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Threshold partitioning
for iterative aggregation – disaggregation method
Ivana Pultarova
Czech Technical University in Prague, Czech Republic
ILAS 2004
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We consider column stochastic irreducible matrix B of type N × N.
The Problem is to find stationary probability vector xp, || xp || = 1,
Bx p  x p .
We explore the iterative aggregation-disaggregation method (IAD).
Notation:
 Spectral decomposition of B, B = P + Z, P 2 = P, ZP = PZ = 0, r(Z) < 1
(spectral radius).
 Number of aggregation groups n, n < N.
 Restriction matrix R of type n × N. The elements are 0 or 1, all column sums
are 1.
 Prolongation N × n matrix S(x) for any positive vector x :
(S(x))ij := xi iff (R)ji = 1, then divide all elements in each column with the
sum of the column.
 Projection N × N matrix P(x) = S(x) R.
 || . || denote 1-norm.
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Iterative aggregation disaggregation algorithm:
step 1. Take the first approximation x0  RN, x0 > 0, and set k = 0.
s
step 2. Solve RB S(xk) zk+1 = zk+1, zk+1 Rn, || zk+1 || = 1, for
(appropriate) integer s, (solution on the coarse level).
step 3. Disaggregate xk+1,1 = S(xk) zk+1.
t
step 4. Compute xk+1 = B xk+1,1 for appropriate integer t,
(smoothing on the fine level).
step 5. Test whether || xk+1 – xk|| is less then a prescribed tolerance. If not,
increase k and go to step 2. If yes, consider xk+1 be the solution of
the problem.
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Propositon 1.
If s = t then the computed approximations xk, k = 1,2,…, follow the formulae
s
a) B P(xk) xk+1 = xk+1,
s
b) xk+1 = (I – Z P(xk))-1 xp,
c) xk+1 – xp = J(xk) (xk – xp),
where J(x) = Bs(I – P(x) Zs)-1(I – P(x))
and also J(x) = Bs(I – P(x) + P(x) J(x)).
Proposition 2.
s
Let V be a global core matrix associated with B . Then
J(x) = V(I – P(x) V)-1(I – P(x)) and J(x) = V(I – P(x) + P(x) J(x)).
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Note.
s
The global core matrix V is here ηP + Z . Using Z k→ 0 for k → ∞,
s
we have V = ηP + Z ≥ 0 for a given η and for a sufficiently large s.
s
s
This is equivalent to B = P + Z ≥ (1- η) P.
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Local convergence.
It is known that for arbitrary integers t ≥ 1 and s ≥ 1 there exists
a neighborhood Ω of xp such, that if xk Ω then xr  Ω, r = k +1, k + 2,…,
and that
k
|| xk+1 - xp|| ≤ c α || xk - xp||,
where c  R and α ≤ min{|| Vloc ||μ, ||(I-P(xp))Z(I-P(xp))||μ}, where ||.||μ is
some special norm in RN.
Here, Vloc is a local core matrix associated with B.
Thus, the local convergence rate of IAD algorithm is the same or better
comparing with the Jacobi iteration of the original matrix B.
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Global convergence.
From Proposition 2 we have
||J(xk)|| ≤ ||V|| ||I – P(xk)|| + ||V|| ||P(xk)|| ||J(xk)||,
i.e.
||J(xk)|| (1 – η) < 2η.
So that the sufficient condition for the global convergence of IAD is
η < 1/3, i.e. the relation
B s > (2/3) P
is the sufficient condition for the global convergence of IAD method.
(It also means r(Z s) ≤ 1/3. B s ≥ (2/3) P is equivalent to P/3 + Z s ≥ 0.
Then P + 3Z s ≥ 0 is a spectral decomposition of an irreducible column
stochastic matrix and then r(Z s) ≤ 1/3.)
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In practical computation of large problems we cannot verify the validity of
s
relation B ≥ η P > 0 to estimate the value of s.
k
But, we can predict the constant k for which B > 0. The value is known to
be less than or equal to N 2- 2 N + 2.
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We propose a new method for achieving B s ≥ ηP > 0 with some η > 0.
Let I – B = M – W be a regular splitting, M -1 ≥ 0, W ≥ 0. Then the solution
of Problem is identical with solution of
(M – W) x = 0.
Denoting Mx = y and setting y := y / || y ||, we have
(I – WM -1) y = 0,
where WM -1 is column stochastic matrix.
Thus, the solution of the Problem is transformed to the solution of
WM -1 y = y,
|| y || = 1,
for any regular splitting M, W of the matrix I – B.
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The good choice of M, W.
According to IAD algorithm we will use a block diagonal matrix M which is
composed of blocks M1, … Mn , each of them invertible.
To achieve (WM -1) s > 0 for low s, we need
 Mi-1> 0, i =1,…, n,
 nnz (WM -1) >> nnz (B), (number of nonzeros).
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Algorithm of a good partitioning.
step 1. For an apropriate threshold τ, 0 < τ < 1, use Tarjan’s parametrized
algorithm to find the irreducible diagonal blocks Bi, i = 1,…,n,
of the permuted matrix B, (we now suppose “B := permuted B”).
step 2. Compose the block diagonal matrix BTar from the blocks Bi,
i = 1,…,n, and set
M = I – BTar / 2 and W = M – (I – B).
Properties of WM -1 .
 WM -1 is irreducible.
 Diagonal blocks of WM -1 are positive.
 (WM -1) s is positive for s ≤ n2 - 2n + 3, n is the number of aggregation
groups. (n = 3 → s = 2)
 The second largest eigenvalue of the aggregated n × n matrix is
approximately the same as that of WM -1.
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Example 1.
 B1


