Scientific Computing Tool

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1 Scentfc Comptng Tool akmar Poondla Department of Mathematcs and Compter Scence Soth Dakota School of Mnes and Technolog Emal: Hanng L Department of Mathematcs and Compter Scence Soth Dakota School of Mnes and Technolog Emal: hl9834@hotmal.com Jeff.S.McGogh Department of Mathematcs and Compter Scence Soth Dakota School of Mnes and Technolog Emal: eff.mcgogh@sdsmt.ed BSTRCT The sccess of hgh performance comptng n modelng scentfc and engneerng applcatons motvates the development of ambtos applcatons. n effcent solton for solvng scentfc problems on a clster has been and s stll of hgh nterest. In ths paper we present the Scentfc Comptng Tool. The Scentfc Comptng Tool provdes a Graphcal User Interface GUI n whch the ser can enter geometrc shapes n one two or three dmensons and nformaton abot the shapes. The tool wll then take the necessar parameters for solvng the lnear ellptc partal dfferental eqatons on a rectlnear doman. The program wll generate a seqental code or parallel code as reqested b the ser for a fnte dfference appromaton defned on the doman sng the teratve methods Jacob Gass Sedel Sccessve Over Relaaton methods. The seral code s mplemented n C. The parallel code s mplemented n C sng POOM lbrar. Ths tool allows the ser to comple rn and generate a graph n the GUI sng Gnplot. Kewords: Scentfc Comptng Doman Decomposton Ellptc PDE POOM Jacob Gass and SOR Graphcal User Interface.

2 Introdcton The compter scence department at SD School of Mnes and Technolog started a Beowlf Clster Proect Clster Comptng and Vsalzaton n sprng 00. Ths proect ams at sng off-the-shelf components to bld a parallel compter to perform the nmeros comple operatons reqred for vsalzaton of large data sets. We decded to come p wth a tool that smplfes the code development n Scentfc Comptng. The prpose of Scentfc Comptng Tool s to create a Graphcal User Interface n whch o can enter n geometrc shapes n one two or three dmensons. The tool then wll take the necessar parameters for solvng a lnear ellptc partal dfferental eqaton on a rectlnear doman and generate code for a fnte dfference appromaton defned on the doman sng the statonar teratve methods Jacob Gass Sedel and Sccessve Over Relaaton methods. Snce doman decomposton methods are based on parttonng of the doman of the phscal problem and each sb doman can be handled ndependentl ths tool s ver effectve on a clster. Ths tool streamlnes the development of scentfc codes. Ths ncldes the process of modelng dscretzaton solvng and parallelzaton of problems from varos felds of applcatons[5]. The varos felds of Scentfc Comptng nclde weather predcton Sesmc data processng astrophscs Nclear Engneerng and Image processng. Ths tool s readl accessble to scentfc applcaton developers whose backgrond does not nclde compter scence. Ths tool leverages estng sophstcated codes lke POOM to acheve ths. The Parallel Obect Orented Methods and pplcatons POOM Toolkt s an open-sorce software for wrtng hgh performance Scentfc Comptng Programs on parallel compters whch was orgnall developed b scentsts at dvanced Comptng Laborator at Los lamos Natonal Laborator LNL and s mantaned b CodeSorcer LLC [89]. lso a Cheetah messagng lbrar provded b LNL an nderlng messagng lbrar Messagng Passng Interface MPI from rgonne Natonal Laborator Comptaton Insttte as well as MM Shared Memor Lbrar are copled wth POOM n order to take the advantages of hgh performance parallel comptng of POOM over mltple processors or clster sstem The scentfc comptng tool provdes an eas-to-se nterface and s a far better means of commncaton than tet-based alternatves. Etensve se of vsal navgaton featres sch as bttons mens and trees and nttve manplaton of data wll make ths tool convenent for the scentfc comptng world. The tools needed to bld and test Java programs are avalable wthot charge. Sn makes the Java Development Kt JDK avalable over the Internet at where an ndvdal can download t. The JDK--whch ncldes the Java compler and nterpreter among other tools s ndobtedl ver smple to se.

