这篇文章考虑了稀疏线性系统的快速求解这个非常重要的问题。
截至 2015 年 11 月，该cuSPARSE库提供了基于 LU 分解的稀疏线性系统解决方案的例程，特别是
cusparse<t>csrilu02
和
cusparse<t>csrsv2_solve
此外，cuSPARSE提供
cusparse<t>csrcolor
它实现了图形着色。在不完全 LU 分解中使用图着色在
图着色：不完全 LU 分解的更多并行性
和
M.Naumov, P.Castonguay, J. Cohen，“并行图着色与 GPU 上不完全 LU 分解的应用”，NVIDIA 研究技术报告，2015 年 5 月。
这个想法是将图形着色算法应用于与系统的系数矩阵相关联的行依赖图，然后相应地重新排序系统方程，以便 LU 分解例程可以提取更多的并行度。
下面，请使用上述想法找到一个完整的示例：
#include <stdio.h>
#include <stdlib.h>
#include <iostream>
#include <assert.h>
#include "Utilities.cuh"
#include <cuda_runtime.h>
#include <cusparse_v2.h>
#define BLOCKSIZE 256
/**************************/
/* SETTING UP THE PROBLEM */
/**************************/
void setUpTheProblem(double **h_A_dense, double **h_x_dense, double **d_A_dense, double **d_x_dense, const int N) {
// --- Host side dense matrix
h_A_dense[0] = (double*)calloc(N * N, sizeof(*h_A_dense));
// --- Column-major ordering
h_A_dense[0][0] = 0.4612f; h_A_dense[0][4] = -0.0006f; h_A_dense[0][8] = 0.f; h_A_dense[0][12] = 0.0f;
h_A_dense[0][1] = -0.0006f; h_A_dense[0][5] = 0.f; h_A_dense[0][9] = 0.0723f; h_A_dense[0][13] = 0.04f;
h_A_dense[0][2] = 0.3566f; h_A_dense[0][6] = 0.0723f; h_A_dense[0][10] = 0.f; h_A_dense[0][14] = 0.0f;
h_A_dense[0][3] = 0.0f; h_A_dense[0][7] = 0.0f; h_A_dense[0][11] = 1.0f; h_A_dense[0][15] = 0.1f;
h_x_dense[0] = (double *)malloc(N * sizeof(double));
h_x_dense[0][0] = 100.0; h_x_dense[0][1] = 200.0; h_x_dense[0][2] = 400.0; h_x_dense[0][3] = 500.0;
// --- Create device arrays and copy host arrays to them
gpuErrchk(cudaMalloc(&d_A_dense[0], N * N * sizeof(double)));
gpuErrchk(cudaMemcpy(d_A_dense[0], h_A_dense[0], N * N * sizeof(double), cudaMemcpyHostToDevice));
gpuErrchk(cudaMalloc(&d_x_dense[0], N * sizeof(double)));
gpuErrchk(cudaMemcpy(d_x_dense[0], h_x_dense[0], N * sizeof(double), cudaMemcpyHostToDevice));
}
/************************/
/* FROM DENSE TO SPARSE */
/************************/
void fromDenseToSparse(const cusparseHandle_t handle, double *d_A_dense, double **d_A, int **d_A_RowIndices, int **d_A_ColIndices, int *nnz,
cusparseMatDescr_t *descrA, const int N) {
cusparseSafeCall(cusparseCreateMatDescr(&descrA[0]));
cusparseSafeCall(cusparseSetMatType (descrA[0], CUSPARSE_MATRIX_TYPE_GENERAL));
cusparseSafeCall(cusparseSetMatIndexBase(descrA[0], CUSPARSE_INDEX_BASE_ZERO));
nnz[0] = 0; // --- Number of nonzero elements in dense matrix
const int lda = N; // --- Leading dimension of dense matrix
// --- Device side number of nonzero elements per row
int *d_nnzPerVector; gpuErrchk(cudaMalloc(&d_nnzPerVector, N * sizeof(int)));
cusparseSafeCall(cusparseDnnz(handle, CUSPARSE_DIRECTION_ROW, N, N, descrA[0], d_A_dense, lda, d_nnzPerVector, &nnz[0]));
// --- Host side number of nonzero elements per row
int *h_nnzPerVector = (int *)malloc(N * sizeof(int));
gpuErrchk(cudaMemcpy(h_nnzPerVector, d_nnzPerVector, N * sizeof(int), cudaMemcpyDeviceToHost));
printf("Number of nonzero elements in dense matrix = %i\n\n", nnz[0]);
