) using finite difference to approximate the derivatives results in a large and very ill-conditioned linear system in the following form:Ax=b,where ACn×n is a complex-valued non-Hermitian and nonsymmetric matrix, b is a known right-hand side, and x is unknown to be determined.

The already existing two typical and representative complex nonsymmetric linear systems obtained from the above-mentioned problem, denoted as M3D2 and M4D2, are two instances of a computational chemistry model problem listed in the following table (refer to [2]).

M3D2 and M4D2 have recently received intensive attention in the literature. Day and Heroux [2] showed that 2-by-2 block formulations obtained equating the real and imaginary parts of Eq. (2) can help nonsymmetric Krylov subspace methods perform reasonably well with standard incomplete LU (ILU) factorizations, competitively with ILU-preconditioned Krylov methods applied to the original complex system. Motivated in part by these work of Day and Heroux, Benzi and Bertaccini [3] further considered real-valued preconditioned iterative methods for the solution of complex linear systems, with an emphasis on symmetric (non-Hermitian) problems. The present paper is devoted to the development of iterative solutions to such problems. The focus of this paper is on experiments of various iterative methods applied to the above-mentioned problems in both situations of no preconditioning and simple diagonal preconditioning, hopefully to demonstrate the competitiveness of our recently proposed Lanczos biconjugate A-orthonormalization methods [4], [5], [6] to other classic and popular iterative methods [7] in the field of molecular dynamics.

In this study we consider the complex formulation (2). For simplicity and convenience of comparison, diagonal preconditioning here means row diagonal scaling of the original coefficient matrix and then applying iterative methods to the diagonally scaled matrix. For other elegant preconditioning techniques to improve the performance and reliability of Krylov subspace methods, refer to the outstanding survey by Benzi [8] and the distinguished book by Saad [9].

The paper is outlined as follows: In Section 2, the Lanczos biconjugate A-orthonormalization methods including the BiCOR, CORS, BiCORSTAB algorithms are briefly introduced. The spectral properties of the involved matrices are investigated in Section 3. The results of numerical experiments, carried out with machine precision 10−16 in double precision floating point arithmetic in MATLAB 7.0.4 with a PC-Pentium (R) D CPU 3.00 GHz, 1 GB of RAM, are presented and discussed in Section 4. Finally, conclusions are summarized in Section 5.

Section snippets

Lanczos biconjugate A-orthonormalization methods for linear systems

In this section, the pseudocodes for the preconditioned BiCOR, CORS and BiCORSTAB methods with a left preconditioner M can be represented by Algorithm 1, Algorithm 2, Algorithm 3, respectively. It can be observed that one iteration of the unpreconditioned BiCOR method requires one matrix–vector product by A and one by AH while the unpreconditioned CORS and BiCORSTAB methods both require two matrix–vector products only by A instead of AH.

With respect to detailed derivations of these three novel

Spectral investigation

In this section, we have a brief investigation of the spectral properties of the original M3D2 complex matrix and the corresponding one with diagonal preconditioning with the eig function of MATLAB, as shown in Fig. 1. It is out of memory for computing the spectrum of the M4D2 complex matrix with the eig function of MATLAB.

It is observed in Fig. 1 that the eigenvalues of the M3D2 matrix with diagonal preconditioning are more clustered than those of the original M3D2 matrix. As we can see in the

Numerical experiments

The results of the numerical tests using different iterative methods are presented in this section. The involved methods include the family of the Lanczos biconjugate A-orthonormalization methods [4], their state-of-the-art counterparts related to the CBiCG method [10] such as the CGS [11] and BiCGSTAB methods [12] while related to BiCR method [13] such as the CRS method [14], the SCGS method [14], and the GMRES method [15] as well as the recently designed IDR (s) family of algorithms [16]. For

Concluding remarks

In this comparative study we have shown and analyzed experiment results on iterative solution of two complex-valued nonsymmetric systems of linear equations arising from a computational chemistry model problem proposed by Sherry Li of NERSC. The good performance of our recently proposed Lanczos biconjugate A-orthonormalization methods [4], [5] to other classic and popular iterative methods [7] is demonstrated again to some extent.

This family of solvers shows good convergence properties, is

Acknowledgments

The authors thank Professor Basil Nyaku, the Associate Editor and the anonymous reviewers for their valuable comments and suggestions, leading to a clearer exhibition of the present numerical experiments. Many thanks to Professor Xiaoye Sherry Li for her kind help and providing us with the excellent paper [1], motivating us to proceed with this study. This research was supported by 973 Program (2007CB311002), NSFC (10926190, 60973015), Sichuan Province Sci. & Tech. Research Project (2009HH0025).

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