3 Descrpton of the problem Partal Dfferental Eqatons descrbe the modelng of phscal processes takng place n or srrondngs. Partal Dfferental Eqaton PDE s a tpe of eqaton n whch the nknown can represent some of the thngs lke the temperatre or the shape of the wave or stress n a bent pece of metal. PDE s an eqaton nvolvng one or more partal dervatves of an nknown fncton of mltple varables [3]. general second-order partal dfferental eqaton looks lke ths. where a b c d e f and g can be fnctons of both the ndependent varables and and the dependent varable. Ths eqaton s sad to be ellptc when 0 4 < ac b parabolc when 0 4 = ac b and hperbolc when. 0 4 > ac b Ellptc partal dfferental eqatons arse sall from eqlbrm or stead-state problems and ther soltons [3]. The goal s to solve ellptc partal dfferental eqaton n the doman = Ω sbect to the Drchlet bondar condton =G on Ω. Ths problem can be solved sng the method of fnte dfferences. The method of fnte dfferences s eas to mplement and t gves good reslts of bondar vale problems. In ths method the dervatves appearng n the eqaton and the bondar condtons are replaced b ther fnte dfference appromatons []. Then the gven eqaton s changed to a dfference eqaton whch s solved b teratve procedres. B replacng the gven dervatves wth ther fnte dfference appromatons we get 1 1 h 1 1 k h 1 1 k 1 1 hk where h s the step sze between dscrete ponts and s eqal to N a b and k s the dscrete ponts and s eqal to M c d. 0 = g f e d c b a 0 = g f e d c b a

4 The above eqaton can be solved b dvdng Ω nto NM cells and applng central dfference formla at each of N 1 M 1 nteror grd ponts. Fgre 1 For the grd pont n the th row and th colmn of the grd gves the formla g4h k = 0 Eqaton g = 9 4h k where 1 N 1 N = 4ak = 4ch = bhk dhk ehk 9 5 = 4 fh k = 4ak = bhk 8ch dhk 8ak 6 3 = bhk = 4ch 7 ehk = bhk Eqaton1 can be solved sng statonar methods Jacob Gass-Sedel Sccessve Over Relaaton and non-statonar methods lke Congate Gradent Method and GMRES.

5 Estng pproach Grd based problems are common n scentfc comptng. n effcent solton to solve the partal dfferental eqatons has been and s stll of nterest. The followng are the tools that are commonl sed b scentfc comptng world. Matlab: Matlab s a hgh level nterpreted programmng langage generall sed for hgh performance nmercal comptaton and vsalzaton The Matlab PDE solver pdepe solves ntal-bondar vale problems for sstems of parabolc and ellptc PDEs n the one space varable and tme t. Mathematca: Mathematca[9] software developed b the compan Wolfram Research s a nmerc and smbolc calclaton sstem that ncorporates an ecellent programmng langage and the capact of ntegratng calclatons graphcs and tet n oneself docment electronc called notebook. Maple: The Maple sstem s an advanced mathematcal problem solvng and programmng envronment. PETSc: PETSc s a software lbrar that provdes data strctres and rotnes for the solvng scentfc applcatons modeled b partal dfferental eqatons on parallel or seral compters. To provde portablt across networks of workstatons PETSC ses cstomzed message-passng sstem. POOM: POOM Parallel Obect-Orented Methods and pplcatons s a collecton of templated C classes for wrtng parallel PDE solvers sng fnte-dfference and partcle methods. It provdes a varet of tools n spportng scentfc comptng wth featres of [6] Contaners and other abstractons for scentfc comptaton Spport for a varet of comptaton modes sch as data parallel stencl-based comptatons and laz evalaton as well as parallel and dstrbted comptaton programs wrtng. Robots of all nter process commncaton for parallel and dstrbted rotnes ot of order eecton and loop rearrangement for effcent program eecton. Wth POOM hgh-level abstractons the programs wrtten n POOM are mch shorter than conventonal FORTRN or C programs. The code s also easer to debg. The toolkt s compatble wth most of C complers. Lmtatons n estng software Solvng scentfc applcatons generall nvolves a great deal of arthmetc. One of the maor problems faced b the programmer s to convert the mathematcal concept nto effcent algorthm developed n some programmng langage. Thogh the estng software has come p wth a programmng envronment stll the ser s epected to wrte hs code for a specfed npt. Ths becomes an phll task to wrte a parallel code whch reqres a dfferent algorthm.