for (int i = 0; i < N; ++i) printf("Number of nonzero elements in row %i = %i \n", i, h_nnzPerVector[i]);
printf("\n");
// --- Device side sparse matrix
gpuErrchk(cudaMalloc(&d_A[0], nnz[0] * sizeof(double)));
gpuErrchk(cudaMalloc(&d_A_RowIndices[0], (N + 1) * sizeof(int)));
gpuErrchk(cudaMalloc(&d_A_ColIndices[0], nnz[0] * sizeof(int)));
cusparseSafeCall(cusparseDdense2csr(handle, N, N, descrA[0], d_A_dense, lda, d_nnzPerVector, d_A[0], d_A_RowIndices[0], d_A_ColIndices[0]));
// --- Host side sparse matrix
double *h_A = (double *)malloc(nnz[0] * sizeof(double));
int *h_A_RowIndices = (int *)malloc((N + 1) * sizeof(*h_A_RowIndices));
int *h_A_ColIndices = (int *)malloc(nnz[0] * sizeof(*h_A_ColIndices));
gpuErrchk(cudaMemcpy(h_A, d_A[0], nnz[0] * sizeof(double), cudaMemcpyDeviceToHost));
gpuErrchk(cudaMemcpy(h_A_RowIndices, d_A_RowIndices[0], (N + 1) * sizeof(int), cudaMemcpyDeviceToHost));
gpuErrchk(cudaMemcpy(h_A_ColIndices, d_A_ColIndices[0], nnz[0] * sizeof(int), cudaMemcpyDeviceToHost));
printf("\nOriginal matrix in CSR format\n\n");
for (int i = 0; i < nnz[0]; ++i) printf("A[%i] = %f ", i, h_A[i]); printf("\n");
printf("\n");
for (int i = 0; i < (N + 1); ++i) printf("h_A_RowIndices[%i] = %i \n", i, h_A_RowIndices[i]); printf("\n");
for (int i = 0; i < nnz[0]; ++i) printf("h_A_ColIndices[%i] = %i \n", i, h_A_ColIndices[i]);
}
/******************/
/* GRAPH COLORING */
/******************/
__global__ void setRowIndices(int *d_B_RowIndices, const int N) {
const int tid = threadIdx.x + blockDim.x * blockIdx.x;
if (tid == N) d_B_RowIndices[tid] = N;
else if (tid < N) d_B_RowIndices[tid] = tid;
}
__global__ void setB(double *d_B, const int N) {
const int tid = threadIdx.x + blockDim.x * blockIdx.x;
if (tid < N) d_B[tid] = 1.f;
}
void graphColoring(const cusparseHandle_t handle, const int nnz, const cusparseMatDescr_t descrA, const double fractionToColor, double *d_A,
const int *d_A_RowIndices, const int *d_A_ColIndices, double **d_B, int **d_B_RowIndices, int **d_B_ColIndices,
cusparseMatDescr_t *descrB, const int N) {
cusparseColorInfo_t info; cusparseSafeCall(cusparseCreateColorInfo(&info));
int ncolors;
int *d_coloring; gpuErrchk(cudaMalloc(&d_coloring, N * sizeof(double)));
gpuErrchk(cudaMalloc(&d_B_ColIndices[0], N * sizeof(double)));
cusparseSafeCall(cusparseDcsrcolor(handle, N, nnz, descrA, d_A, d_A_RowIndices, d_A_ColIndices, &fractionToColor, &ncolors, d_coloring,
d_B_ColIndices[0], info));
int *h_coloring = (int *)malloc(N * sizeof(double));
int *h_B_ColIndices = (int *)malloc(N * sizeof(double));
gpuErrchk(cudaMemcpy(h_coloring, d_coloring, N * sizeof(double), cudaMemcpyDeviceToHost));
gpuErrchk(cudaMemcpy(h_B_ColIndices, d_B_ColIndices[0], N * sizeof(double), cudaMemcpyDeviceToHost));
for (int i = 0; i < N; i++) printf("h_coloring = %i; h_B_ColIndices = %i\n", h_coloring[i], h_B_ColIndices[i]);
gpuErrchk(cudaMalloc(&d_B_RowIndices[0], (N + 1) * sizeof(int)));
int *h_B_RowIndices = (int *)malloc((N + 1) * sizeof(double));
