Elsevier

Parallel Computing

Volume 74, May 2018, Pages 34-48
Parallel Computing

Block SS–CAA: A complex moment-based parallel nonlinear eigensolver using the block communication-avoiding Arnoldi procedure

https://doi.org/10.1016/j.parco.2017.11.007Get rights and content

Highlights

  • This paper proposes a novel complex moment-based parallel nonlinear eigensolver.

  • The proposed method shows a good scalability based on its hierarchical parallelism.

  • The communication-avoiding Arnoldi procedure leads small cost and high accuracy.

  • The proposed method shows higher performance compared with traditional methods.

Abstract

Complex moment-based parallel eigensolvers have been actively studied owing to their high parallel efficiency. In this paper, we propose a block SS–CAA method, which is a complex moment-based parallel nonlinear eigensolver that makes use of the block communication-avoiding Arnoldi procedure. Numerical experiments indicate that the proposed method has higher performance compared with traditional complex moment-based nonlinear eigensolvers, i.e., the block SS–Hankel and Beyn methods.

Introduction

In this paper, we consider complex moment-based parallel eigensolvers that compute all eigenvalues located in a certain region of a nonlinear eigenvalue problem (NEP) and their corresponding eigenvectors: T(λi)xi=0,xiCn{0},λiΩC,where the matrix-valued function T:ΩCn×n is holomorphic in some open domain Ω. We write this as TH(Ω,Cn×n). Here, we also assume that the target eigenvalues λi ∈ Ω are simple and that the number of the target eigenpairs m is less than the size of the problem n.

There are several algorithms for solving NEPs including Newton-type methods [1], [2], projection methods such as Arnoldi and Jacobi-Davidson methods [3], [4], rational Krylov methods [5], [6] and complex moment-based methods [7], [8], [9], [10], [11]. Newton-type methods show quadratic convergence; however, they require good initial approximations for both eigenvalues and eigenvectors. Projection methods and rational Krylov methods have been well studied and are considered very reasonable choices for computing few eigenpairs. They can efficiently compute some eigenvalues; however, it is difficult to guarantee that all the eigenvalues within the target region are obtained. In addition, projection methods encounter another difficulty in parallel computing because they are essentially based on sequential procedure.

In contrast, complex moment-based methods compute all eigenvalues within the target region using a contour integral without good initial approximations. Regarding parallel computing efficiency, complex moment-based methods have a big advantage compared with other methods because the most time-consuming part of these methods is the contour integral, which is suitable for parallel computing rather than the sequential procedure. Based on the contour integral, complex moment-based methods have higher level hierarchical parallelism than others. Thanks to this high-level hierarchical parallelism, complex moment-based methods achieve higher scalability than Krylov-type methods [12]. It is noted that rational Krylov methods also have same parallelization potential as complex moment-based methods.

In this paper, to develop the complex moment-based methods, we propose a novel complex moment-based parallel eigensolver for NEP (1) that makes use of the block communication-avoiding Arnoldi procedure [13], which we call a block SS–CAA method. The proposed method reduces the target NEP (1) to a standard eigenvalue problem (SEP) as well as traditional complex moment-based nonlinear eigensolvers, i.e., the block SS–Hankel method [7], [8] and the Beyn method [10]. The proposed method is expected to achieve a higher accuracy than the block SS–Hankel method and a smaller elapsed time than the Beyn method. This point is evaluated here through numerical experiments (Section 4).

The remainder of this paper is organized as follows. Section 2 briefly describes complex moment-based nonlinear eigensolvers with hierarchical parallelism. In Section 3, we present the block SS–CAA method using the block communication-avoiding Arnoldi procedure. The performed numerical experiments are reported in Section 4. The paper concludes with Section 5.

Throughout, we use the following notations. As V=[v1,v2,,vL]Cn×L, we define the range space of the matrix V by R(V):=span{v1,v2, ,vL}. In addition, as ACn×n, Kk(A,V) and Bk(A,V) are the block Krylov subspaces Kk(A,V):=R([V,AV,A2V,,Ak1V]),Bk(A,V):={i=0k1AiVαi|αiCL×L}.We also use MATLAB notations. The submatrix comprising elements in the ith through jth rows and the pth through qth columns is given by A(i: j, p: q). Let A, B, C and D represent matrices, and set [A,B;C,D]:=[ABCD].

Section snippets

Complex moment-based nonlinear eigensolvers

Sakurai and Sugiura have proposed a complex moment-based eigensolver for solving interior generalized eigenvalue problems (GEPs), the SS–Hankel method [14], which is based on Cauchy’s integral formula and constructs certain complex moment matrices using a contour integral. The most time-consuming part in the SS–Hankel method involves solving linear systems at each quadrature point. However, as these linear systems can be independently solved, the SS–Hankel method shows good scalability. For

Novel complex moment-based nonlinear eigensolver using the block communication-avoiding Arnoldi procedure

As noted in Section 2, Step 3 is also time-consuming in parallel computing and the block SS–Hankel and Beyn methods have a significant advantage over the other methods regarding the cost for Step 3. However, these methods have other difficulties. The accuracy of the block SS–Hankel method is worse than that of the other methods. The Beyn method needs to solve linear systems with a much larger number of right-hand sides.

To remedy these difficulties, in this section, we propose a novel complex

Numerical experiments

In this section, we evaluate the performance of the block SS–CAA method (Algorithm 3), as compared with that of the block SS–Hankel (Algorithm 1) and Beyn (Algorithm 2) methods.

In all numerical experiments, we let the quadrature points be located on a circle with center γ and radius ρ, i.e., zj=γ+ρ(cos(θj)+isin(θj)),θj=2πN(j12),j=1,2,,N.The corresponding weights are set as ωj=ρN(cos(θj)+isin(θj)),j=1,2,,N.

Conclusions

In this paper, we proposed the block SS–CAA method, which is a novel complex moment-based parallel eigensolver using the block communication-avoiding Arnoldi procedure, for the nonlinear eigenvalue problems (1). The block SS–CAA method has competitive advantages over the traditional complex moment-based nonlinear eigensolvers.

  • From the theoretical aspect, we observed that

    • -

      The block SS–CAA method reduces the target NEP to SEP as well as the block SS–Hankel and Beyn methods. This is a significant

Acknowledgements

The authors are grateful to anonymous referees for useful comments. This research was supported partly by JST/ACT-I (No. JPMJPR16U6), JST/CREST, MEXT KAKENHI (No. 17K12690) and University of Tsukuba Basic Research Support Program Type A. This research in part used computational resources of COMA provided by Interdisciplinary Computational Science Program in Center for Computational Sciences, University of Tsukuba.

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