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CUDA Sparse Solve: cuSPARSE Linear System Solver

December 25, 202515 minBy RightNow AI Team

Introduction

Sparse matrices have mostly zero entries, common in graph algorithms, PDEs, and network problems. Direct sparse solvers use specialized factorizations; iterative solvers use matrix-vector products. cuSPARSE provides sparse BLAS operations; cuSOLVER provides sparse direct solvers. Choice depends on matrix structure, sparsity pattern, and number of solves.

Common Performance Issues

Using dense solver - O(n³) instead of O(nnz) per iteration
Wrong sparse format - CSR vs CSC vs COO have different use cases
No preconditioning - iterative methods converge slowly
Not reusing symbolic factorization - repeated analysis phase

Optimization Techniques

1. cuSOLVER Sparse Direct

Sparse LU/Cholesky for direct solve with fill-in minimization.

2. ILU Preconditioning

Incomplete LU for fast approximate solve in iterative methods.

3. Reordering

AMD/RCM reordering to minimize fill-in during factorization.

Implementation Comparison

Before (Naive Implementation)

Dense conversion destroys sparsity benefits—O(n²) memory and O(n³) time.

cuda

// Converting sparse to dense - terrible for large sparse matrices!
void sparse_solve_naive(int* rowPtr, int* colIdx, float* values,
                        float* b, float* x, int n, int nnz) {
    // Convert to dense (O(n²) memory!)
    float* d_A_dense;
    cudaMalloc(&d_A_dense, n * n * sizeof(float));
    sparse_to_dense(rowPtr, colIdx, values, d_A_dense, n);

    // Dense solve (O(n³) time!)
    cusolverDnSgetrf(...);
    cusolverDnSgetrs(...);
}

After (Optimized Implementation)

Sparse direct solve exploits structure; iterative with ILU for very large systems.

cuda

void sparse_solve_direct(cusolverSpHandle_t solver,
                          int n, int nnz,
                          int* d_rowPtr, int* d_colIdx, float* d_values,
                          float* d_b, float* d_x) {
    cusparseMatDescr_t descr;
    cusparseCreateMatDescr(&descr);

    // Analyze sparsity pattern (do once, reuse for multiple solves)
    csrluInfo_t info;
    cusolverSpCreateCsrluInfo(&info);
    cusolverSpXcsrluAnalysis(solver, n, nnz, descr, d_rowPtr, d_colIdx, info);

    // Get buffer size
    size_t buffer_size;
    cusolverSpScsrluBufferInfo(solver, n, nnz, descr, d_values, d_rowPtr, d_colIdx,
                                info, &buffer_size, &buffer_size);
    void* d_buffer;
    cudaMalloc(&d_buffer, buffer_size);

    // Numeric factorization
    cusolverSpScsrluFactor(solver, n, nnz, descr, d_values, d_rowPtr, d_colIdx,
                           info, 1e-12, d_buffer);

    // Solve
    cusolverSpScsrluSolve(solver, n, d_b, d_x, info, d_buffer);
}

// Iterative solve with ILU preconditioner
void sparse_solve_iterative(cusparseHandle_t sparse, cublasHandle_t blas,
                            int n, int nnz, int* d_rowPtr, int* d_colIdx, float* d_values,
                            float* d_b, float* d_x, int max_iter, float tol) {
    // Setup ILU(0) preconditioner
    csrilu02Info_t iluInfo;
    cusparseCreateCsrilu02Info(&iluInfo);
    cusparseScsrilu02(sparse, n, nnz, descr, d_values, d_rowPtr, d_colIdx, iluInfo, ...);

    // Conjugate gradient with ILU preconditioning
    pcg_solve(sparse, blas, d_rowPtr, d_colIdx, d_values, d_b, d_x, iluInfo, max_iter, tol);
}

Performance Results

Metric	Naive	Optimized	Improvement
n=100k, nnz=1M	OOM	450ms (direct)	Feasible
n=1M, nnz=10M (5-point stencil)	OOM	2.1s (CG+ILU)	Feasible
Memory n=100k sparse	40GB (dense)	50MB (sparse)	800x less

Frequently Asked Questions

When to use direct vs iterative sparse solvers?

Direct: moderate size (n<100k), multiple RHS, need exact solution, good fill-in pattern. Iterative: very large (n>1M), single RHS, tolerance acceptable, matrix is SPD or has good spectrum.

What sparse format should I use?

CSR (Compressed Sparse Row) for row operations, SpMV. CSC for column operations. COO for construction/modification. CSR is most common for solving. Convert COO→CSR for computation.