setRowIndices<<<iDivUp(N + 1, BLOCKSIZE), BLOCKSIZE>>>(d_B_RowIndices[0], N);
gpuErrchk(cudaMemcpy(h_B_RowIndices, d_B_RowIndices[0], (N + 1) * sizeof(int), cudaMemcpyDeviceToHost));
printf("\n"); for (int i = 0; i <= N; i++) printf("h_B_RowIndices = %i\n", h_B_RowIndices[i]);
gpuErrchk(cudaMalloc(&d_B[0], N * sizeof(double)));
double *h_B = (double *)malloc(N * sizeof(double));
setB<<<iDivUp(N, BLOCKSIZE), BLOCKSIZE>>>(d_B[0], N);
gpuErrchk(cudaMemcpy(h_B, d_B[0], N * sizeof(double), cudaMemcpyDeviceToHost));
printf("\n"); for (int i = 0; i < N; i++) printf("h_B = %f\n", h_B[i]);
// --- Descriptor for sparse mutation matrix B
cusparseSafeCall(cusparseCreateMatDescr(&descrB[0]));
cusparseSafeCall(cusparseSetMatType (descrB[0], CUSPARSE_MATRIX_TYPE_GENERAL));
cusparseSafeCall(cusparseSetMatIndexBase(descrB[0], CUSPARSE_INDEX_BASE_ZERO));
}
/*************************/
/* MATRIX ROW REORDERING */
/*************************/
void matrixRowReordering(const cusparseHandle_t handle, int nnzA, int nnzB, int *nnzC, cusparseMatDescr_t descrA, cusparseMatDescr_t descrB,
cusparseMatDescr_t *descrC, double *d_A, int *d_A_RowIndices, int *d_A_ColIndices, double *d_B, int *d_B_RowIndices,
int *d_B_ColIndices, double **d_C, int **d_C_RowIndices, int **d_C_ColIndices, const int N) {
// --- Descriptor for sparse matrix C
cusparseSafeCall(cusparseCreateMatDescr(&descrC[0]));
cusparseSafeCall(cusparseSetMatType (descrC[0], CUSPARSE_MATRIX_TYPE_GENERAL));
cusparseSafeCall(cusparseSetMatIndexBase(descrC[0], CUSPARSE_INDEX_BASE_ZERO));
const int lda = N; // --- Leading dimension of dense matrix
// --- Device side sparse matrix
gpuErrchk(cudaMalloc(&d_C_RowIndices[0], (N + 1) * sizeof(int)));
// --- Host side sparse matrices
int *h_C_RowIndices = (int *)malloc((N + 1) * sizeof(int));
// --- Performing the matrix - matrix multiplication
int baseC;
int *nnzTotalDevHostPtr = &nnzC[0];
cusparseSafeCall(cusparseSetPointerMode(handle, CUSPARSE_POINTER_MODE_HOST));
cusparseSafeCall(cusparseXcsrgemmNnz(handle, CUSPARSE_OPERATION_NON_TRANSPOSE, CUSPARSE_OPERATION_NON_TRANSPOSE, N, N, N, descrB, nnzB,
d_B_RowIndices, d_B_ColIndices, descrA, nnzA, d_A_RowIndices, d_A_ColIndices, descrC[0], d_C_RowIndices[0],
nnzTotalDevHostPtr));
if (NULL != nnzTotalDevHostPtr) nnzC[0] = *nnzTotalDevHostPtr;
else {
gpuErrchk(cudaMemcpy(&nnzC[0], d_C_RowIndices + N, sizeof(int), cudaMemcpyDeviceToHost));
gpuErrchk(cudaMemcpy(&baseC, d_C_RowIndices, sizeof(int), cudaMemcpyDeviceToHost));
nnzC -= baseC;
}
gpuErrchk(cudaMalloc(&d_C_ColIndices[0], nnzC[0] * sizeof(int)));
gpuErrchk(cudaMalloc(&d_C[0], nnzC[0] * sizeof(double)));
double *h_C = (double *)malloc(nnzC[0] * sizeof(double));
int *h_C_ColIndices = (int *)malloc(nnzC[0] * sizeof(int));
cusparseSafeCall(cusparseDcsrgemm(handle, CUSPARSE_OPERATION_NON_TRANSPOSE, CUSPARSE_OPERATION_NON_TRANSPOSE, N, N, N, descrB, nnzB,
d_B, d_B_RowIndices, d_B_ColIndices, descrA, nnzA, d_A, d_A_RowIndices, d_A_ColIndices, descrC[0],
