SCHOOL NAVAL POSTORADUATE. Monterey, California. ckh SJAN D 00F. Beny Neta. November 1988

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1 NPS NAVAL POSTORADUATE Monterey, California SCHOOL 00F N SJAN D ckh SOLUTION OF LINEAR INITIAL VALUE PROBLEMS ON A HYPERCUBE Beny Neta November 1988 Approved for public release; distribution unlimited Prepared for: Texas Tech University Lubbock, TX

2 NAVAL POSTGRADUATE SCHOOL MONTEREY, CALIFORNIA Rear Admiral R. C. Austin Superintendent Harrison Shull Provost This report was prepared in conjunction with research conducted for the National Science Foundation and funded through Texas Tech University. Reproduction of all or part of this report is authorized. Prepared by: BENY NETA* Associate Professor of Mathematics Reviewed by: Released by: HAROLD M. FREDRICKSEN Chairman Dean Policy Sciences *Formerly associated with Texas Tech University

3 UNCLASSIFIED REPORT DOCUMENTATION PAGE i4 RiPU 1 SiE.URIIY LtAS ta,(aiion to N h= Ki.ii M... I-: i!, SIELUNiIY LLA )I.ILAhIUN AUIHOHIIY J LIIKINIUIIUN, AVALAdiLIIY O i'0ort 'D Oi..AS I4.ATIONI UWNIRAOIN i SLM DUIA Approved for public release; distribution unlimited 4 PtLHfOHMIN OHLiANiZAIION IEPOHI NuMItstk MONIIUiNj. UHuA1NtAIIUN R4LP091 NUMdtKILS) NPS ~~~~~(it NPS NAAML Or POitUHrMING OiiANiZA IIUN 0 D L.tl-I'l(i,YMdUL ;a NAMIE O" MUNIrT..) IN, ORjANiLArION I,ppicabe) Na i n l S e ce F u d t o,b53 National Science Foundation 6C AODESS (City. State. and ZIPCOce) /b AODORISS (Cory. Stare. adrl ZIP Cooe Washington, D.C ba I,,AN'E Oi FUNDINGiSPONSO.ihNG ISD OFFICE SYMBOL 9 PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER ORGAt.iZATIONj (it applicable) Texas Technical University 1 6 AUORES (Cty. State. and ZIP Cooe) 10 SOIuRCE 0; FLNDING NUMBERS PROGRAM6 PRO.ECT TASK WORK. UNIT Lubbock, TX ELEMENT NO NO NO. ACCESSION NO.1 1IEI i (incilul SeCUnrY CliaSsetiCtion) Solution of Linear Initial Value Problems on a Hypercube -1i2 PERSONAL AUTHOR(S) Beny Neta and C. P. Katti :i3a TYPE OF REPORT 13b TIME COVERED II QATEOFREPC5 ejr.month.,day) 15 PAGE COUNT :Technical Report I FROM 8/1/86 TO 9/30/88 ' ovember 12 '16 SUPPLEMENTARY NOTATION (OSATI CODES 18 SUBJECT TERMS (Continue on reverse if necessary and identity by block number) VELD I GROUP SUB-GROUP initial value problems, parallel scheme, hypercube, * Ibox scheme, recursive doubling technique 1 19 ABSTRACT (Continue on reverse It necessary and identity by block number) There are many articles discussing the solution of boundary value problems on various parallel machines. The solution of initial value problems does not lend itself to parallelism, since in this case one uses methods that are sequential in nature. Here we develop a parallel scheme for initial value problems cased on the box scheme and a modified recursive doubling technique. Fully implicit Runge Kutta methods were discussed by Jackson and Norsett (1986) and Lie (1987). capabilities. Lie assumes that each processor of the parallel computer having vector 20 DISIRIUTIONIAVAILABILITY OF ABSTRACT 21 ABSTRACT SECURITY CLASSIFICATION ERIJC(ASSIFIEDtJNLIMITED 0 SAME AS RPT 0 OTIC USERS UNCLASSIFIED 91a NAME OF ESPUNSiBLE INDIVIDUAL Area Code) 22c. OFICI 53NEPHdEIre SYMBOL Beny Neta ( _3n2 5 3N 0 DD FORM 1473, 64 MAR BIAPH@ostionlhyo@use8ufltSi@ifIust$d SfCURITY CLASSIFICATION OF THIS PAGE All other edition ae obsolete. UNCLASSIFIED,..,:...,.:... ;.,,,...,.,.....':,,.',,....,... =.UNCLASSIFIED ='-.,,,,r-l

