Parallel Numerics. 1 Preconditioning & Iterative Solvers (From 2016)

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Cornelius Leonard
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1 Technsche Unverstät München WSe 6/7 Insttut für Informatk Prof. Dr. Thomas Huckle Dpl.-Math. Benjamn Uekermann Parallel Numercs Exercse : Prevous Exam Questons Precondtonng & Iteratve Solvers (From 6) Gven s the system of lnear equatons Ax = b wth A =, b = The exact soluton of ths system s x = (, 6, ) T and does not have to be calculated separately. a) Apply the statc SPAI algorthm to determne a part of precondtoner M for matrx A wth sparsty pattern J = {}. ( credts) Hnt: You can use the normal equatons to solve a least squares problem.. mn A M e = mn M (J ) M = mn M M M By applyng the Normal Equatons, we solve the least squares problem: (,, ) M = (,, ) M =. M =.

2 b) Combne your result of subtask a) wth.4.5 to get an actual rght-precondtoner M. Precondton Ax = b wth M and calculate two steps of the Jacob teraton of the precondtoned system. Use y () = (,, ) T as start vector. ( 4 credts) (If you dd not get any result n subtask a), then use M = (,,.6) T.) Â =.4.5. = r () = b Ây() =.8.5 =..5 y () = y () + D r () = + 4 =.9.4 r () 9 = b Ây() 5 =.8.5 =..5 y () = y () + D r () = + 4 = c) Wthout precondtonng, Jacob gves the followng sequence of solutons x () for Ax = b (agan wth a start vector x () = (,, ) T ): x () =, x () = 4, x () = 6,..., x () = ,... Compare ths result wth the actual soluton x and your soluton for the precondtoned system from subtask b). What s the reason for your observaton? ( credt) (If you dd not get any result n subtask b), then just compare the result from above wth the actual soluton x.) Obvously, the sequence x (), x (),... dverges whle the sequence from subtask b) converges. The reason for ths behavor s the usage of the relatvely lmted Jacob method and the precondtoner M: M mproves the propertes of A (better condton number) so that even the relatvely lmted Jacob method solves the system successfully. Sparse Gauss Elmnaton (From 5) Gven s the sparse symmetrc 5 5 matrx

3 4 A = a) Let A be an adjacency matrx. Draw the correspondng graph G(A)! ( credt) n n 5 n n 4 n b) Use Gaussan elmnaton wthout any pvotng to elmnate the frst column of A! What can be observed after ths elmnaton regardng the remanng matrx elements n comparson to the orgnal matrx A? ( credts) There s massve fll n whle elmnatng the frst column. After ths elmnaton step, there are no zeros at all n the resultng matrx whch means the matrx s not sparse any more. c) The multple mnmum degree reorderng s one way to do pvotng for sparse matrces. It works as follows: Defne r m := number of nonzero entres n row m. Choose pvot ndex by r = mn m r m. If ths holds for multple, then chose one of these randomly. Do the pvot permutaton and the elmnaton.

4 Use Gaussan elmnaton wth multple mnmum degree reorderng to elmnate the frst column of A! What can be observed after ths elmnaton regardng the remanng matrx elements n comparson to the result of subtask b)? ( credts) Hnt: Snce A s a symmetrc matrx, rows and columns have to be nterchanged when pvotng to keep the symmetrc structure. Snce r = r = r 4 = r 5 = all have the same mnmum degree, one of these rows can be chosen randomly. We chose = : The fll n s much smaller than wthout pvotng. Wth multple mnmum degree reorderng, there are much more zeros n the transformed matrx then wthout (6 vs. ) whch also have a sparsty pattern. d) Brefly descrbe how multple mnmum degree reorderng can be used to parallelze Gaussan elmnaton for sparse matrces! ( credt) Assumng there are multple where r = mn m r m holds, then the elmnaton can be done n parallel where every processng element s dong the elmnaton wth a dfferent as pvot element. To do so, the correspondng nodes n n G(A) must not be neghbors. Ths approach can also be appled when havng dfferent degrees but ths can lead to load mbalances. Parallel Jacob (From ) From the lecture and the tutorals you are famlar wth the Jacob statonary teratve method for solvng a lnear system Ax = b where A s an n n matrx, b the gven rght-hand sde vector and x the soluton vector to be computed. Gve a parallel mplementaton of Jacob s method by answerng the followng questons. Assume that a block dstrbuton of all vectors and a block-row dstrbuton of A s avalable on all processng elements (PEs). The followng varables used n the source code are place holders for: p number of processng elements (PEs) n dmenson of the underlyng problem my_rank rank ID of current PE max_ter maxmum number of teratons to perform A_local local block-row dstrbuton of A b_local, x_local local block dstrbuton of b and x, respectvely a) Several data has to be avalable on all processng elements (PEs). Gve the MPI/C-code to make the dmenson of the underlyng problem (varable n) and the maxmum number of teratons (varable max_ter) avalable on all PEs. Provde only the requested source