d_C[0], d_C_RowIndices[0], d_C_ColIndices[0]));
double *h_C_dense = (double*)malloc(N * N * sizeof(double));
double *d_C_dense; gpuErrchk(cudaMalloc(&d_C_dense, N * N * sizeof(double)));
cusparseSafeCall(cusparseDcsr2dense(handle, N, N, descrC[0], d_C[0], d_C_RowIndices[0], d_C_ColIndices[0], d_C_dense, N));
gpuErrchk(cudaMemcpy(h_C , d_C[0], nnzC[0] * sizeof(double), cudaMemcpyDeviceToHost));
gpuErrchk(cudaMemcpy(h_C_RowIndices, d_C_RowIndices[0], (N + 1) * sizeof(int), cudaMemcpyDeviceToHost));
gpuErrchk(cudaMemcpy(h_C_ColIndices, d_C_ColIndices[0], nnzC[0] * sizeof(int), cudaMemcpyDeviceToHost));
printf("\nResult matrix C in CSR format\n\n");
for (int i = 0; i < nnzC[0]; ++i) printf("C[%i] = %f ", i, h_C[i]); printf("\n");
printf("\n");
for (int i = 0; i < (N + 1); ++i) printf("h_C_RowIndices[%i] = %i \n", i, h_C_RowIndices[i]); printf("\n");
printf("\n");
for (int i = 0; i < nnzC[0]; ++i) printf("h_C_ColIndices[%i] = %i \n", i, h_C_ColIndices[i]);
gpuErrchk(cudaMemcpy(h_C_dense, d_C_dense, N * N * sizeof(double), cudaMemcpyDeviceToHost));
for (int j = 0; j < N; j++) {
for (int i = 0; i < N; i++)
printf("%f \t", h_C_dense[i * N + j]);
printf("\n");
}
}
/******************/
/* ROW REORDERING */
/******************/
void rowReordering(const cusparseHandle_t handle, int nnzA, cusparseMatDescr_t descrB, double *d_B, int *d_B_RowIndices, int *d_B_ColIndices,
double *d_x_dense, double **d_y_dense, const int N) {
gpuErrchk(cudaMalloc(&d_y_dense[0], N * sizeof(double)));
const double alpha = 1.;
const double beta = 0.;
cusparseSafeCall(cusparseDcsrmv(handle, CUSPARSE_OPERATION_NON_TRANSPOSE, N, N, nnzA, &alpha, descrB, d_B, d_B_RowIndices, d_B_ColIndices, d_x_dense,
&beta, d_y_dense[0]));
double *h_y_dense = (double*)malloc(N * sizeof(double));
gpuErrchk(cudaMemcpy(h_y_dense, d_y_dense[0], N * sizeof(double), cudaMemcpyDeviceToHost));
printf("\nResult vector\n\n");
for (int i = 0; i < N; ++i) printf("h_y[%i] = %f ", i, h_y_dense[i]); printf("\n");
}
/*****************************/
/* SOLVING THE LINEAR SYSTEM */
/*****************************/
void LUDecomposition(const cusparseHandle_t handle, int nnzC, cusparseMatDescr_t descrC, double *d_C, int *d_C_RowIndices, int *d_C_ColIndices,
double *d_x_dense, double **d_y_dense, const int N) {
/******************************************/
/* STEP 1: CREATE DESCRIPTORS FOR L AND U */
/******************************************/
cusparseMatDescr_t descr_L = 0;
cusparseSafeCall(cusparseCreateMatDescr (&descr_L));
cusparseSafeCall(cusparseSetMatIndexBase(descr_L, CUSPARSE_INDEX_BASE_ZERO));
cusparseSafeCall(cusparseSetMatType (descr_L, CUSPARSE_MATRIX_TYPE_GENERAL));
cusparseSafeCall(cusparseSetMatFillMode (descr_L, CUSPARSE_FILL_MODE_LOWER));
cusparseSafeCall(cusparseSetMatDiagType (descr_L, CUSPARSE_DIAG_TYPE_UNIT));
cusparseMatDescr_t descr_U = 0;
cusparseSafeCall(cusparseCreateMatDescr (&descr_U));
cusparseSafeCall(cusparseSetMatIndexBase(descr_U, CUSPARSE_INDEX_BASE_ZERO));