6 Scentfc Comptng Tool Ths tool generates code a seral code and parallel code for a ser specfed npt to solve a general ellptc partal dfferental eqatons. The seral code s mplemented n C. The parallel code s mplemented n C sng POOM lbrar. The ser nterface provdes a smlar look and feel to a GUI based program and s mplemented sng Java Swng for ncreased portablt to other operatng sstems. To the greatest degree possble the laot and fnctonalt of mens dalog boes and toolbars follows the standard GUI gdelnes for Software Desgn. The followng are the mportant classes sed n or sstem for generatng the code to solve ellptc partal dfferental eqatons. Start: Ths s the startng pont of the tool. Ths brngs p the splash screen and provdes the ser the detals of the sstem he s rnnng and t drects to the pdetool. Pdetool: Ths s the heart of the tool. Ths class represents the vew of the herarchcal data of the PDE. Fgre s the precedng fgre shows the nodes of the tree contan Ellptc Parabolc Hperbolc and ellptc node contanng the chldren one dmensonal two dmensonal three-dmensonal. When the ser rght clcks one dmensonal two dmensonal threedmensonal the correspondng sb nodes contanng lne rectangle cbe are obtaned. When the ser clcks the node lne rectangle ellpse crcle and cbe the correspondng obect s created n an nternal frame. The nternal frame can be reszed and moved throgh ot the wndow. When the ser clcks lne/rectangle/cbe a wzard s created that takes n the necessar npts for handlng the PDE problem. Ths s handled n the host class.

7 Host: Ths contans a handle to an arra of panels each representng one-step wzard process. Ths also captres the npt gven b the ser. Frame: Based on the npt entered b the ser ths class wrtes them n a sntactcall correct format and calls the correspondng solver to solve the partclar problem. The followng s the class dagram of the scentfc comptng tool. Fgre 3

8 parallel sample code generated b se of scentfc comptng tool for solvng a varable Posson eqaton wth varable coeffcents and bondar condtons s shown below. #nclde "Pooma/rras.h" #nclde <ostream> #nclde <fstream.h> #defne N 10 #defne N 10 #defne strde #defne Iteraton 00 #defne hsqare doblen-1.0/doblen*n #defne ksqare doblen-1.0/doblen*n //Defne the coeffcents a b and c. nlne doble doble doble; nlne doble Bdoble doble; nlne doble Cdoble doble; nlne doble doble doble {retrn ; nlne doble Bdoble doble {retrn ; nlne doble Cdoble doble {retrn ; //Bondar condtons to be appled. nlne doble g1dobledoble; nlne doble gdobledoble; nlne doble g3dobledoble; nlne doble g4dobledoble; nlne doble g1doble doble { retrn ; nlne doble gdoble doble { retrn ; nlne doble g3doble doble { retrn ; nlne doble g4doble doble { retrn ; // ppl a Jacob teraton on the gven doman. vod ppljacob const rra<> & V // the doman to be solved. const rra<> & a // the doman to be solved. const rra<> & b // the doman to be solved. const rra<> & c // the doman to be solved. const Range<1> & I // as sbscrpt const Range<1> & J // as sbscrpt { VIJ = 1.0/.0*ksqare*aIJ hsqare*bij * ksqare*aij*vi1j VI-1J hsqare*bij*vij1 VIJ-1 - cij*hsqare*ksqare ; //Calclate the sm of sqares errors n a D rra. //No modfcaton needs below. template<class ValeTpe class EngneTag> ValeTpe sm_sqr const rra< ValeTpe EngneTag> & { ValeTpe sm = 0.0; nt begn_0 =.frst0 end_0 =.last0 begn_1 =.frst1 end_1 =.last1; // Mst block before scalar loop. Pooma::blockndEvalate; nt man nt argc // argment cont char *argv[] // argment lst { ofstream osf; // Intalze Pooma. Pooma::ntalzeargc argv; // The arra to be solved zero ot. rra<> VN N; V = 0.0; // The rght hand sde fncton of the eqaton f: //Intalzaton rra<> an N; a = 0.0; rra<> bn N; b = 0.0; rra<> cn N; c = 0.0; // Mst block before scalar code. Pooma::blockndEvalate; //Defne f fornt = 0; < N; fornt = 0; < N; { a = 10; //coeffcent of b = B10; //coeffcent of c = C00; //rght hand sde fncton cn/n/ = doble -N/.0; fornt = 0; < N; { V0 = g100; VN-1 = g00; fornt = 0; < N; { V0 = g300; VN-1 = g400; // The nteror doman now wth nmber of strde. // No modfcaton s needed below at ths tme. Range<1> I1 N-strde1 strde J1 N-strde1 strde; // Iterate tll converged or mamm Iteraton steps. doble SSErr = 0.01; // anthng greater than threshold nt teraton; for teraton=0; teraton < Iteraton && SSErr > 1e-6; teraton { //Red block ppljacob V a b c I J; ppljacob V a b c I1 J1; //Black block

