Bootstrap AMG for Markov Chain Computations

Transcription

1 Bootstrap AMG for Markov Chan Computatons Karsten Kahl Bergsche Unverstät Wuppertal May 27, 200

2 Outlne Markov Chans Subspace Egenvalue Approxmaton Ingredents of Least Squares Interpolaton Egensolver Bootstrap Setup Bootstrap AMG for Markov Chan Computatons, Karsten Kahl 2/6

3 Fnte homogeneous Markov Chans States s,..., s n, n N Transton probabltes p(s j s ) Transton matrx T = (t j ),j R n n t j = p(s j s ) s 7 s 2 Example: T = s s 2 5 Bootstrap AMG for Markov Chan Computatons, Karsten Kahl /6

4 Propertes of Markov Chans Column-stochastc transton matrces T, t T = t, t = ( ) T rreducble exstence of unque steady-state soluton Tx = x, x > 0, x = spec r (B) U (0) = {z C, z < } Equvalent formulaton as system of lnear equatons Ax = (I T ) x = 0 t A = 0, span({x}) = null r (A), spec r (A) U () Bootstrap AMG for Markov Chan Computatons, Karsten Kahl /6

5 Power Method Iteratve method for computaton of steady-state soluton x k+ = Tx k = (I A) x k = x k Ax k Power method for T Rchardson teraton for A Wth = λ > λ 2... λ n 0 egenvalues of T x k x 2 = c λ 2 k Slow convergence for λ 2, yet fast reducton of components of x k correspondng to λ << Bootstrap AMG for Markov Chan Computatons, Karsten Kahl 5/6

6 Subspace Egenvalue Approxmaton Ingredents of Least Squares Interpolaton Egensolver Bootstrap Setup Approxmaton of Egenpar Tx = λ max x n a subspace V Bass v,..., v k of V, V = ( v v k ) Galerkn formulaton V t TVy = λ max V t Vy x = Vy x, λ max λ max Petrov-Galerkn formulaton W t TVy = λ max W t Vy x = Vy x, λ max λ max Specal case: Krylov subspaces V = K k Orthonormal bass v,..., v k, V t V = I B-orthogonal bass v,..., v k and w,..., w k, W t V = I Bootstrap AMG for Markov Chan Computatons, Karsten Kahl 6/6

7 Subspace Egenvalue Approxmaton Ingredents of Least Squares Interpolaton Egensolver Bootstrap Setup Ingredents of (Needed, Gven) Sparse lnear system of equatons Au = f Herarchy of sparse systems of lnear equatons A l u l = f l Approprate smoothng teratons S l Defnton of restrcton and nterpolaton R l+ l : V l V l+ and P l l+ : V l+ V l Defnton of operator herarchy A = A 0, A,..., A L A l+ = R l+ l A l P l l+ Columns of P l l+ form sparse bass for subspace V l+ n V l complementary to the smoother Bootstrap AMG for Markov Chan Computatons, Karsten Kahl 7/6

8 Subspace Egenvalue Approxmaton Ingredents of Least Squares Interpolaton Egensolver Bootstrap Setup Least Squares Interpolaton Computaton of nterpolaton weghts (p ) j, F, j C L C (p ) = k s= ω s (u (s) (p ) j u (s) j ) 2 mn p j C Test vectors u (),..., u (k) R n, Set of nterpolatory ponts C C for F ( Weghts ω s R + to bas LS ft, e.g., ω s = u (s) Au (s) Splttng of varables Ω = F C, Interpolaton P from C to Ω, ( ) Pfc P =, p I j 0, F, j C ) 2 Bootstrap AMG for Markov Chan Computatons, Karsten Kahl 8/6

9 Subspace Egenvalue Approxmaton Ingredents of Least Squares Interpolaton Egensolver Bootstrap Setup Egensolver Bootstrap Setup Test vectors ntally postve random, slghtly smoothed S η u (),..., S η u (k) MG herarchy wth nterpolaton Pl+ l, averagng restrcton R l+ l ( t R l+ l = t ) A l+ = R l+ l A l Pl+ l B l+ = R l+ l B l Pl+ l Bootstrap Multgrd Egensolver f Coarsest level then V L = {v (L) } =,...,kv A L v (L) = λ (L) B L v (L) else v (l) = Pl+ l v (l+) = λ (l+) λ (l) ( ) Relax A l λ (l) B l Update λ (l) end f v (l) = 0 = A l v (l),v (l) 2 B l v (l),v (l) 2 New approxmaton of steady-state vector, enhancement of TV set, qualty control of MG herarchy Bootstrap AMG for Markov Chan Computatons, Karsten Kahl 9/6

10 Subspace Egenvalue Approxmaton Ingredents of Least Squares Interpolaton Egensolver Bootstrap Setup Bootstrap Multgrd Egensolver Cyclng Strateges Relax on Au = 0, u U Compute V, s.t., Av = λbv, v V Relax on Av = λbv, v V Relax on Au = 0, u U and Av = λbv, v V Bootstrap AMG for Markov Chan Computatons, Karsten Kahl 0/6

11 Unform 2D Network 2 2 N MLE parnold 7 V 8 V 0 V 9 V 2 Accuracy 0 8 ω-jacob smoother wth ω =.7 TV sets U = V = 6 V(2, 2)-cycle, calber 2 Coarsest grd 5 5 Bootstrap AMG for Markov Chan Computatons, Karsten Kahl /6

12 Tandem Queueng Network µ y µ y µ µ µ y µ y µ x µ x µ x µ µ x µ y µ x µ µ x µ y µ =, µ x = 0, µ y = 0 N MLE parnold 6 V 6 V 6 V 7 V Accuracy 0 8 ω-jacob smoother wth ω =.7 TV sets U = V = 6 V(2, 2)-cycle, calber 2 Coarsest grd 5 5 Bootstrap AMG for Markov Chan Computatons, Karsten Kahl 2/6

13 Random Planar Graph (a) Random planar graph, n = 256 (b) Coarsenng, n = 256 p(j ) = d out (j) Bootstrap AMG for Markov Chan Computatons, Karsten Kahl /6

14 Random Planar Graph N MLE parnold 8 V 0 V 0 V V Accuracy 0 8 ω-jacob smoother wth ω =.7 TV sets U = V = 6, calber 2 V(2, 2)-cycle, -grd method Bootstrap AMG for Markov Chan Computatons, Karsten Kahl /6

15 Conclusons and Outlook Conclusons Egensolver Reuse of AMG herarchy as a precondtoner Promsng and scalng results for smple Markov Chans Outlook More complex Markov Chans Theory for LS nterpolaton and MG Egensolver Effcent (parallel) mplementaton of Bootstrap AMG Bootstrap AMG for Markov Chan Computatons, Karsten Kahl 5/6

16 Thank you for your attenton! Bootstrap AMG for Markov Chan Computatons, Karsten Kahl 6/6