1. RKPACK A numerical package for solving large eigenproblems
Che-Rung Lee

2. University of Maryland, College Park
Outline
Introduction
RKPACK
Experiments
Conclusion
2019/7/28
University of Maryland, College Park

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Introduction
The residual Krylov method
Shift-invert enhancement
Properties and examples
2019/7/28
University of Maryland, College Park

4. The residual Krylov method
Basic algorithm
Let be a selected eigenpair approximation of A.
Compute the residual
Use r in subspace expansion.
2019/7/28
University of Maryland, College Park

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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.
2019/7/28
University of Maryland, College Park

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Example
A 100x100 matrix with eigenvalues 1, 0.95, …,
2019/7/28
University of Maryland, College Park

7. Shift-invert enhancement
Algorithm: (shift value =  )
Let be a selected eigenpair approximation of A.
Compute the residual
Solve the equation .
Use v in subspace expansion.
Equation in step 3 can be solved in low accuracy, such as 103.
2019/7/28
University of Maryland, College Park

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Example
100
10-5
10-10
10-15
The same matrix
Shift value is 1.3
Linear systems are solved to 10 3.
2019/7/28
University of Maryland, College Park

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RKPACK
Features
Computation modes
Memory requirement
Time complexity
2019/7/28
University of Maryland, College Park

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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
2019/7/28
University of Maryland, College Park

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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.
2019/7/28
University of Maryland, College Park

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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
2019/7/28
University of Maryland, College Park

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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 (n) + kO(nm) + kO(m3) + kg(n, )
g(n, ) : the time for solving linear system to the precision .
2019/7/28
University of Maryland, College Park

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Experiments
Test problem
Performance of RKPACK
The inexact residual Krylov method
The successive inner-outer process
2019/7/28
University of Maryland, College Park

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Test problem
Let A be a 1000010000 matrix with first 100 eigenvalues 1, 0.95, …, , 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.
2019/7/28
University of Maryland, College Park

16. 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
2019/7/28
University of Maryland, College Park

17. 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
RKPACK
Iteration 68
153
ETime
623.46
MVM
14552
4932
Prec
1013
103
2019/7/28
University of Maryland, College Park

18. 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.
 is the desired precision of computed eigenpairs
m is the maximum dimension of subspace
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University of Maryland, College Park

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Experiment and result
Compute six smallest eigenpairs.
The required precision (using GMRES)
Etime: second
MVM: 6282
Iteration: 67
10-2
10-4
10-6
10-8
10-10
2019/7/28
University of Maryland, College Park

20. 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.
Etime :
MVM : 13307
Iteration : 163
100
10-2
10-4
10-6
10-8
10-10
10-12
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Conclusion
Summary
Future work
2019/7/28
University of Maryland, College Park

23. University of Maryland, College Park
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.
2019/7/28
University of Maryland, College Park

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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 methods, residual power method …
2019/7/28
University of Maryland, College Park