Transcript RKPACK A numerical package for solving large eigenproblems Che-Rung Lee Outline Introduction  RKPACK  Experiments  Conclusion  2015/11/6 University of Maryland, College Park.

RKPACK
A numerical package for
solving large eigenproblems
Che-Rung Lee
Outline
Introduction
 RKPACK
 Experiments
 Conclusion
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University of Maryland, College Park
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Introduction
The residual Krylov method
 Shift-invert enhancement
 Properties and examples

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University of Maryland, College Park
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The residual Krylov method

Basic algorithm
Let (  , z ) be a selected eigenpair
approximation of A.
2. Compute the residual r  Az  z .
3. Use r in subspace expansion.
1.
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Properties

The selected approximation (candidate)
can converge even with errors.
allowed error || f || must be less than  ||r||,
for a constant  <1.
 The

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 =  )
Let (  , z ) be a selected eigenpair
approximation of A.
2. Compute the residual r  Az  z .
3. Solve the equation ( A  I )v  r .
4. Use v in subspace expansion.
1.

Equation in step 3 can be solved in low
accuracy, such as 103.
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Example



The same
matrix
Shift value is
1.3
Linear
systems are
solved to
10 3.
100
10-5
10-10
10-15
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5
10
15
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25
30
35
40
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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
subspace
 Use the Krylov-Schur restarting algorithm
 Use reverse communication
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Computation modes

Two computation modes
 The

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 A
 m: 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 mode
 kf

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(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 1000010000 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 1013.
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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
Etime
MVM (Iteration)
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RKPACK
25.93
24.41
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 ARPACK
 Prec:
precision requirement of solution
ARPACK
Iteration
68
153
4246.77
623.46
MVM
14552
4932
Prec
1013
103
ETime
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Inexact residual Krylov method
Allow increasing errors in the computation
 Use the normal mode with matrix A1.

 The
required precision of solving A1.
 
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
m r

 is the desired precision of computed eigenpairs

m is the maximum dimension of subspace
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Experiment and result
Compute six smallest eigenpairs.
 The required precision (using GMRES)

13
10

20 r
10-2
10-4
Etime: 910.11 second
 MVM: 6282
 Iteration: 67
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10-8
10-10
20
30
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50
60
70
80
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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)
103,106,109,1012. 10

0
10-2
Etime : 1188.12
 MVM : 13307
 Iteration : 163

10-4
10-6
10-8
10-10
10-12
20
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80
100
120
140
160
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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

Parallelization
 Data
…
parallelism
Block version of the residual Krylov method
 Other eigenvector approximations

 Refine

Ritz vector or Harmonic Ritz vector
New algorithms
 Inexact
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methods, residual power method …
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