COSC 6374 Parallel Computation

Transcription

1 COSC 67 Parallel Comptaton Partal Derental Eqatons Edgar Gabrel Fall 0 Nmercal derentaton orward derence ormla From te denton o dervatves one can derve an appromaton or te st dervatve Te same ormla can be obtaned rom te Taylor seres, e.g. lm 0 y n n n

2 Center Derence Formla A better ormla s derved lookng at te ollowng two terms Sbtractng eqaton : rom : leads to s qadratc n te error term!! [...] ] [ y n n n : : Center Derence Formla or nd Dervatves Etend : and : by an addtonal term Addng bot eqatons leads to!!!! [...] ] [

3 Nmercal derentaton - smmary Forward derence ormla: Center derence ormla or te st dervatve: [ ] Center derence ormla or te nd dervatve: [ ] Derental eqatons - termnology Derental eqatons: eqatons contanng te dervatve o a ncton as a varable An ordnary derental eqaton ODE only contans nctons o one ndependent varable A partal derental eqaton PDE contans nctons o mltple ndependent varables and ter partal dervatves Te order o a derental eqaton s tat o te gest dervatve tat t contans Te goal s to nd a ncton yt wose dervatves lll te gven derental eqatons, e.g. n n y t t, y, y, y,..., y

4 Fnte Derences Approac or Solvng Derental Eqatons I te analytc solton o te DE can not be determned, calclate an appromate solton n dscrete locatons Replace te dervatves n te DE by an accordng appromaton ormla y t [ y t y t ] y t [ y t y t y t ] Eample I Solve te ollowng two pont bondary vale problem sng te nte derence metod d y dy d d y 0 y Lets assme te ponts o nterest are eqally spaced b a a 0 n n e.g. or =0., te mes ponts are 0 0, 0., 0., 0.6, 0.8, 5.0 De to te bondary vales: y - y are nknown y 0 y 0 y 5 y 5

5 Eample II Dscrete verson o te ODE sng central derences: y 0 0 y y y y y 0 0 y y y y y y y 5 y y y 0y y 0 : Eample III 0y0 50y 0y y 0y 00. : 50y 0y : 0y 50y 0y : 0y 50y 0y 6 : 0y 50y 68 or y y 0 y 50y 6 68 A y b 5

6 Solvng Ay=b sng an teratve solver e.g. te B-CGSTAB algortm Gven A,b and an ntal gess y 0 r0 b Ay 0 Gven rˆ sc tat rˆ T 0 r v p or =,, rˆ T 0 r p r v p v Ap T rˆ 0 v s r v t As T t s T t t y y p s r s t Matr-vector mltplcaton Scalar prodct Scalar prodct n parallel Scalar prodct: s N 0 a[ ]* b[ ] Parallel algortm s N / 0 N / a[ ]* b[ ] N N / a[ ]* b[ ] N / alocal[ ]* blocal[ ] alocal[ ]* blocal[ ] 0 0 rank0 Process wt rank=0 a 0... N b 0... N a N... N b N... N rank reqres commncaton between te processes Process wt rank= 6

7 Matr-vector prodct n parallel rs rs 0 rs 50 rs Process 0 Process rs rs 0 50 rs rs Process 0 needs Process needs Matr vector prodct n parallel II Introdcton o gost cells Process zero Process one Lookng at te sorce code, e.g p r v Ap p v snce te vector sed n te matr-vector mltplcaton canges every teraton, yo ave to pdate te gost cells beore every matr-vector mltply operaton 7

8 8 Matr vector prodct n parallel III so te parallel algortm or te same area s: v p r p Ap v Update te gost-cells o p, e.g - Process 0 sends p to Process - Process sends p to Process 0 D Eample - Laplace eqaton I -D Laplace eqaton Central dscretzaton leads to 0,, y y y 0,,,,,, j j j j j j,j -,j +,j,j+,j-

