Mesh computation in Java

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Mesh computation
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Consider a simple mesh computation that solves the 1D heat diffusion equation The details of this problem and its solution using OpenMP are presented in the Geometric Decomposition pattern We reproduce this solution in Fig 527 This program would work well on most shared memory computers A careful analysis of the program performance, however, would expose two performance problems First, the single directive required to protect the swapping of the shared pointers adds an extra barrier, thereby greatly increasing the synchronization overhead Second, on NUMA computers, memory access overhead is likely to be high because we've made no effort to keep the arrays near the PEs that will be manipulating them We address both of these problems in Fig 528 To eliminate the need for the single directive, we modify the program so each thread has its own copy of the pointers uk and ukp1 This can be done with a private clause, but to be useful, we need the new, private copies of uk and ukp1 to point to the shared arrays comprising the mesh of values We do this with the firstprivate clause applied to the parallel directive that creates the team of threads The other performance issue we address, minimizing memory access overhead, is more subtle As discussed earlier, to reduce memory traffic in the system, it is important to keep the data close to the PEs that will work with the data On NUMA computers, this corresponds to making sure the pages of memory are allocated and "owned" by the PEs that will be working with the data contained in the page The most common NUMA page placement algorithm is the "first touch" algorithm, in which
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the PE first referencing a region of memory will have the page holding that memory assigned to it So a very common technique in OpenMP programs is to initialize data in parallel using the same loop schedule as will be used later in the computations
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Figure 527 Parallel heat-diffusion program using OpenMP This program is described in the Examples section of the Geometric Decomposition pattern
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#include <stdioh> #include <stdlibh> #include <omph> #define NX 100 #define LEFTVAL 10 #define RIGHTVAL 100 #define NSTEPS 10000 void initialize(double uk[], double ukp1[]) { uk[0] = LEFTVAL; uk[NX-1] = RIGHTVAL; for (int i=1; i < NX-1; ++i) uk[i] = 00; for (int i = 0; i < NX; ++i) ukp1 [i] = uk[i]; } void printValues(double uk[], int step) { /* NOT SHOWN */ } int main(void) { /* pointers to arrays for two iterations of algorithm */ double *uk = malloc(sizeof(double) * NX); double *ukp1 = malloc(sizeof(double) * NX); double *temp; int i,k; double dx = 10/NX; double dt = 05*dx*dx; #pragma omp parallel private (k, i) { initialize(uk, ukp1); for (k = 0; k < NSTEPS; ++k) { #pragma omp for schedule(static) for (i = 1; i < NX-1; ++i) { ukp1[i]=uk[i]+ (dt/(dx*dx))*(uk[i+1]-2*uk[i]+uk[i-1]); } /* "copy" ukp1 to uk by swapping pointers */ #pragma omp single {temp = ukpl; ukpl = uk; uk = temp;} }
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We do this by first changing the initialization slightly so the initialization loop is identical to the
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computation loop We then use the same loop parallelization directive on the initialization loop as on the computational loop This doesn't guarantee an optimal mapping of memory pages onto the PEs, but it is a portable way to improve this mapping and in many cases come quite close to an optimal solution
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Figure 528 Parallel heat-diffusion program using OpenMP, with reduced thread management overhead and memory management more appropriate for NUMA computers
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#include <stdioh> #include <stdlibh> #include <omph> #define NX 100 #define LEFTVAL 10 #define RIGHTVAL 100 #define NSTEPS 10000 void initialize(double uk[], double ukp1[]) { int i; uk[0] = LEFTVAL; uk[NX-1] = RIGHTVAL; ukp1[NX-1] = 00; #pragma omp for schedule(static) for (i = 1; i < NX-1; ++i){ uk[i] = 00; ukp1[i] = 00; } } void printValues(double uk[], int step) { /* NOT SHOWN */ } int main(void) { /* pointers to arrays for two iterations of algorithm */ double *uk = malloc(sizeof(double) * NX); double *ukp1 = malloc(sizeof(double) * NX); double *temp; int i,k; double dx = 10/NX; double dt = 05*dx*dx; #pragma omp parallel private (k, i, temp) firstprivate(uk, ukpl) { initialize(uk, ukpl); for (k = 0; k < NSTEPS; ++k) { #pragma omp for schedule(static) for (i = 1; i < NX-1; ++i) { ukpl[i]=uk[i]+ (dt/(dx*dx))*(uk[i+1]-2*uk[i]+uk[i-1]); } /* "copy" ukpl to uk by swapping pointers */ temp = ukpl; ukpl = uk; uk = temp; }
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