9 for nt = begn_0; <= end_0; { for nt = begn_1; <= end_1; { ValeTpe vale =.read ; sm = vale * vale; retrn sm; ppljacob Va b c I1 J; ppljacob Va b c I J1; //Compte resdal. SSErr = sm_sqr VI1J VI-1J*ksqare*aIJ VIJ1 VIJ-1*hsqare*bIJ - cij*hsqare*ksqare.0*ksqare*aij hsqare*bij*vij ; // Prnt ot the reslt. std::cot << "Iteratons = " << teraton << std::endl; std::cot << "Resdal = " << SSErr << std::endl; std::cot << "Solved doman sze n " << N << " " << N << "." << std::endl; //Wrte the reslt to fle. osf.open "reslt.tt"; osf << V << std::endl; osf.close ; // Clean p and report sccess. Pooma::fnalze; retrn 0; The entre code generaton was dvded nto three steps. The flow dagram shown below descrbes how the code generaton was handled effectvel wthn the scentfc comptng tool. Step I Defne program prototpe & header fles Step II Translate the ser npts to a sntactcall correct wa that can be nderstood b the compler Step III Create sorce code sng a template that handles the above npts.

10 Reslts and Dscsson The goal of ths proect s to come p wth a small prototpe n bldng a generc tool that s stable for scentfc comptng on a clster. s the frst step we tested the tool for the lnear ellptc partal dfferental eqatons. The generated code was compled and rn on the clster. In the case of parallel code eecton a clster wth 4 nodes rnnng Pentm III 440 MHz processor and 56 MB RM was sed for testng performance. 10/100 Base-T Faster Ethernet card s sed for commncatons between nodes. Fgre 4 llstrates the performance of the three solvers of sng dfferent algorthms for solvng a Posson eqaton on a rectanglar grd wth seqental code mplemented n C. smple method for solvng the above eqaton s to teratvel appl the eqaton ntl there s lttle change n. Ths s the Jacob teratve method for solvng the eqaton. modfcaton of ths process called the Gass-Sedel method ma fnd the solton n fewer teratons. s descrbed n Introdcton to Parallel lgorthms and rchtectres page 98 ths method sggests that we can calclate the new vales of all the even-part ponts sng eqaton 1; then calclate the new vales for all of the odd-part ponts sng the vales st calclated for the even part ponts[1]. The part of the pont /N /N s defned to be even f s even and odd f s odd. Ths wll case the method to converge n fewer teratons bt ma not be benefcal for small matrces snce ths reqres two commncatons per teraton nstead of one for Jacob s method. Gass- Sedel teraton method can be frther be mproved b changng the convergence rate sng the Sccessve over relaaton SOR method. pplcaton of SOR to the above set of eqaton gves m 1 = 1 m m 1 ω ω Where 1 < ω <..0 The vale of ω s normall. π 1 sn N N Where N N are the sze of the grd along the and -as.

Comparson of Performance of 3 Statonar Iteratve Methods 350 300 50 Jacoban Gass Sedel Sccessve over Relaaton Tme seconds 00 150 100 50 0 0 000 4000 6000 8000 10000 Nmber of Grd Ponts Fgre 4.

11 Comparson of Performance of 3 Statonar Iteratve Methods Jacoban Gass Sedel Sccessve over Relaaton Tme seconds Nmber of Grd Ponts Fgre 4. Comparson of performance of three algorthms mplemented n the seral code. The followng s a graph generated b the tool for solvng = 0 n the doman Ω=0101 sbect to the drchlet bondar condton = snh*sn10* on Ω..

