Error propagation formula
of multi-level iterative aggregation-disaggregation methods
with arbitrary number of levels
applied to non-symmetric problems
(Algebraic multigrid for stochastic matrices)
Ivana Pultarová
Department of Mathematics, Faculty of Civil Engineering
Czech Technical University in Prague
CMAM 2012, Berlin
(CTU in Prague)
Algebraic MG for stochastic matrices
CMAM 2012
1 / 19
Outline
1
Algebraic multigrid (AMG)
2
Stochastic matrices, Markov chains, stationary probability distribution vector
3
Iterative aggregation - disaggregation (IAD) methods
4
Convergence and divergence of IAD
5
Conclusions and open questions
(CTU in Prague)
Algebraic MG for stochastic matrices
CMAM 2012
2 / 19
Algebraic multigrid (AMG)
Solve Ax = b, where A ∈ RN×N is
symmetric and positive definite
sparse
A1 ≈ 0, where 1 is all ones vector and 0 is a zero vector
Approximations x n and error vectors en = x n − x. The matrix iteration methods (Jacobi,
Gauss-Seidet, etc.) ”smooth” the error:
high frequency components v n of the error are annilihated faster than that with small
frequencies,
then ||r n || = ||b − Ax n || = ||A(x − x n )|| = ||Aen || ||en || and
from
v T Av ≡
X
1X
−Aij (vi − vj )2 +
Aij vi2 ,
2
i,j
i,j
we get
T
en Aen =
X
1X
−Aij (ein − ejn )2 (ein )2 Aii .
2
i,j
Thus there is almost no difference between ”strongly connected” components of the error
vector en .
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Algebraic MG for stochastic matrices
CMAM 2012
3 / 19
Aggregation based AMG
Let rows of R ∈ RNc ×N be (sparse approximations of) low frequency vectors of A, Nc < N.
Let the coarse matrix and the coarse right hand side be
Ac = RAR T ,
rc = Rr n ,
n
then Ac uc = rc is a restriction of the problem to a ”coarser mesh” and xnew
= x n + R T uc is a
better approximation to x.
Algorithm of one cycle of AMG for Ax = b for a given x n
1. Solve (coarse, small) Nc × Nc problem RAR T z = R(b − Ax n ).
2. Prolong z and add to the current approximation: x n+1,0 = x n + R T z.
3. Apply ν steps of (large) basic iteration with T : x n+1,i = T ν (x n+1,i ) + b.
Error propagation matrix for x n+1 − x = JMG (x n − x),
JMG = T I − R T (RAR T )−1 RA .
(CTU in Prague)
Algebraic MG for stochastic matrices
CMAM 2012
4 / 19
1
Algebraic multigrid (AMG)
2
Stochastic matrices, Markov chains, stationary probability distribution vector
3
Iterative aggregation - disaggregation (IAD) methods
4
Convergence and divergence of IAD
5
Conclusions and open questions
(CTU in Prague)
Algebraic MG for stochastic matrices
CMAM 2012
5 / 19
Problem description.
We assume an irreducible N × N column stochastic matrix B, i.e.
B≥0
and
eT B = eT .
Find stationary probability distribution vector (Perron eigenvector) of a column stochastic
matrix, i.e. find x such that
Bx = x,
eT x = 1
or
(I − B)x = 0,
eT x = 1.
Perron-Frobenius theorem.
Solution
Direct solvers.
Numerical solution. Power method, Jacobi m., Gauss-Seidel m., their block modifications.
Iteration matrix T = M −1 W , where I − B = M − W , is a regular splitting (M −1 ≥ 0, W ≥ 0).
Krylov subspace methods.
Multilevel methods, based on aggregation of states: iterative aggregation - disaggregation
(IAD) methods. Motivation from PDEs, but no symmetry here.
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Algebraic MG for stochastic matrices
CMAM 2012
6 / 19
Example. Find stationary probability distribution of the random process with five states
characterized by probabilities Bij of transition from j to i.
1
0
 0.9

B= 0
 0
0.1

0.3
0
0.7
0
0
0
0
0.7
0.1
0.2
0.6
0
0.3
0.1
0
0
0
0
1
0

0.3


.