B
...

0

0

B1 ... 0 
.... ... ... 

...  B1 
T
...
 0 0... 

 1 0... 0
B1  
... ...
...

 0 0... 1  
1

0
...

0
 0... ...  


   ... ... ...
 0... ... 0 


Matrix B is composed from n × n blocks of size m. We set ε = 0.01,
δ = 0.01. Then B is normalized.
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Example 1.
a) IAD method for WM -1
and threshold Tarjan’s
block matrix M, s = 1,
r(ZWM) = 0.9996. (Exact
solution – red, the last of
approximations - black circles).
b) Power iterations for
WM -1 and the same M as
in a), s = 1, r(ZWM) =
0.9996. (Exact solution – red,
the last of 500 approximations black circles. No local
convergence effect.).
c) Rates of convergence
of a) and b).
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Example 2.
 B1


B
...

0

0

B1 ... 0 
.... ... ... 

...  B1 
T
...
 0 0... 

 1 0... 0
B1  
... ...
...

 0 0... 1  
1

0
...

0
 0... ...  


   ... ... ...
 0... ... 0 


Matrix B is composed from n × n blocks of size m. We set ε = 0.01,
δ = 0.01. Then B := B + C (10% of C are 0.1) and normalized.
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Example 2.
IAD for B and
WM -1.
Power method
for B and
WM -1.
Convergence
rate for IAD
and power
method.
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Example 2.
Another
random entries.
a)
IAD for B and
WM -1.
b)
Power method
for B and
WM -1.
c)
Convergence
rate for IAD
and power
method.
ILAS 2004
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I. Marek and P. Mayer
Convergence analysis of an aggregation/disaggregation iterative method for computation
stationary probability vectors
Numerical Linear Algebra With Applications, 5, pp. 253-274, 1998
I. Marek and P. Mayer
Convergence theory of some classes of iterative aggregation-disaggregation methods for
computing stationary probability vectors of stochastic matrices
Linear Algebra and Its Applications, 363, pp. 177-200, 2003
G. W. Stewart
Introduction to the numerical solutions of Markov chains, 1994
A. Berman, R. J. Plemmons
Nonnegative matrices in the mathematical sciences, 1979
G. H. Golub, C. F. Van Loan
Matrix Computations, 1996
ETC.
ILAS 2004
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