9 -D Eample: Laplace eqaton II Parallel doman decomposton Data ecange at process bondares reqred Halo cells / Gost cells Copy o te last row/colmn o data rom te negbor process Eample -D Laplace eqaton IV Process mappng and determnng negbor processes np : no o procs n drecton y npy : no o procs n y drecton n rank n n let rgt n p down rank rank np rank np ,,,, ,,,, 0 0,0,0,0,0 At bondares: set te rank o te accordng negbor to MPI_PROC_NULL a message sent to MPI_PROC_NULL wll be gnored by te MPI lbrary Easer: se cartesan topology nctons 9

10 Data storage wt gost-cells,j s stored n a matr!!assmng C!! nlocal : no o local ponts n drecton nylocal : no o local ponts n y drecton Dmenson o on an nner process = not beng at a bondary: n local, n ylocal wt : n local,: n contanng te local data ylocal gostcells need to be taken nto accont or loops, e.g. or =; < nlocal+; ++ { or j=; j<nylocal+; j++ { [][j] = } } y rom n p to n p to n let rom n let n nylocal + n nylocal to n rgt rom n rgt, n ylocal to be sent to n let 0 n local, n ylocal to be sent to n rgt n local, to be sent to n down n local, n ylocal to be sent to n p 0, n ylocal to be receved rom n let n local +, n ylocal to be receve rom n rgt n local, 0 to be receved rom n down n local, n ylocal + to be receved rom n p 0 to n down rom n down n local + n local 0

11 Commncaton n y-drecton MPI_Reqest req[8]; MPI_Irecv&[][nylocal+], nlocal, MPI_DOUBLE, np, tag, comm, &req[0]; MPI_Irecv&[][0], nlocal, MPI_DOUBLE, ndown, tag, comm, &req[]; MPI_Isend&[][nylocal], nlocal, MPI_DOUBLE, np, tag, comm, &req[]; MPI_Isend&[][], nlocal, MPI_DOUBLE, ndown, tag, comm, &req[]; // Watall mgt be postponed ntl commncaton // n -drecton as also been posted MPI_Watall, req, MPI_STATUSES_IGNORE; Commncaton n -drecton I Problem: te data wc we ave to send s not contgos n te memory Logcal vew o te matr Layot n memory o te same matr n C

12 Commncaton n -drecton II How to mplement te alo-cell ecange n - drecton? Send/Recv every element n a separate message + tecncally correct - very slow - derved datatypes - copy te data nto a separate vector/array and send ts array + works a more general nterace s provded by MPI to pack data nto a contgos ber beore sendng Commncaton n -drecton III doble *sblet, *sbrg, *rblet, *rbrgt; /* allocate te temporary bers */ sblet = malloc nylocal * szeodoble; sbrgt = malloc nylocal * szeodoble; rblet = malloc nylocal * szeodoble; rbrgt = malloc nylocal * szeodoble; sblet == NULL sbrgt == NULL rblet == NULL rbrgt == NULL { prnt Cold not allocate memory!\n ; MPI_Abort MPI_COMM_WORLD, ; } /* Pack te data beore sendng */ or j=0, =; <nylocal+; ++, j++ { sbrgt[j] = [nlocal][]; sblet[j] = [][]; }

13 Commncaton n -drecton IV /* Eecte now te real commncaton */ MPI_Irecvrblet, nylocal, MPI_DOUBLE, nlet, tag, comm, &req[0]; MPI_Irecvrbrgt, nylocal, MPI_DOUBLE, nrgt,tag, comm, &req[]; MPI_Isendsblet, nylocal, MPI_DOUBLE, nlet, tag, comm, &req[]; MPI_Isendsbrgt, nylocal, MPI_DOUBLE, nrgt, tag, comm, &req[]; MPI_Watall, req, MPI_STATUSES_IGNORE; /* Unpack te receved data */ or j=0, =; <nylocal+; ++, j++ { [nlocal+][] = rbrgt[j]; [0][] = rblet[j]; } ree rblet ; ree rbrgt ; ree sblet ; ree sbrgt ; Derved data types vs. pack/npack Advantages o derved datatypes: avods temporary bers smpler code potentally aster Advantages o pack/npack mgt lead to perormance advantages te same packed ber as to be sent to mltple targets many sers nd pack/npack nttve smlar to smply copyng te data tems nto a temporary ber