2
0.9

0.6
0.1
0.3
0.7
0.1390
 0.1251 


Vector x is x ≈  0.4609 .
 0.1691 
0.1061

1
4
0.1
1
3
5
0.2
0
0
Applications: queuing networks; reliability of systems, e.g. in railway safety; Google computation
- PageRank; probabilities of genus appearance (very large scale); etc.
(CTU in Prague)
Algebraic MG for stochastic matrices
CMAM 2012
7 / 19
1
Algebraic multigrid (AMG)
2
Stochastic matrices, Markov chains, stationary probability distribution vector
3
Iterative aggregation - disaggregation (IAD) methods
4
Convergence and divergence of IAD
5
Conclusions and open questions
(CTU in Prague)
Algebraic MG for stochastic matrices
CMAM 2012
8 / 19
Notation
Set of events {1, 2, 3, . . . , N} divided into Nc pairwise disjoint aggregation groups G1 , G2 , . . . , GNc .
Reduction R and prolongation S(x) matrices are e.g. for G1 = {1, 2}, G2 = {3, 4, 5}
1/3
 2/3

S(y) =  0
 0
0

R=
1
0
1
0
0
1
0
1
0
1
,
0
0
2/6
3/6
1/6

2
4 

1 

y=
 2 .
12  3 
1




,

if
Aggregated (coarse) matrix RBS(y) =
=
1
0
1
0
0
1
0
1


0


1

0
0
0
1
0
0
0.3
0
0.4
0.3
0
0.5
0.5
0
0
0.1
0
0
0
0.9
0
1
0
0
0





1/3
2/3
0
0
0
0
0
2/6
3/6
1/6

 1/5

=
4/5

23/60
37/60
.
Projection P(y) = S(y)R .
(CTU in Prague)
Algebraic MG for stochastic matrices
CMAM 2012
9 / 19
Algorithm of one cycle of the IAD method for Bx = x for a given x n
1. Solve (coarse) Nc × Nc problem RBS(x n )z = z.
2. Prolong z to the original size x n+1,0 = S(x n )z,
3. Apply ν steps of (large) basic iteration, x n+1 = T ν x n+1,0 .
Error propagation formula
x n+1 − x = J(x n )(x n − x),
where
−1
J(x n ) = T ν I − P(x n )(B − xeT )
I − P(x n ) ,
P(x n ) = S(x n )R.
Note:
In general ρ(J(x n )IAD ) is not less than one even for converging sequences.
Only for proving the local convergence.
(CTU in Prague)
Algebraic MG for stochastic matrices
CMAM 2012
10 / 19
Symmetric vs. nonsymmetric MG
. . . it means, MG for symmetric and nonsymmetric marices.
Theorem. The MG method converges for symmetric matrix A for any kind of aggregation and for
any number of smoothing steps within one multigrid cycle.
No analogous statement has been proved for the nonsymmetric case - stochastic matrices.
However, the MG algorithms for nonsymmetric problems exist and ”mostly they converge.” The
heuristic explanation of their fast convergence is based on similarity with the symmetric case.
W. J. Stewart 1994; P. Buchholz, T. Dayar, G. Horton, S. T. Leutenegger, U. R. Krieger,
A. N. Langville, I. Marek, P. Mayer, C. D. Meyer;
E. Treister, I. Yavneh, 2011;
H. De Sterck, T. A. Manteuffel, S. F. McCormick, Q. Nguyen and J. Ruge, 2008 - 2012;
...
But, there are differences between symmetric and nonsymmetric MG.
(CTU in Prague)
Algebraic MG for stochastic matrices
CMAM 2012
11 / 19
1
Algebraic multigrid (AMG)
2
Stochastic matrices, Markov chains, stationary probability distribution vector
3
Iterative aggregation - disaggregation (IAD) methods
4
Convergence and divergence of IAD
5
Conclusions and open questions
(CTU in Prague)
Algebraic MG for stochastic matrices
CMAM 2012
12 / 19
Small example. Doubly stochastic matrix