cusparseSafeCall(cusparseSetMatType (descr_U, CUSPARSE_MATRIX_TYPE_GENERAL));
cusparseSafeCall(cusparseSetMatFillMode (descr_U, CUSPARSE_FILL_MODE_UPPER));
cusparseSafeCall(cusparseSetMatDiagType (descr_U, CUSPARSE_DIAG_TYPE_NON_UNIT));
/**************************************************************************************************/
/* STEP 2: QUERY HOW MUCH MEMORY USED IN LU FACTORIZATION AND THE TWO FOLLOWING SYSTEM INVERSIONS */
/**************************************************************************************************/
csrilu02Info_t info_C = 0; cusparseSafeCall(cusparseCreateCsrilu02Info (&info_C));
csrsv2Info_t info_L = 0; cusparseSafeCall(cusparseCreateCsrsv2Info (&info_L));
csrsv2Info_t info_U = 0; cusparseSafeCall(cusparseCreateCsrsv2Info (&info_U));
int pBufferSize_M, pBufferSize_L, pBufferSize_U;
cusparseSafeCall(cusparseDcsrilu02_bufferSize(handle, N, nnzC, descrC, d_C, d_C_RowIndices, d_C_ColIndices, info_C, &pBufferSize_M));
cusparseSafeCall(cusparseDcsrsv2_bufferSize (handle, CUSPARSE_OPERATION_NON_TRANSPOSE, N, nnzC, descr_L, d_C, d_C_RowIndices, d_C_ColIndices, info_L, &pBufferSize_L));
cusparseSafeCall(cusparseDcsrsv2_bufferSize (handle, CUSPARSE_OPERATION_NON_TRANSPOSE, N, nnzC, descr_U, d_C, d_C_RowIndices, d_C_ColIndices, info_U, &pBufferSize_U));
int pBufferSize = max(pBufferSize_M, max(pBufferSize_L, pBufferSize_U));
void *pBuffer = 0; gpuErrchk(cudaMalloc((void**)&pBuffer, pBufferSize));
/************************************************************************************************/
/* STEP 3: ANALYZE THE THREE PROBLEMS: LU FACTORIZATION AND THE TWO FOLLOWING SYSTEM INVERSIONS */
/************************************************************************************************/
int structural_zero;
cusparseSafeCall(cusparseDcsrilu02_analysis(handle, N, nnzC, descrC, d_C, d_C_RowIndices, d_C_ColIndices, info_C, CUSPARSE_SOLVE_POLICY_NO_LEVEL, pBuffer));
cusparseStatus_t status = cusparseXcsrilu02_zeroPivot(handle, info_C, &structural_zero);
if (CUSPARSE_STATUS_ZERO_PIVOT == status){ printf("A(%d,%d) is missing\n", structural_zero, structural_zero); }
cusparseSafeCall(cusparseDcsrsv2_analysis(handle, CUSPARSE_OPERATION_NON_TRANSPOSE, N, nnzC, descr_L, d_C, d_C_RowIndices, d_C_ColIndices, info_L, CUSPARSE_SOLVE_POLICY_NO_LEVEL, pBuffer));
cusparseSafeCall(cusparseDcsrsv2_analysis(handle, CUSPARSE_OPERATION_NON_TRANSPOSE, N, nnzC, descr_U, d_C, d_C_RowIndices, d_C_ColIndices, info_U, CUSPARSE_SOLVE_POLICY_USE_LEVEL, pBuffer));
/************************************/
/* STEP 4: FACTORIZATION: A = L * U */
/************************************/
int numerical_zero;
cusparseSafeCall(cusparseDcsrilu02(handle, N, nnzC, descrC, d_C, d_C_RowIndices, d_C_ColIndices, info_C, CUSPARSE_SOLVE_POLICY_NO_LEVEL, pBuffer));
status = cusparseXcsrilu02_zeroPivot(handle, info_C, &numerical_zero);
if (CUSPARSE_STATUS_ZERO_PIVOT == status){ printf("U(%d,%d) is zero\n", numerical_zero, numerical_zero); }