4 SOLUTION OF LINEAR INITIAL VALUE PROBLEMS ON A HYPERCUBE C. P. Katti Jawaharlal Nehru University School of Computer and Systems Sciences New Delhi, India and B. Neta Code 53Nd Monterey, CA 'Abstract There are many articles discussing the solution of boundary value problems on various parallel machines. The solution of initial value problems does not lend itself to parallelism, since in this case one uses methods that are sequential in nature. Here we develop a parallel scheme for initial value problems based on the box scheme and a modified recursive doubling technique. Fully implicit Runge Kutta methods were discussed by Jackson and Norsett (1986) and Lie (1987). Lie assumes that each processor of the parallel computer having vector capabilities. ( J 1 Introduction We consider the solution of linear initial value problems on a hypercube. "By a hypercube we intend a distributed memory MIMD computer with communication between processors... via a communication network having the topology of a p-dimensional cube, with the vertices considered as processors and the edges as communication links" (Keller and Nelson, 1987). See also Fox (1984, 1985, 1

5 1987) and Fox and Otto (1984). Our method of solution is based on the box scheme to discretize the system of initial value problems y' = Ay + f(x) y(a) = yo where y and f are n-dimensional vectors and A is an n x n matrix. The resulting system of equations is solved by a modified version of the recursive doubling technique (see Stone, 1973). In the next section, the discretization is described and the resulting system of equation is given. Section 3 will describe the modified recursive doubling technique and its application to our system. It will be interesting to experiment with the method and compare the results to a sequential initial value solver of the same order. 2 The Single Step Method Consider the system of initial value problems y =A(x)y + f(x), a < z < b y(a) =y' (1) where Y = (Y1,...,y),(X)) T A =(Y'10,.'-'n0) and A --- aj (), 1< i, j :n. Let where xj = a+ jh, j=o,1,...,m (2) b-a(3 h - 3)

6 be a uniform mesh. The box scheme (see e.g. Keller, 1976), applied to (1) yields S,; +h{a,+ (y,++y,)/2+ f+l},,(4) Y0O : Yo where A 3 1 A a+(i+ ) h) and yj is the approximation to y(xj). Let {j,, i = 1,2,...,s} be a strictly increasing sequence such that j, > 0 and j, = m. We shall compute the solution at the points xi = xj. Let 4pi be n x n matrices defined for each i Op, = z ;+ z 2I A;+_, i 1, 2,..., s, (5) 2=1- + where jo = 0 and h is sufficiently small so that I - Aj+ are nonsingular enniglr Similarly let the n-vector pi be +h (I _ h Aj,_ fj,_!, i = 1,2,...,s, (6) where go = 0 (7) and (2 2,+ 2(8)[( j = 0,...,ji - ji_1-2. Then it can be easily shown as in Keller and Nelson (1987), that ;Ion For Yji yi- + vi, i =1,2,...,s. (9) - 3 CoP' 1 V(M ~ o~ E G) o 0... %" S C 0 -. (