1/2 0
B =  1/2 0
0
1

1/2
1/2  ,
0


1/3
x =  1/3 
1/3
Power method yields a convergent sequence x k for all starting x 0 . Note ρ(B − xeT ) = 1/2.
The IAD method with T = B, ν = 1 and aggregation


0 1/2
1/2

B=
1/2
0 1/2 
1
0
0
yields divergent (oscillating) sequence x k for almost all starting x 0 . Note ρ(J(x)) = 1.
The IAD method with T = B, ν = 1 and aggregation


1/2 0
1/2

1/2 0
1/2 
B=
0
1
0
yields exact solution after the second step x 2 = x for any starting x 0 . Note ρ(J(x)) = 0.
(CTU in Prague)
Algebraic MG for stochastic matrices
CMAM 2012
13 / 19
Small example. Doubly stochastic matrix

1/2 0
B =  1/2 0
0
1

1/2
1/2  ,
0


1/3
x =  1/3 
1/3
Power method yields a convergent sequence x k for all starting x 0 . Note ρ(B − xeT ) = 1/2.
The IAD method with T = B, ν = 1 and aggregation


0 1/2
1/2

B=
1/2
0 1/2 
1
0
0
yields divergent (oscillating) sequence x k for almost all starting x 0 . Note ρ(J(x)) = 1.
The IAD method with T = B, ν = 1 and aggregation


1/2 0
1/2

1/2 0
1/2 
B=
0
1
0
yields exact solution after the second step x 2 = x for any starting x 0 . Note ρ(J(x)) = 0.
(CTU in Prague)
Algebraic MG for stochastic matrices
CMAM 2012
13 / 19
Small example. Doubly stochastic matrix

1/2 0
B =  1/2 0
0
1

1/2
1/2  ,
0


1/3
x =  1/3 
1/3
Power method yields a convergent sequence x k for all starting x 0 . Note ρ(B − xeT ) = 1/2.
The IAD method with T = B, ν = 1 and aggregation


0 1/2
1/2

B=
1/2
0 1/2 
1
0
0
yields divergent (oscillating) sequence x k for almost all starting x 0 . Note ρ(J(x)) = 1.
The IAD method with T = B, ν = 1 and aggregation


1/2 0
1/2

1/2 0
1/2 
B=
0
1
0
yields exact solution after the second step x 2 = x for any starting x 0 . Note ρ(J(x)) = 0.
(CTU in Prague)
Algebraic MG for stochastic matrices
CMAM 2012
13 / 19
Small example. Doubly stochastic matrix

1/2 0
B =  1/2 0
0
1

1/2
1/2  ,
0


1/3
x =  1/3 
1/3
Power method yields a convergent sequence x k for all starting x 0 . Note ρ(B − xeT ) = 1/2.
The IAD method with T = B, ν = 1 and aggregation


0 1/2
1/2

B=
1/2
0 1/2 
1
0
0
yields divergent (oscillating) sequence x k for almost all starting x 0 . Note ρ(J(x)) = 1.
The IAD method with T = B, ν = 1 and aggregation


