Applications of MPI I

Transcription

1 Applications of MPI I N-Dimensional Integration Using Monte Carlo Techniques Gürsan ÇOBAN

2 Outline Numerical Integration Techniques Monte Carlo Techniques for Numerical Integration Some MPI Examples (3D) Using external library with domain decomposition (CUBA library) Comparisons of different techniques 2

3 Numerical Integration Techniques Consider the definite integral Numerical Integration Methods - Newton-Cotes Rules (interpolation) - Quadrature 3

4 Numerical Integration Techniques Consider the definite integral Numerical Integration Methods - Monte Carlo Integration (random sampling) F(x) 4

5 Numerical Integration Techniques Consider the definite integral Partition the interval into small subintervals Construct a polynomial approximation to f in each subinterval. Approximate the integral over the subinterval by the values of polynomial Sum these approximated values 5

6 Numerical Integration Techniques Decomposition of computing domain x x x x x x n x? x x x x x x n x 6

7 Numerical Integration Techniques N-dimensional integration f ( x, x2,, xn) dxdx2 dx n Multidimensional integrals can also be solved by using listed numerical methods Especially interpolation based methods have implicit numerical structures which exaggerates the number of unknowns. These are expensive methods for high dimensions 7

8 Numerical Integration Techniques For multidimensional Integral, if we divide each dimension into 5 subintervals we have, subdomains Quadrature based methods still suffer from high dimension operations (nested structure). 0 f ( x) f ( x) dx 0 0 i 0 w i f ( x, z f ( x ) i,, z 9 ) dz dz 9 8

9 Numerical Integration Techniques Monte Carlo Integration with hit or miss method (Method I) d - Let be a set that contains has known volume V - Generate n randomly distributed points in. nˆ is the number of these points that also lie in. The estimate of the volume is given by, nˆ V V n which corresponds to a multidimensional integration. 9

10 Numerical Integration Techniques Monte Carlo Integration with sample mean (Method II) - The method is based on the mean-value theorem of calculus which states that the definite integral is determined by the average value of the integrand f(x) in the range a x b. y x x x2 x3 0

11 Numerical Integration Techniques N computing Pi from the area of a quarter-circle. x 2 y 2 ( equation ) x 2 y 2 for(i=0; i<n;i++) { x=pow((rand()%000)/000.0,2.0); y=pow((rand()%000)/000.0,2.0); if(sqrt(x+y)<=) count_inside++; } pi = 4*(count_inside)/n;

12 Numerical Integration Techniques P0 Domain Decompositon P P2 P3 Computation Parallel Algorithm for Integration - The domain is divided into #procs subdomains. - Each processor computes in its own subdomain. - Finally the local results are summed by master processor to obtain the overall integral. Reduction P0 2

13 Parallel Code (Newton-Cotes) Domain Decomposition of D integration Filling the blanks in mpi_pi.c program with following conditions: I. Master MPI process (rank 0) sends the number of intervals n to all worker processors. Use MPI_Send and MPI_Recv in an iterative manner. II. Master collects the locally stored mypi values. Then master sums all mypi and stores the final sum in variable pi. Use MPI_Send and MPI_Recv in an iterative manner. 3

14 Parallel Code (Method I) P Estimation of volume of sphere and Pi in 3-D Filling the blanks in MCpi_3d_par.c program with following conditions: P0 P2 I. All processors should know the boundaries of the domain of integration. MyFloor MyCeiling II. Master collects the locally stored mycountinside values. Then master sums all mycountinside and stores the final sum in variable CountInside. Use MPI_Send and MPI_Recv in an iterative manner. 4

15 Parallelization of Scientific Libraries What are Scientific Libraries - GNU Scientific Library - Numerical Linear Algebra (Blas, Lapack, Super-LU) - Netlib Software Repository for Fortran - Cuba Numerical Integration Library For general purpose numerical routines that are statically (or dynamically) linked to the existing software. Various high precision routines that reduce code development time. 5

16 Parallelization of Cuba Library The Cuba library offers a choice of four independent routines for multidimensional numerical integration: Vegas, Suave, Divonne, and Cuhre. Routine Basic integration method Algorithm type Variance reduction Vegas Sobol quasi-random sample or Mersenne Twister pseudo-random sample Monte Carlo Monte Carlo importance sampling Suave Sobol quasi-random sample or Mersenne Twister pseudo-random sample Monte Carlo Monte Carlo globally adaptive subdivision + importance sampling Divonne Korobov quasi-random sample or Sobol quasi-random sample or Mersenne Twister pseudo-random sample or cubature rules Monte Carlo Monte Carlo Monte Carlo deterministic stratified sampling, aided by methods from numerical optimization Cuhre cubature rules deterministic globally adaptive subdivision 6

17 Parallelization of Cuba Library Cuba library static library creation consists of following steps: wget tar xvf Cuba-3.0.tar.gz cd Cuba-3.0./configure make The static library libcuba.a is created under Cuba-3.0 directory 7

18 Parallelization of Cuba Library Run the sample program cd Cuba-3.0/demo/ cp Cuba-3.0/cuba.h. mpicc -o demo.out demo-c.c../libcuba.a./demo.out 8

19 Parallelization of Cuba Library h Parallelize the sample code mpi_cuba.c. Apply initialization Apply domain decomposition (choose strategy) Write a send-recv block which collects local sums in an iterative manner. n n n n f ( x,, xn, xn) dx dx n dx n P 2 P N- P N local_a local_b 9

20 Değerlendirme Genel Değerlendirme Genel değerlendirme (her bir çalıştayın son günü): Günlük değerlendirme formları: 20