12 s a comparson Fgre 5 shows the performance of eecton of a parallel code b applng Jacob Iteraton Relaaton method to solve the same problem over 4 nodes clster sstem. One ma wonder how ths parallel code cold be so effcent t mght have been too good to be tre. However wth carefl eamnaton of the sample parallel code one can observe that the mplementaton of Jacob Iteraton Relaaton algorthm has taken the benefts of data parallel epresson on POOM. s t has been show n the sample code both parameters and varable nknown were defned as rra obects. The ntalzaton of these arra obects was done wth sngle statement bt none for loop presents. The Range obects shown n the sample code one dmenson n ths case beneft the representaton of nde seqences wth non-nt strdes whch provdes an effcent wa to defne non-adacent arra elements n parallel fashon. The Red/Black block operatons wthn teraton loop frther mprove the parallel performance on comptng t redces the amont of memor that a program reqres [9]. Comparson of Performance of Parallel vs. Seqental Comptaton Tme seconds Parallell Code Seqental Code Nmber of Grd Ponts Fgre 5. Comparson of performance of parallel verss seqental code. nother mportant pont shold be addressed here regardng POOM hghl abstractve obects s that t ses a tpe of obect called Engne a ke to the POOM hgh performance [7]. n engne s an abstractve obect mplemented n POOM whch performs the low-level vale storage comptaton and element wse access for a contaner [6]. The fncton sm_sqr shown n the sample code takes the advantages of Engnes. It s completel dfferent form other langages that se whole-arra operatons whch sall reqres temporar arra for holdng data. In fact drng the comptng process the arras do not store data n POOM bt act as handles on an engne. The engne knows how to evalate and retrn vales on the gven sets of ndces

13 correspondng the predefned arras. That s engne can reference data storage drectl and translate a set of ndces nto a vale b lookng p the vale based on the ndces n memor. n nterestng observaton from Fgre 5 s the overhead of nterprocess commncaton among the nodes of clster for parallel code eecton. Intall the grd sze s relatvel small sch that seqental code eecton s mch faster than parallel code becase the commncaton overhead can t be elmnated for the parallel code. In fact for a small grd sze the parallel code s not the frst choce for effcenc concern. Conclsons & Ftre Enhancements In ths proect t s am to develop a tool that smplfes the code development n scentfc comptng. The tool can generate both seral and parallel code for a general ellptc PDE problem accordng to the ser specfed npt n a fashon of less tme consmng wth ser-frendl nterface. Wth some smple modfcatons for eample addng a node to the tree strctre t can be etended to solve dfferent knd of problems. Crrent work that we have done s attemptng to solve lnear ellptc partal dfferental eqatons sng the statonar methods. Crrentl we are workng on the nonstatonar methods sch as Congate Gradent method and GMRES and the wll be added n the net release. Ths tool solves ellptc partal dfferental eqatons on a rectlnear doman. Ths can be etended to lnear two-dmensonal frst order sstems of ellptc partal-dfferental eqatons PDE s and assocated bondar condtons over a fnte non of rectangles. Detals can be fond n [8]. Scentfc Comptng Tool crrentl can generate parallel code that ses data parallel mechansm. Wth the clster sstem t wold be more effcent that machne parallelsm cold be ntrodced wth the consderaton of load balancng over all nodes of whole clster sstem. In order to have fll benefcar of POOM the tool shold be able to spport parallel comptaton over a fll dstrbted clster sstem. References 1. Thomson L.F. Introdcton to Parallel lgorthms and rchtectres: rras Trees and Hpercbes. San Mateo Calforna: Morgan Kafmann Grewal B.S. Hgher Engneerng Mathematcs Khanna Pblcatons Dchatea P. and ZachMann D.W. Partal Dfferental Eqatons Scham s otlne seres n mathematcs Pblsher Naghton P. and Schldt H. The Complete Reference Java 3 rd. edton Tata McGraw-Hll Norton C.D. Thess work on Obect orented paradgms n scentfc comptng Department of Compter Scence Rensselaer Poltechnc Insttte Tro New York US. 6. Oldham J.D. POOM C Toolkt for Hgh Performance Parallel Scentfc Comptng Codesorcer LLC. March 1 st. 001.

14 7. Wllams T.J. Renders J.W. Hmphre W.F. and Cmmngs J.C. POOM User Gde Parallel Obect-Orented Methods and pplcatons Los lamos Natonal Laborator HOHN.M. On the Solton of Med Bondar Vale Problems n Elastct. Ph.D. thess Department of Mathematcs Unverst of Utah SaltLake Ct UT USDec Edcatonal applcatons of Mathematca cknowledgements We wold lke to thank Dr.Gregg Stbbendeck for the valable sggestons n desgnng the nterface. We wold lke to thank the Beowlf Clster grop at SDSM&T for ther spport. We wold lke to thank all the ndvdals who are drectl or ndrectl nvolved n the sccessfl completon of ths proect.

Modeling Local Uncertainty accounting for Uncertainty in the Data

Modeling Local Uncertainty accounting for Uncertainty in the Data Modelng Local Uncertanty accontng for Uncertanty n the Data Olena Babak and Clayton V Detsch Consder the problem of estmaton at an nsampled locaton sng srrondng samples The standard approach to ths problem