1/2 0
1/2

1/2 0
1/2 
B=
0
1
0
yields exact solution after the second step x 2 = x for any starting x 0 . Note ρ(J(x)) = 0.
(CTU in Prague)
Algebraic MG for stochastic matrices
CMAM 2012
13 / 19
NCD Markov chains
Nearly completely decomposable (NCD) Markov chains:
Diagonal blocks of much larger magnitude than the off-diagonal blocks; their largest eigenvalues
close to one; other eigenvalues separated from one.
Sufficient for global convergence of IAD [W. J. Stewart, 1994].
Main drawback: hard to recognize whether a matrix is NCD or not.
Non-zero pattern of B
B has a positive row or column or diagonal, and T = αB + (1 − α)I, ν = 1 are sufficient for local
convergence. [I. Marek, I. Pultarová, 2006].
Special choice of groups (”1,1,. . . ,1,N1 ”) and T = B, ν = 1: necessary and sufficient cond. for
global convergence. Estimate of the asymptotic rate of convergence. [I. Ipsen, S. Kirkland, 2006].
General choice of groups and T = B, ν = 1: necessary and sufficient cond. for local convergence.
Estimate of the asymptotic rate of convergence. [I. Pultarová, 2008].
(Proofs based on relations λ2 (B) ≤ τ (B) = 12 maxi,j ||Bei − Bej ||1 for localization of spectra
[E. Seneta, 1984] and on stochastic complement formulation.)
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Algebraic MG for stochastic matrices
CMAM 2012
14 / 19
B non-symmetric
For any K > 0 there exists (even doubly) stochastic B of size less than 4K that ρ(J(x)) > K .
[I. Pultarová, I. Marek, 2011].
Examples constructed for T = B, ν = K , five-diagonal permutation stochastic matrices B of type
0
0
5
5
obtained from
10
10
15
15
by Cuthill-McKee algorithm.
20
20
0
5
10
nz = 20
15
0
20
5
10
nz = 20
15
20
Multilevel IAD, i.e. more than two levels
Reduction and prolongation matrices between levels k and k + 1 denoted by Rk and Sk (y).
Then for the m-levels IAD algorithm we have projections
Pjk (y) = Sj (y)Sj+1 (y) . . . Sk −1 (y)Rk−1 . . . Rj+1 Rj ,
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Algebraic MG for stochastic matrices
j < k.
CMAM 2012
15 / 19
Theorem. The error in the cycle n of a multi-level IAD methods with an arbitrary number of levels
L ≥ 2 and with one pre-smoothing step and with one post-smoothing step in every level,
µm = νm = 1, m = 1, 2, . . . , L − 1, is x n+1 − x = J (x n ) (x n − x), where
J(x n )
=
T
L−1
Y
L−1
X
k=2
k =1
(Pk T )(I − PL Z )−1
+T
L−2
m
XY
(Pk T )
m=1 k =2
m
X
(Pk − Pk +1 )Mk−1
(Pk − Pk +1 )Mk−1 ,
k =1
where M0 = T and
Mk = T +
k
X
TPj (T − I) T ,
j=2
for k = 1, 2, . . . , L − 2, P1 = I and
Pk = P(u 1 , u 2 , . . . , u k −1 )1k = S(u 1 )1 . . . S(u k−1 )k −1 Rk−1 . . . R1 ,
for k = 2, 3, . . . , L, where u 1 = Tx n , u 2 = R1 T 2 x n , u 3 = R2 R1 TP2 T 2 x n and
u k = Rk −1 . . . R1 TPk −1 TPk −2 . . . TP3 TP2 T 2 x n ,
for k = 4, . . . , L − 1.
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[I. Pultarová, 2011]
Algebraic MG for stochastic matrices
CMAM 2012
16 / 19
Multi-level IAD with m > 2 levels
Error propagation formula available (previous slide).
Consider these hypotheses:
If the number of basic iteration is increased, then the convergence is faster.
If m-level IAD is locally convergent then (m + 1)-level IAD is locally convergent.
If (m + 1)-level IAD is locally convergent then m-level IAD is locally convergent.
(CTU in Prague)
Algebraic MG for stochastic matrices
CMAM 2012
17 / 19
Multi-level IAD with m > 2 levels
Error propagation formula available (previous slide).
Consider these hypotheses:
If the number of basic iteration is increased, then the convergence is faster.
If m-level IAD is locally convergent then (m + 1)-level IAD is locally convergent.
If (m + 1)-level IAD is locally convergent then m-level IAD is locally convergent.
(CTU in Prague)
Algebraic MG for stochastic matrices
CMAM 2012
