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===== Source Code ====={| class="wikitable mw-collapsible mw-collapsed"! poissan.h|-|<source>#ifndef POISSON_H#define POISSON_H#include <fstream> namespace DPS{ class Poisson { size_t nRowsTotal; size_t nColumns; float* data; int bufferSide;  void update (size_t startRow, size_t endRow, const float wx, const float wy); void bufferSwitch(){ bufferSide = 1 - bufferSide; };  public: Poisson(std::ifstream& ifs); Poisson(const size_t r, const size_t c, float* d); ~Poisson(){ delete[] data; }; float* operator()(const size_t iteration, const float wx, const float wy); float* operator()(const size_t iteration){ return operator()(iteration,0.1,0.1); } void show(std::ostream& ofs) const; };}#endif </source>|}{| class="wikitable mw-collapsible mw-collapsed"! poissan.cpp|-|<source>#include <cstring>#include <cstdlib>#include <iomanip>#include <iostream>#include <string>#include "poisson.h" namespace DPS{ Poisson::Poisson(std::ifstream& ifs){ std::string line; bufferSide = 0;  /* find number of columns */ std::getline(ifs,line); for (size_t i = 0 ; i < line.size() ; i++){ if(line[i]==' ') nColumns++; } nColumns++;  /* find number of rows */ nRowsTotal++; /* already fetched one */ while(std::getline(ifs,line)) nRowsTotal++; ifs.clear();  try{ data = new float[nColumns * nRowsTotal * 2]; } catch (...){ throw std::runtime_error("Failed to Allocate Memory"); }  /* readin data */ ifs.seekg(0,ifs.beg); std::cout << ifs.tellg() << std::endl; for (size_t i = 0 ; i < nRowsTotal * nColumns ; i++) { ifs >> data[i]; }  std::memset(data+nRowsTotal*nColumns,0,nRowsTotal*nColumns*sizeof(float)); }  Poisson::Poisson(const size_t r, const size_t c, float* d){ bufferSide = 0; nRowsTotal = r; nColumns = c; try{ data = new float[r*c*2]; } catch (...){ throw std::runtime_error("Failed to Allocate Memory"); } std::memcpy(data,d,r*c*sizeof(float)); std::memset(data+r*c,0,r*c*sizeof(float)); }  void Poisson::update (size_t startRow, size_t endRow, const float wx, const float wy){ float* x_new = data + (1-bufferSide)*nRowsTotal*nColumns; float* x_old = data + bufferSide*nRowsTotal*nColumns; for (size_t i = startRow; i <= endRow; i++) for (size_t j = 1; j < nColumns - 1; j++) x_new[i * nColumns + j] = x_old[i * nColumns + j] + wx * (x_old[(i + 1) * nColumns + j] + x_old[(i - 1) * nColumns + j] - 2.0f * x_old[i * nColumns + j]) + wy * (x_old[i * nColumns + j + 1] + x_old[i * nColumns + j - 1] - 2.0f * x_old[i * nColumns + j]); }  float* Poisson::operator()(const size_t nIterations, const float wx, const float wy){ for (size_t i = 0; i < nIterations; i++) { update(0, nRowsTotal-1, wx, wy); bufferSwitch(); } return data; }  void Poisson::show(std::ostream& ofs) const{ ofs << std::fixed << std::setprecision(1); for (size_t j = 0; j < nColumns ; j++) { for (size_t i = 0 ; i < nRowsTotal ; i++) ofs << std::setw(8) << data[ bufferSide*nColumns*nRowsTotal + i * nColumns + j]; ofs << std::endl; } }} </source>|}{| class="wikitable mw-collapsible mw-collapsed"! main.cpp|-|<source>// based on code from LLNL tutorial mpi_heat2d.c// Master-Worker Programming Model// Chris Szalwinski - 2018/11/13// Adopted by Tony Sim - 2019/02/16#include <iostream>#include <fstream>#include <iomanip>#include <cstdlib>#include <stdexcept> #include "poisson.h" // solution constantsconst size_t NONE = 0;const size_t MINPARTITIONS = 1;const size_t MAXPARTITIONS = 7;// weightsconst float wx = 0.1f;const float wy = 0.1f; int main(int argc, char** argv) { if (argc != 4) { std::cerr << "*** Incorrect number of arguments ***\n"; std::cerr << "Usage: " << argv[0] << " input_file output_file no_of_iterations\n"; return 1; }  std::ifstream input(argv[1]); std::ofstream output(argv[2]); std::ofstream temp("init.csv");  if(!input.is_open()){ std::cerr << "Invalid Input File" << std::endl; return 2; } if(!output.is_open()){ std::cerr << "Invalid Output File" << std::endl; return 2; }  DPS::Poisson* p = nullptr; try{ p = new DPS::Poisson(input); } catch(std::exception& e){ std::cerr << "Error: " << e.what() << std::endl; }  p->show(temp);  size_t nIterations = std::atoi(argv[3]);  (*p)(nIterations);  // write results to file p->show(output);  delete p; } </source>|}===== Introduction =====The presented code simulates heat map using Jacobi's method for Poisson's equation. It is represented in a 2D array, and each element updates its value based on the adjacent elements at a given moment. Each iteration represent one instance in time. By repeating the calculation over the entire array through multiple iterations, we can estimate the state of the heat transfer after a given time interval. ===== Profiling =====The profiling was conducted using a data set of 79 rows and 205 columns over 150000 iterations.{| class="wikitable mw-collapsible mw-collapsed"! Flat profile|-| Flat profile: Each sample counts as 0.01 seconds. % cumulative self self total time seconds seconds calls us/call us/call name 98.57 2.75 2.75 150000 18.33 18.33 DPS::Poisson::update(unsigned long, unsigned long, float, float) 0.00 2.75 0.00 1 0.00 0.00 _GLOBAL__sub_I__ZN3DPS7PoissonC2ERSt14basic_ifstreamIcSt11char_traitsIcEE 0.00 2.75 0.00 1 0.00 0.00 _GLOBAL__sub_I_main  |}{| class="wikitable mw-collapsible mw-collapsed"! Call graph|-| Call graph  granularity: each sample hit covers 2 byte(s) for 0.36% of 2.75 seconds index % time self children called name 2.75 0.00 150000/150000 DPS::Poisson::operator()(unsigned long, float, float) [2][1] 100.0 2.75 0.00 150000 DPS::Poisson::update(unsigned long, unsigned long, float, float) [1]----------------------------------------------- <spontaneous>[2] 100.0 0.00 2.75 DPS::Poisson::operator()(unsigned long, float, float) [2] 2.75 0.00 150000/150000 DPS::Poisson::update(unsigned long, unsigned long, float, float) [1]----------------------------------------------- 0.00 0.00 1/1 __libc_csu_init [21][10] 0.0 0.00 0.00 1 _GLOBAL__sub_I__ZN3DPS7PoissonC2ERSt14basic_ifstreamIcSt11char_traitsIcEE [10]----------------------------------------------- 0.00 0.00 1/1 __libc_csu_init [21][11] 0.0 0.00 0.00 1 _GLOBAL__sub_I_main [11]-----------------------------------------------  Index by function name  [10] _GLOBAL__sub_I__ZN3DPS7PoissonC2ERSt14basic_ifstreamIcSt11char_traitsIcEE (poisson.cpp) [11] _GLOBAL__sub_I_main (main.cpp) [1] DPS::Poisson::update(unsigned long, unsigned long, float, float)  |} =====Analysis=====given 98.57 percent of time is spent on the update() function, it is considered the hotspot.Total time taken was 2.75. If we consider a GPU environment with 1000 cores, we can estimate the following speedup:S1000 = 1/(1-.9857 + .9857/1000) = 65.00In fact, the speed will decrease from 2.75 seconds to 0.0450 seconds. As each iteration depends on the product of the previous iteration, there is a dependency resolution that might hamper the parallel process.Consideration may also be extended to resolving ghost cells across different SMX while using the device global memory as the transfer pipeline. ==== Robert Subject: ========= Multi Sampling Anti Aliasing =====