Iterative Solver

Krylov methods for sparse systems

→

Conjugate Gradient

Classic iterative method for SPD sparse

→

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CUDA sparse solvecuSPARSECSR matrixsparse LUiterative solverILU preconditioner

Introduction

Implementation Comparison

Before (Naive Implementation)

Dense conversion destroys sparsity benefits—O(n²) memory and O(n³) time.

cuda

// Converting sparse to dense - terrible for large sparse matrices!
void sparse_solve_naive(int* rowPtr, int* colIdx, float* values,
                        float* b, float* x, int n, int nnz) {
    // Convert to dense (O(n²) memory!)
    float* d_A_dense;
    cudaMalloc(&d_A_dense, n * n * sizeof(float));
    sparse_to_dense(rowPtr, colIdx, values, d_A_dense, n);

    // Dense solve (O(n³) time!)
    cusolverDnSgetrf(...);
    cusolverDnSgetrs(...);
}

After (Optimized Implementation)

Sparse direct solve exploits structure; iterative with ILU for very large systems.

cuda

void sparse_solve_direct(cusolverSpHandle_t solver,
                          int n, int nnz,
                          int* d_rowPtr, int* d_colIdx, float* d_values,
                          float* d_b, float* d_x) {
    cusparseMatDescr_t descr;
    cusparseCreateMatDescr(&descr);

    // Analyze sparsity pattern (do once, reuse for multiple solves)
    csrluInfo_t info;
    cusolverSpCreateCsrluInfo(&info);
    cusolverSpXcsrluAnalysis(solver, n, nnz, descr, d_rowPtr, d_colIdx, info);

    // Get buffer size
    size_t buffer_size;
    cusolverSpScsrluBufferInfo(solver, n, nnz, descr, d_values, d_rowPtr, d_colIdx,
                                info, &buffer_size, &buffer_size);
    void* d_buffer;
    cudaMalloc(&d_buffer, buffer_size);

    // Numeric factorization
    cusolverSpScsrluFactor(solver, n, nnz, descr, d_values, d_rowPtr, d_colIdx,
                           info, 1e-12, d_buffer);

    // Solve
    cusolverSpScsrluSolve(solver, n, d_b, d_x, info, d_buffer);
}

// Iterative solve with ILU preconditioner
void sparse_solve_iterative(cusparseHandle_t sparse, cublasHandle_t blas,
                            int n, int nnz, int* d_rowPtr, int* d_colIdx, float* d_values,
                            float* d_b, float* d_x, int max_iter, float tol) {
    // Setup ILU(0) preconditioner
    csrilu02Info_t iluInfo;
    cusparseCreateCsrilu02Info(&iluInfo);
    cusparseScsrilu02(sparse, n, nnz, descr, d_values, d_rowPtr, d_colIdx, iluInfo, ...);

    // Conjugate gradient with ILU preconditioning
    pcg_solve(sparse, blas, d_rowPtr, d_colIdx, d_values, d_b, d_x, iluInfo, max_iter, tol);
}

Metric

Naive

Optimized

Improvement

n=100k, nnz=1M

OOM

450ms (direct)

Feasible

n=1M, nnz=10M (5-point stencil)

OOM

2.1s (CG+ILU)

Feasible

Memory n=100k sparse

40GB (dense)

50MB (sparse)

800x less

Frequently Asked Questions

When to use direct vs iterative sparse solvers?

Direct: moderate size (n<100k), multiple RHS, need exact solution, good fill-in pattern. Iterative: very large (n>1M), single RHS, tolerance acceptable, matrix is SPD or has good spectrum.

What sparse format should I use?

CSR (Compressed Sparse Row) for row operations, SpMV. CSC for column operations. COO for construction/modification. CSR is most common for solving. Convert COO→CSR for computation.

CUDA Sparse Solve: cuSPARSE Linear System Solver

Introduction

Common Performance Issues

Optimization Techniques

1. cuSOLVER Sparse Direct

2. ILU Preconditioning

3. Reordering

Implementation Comparison

Before (Naive Implementation)

After (Optimized Implementation)

Performance Results

Frequently Asked Questions

When to use direct vs iterative sparse solvers?

What sparse format should I use?

Related Guides

CUDA Sparse Solve: cuSPARSE Linear System Solver

Introduction

Common Performance Issues

Optimization Techniques

1. cuSOLVER Sparse Direct

2. ILU Preconditioning

3. Reordering

Implementation Comparison

Before (Naive Implementation)

After (Optimized Implementation)

Performance Results

Frequently Asked Questions

When to use direct vs iterative sparse solvers?

What sparse format should I use?

Related Guides