17 / 19
Multi-level IAD with m > 2 levels
Error propagation formula available (previous slide).
Consider these hypotheses:
If the number of basic iteration is increased, then the convergence is faster. NO
If m-level IAD is locally convergent then (m + 1)-level IAD is locally convergent.
If (m + 1)-level IAD is locally convergent then m-level IAD is locally convergent.
(CTU in Prague)
Algebraic MG for stochastic matrices
CMAM 2012
17 / 19
Multi-level IAD with m > 2 levels
Error propagation formula available (previous slide).
Consider these hypotheses:
If the number of basic iteration is increased, then the convergence is faster. NO
If m-level IAD is locally convergent then (m + 1)-level IAD is locally convergent.
If (m + 1)-level IAD is locally convergent then m-level IAD is locally convergent.
(CTU in Prague)
Algebraic MG for stochastic matrices
CMAM 2012
17 / 19
Multi-level IAD with m > 2 levels
Error propagation formula available (previous slide).
Consider these hypotheses:
If the number of basic iteration is increased, then the convergence is faster. NO
If m-level IAD is locally convergent then (m + 1)-level IAD is locally convergent. NO
If (m + 1)-level IAD is locally convergent then m-level IAD is locally convergent.
(CTU in Prague)
Algebraic MG for stochastic matrices
CMAM 2012
17 / 19
Multi-level IAD with m > 2 levels
Error propagation formula available (previous slide).
Consider these hypotheses:
If the number of basic iteration is increased, then the convergence is faster. NO
If m-level IAD is locally convergent then (m + 1)-level IAD is locally convergent. NO
If (m + 1)-level IAD is locally convergent then m-level IAD is locally convergent.
(CTU in Prague)
Algebraic MG for stochastic matrices
CMAM 2012
17 / 19
Multi-level IAD with m > 2 levels
Error propagation formula available (previous slide).
Consider these hypotheses:
If the number of basic iteration is increased, then the convergence is faster. NO
If m-level IAD is locally convergent then (m + 1)-level IAD is locally convergent. NO
If (m + 1)-level IAD is locally convergent then m-level IAD is locally convergent. NO
(CTU in Prague)
Algebraic MG for stochastic matrices
CMAM 2012
17 / 19
1
Algebraic multigrid (AMG)
2
Stochastic matrices, Markov chains, stationary probability distribution vector
3
Iterative aggregation - disaggregation (IAD) methods
4
Convergence and divergence of IAD
5
Conclusions and open questions
(CTU in Prague)
Algebraic MG for stochastic matrices
CMAM 2012
18 / 19
Conclusions
Necessary and sufficient conditions for local convergence of some IAD methods.
Error propagation formula for any number of levels.
No relation between convergence of m-level and (m + 1)-level IAD methods.
Open questions
Local convergence criteria for m ≥ 3 levels.
Appropriate ordering and aggregation of states.
Main question. Does the local convergence always imply the global convergence?
(CTU in Prague)
Algebraic MG for stochastic matrices
CMAM 2012
19 / 19
Conference
Preconditioning of Iterative Methods - Theory and Applications 2013, (PIM 2013)
Prague, Czech Republic, July 1 - 5, 2013.
In honor of Ivo Marek on the occasion of his 80th birthday.
Themes:
Preconditioning of sparse matrix problems, symmetric or non-symmetric,
arising in large-scale real applications.
Multilevel preconditioning techniques, including multigrid, algebraic multilevel, and domain
decomposition methods for partial differential equations.
Multilevel solution of characteristics of Markov chains.
Other related topics.
Deadline of Abstract Submission: February 23, 2013
Early Payment Deadline: April 27, 2013
E-mail address: [email protected]
Web page: http://pim13.fsv.cvut.cz
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Algebraic MG for stochastic matrices
CMAM 2012
19 / 19