5.3 Preconditioning. - M is easy to deal with in parallel (reduced approximate direct solver)

Transcription

1 5.3 Precoditioig Direct solvers: Sequetial, loosig sparsity terative solvers: easy parallel ad sparse, but possibly slowly coverget Combiatio of both methods: clude precoditioer i the form - x - b, such that - is easy to deal with i parallel (reduced approximate direct solver) - spectrum of - is much better clustered Or iclude precoditioer - i the form x b, such that - is easy to deal with i parallel (reduced approximate iverse) - spectrum of is much better clustered

2 Statioary Precoditioers Geeral both sided precoditioig: x b (K y) b ad K y x Statioary iteratio to splittig N : Covergece depeds o - < hat is exactly a coditio for a good precoditioer: spectrum clustered aroud, - f splittig leads to fast covergece, tha is also a good precoditioer. this sese, pcg with statioary precoditioer ca be see as a accelaratio of the statioary method with splittig -N.

3 Statioary Precoditioers Jacobi splittig with D diag() gives Jacobi precoditioer : D Gauss-Seidel splittig D L leads to Gauss-Seidel precoditioer Relaxatio: x,ew : ( ω ) x - + ω x covex combiatio of old ad ew iterate (Jacobi or GS) Symmetrizatio: first iteratio with precoditioer secod iteratio with ew : + Special case damped GS SSOR: D ω ω L D ω D ω L 3

4 LU Precoditioer dea: pply Gauss-Eliatio algorithm, but oly o allowed patter! complete LU factorizatio LU Reduce i the OR-loops the idices to the idices with - allowed patter, e.g. LU(0) for patter of - values that are ot to small, LU for LU with treshold Leads to approximate LU factorizatio L U + R, precoditioer L U with all igored fill-i etries collected i R. LU: collect all igored fill-i etries o the related mai diagoal elemets maitais the row sum or the actio o (,,,) 4

5 Overview explicit precoditioers LU ad statioary methods use approximatios o itself. he resultig precoditioers are give by triagular matrices L, that have to be solved i each iteratio step: L - x! Strogly sequetial! Jacobi easy to parallelize, but slow covergece. Questio: How to derive precoditioers that lead to fast covergece ad are easy to parallelize? dea: id approximatios o -. he the solutio of the liear system give by the precoditioer, is oly x, a matrix vector product! 5

6 Parallel Precoditioig id precoditioer, that satisfies three coditios: (i) he computatio of is fast i parallel (ii) he applicatio x i each iteratio step is easy i parallel (iii) he spectrum or is clustered fast covergece Examples: or GS is (i) ad (iii) OK, but ot (ii) or Jacobi is (i) ad (ii) OK, but ot (iii) or LU is (iii) OK, but ot (i) ad (ii) 6

7 7 Polyomial Precoditioers q 0... ) ( 0 γ γ γ γ Characteristic polyomial for : Better approximatio by fidig regio S i R or C that cotais early all eigevalues, ad the fid polyomial p that is ear the iverse i S Solutio: Normalized Chebyshev polyomials dvatage of polyomial precoditioer: better i parallel Disadvatage: Not-optimal approximatio i the same Krylov subspace ) (, ) ( max ) ( λ λ λ p P S p p x Gives polyomial represetatio for - (γ 0 0) : ( ) ) (... 0 p γ γ γ γ herefore, it maes sese to approximate - by a polyomial i.

8 Sparse approximate verses Other approach for approximatig - by orm imizatio: over some sparsity patter P. Choice of the orm? - aalytic (to allow the explicit solutio of this problem) - easy to compute (i parallel) Optimal orm: robeius orm : a i, i,, trace ( ) <, > 8

9 SP i parallel irst, we choose the patter P i a static way a priori, e.g. as the patter of ( ) e e Hece, to imize the robeius orm, we have to solve Least Squares problems i the sparse colums of. his ca be doe fully i parallel! But costs for LS problems? 9

10 SP ad LS e 0 * 0 e * 0 0 ( J ) e Deote by J the set of allowed idices i the -th colum of. * * - * J ) ( J ) (:, 0

11 SP ad Sparse LS (:,J ) is a sparse rectagular matrix. ** idex set, shadow of J. * * We ca reduce the sparse LS to (,J ): * * * * (, J ) ( J ) e( ) ~ ~ e~ Solve small LS problem by Householder QR for (,J ) ~

12 Computig J e Delete superfluous zeros i Least Squares Problem: or idex set J i eep oly ( :, J ) ( :, J ) eep oly ozero rows (, J ) Solve small Least Squares problem i (, J ), e.g. by QR-decompositio, Householder method.

13 Sparsity Patter? - will be o more sparse! s a priori choice of a good approximate sparsity patter for we ca choose the patter of - - ( ) for some, - ε with sparsified - a combiatio of above ε by sparsificatio of : delete all etries with i, < ε 3

14 Dyamic Patter idig (SP) - Start with thi approximate patter J for - Compute optimal colum,opt (J ) by LS - id ew etry for such that,opt (J )+ λ e has smaller residual i the robeius orm. ( ) e ( ) r + λ( r ) + λ + λe ( e ) + λe Choose idex with r 0 ad λopt with r ( r ) r 4

15 Bloc SP Partitio the give matrix i small blocs ( x or 3 x 3) ad apply the robeius orm imizatio with blocwise patter. dvatage: Uderlyig bloc structure will also appear i the patter of - improved patter terative SP Start with patter of Costruct relative to ew matrix Costruct 3 relative to ew matrix.. dvatage: Cheaper, but iferior approximatio 5

16 actorized SP, SP pproximate the iverse Cholesy factor of L L ( L L ) L L L L L L L L Normal equatios give ( J, J ) L( J ) α e trace( L L) he same L results from (/ ) det( L L) 6

17 SP ad arget atrices P ssume, P is a good sparse precoditioer for. mprove P by computig ad solvig the above robeius orm imizatio. f is give by two parts, e.g. a advectio part ad a diffusio part we ca choose P as Laplacia relative to the diffusio part - easy to solve - ad the we add for improvig P relative to the advectio part 7

18 8 SP Probig v B u C B C ρ ρ 0 0 e e B C ρ ρ or example Geeralize robeius orm imizatio to Origial SP exteded by a additioal orm imizatio to deliver especially good results o vector e. ( ) ( ) B C W B C e ρ ( ) ( ) W e ρ Similarly

19 Probig id precoditioer of special form (tridiag, bad) for precoditioig a matrix that is ot give explicitly, but oly by its actio o certai vectors, e.g. e S f. Example: Schur complemet S. Choose e.g. e(,,,), precoditioer as diagoal matrix Ddiag(d,,d ). he we have to satisfy e D e S f. fter the computatio of f the solutio is give by d f. Disadvatage: Ca use oly very special patter for ad probig vectors e. SP regularizatio for the geeral probig method with ay patter ad ay e: ~ S + ρ e e S with ay sparse approximatio S ~ of S 9