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A LargeGrained Parallel Algorithm for Nonlinear Eigenvalue Problems Using Complex Contour Integratio

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Title: A LargeGrained Parallel Algorithm for Nonlinear Eigenvalue Problems Using Complex Contour Integratio


1
A Large-Grained Parallel Algorithm for Nonlinear
Eigenvalue Problems Using Complex Contour
Integration
  • Takeshi Amako, Yusaku Yamamoto and Shao-Liang
    Zhang
  • Dept. of Computational Science Engineering
  • Nagoya University, Japan

2
Outline of the talk
  • Introduction
  • The nonlinear eigenvalue problem
  • Existing algorithms
  • Our objective
  • The algorithm
  • Formulation as a nonlinear equation
  • Application of Kravanja et als method
  • Detecting and removing spurious eigenvalues
  • Numerical results
  • Accuracy of the computed eigenvalues
  • Parallel performance
  • Conclusion

3
Introduction
  • The nonlinear eigenvalue problem
  • Given A(z) ? Cnn , z complex parameter
  • Find z1 ? C such that A(z1) x 0 has a nonzero
    solution x x1.
  • z1 and x1 are called the eigenvalue and the
    corresponding eigenvector, respectively.
  • Examples
  • A(z) A zB z2C quadratic eigenvalue
    problem
  • A(z) A zB ezC general nonlinear
    eigenvalue problem
  • Applications
  • Electronic structure calculation
  • Nonlinear elasticity
  • Theoretical fluid dynamics

4
Existing algorithms
  • Multivariate Newtons method and its variants
  • Locally quadratic convergence
  • Requires good initial estimate both for z1 and
    x1.
  • Nonlinear Arnoldi methods
  • Nonlinear Jacobi-Davidson methods
  • Efficient for large sparse matrices
  • Not suitable for finding all eigenvalues within a
    specified region of the complex plane

5
Our objective
  • Let
  • G closed Jordan curve on the complex plane,
  • A(z) ? Cnn analytical function of z in G.
  • We propose an algorithm that
  • can find all the eigenvalues within G, and
  • has large-grain parallelism.

Im z
Assumption In the following, we mainly
consider the case where G is a circle centered at
the origin and with radius r.
G
Re z
O
r
Related work Sakurai et al. propose an
algorithm for linear generalized eigenvalue
problems
6
Our approach
  • The basic idea
  • Let f(z) det(A(z)).
  • Then f(z) is an analytical function of z in G and
    the eigenvalues of A(z) are characterized as the
    zeros of f(z).
  • Use Kravanjas method (Kravanja et al., 1999) to
    find the zeros of an analytic function.

7
Finding zeros of f(z)
  • Let
  • z1, z2, ..., zm zeros of f(z) in G, and
  • n1, n2, ..., nm their multiplicity.
  • Then f(z) can be written as
  • Define the complex moments by
  • Then

f(z) g(z)
analytical and nonzero in G
analytical in G
8
Finding zeros of f(z) (cont'd)
  • To extract information on zk from mp, define
    the following matrices
  • Then it is easy to see that

9
Finding zeros of f(z) (cont'd)
  • Noting that Vm and Dm are nonsingular, we have
    the following equivalence relation
  • That is, we can find the zeros of f(z) in G by
  • computing the complex moments m0, m1 , ...,
    m2m-1,
  • constructing Hm and Hmlt, and
  • computing the eigenvalues of Hmlt lHm.

l is an eigenvalue of Hmlt lHm ? l is an
eigenvalue of Lm lI ? ?k, l zk
10
Application to the nonlinear eigenvalue problem
  • In our case, f(z) det(A(z)) and
  • By applying the trapezoidal rule with K points,
    we have
  • where

G
11
The algorithm
12
Detecting and removing spurious eigenvalues
  • Usually, we do not know m, the number of
    eigenvalues of A(z) in G, in advance and use some
    estimate M instead.
  • When M gt m, the eigenvalues of Hmlt lHm include
    spurious solutions that do not correspond to an
    eigenvalue of A(z).
  • To detect them, we compute the corresponding
    eigenvector by inverse iteration and evaluate the
    relative residual defined by
  • Of course, this quantity can also be used to
    check the accuracy of the computed eigenvalues.

relative residual
13
Numerical results
  • Test problem
  • A(z) A zI eB(z), where
  • A(z) real random nonsymmetric matrix
  • B(z) antidiagonal matrix with antidiagonal
    elements ez
  • e parameter to specify the strength of
    nonlinearity
  • Parameters
  • n 500, 1000, 2000
  • e 0, 104, 103, 102, 101
  • Computational environment
  • Fujitsu HPC2500 (SPARC 64IV), 1-16 processors
  • Program written with C and MPI
  • LAPACK routines were used to compute (A(z))1 and
    to compute the eigenvalues of Hmlt lHm.

14
Accuracy of the computed eigenvalues
  • Parameters
  • n 500 and e 0.1
  • r 0.85, K 128 and M 11.
  • There are 7 eigenvalues in G.
  • Results
  • Our algorithm succeeded in locating all the
    eigenvalues in G.
  • The relative residuals were all under 1010.
  • Similar results for other cases.

Im z
Re z
15
Effect of K and M on the accuracy
  • Effect of the number of sample points K
  • Usually K128 gives sufficient accuracy.
  • Effect of the Hankel matrix size M
  • It is better to take M a few more than the number
    of eigenvalues within G (7 in this case).
  • This is to mitigate the perturbation from
    eigenvalues outside G.

K
M
Residuals as a function of K.
Residuals as a function of M.
16
Detecting and removing spurious eigenvalues
  • Parameters
  • n 1000 and e 0.01
  • r 0.7, K 128 and M 10.
  • There are 9 eigenvalues in G.
  • Eigenvalues of Hmlt lHm
  • 10 eigenvalues were found within G.
  • For 9 of the eigenvalues, the residual was less
    than 1011.
  • For one eigenvalue, the residual was 102.

Im z
Re z
17
Parallel performance
  • Performance on Fujitsu HPC2500
  • Matrix size n 500, 1000, 2000
  • Number of processors P 1, 2, 4, 8, 16

Almost linear speedup was obtained in all cases
due to large-grain parallelism.
Execution time (sec)
Number of processors
18
Parallel performance (cont'd)
  • Performance in a Grid environment
  • Matrix size n 1000
  • Machine Intel Xeon Cluster
  • Master-worker type parallelization using OmniRPC
    (GridRPC)

Good scalability was obtained for up to 14
processors.
20000
16
Execution time
Speedup
14
13000
12
10
10000
8
6
03000
4
2
00000
0
Number of processors
2
4
6
8
10
12
14
19
Summary of this study
  • We proposed a new algorithm for the nonlinear
    eigenvalue problem based on complex contour
    integration.
  • Our algorithm can find all the eigenvalues within
    a closed curve on the complex plane. Moreover, it
    has large-grain parallelism and is expected to
    show excellent parallel performance.
  • These advantages have been confirmed by numerical
    experiments.

20
Future work
  • Performance evaluation on large-scale grid
    environments.
  • Application to practical problems.
  • Computation of scaling exponent in theoretical
    fluid dynamics
  • Development of an efficient algorithm for
    computing
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