1. WORKSHOP ERCIM 20041 Global convergence for iterative aggregation – disaggregation method Ivana Pultarova Czech Technical University in Prague, Czech Republic

2. WORKSHOP ERCIM 20042 We consider N × N column stochastic irreducible matrix B, not cyclic. The Problem is to find stationary probability vector x p, || x p || = 1, B x p = x p We explore the iterative aggregation-disaggregation method (IAD). Notation:  ||. || denotes 1-norm.  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 := x i 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.

3. WORKSHOP ERCIM 20043 Iterative aggregation disaggregation algorithm: step 1. Take the first approximation x 0 R N, x 0 > 0, and set k = 0. step 2. Solve R B s S(x k ) z k+1 = z k+1, z k+1 R n, || z k+1 || = 1, for (appropriate) integer s, (solution on the coarse level). step 3.Disaggregate x k+1,1 = S(x k ) z k+1. step 4. Compute x k+1 = T t x k+1,1 for an appropriate integer t, (smoothing on the fine level). Block-Jacobi, block-Gauss-Seidel, T = B… step 5.Test whether || x k+1 – x k || is less then a prescribed tolerance. If not, increase k and go to step 2. If yes, consider x k+1 be the solution of the problem.

4. WORKSHOP ERCIM 20044 Propositon. The computed approximations x k, k = 1,2,…, follow the formula x k+1 – x p = J(x k ) (x k – x p ), where J(x) = T t (I – P(x) Z s ) -1 (I – P(x)). If t > s and T = B then J(x) = B t-s K(x), where K(x) = B s (I – P(x) + P(x) K(x)).

5. WORKSHOP ERCIM 20045 Example. Let T = B, s = t = 1 and B =. Then r(Z) = 0. For x 0 = [1/12, 10/12, 1/12] T it is r(J(x 0 )) = 2.1429 while for x 0 = [10/12, 1/12, 1/12] T it is r(J(x 0 )) = 0.0732.

6. WORKSHOP ERCIM 20046 Global convergence. When B s ≥ η > ηP and T = B s, then for the global core matrix V corresponding to B s J(x) = V t (I – P(x) V ) -1 (I – P(x)) = V t-1 K(x) and ||K(x)|| ≤ ||V|| ||I – P(x)|| + ||V|| ||P(x)|| ||K(x)||, thus ||K(x)|| < 2(1 – η) / η. So that the sufficient condition for the global convergence of IAD is (1 > ) η > 2/3, i.e. B s > (2/3) P. In the opposite case, the value of t in J(x) = B t-1 K(x) = V t-1 K(x), can be easily estimated to ensure || J(x) || < 1, t ≥ log (η/2) / log (1- η).

7. WORKSHOP ERCIM 20047 We propose a method for achieving B s ≥ η > 0. Let I – B = M – W be a regular splitting, M -1 ≥ 0, W ≥ 0. Then the solution of Problem is identical with the 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.

8. WORKSHOP ERCIM 20048 The choice of M, W – algorithm of a good partitioning. step 1.For an appropriate threshold τ, 0 < τ < 1, use Tarjan’s parametrized algorithm to find the irreducible diagonal blocks B i, i = 1,…,n, of the (properly) permuted matrix B, (we now suppose “B := permuted B”). step 2. Compose the block diagonal matrix B Tar from the blocks B i, i = 1,…,n, and set M = I – B Tar / 2 and W = M – (I – B).

9. WORKSHOP ERCIM 20049 Properties of WM -1 obtained by the algorithm of a good partitioning:  WM -1 is irreducible.  Diagonal blocks of WM -1 are positive.  (WM -1 ) s is positive for “low” s, s ≤ n + 1, n is the number of the aggregation groups.  The second largest eigenvalue of the aggregated n × n matrix is approximately the same as that of WM -1.

10. WORKSHOP ERCIM 200410 Conclusion.  To achieve the global convergence of IAD method, we consider the original Problem in the form WM -1 y = y, where I – B = M – W and W, M is a (weak) regular splitting of I – B constructed by “the algorithm of a good partitioning”.  When n is the number of aggregation groups, (WM -1 ) n+1 is positive ( > η). Matrix WM -1 can be stored in the factor form.  The number of smoothing steps is given by t ≥ n + log (η/2) / log (1- η).  The computational complexity is equal to the IAD with the block Jacobi smoothing steps, but the global convergence is ensured here.

11. WORKSHOP ERCIM 200411 Example 1. 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.

12. WORKSHOP ERCIM 200412 Example 1. a) IAD for B and WM -1. b) Power method for B and WM -1. c) Convergence rate for IAD and power method.

13. WORKSHOP ERCIM 200413 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.