5 code lnes,.e. no complete program s necessary. Followng source code lnes are requested: MPI Bcast(&n,, MPI INT,, MPI COMM WORLD); MPI Bcast(&max ter,, MPI INT,, MPI COMM WORLD); b) Formulate the component-wse representaton of the Jacob method for x (k+) j. Gve addtonally a pseudocode formulaton to compute the complete vector x (k+). The component-wse representaton: x (k+) j = a jj =, n j ( a j x (k) ) + b j Pseudocode for Jacob wth SAXPY for nner for-loop and GAXPY for outer for-loop: x (k+) = for j =,..., n...for =,..., n...f j...x (k+)...end...end end for =,..., n...x (k+) end = x (k+) = x(k+) +b a x (k) j a j c) Complete the followng routne vod Parallel_Jacob(...) wth own source code at lne. The routne should compute a parallel verson of the statonary Jacob method. Assume that n s a multple of p and that the macro Swap(x,y) n lne s avalable. Note that the teraton s stopped f max_ter s reached or x new x old < 6. :#defne Swap(x,y) {float* temp; temp = x; x = y; y = temp;} : :vod Parallel Jacob(MATRIX T A local, float x local[], float b local[], 4:...nt n, nt max ter, nt p, nt my rank) { 5:...nt local, dag, j, ter num; 6:...float x temp[max DIM], x temp[max DIM]; 7:...float* x old; 8:...float* x new; 9: :.../* Intalze local temporary x */ :...MPI Allgather(b local, n/p, MPI FLOAT, x temp, n/p, MPI FLOAT, :...MPI COMM WORLD); : 4:...x new = x temp; 5:...x old = x temp;

6 6:...ter num = ; 7:...do { 8:...ter num++; 9:.../* Interchange x old and x new */ :...Swap(x old, x new); : :.../* *** nsert code for parallel Jacob here *** */ : 4:...} whle ((ter num < max ter) && (norm(x new,x old,n) >= e-6)); 5:} /* run through own local part of vector */ for ( local = ; local < n/p; local++){... dag = local + my rank * n/p;.../* copy b to x */...x local[ local] = b local[ local];.../* frst sum from :dag- */...for (j = ; j < dag; j++)...x local[ local] = x local[ local] - A local[ local][j]*x old[j];.../* second sum from dag+:n */...for (j = dag+; j < n; j++)...x local[ local] = x local[ local] - A local[ local][j]*x old[j];.../* dvde by dagonal element */...x local[ local] = x local[ local] / A local[ local][ dag]; } /* collect new global vector x new, must be avalable for norm */ MPI Allgather(x local, n/p, MPI FLOAT, x new, n/p, MPI FLOAT, MPI COMM WORLD);

An Application of the Dulmage-Mendelsohn Decomposition to Sparse Null Space Bases of Full Row Rank Matrices

An Application of the Dulmage-Mendelsohn Decomposition to Sparse Null Space Bases of Full Row Rank Matrices Internatonal Mathematcal Forum, Vol 7, 2012, no 52, 2549-2554 An Applcaton of the Dulmage-Mendelsohn Decomposton to Sparse Null Space Bases of Full Row Rank Matrces Mostafa Khorramzadeh Department of Mathematcal