/*********************/
/* STEP 5: L * z = x */
/*********************/
// --- Allocating the intermediate result vector
double *d_z_dense; gpuErrchk(cudaMalloc(&d_z_dense, N * sizeof(double)));
const double alpha = 1.;
cusparseSafeCall(cusparseDcsrsv2_solve(handle, CUSPARSE_OPERATION_NON_TRANSPOSE, N, nnzC, &alpha, descr_L, d_C, d_C_RowIndices, d_C_ColIndices, info_L, d_x_dense, d_z_dense, CUSPARSE_SOLVE_POLICY_NO_LEVEL, pBuffer));
/*********************/
/* STEP 5: U * y = z */
/*********************/
gpuErrchk(cudaMalloc(&d_y_dense[0], N * sizeof(double)));
cusparseSafeCall(cusparseDcsrsv2_solve(handle, CUSPARSE_OPERATION_NON_TRANSPOSE, N, nnzC, &alpha, descr_U, d_C, d_C_RowIndices, d_C_ColIndices, info_U, d_z_dense, d_y_dense[0], CUSPARSE_SOLVE_POLICY_USE_LEVEL, pBuffer));
double *h_y_dense = (double *)malloc(N * sizeof(double));
gpuErrchk(cudaMemcpy(h_y_dense, d_y_dense[0], N * sizeof(double), cudaMemcpyDeviceToHost));
printf("\n\nFinal result\n");
for (int k=0; k<N; k++) printf("x[%i] = %f\n", k, h_y_dense[k]);
}
/********/
/* MAIN */
/********/
int main()
{
// --- Initialize cuSPARSE
cusparseHandle_t handle; cusparseSafeCall(cusparseCreate(&handle));
/*************************************************/
/* SETTING UP THE ORIGINAL LINEAR SYSTEM PROBLEM */
/*************************************************/
const int N = 4; // --- Number of rows and columns
double *h_A_dense; double *h_x_dense;
double *d_A_dense; double *d_x_dense;
setUpTheProblem(&h_A_dense, &h_x_dense, &d_A_dense, &d_x_dense, N);
/************************/
/* FROM DENSE TO SPARSE */
/************************/
//--- Descriptor for sparse matrix A
cusparseMatDescr_t descrA;
int *d_A_RowIndices, *d_A_ColIndices;
double *d_A;
int nnzA;
fromDenseToSparse(handle, d_A_dense, &d_A, &d_A_RowIndices, &d_A_ColIndices, &nnzA, &descrA, N);
/******************/
/* GRAPH COLORING */
/******************/
const double fractionToColor = 0.95;
int *d_B_RowIndices, *d_B_ColIndices;
double *d_B;
int nnzB;
cusparseMatDescr_t descrB;
graphColoring(handle, nnzB, descrA, fractionToColor, d_A, d_A_RowIndices, d_A_ColIndices, &d_B, &d_B_RowIndices, &d_B_ColIndices, &descrB, N);
/*************************/
/* MATRIX ROW REORDERING */
/*************************/
int nnzC;
int *d_C_RowIndices, *d_C_ColIndices;
double *d_C;
cusparseMatDescr_t descrC;
matrixRowReordering(handle, nnzA, nnzB, &nnzC, descrA, descrB, &descrC, d_A, d_A_RowIndices, d_A_ColIndices, d_B, d_B_RowIndices, d_B_ColIndices,
&d_C, &d_C_RowIndices, &d_C_ColIndices, N);
/******************/
/* ROW REORDERING */
/******************/
double *d_y_dense;
rowReordering(handle, nnzA, descrB, d_B, d_B_RowIndices, d_B_ColIndices, d_x_dense, &d_y_dense, N);
/*****************************/
/* SOLVING THE LINEAR SYSTEM */
/*****************************/
double *d_xsol_dense;
LUDecomposition(handle, nnzC, descrC, d_C, d_C_RowIndices, d_C_ColIndices, d_y_dense, &d_xsol_dense, N);
}