multithreading - 特征，并行 ConjugateGradient 失败：更多线程，更多成本

Question

我想在 Eigen 3.3.7 ( gitlab ) 中使用并行 ConjugateGradient 来解决 Ax=b，但它表明更多的线程，更多的计算成本。我测试了这个问题中的代码，并将矩阵维度从 90000 更改为 9000000。这是代码（我将文件命名为 test-cg-parallel.cpp）：

    // Use RowMajor to make use of multi-threading
typedef SparseMatrix<double, RowMajor> SpMat;
typedef Triplet<double> T;

// Assemble sparse matrix from
// https://eigen.tuxfamily.org/dox/TutorialSparse_example_details.html
void insertCoefficient(int id, int i, int j, double w, vector<T>& coeffs,
                       VectorXd& b, const VectorXd& boundary)
{
  int n = int(boundary.size());
  int id1 = i+j*n;
  if(i==-1 || i==n) b(id) -= w * boundary(j); // constrained coefficient
  else  if(j==-1 || j==n) b(id) -= w * boundary(i); // constrained coefficient
  else  coeffs.push_back(T(id,id1,w));              // unknown coefficient
}

void buildProblem(vector<T>& coefficients, VectorXd& b, int n)
{
  b.setZero();
  ArrayXd boundary = ArrayXd::LinSpaced(n, 0,M_PI).sin().pow(2);
  for(int j=0; j<n; ++j)
    {
      for(int i=0; i<n; ++i)
        {
          int id = i+j*n;
          insertCoefficient(id, i-1,j, -1, coefficients, b, boundary);
          insertCoefficient(id, i+1,j, -1, coefficients, b, boundary);
          insertCoefficient(id, i,j-1, -1, coefficients, b, boundary);
          insertCoefficient(id, i,j+1, -1, coefficients, b, boundary);
          insertCoefficient(id, i,j,    4, coefficients, b, boundary);
        }
    }
}

int main()
{
  int n = 3000;  // size of the image
  int m = n*n;  // number of unknowns (=number of pixels)
  // Assembly:
  vector<T> coefficients;          // list of non-zeros coefficients
  VectorXd b(m);                   // the right hand side-vector resulting from the constraints
  buildProblem(coefficients, b, n);
  SpMat A(m,m);
  A.setFromTriplets(coefficients.begin(), coefficients.end());
  // Solving:
  // Use ConjugateGradient with Lower|Upper as the UpLo template parameter to make use of multi-threading
  clock_t time_start, time_end;
  time_start=clock();
  ConjugateGradient<SpMat, Lower|Upper> solver(A);
  VectorXd x = solver.solve(b);         // use the factorization to solve for the given right hand side

  time_end=clock();
   cout<<"time use:"<<1000*(time_end-time_start)/(double)CLOCKS_PER_SEC<<"ms"<<endl;
   return 0;
}

我用gcc 7.4.0编译代码，6核（12线程）Intel Xeon E2186G CPU，编译运行细节如下：

liu@liu-Precision-3630-Tower:~/test$ g++ test-cg-parallel.cpp -O3 -fopenmp -o cg
liu@liu-Precision-3630-Tower:~/test$ OMP_NUM_THREADS=1 ./cg

time use:747563ms

liu@liu-Precision-3630-Tower:~/test$ OMP_NUM_THREADS=4 ./cg

time use: 1.49821e+06ms

liu@liu-Precision-3630-Tower:~/test$ OMP_NUM_THREADS=8 ./cg

time use: 2.60207e+06ms

谁能给我一些建议？非常感谢。

标签： multithreadinggccopenmpeigenlinear-equation