7 Remarks 1. The matrices to be inverted are of order n, the number of equations in the original system (1). 2. The last factor in the product defining Di is the matrix required in computing (pi. 3. The vector j,-1-j,, can be computed by (7) - (8) in the same loop one computes Ti since it requires the same matrices. 3 Parallel Evaluation To solve (9) on a hypercube with p = s processors, one can modify the recursive doubling technique developed by Stone (1973). Let b, = 1y o'+ P, bj = pj, j=,,., and let Y(j) be a function of bj, bj- 1,. ) b. -i l..., y-il. Then the following results can be proved using similar arguments as in Stone(1973). Theorem. Let Y 1 (j) satisfy the recurrence relation YiI(JU) = Y(U) + 'P Yi (J- 1), i,j I (11) with boundary conditions Y(J)=b, ju) 1 Y_0(j) 0, j 0ori<0. (12) Then (i) yi+ 8 (j) = Y(j) + -I O2I-k-a+lYi(j - s) (13) k=j-s+

8 (ii) Y{(U) =.. +k+1 YI(k), i > j > 1, (14) k=1ls=k+l (iii) for 1 Y (j) = y,. (15) Corollary Y 2 (j) = 1(j) + {2j-k-i+} Y(j- i), i,> 1 (16) This corollary provides the recursive doubling algorithm for the solution of (9). Let ]-lpi'-k+l j < i k=1 M(j) = (17) k=j-i+l then (16) can be written as with boundary conditions y 2,(j) = Y(j) + M,(j) Y(j -i) i,j> 1 M 2 (j) = M(j)M(j- ),j>1 M,(U) = %, j"> MI (j) 1, i < 0 or j _< 0. (18) (19) We are now ready to state the algorithm. Algorithm For i = 1 to /2 in steps of i do: Yi(j)=Yi(j)+M,(j)Yi(j-i) i<j _8 Mzi (j) = Mi (j) Mi5(j - i) < j < 5

9 Next i. From our theorem, Y(') = yj, for 1 < j < s, so that Ys is the solution of (9). We note that for each i, the indices pertaining to j are executed simultaneously on s processors. Since i doubles during each iteration, log 2 s iterations are required for computation. Acknowledgements This work was partially supported by the National Science Foundation, through Grants No. INT and INT References 1. H.B. Keller and P. Nelson (1987). Hypercube Implementations of Parallel Shooting, Applied Mathematics and Computations, to appear. 2. G. Fox (1984). Concurrent Processing for Scientific Calculations, Proceedings COMPCON 1984 Conference. pp.70-73, IEEE Computer Society. 3. G.C. Fox (1984). Are Concurrent Processors General Purpose Computers?, IEEE Trans. Nucl. Sci.. NS-32, pp G.C. Fox and S. W. Otto (1984). Algorithm for Concurrent Processors, Physics Today, 37, No. 5, pp H.S. Stone (1973). An Efficient Parallel Algorithm for the Solution of a Tridiagonal Linear System of Equations, J. Assoc. Comput. Mach., 20, pp H. B. Keller (1976). Numerical Solution of Two Point Boundary Value Problems, SIAM, Reg. Conf. Ser. in Applied Math. 7. K. K. Jackson and S. P. Norsett (1986). Parallel Runge-Kutta Methods, unpublished report. 6

10 8. I. Lie (1987). Some Aspects of Parallel Runge-Kutta Methods, Institutt for Numerisk Mathcmatikk, Universitetet i Trondheim, Norway, ISBN , 39 pages. 7

11 DISTRIBUTION LIST Director 2 Defense Tech. Information Center Cameron Station Alexandria, VA Director of Research Administration Code 012 Library 2 Code 0142 Code 53 Center for Naval Analyses 4401 Ford Avenue Alexandria, VA NO. OF COPIES Professor Beny Neta 15 Code 53Nd Dr. C. P. Katti 5 J. Nehru University School of Computer & Systems Sciences New Delhi India Professor Paul Nelson Texas A & M University Department of Nuclear Engineering and Mathematics College Station, TX Professor H. B. Keller Department of Applied Mathematics California Institute of Technology Pasadena, CA

12 Professor H. Dean Victory, Jr. Texas Tech University Lubbock, TX Professor Gordon Latta Code 53Lz Professor Allan Goldstein Code 53Go Professor Arthur Schoenstadt Code 53Zh Department if Mathematics Professor M. M. Chawla, Head III/III/B-I, IIT Campus Hauz Khas, New Delhi India 2