RKPACKA numerical package for solving large eigenproblems

Che-Rung Lee

University of Maryland, College Park 2112/04/19

Outline

Introduction RKPACK Experiments Conclusion

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Introduction

The residual Krylov method Shift-invert enhancement Properties and examples

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The residual Krylov method

Basic algorithm1. Let be a selected eigenpair

approximation of A.

2. Compute the residual .

3. Use r in subspace expansion.

),( z

zAzr

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Properties

The selected approximation (candidate) can converge even with errors.The allowed error || f || must be less than ||

r||, for a constant <1. The residual Krylov method can work with

an initial subspace that contains good Ritz approximations.

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Example

A 100x100 matrix with eigenvalues 1, 0.95, …,0.9599.

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Shift-invert enhancement

Algorithm: (shift value = )1. Let be a selected eigenpair

approximation of A.

2. Compute the residual .

3. Solve the equation .

4. Use v in subspace expansion. Equation in step 3 can be solved in low

accuracy, such as 103.

),( z

zAzr rvIA )(

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Example

The same matrix

Shift value is 1.3

Linear systems are solved to 10 3.

0 5 10 15 20 25 30 35 40

100

10-5

10-10

10-15

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RKPACK

Features Computation modes Memory requirement Time complexity

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Features

Can compute several selected eigenpairs Allow imprecise computational results with

shift-invert enhancement Can start with an appropriate initial subspa

ce Use the Krylov-Schur restarting algorithm Use reverse communication

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Computation modes

Two computation modesThe normal mode:

needs matrix vector multiplication only

The imprecise shift-invert mode: needs matrix vector multiplication and linear

system solving (with low accuracy requirement) can change the shift value

Both can be initialized with a subspace.

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Memory requirement

Use the Krylov-Schur restarting algorithm to control the maximum dimension of subspace

Required memory: O(nm)+O(m2)n: the order of matrix Am: the maximum dimension of subspace

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Time complexity

The normal mode:kf (n)+kO(nm)+kO(m3)

f (n): the time for matrix vector multiplication. k: the number of iterations

The imprecise shift-invert modekf (n) + kO(nm) + kO(m3) + kg(n, )

g(n, ) : the time for solving linear system to the precision .

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Experiments

Test problem Performance of RKPACK The inexact residual Krylov method The successive inner-outer process

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Test problem

Let A be a 1000010000 matrix with first 100 eigenvalues 1, 0.95, …, 0.9599, and the rest randomly distributed in (0.25, 0.75).

Eigenvectors are randomly generated. Maximum dimension of subspace is 20. Stopping criterion: when the norm of

residual is smaller than 1013.

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Performance of the normal mode

Compute six dominant eigenpairs. Compare to the mode 1 of ARPACK

Etime: elapse time (second)MVM: number of matrix vector multiplications Iteration: number of subspace expansions

ARPACK RKPACK

Etime 25.93 24.41

MVM (Iteration) 117 142

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The imprecise shift-invert mode

Compute six smallest eigenvalues.Use GMRES to solve linear system. (shift = 0)

Compare to the mode 3 of ARPACKPrec: precision requirement of solution

ARPACK RKPACK

Iteration 68 153

ETime 4246.77 623.46

MVM 14552 4932

Prec 1013 103

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Inexact residual Krylov method

Allow increasing errors in the computation Use the normal mode with matrix A1.

The required precision of solving A1.

is the desired precision of computed eigenpairs m is the maximum dimension of subspace

rm

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Experiment and result

Compute six smallest eigenpairs. The required precision (using GMRES)

Etime: 910.11 second MVM: 6282 Iteration: 67

r20

10 13

20 30 40 50 60 70 80

10-2

10-4

10-6

10-8

10-10

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Successive inner-outer process

Use the convergence properties of Krylov subspace (superlinear) to minimize total number of MVM. (Golub, Zhang and Zha, 2000)

Divide the process into stages, with increasing precision requirement.

The original algorithm can only compute a single eigenpair

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Experiment and result

Compute six smallest eigenpairs. Four stages with required precision (GMRES)

103,106,109,1012.

Etime : 1188.12 MVM : 13307 Iteration : 163

20 40 60 80 100 120 140 160

100

10-2

10-4

10-6

10-8

10-10

10-12

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Conclusion

Summary Future work

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Summary

The residual Krylov method for eigenproblems allows errors in the computation, and can work on an appropriate initial subspace.

RKPACK can solve eigenproblems rapidly when uses the imprecise shift-invert enhancement, and is able to integrate other algorithms easily.

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Future work

ParallelizationData parallelism

Block version of the residual Krylov method Other eigenvector approximations

Refine Ritz vector or Harmonic Ritz vector New algorithms

Inexact methods, residual power method …

…