Alright, let's figure out how to get that sparse-dense matrix-vector multiplication (Ax) blazing fast on your CUDA GPU—especially since you're running it repeatedly and need top performance. Here's a practical, step-by-step guide with concrete code examples and optimization tips tailored to your 15k×15k, 5% density matrix:

Before diving in, make sure you have:

• A CUDA-capable GPU (compute capability ≥ 3.5, which covers most modern GPUs)
• The right GPU-accelerated Python libraries installed (we'll cover two popular options below)

2.1 CuPy: Drop-in Replacement for Scipy (Lowest Migration Cost) #

CuPy is designed to mirror Scipy's API almost exactly, so you won't have to rewrite much code. It offloads all computations to the GPU automatically.

Setup #

Install CuPy matching your CUDA version (replace cuda12x with your version, e.g., cuda11x for CUDA 11):

pip install cupy-cuda12x

Code Example #

The key here is to transfer data to the GPU once—don't do this inside your loop (PCIe data transfer is way slower than GPU computation):

import cupy as cp
import numpy as np
from scipy.sparse import csc_matrix

# Your existing Scipy CSC matrix and NumPy vector (example values here)
A_scipy = csc_matrix((15000, 15000), density=0.05)
x_np = np.random.rand(15000)

# Move data to GPU ONCE
A_cupy = cp.sparse.csc_matrix(A_scipy)
x_cupy = cp.array(x_np)

# Repeated multiplication (runs entirely on GPU)
for _ in range(1000):
    y_cupy = A_cupy.dot(x_cupy)

# Transfer result back to CPU only if you need it
y_np = cp.asnumpy(y_cupy)

CuPy's sparse dot implementation is optimized for CUDA, so this will give you a huge speedup over Scipy's CPU version for repeated runs.

2.2 PyTorch: Great for Deep Learning Workflows #

If you're already working in the PyTorch ecosystem, or need to integrate this with other GPU-based tasks, PyTorch's sparse tensor support is a solid choice.

Setup #

Install PyTorch with CUDA support:

pip install torch torchvision torchaudio --index-url https://download.pytorch.org/whl/cu121

Code Example #

PyTorch uses COO format for sparse tensors (best supported for GPU operations), so we'll convert your Scipy CSC matrix to COO first:

import torch
import numpy as np
from scipy.sparse import csc_matrix

# Your existing Scipy matrix and NumPy vector
A_scipy = csc_matrix((15000, 15000), density=0.05)
x_np = np.random.rand(15000)

# Convert Scipy CSC to PyTorch sparse CUDA tensor
indices = torch.tensor(np.vstack((A_scipy.row, A_scipy.col)), dtype=torch.long)
values = torch.tensor(A_scipy.data, dtype=torch.float32)
A_torch = torch.sparse_coo_tensor(indices, values, size=A_scipy.shape).cuda()

# Move vector to GPU
x_torch = torch.tensor(x_np, dtype=torch.float32).cuda()

# Repeated multiplication (note: we unsqueeze to make x a column vector for sparse.mm)
for _ in range(1000):
    y_torch = torch.sparse.mm(A_torch, x_torch.unsqueeze(1)).squeeze()

# Transfer back to CPU if needed
y_np = y_torch.cpu().numpy()

• Avoid GPU-CPU Data Transfers in Loops: As mentioned earlier, moving data between CPU and GPU is a major bottleneck. Keep your matrix and vectors on the GPU for all repeated computations.
• Use Lower Precision if Possible: If your use case allows it, switch to float16 (half-precision) or even float8—GPUs process these much faster. For CuPy, use cp.float16; for PyTorch, use torch.float16.
• Batch Your Vectors: If you have multiple x vectors to compute Ax for, stack them into a dense matrix and run a single sparse-dense matrix multiplication instead of multiple vector multiplies. This is far more efficient.
• Pin Host Memory: If you need to frequently send new data from CPU to GPU, use pinned memory (CuPy's cp.cuda.alloc_pinned_memory or PyTorch's torch.cuda.pin_memory()) to speed up transfers.

Always test to make sure you're getting the performance you expect. Use timeit to measure per-iteration times:

import timeit

# Test CuPy performance
def cupy_mult():
    return A_cupy.dot(x_cupy)

print(f"CuPy average time per iteration: {timeit.timeit(cupy_mult, number=1000)/1000:.6f} seconds")

# Test PyTorch performance
def torch_mult():
    return torch.sparse.mm(A_torch, x_torch.unsqueeze(1)).squeeze()

print(f"PyTorch average time per iteration: {timeit.timeit(torch_mult, number=1000)/1000:.6f} seconds")

Compare these times to your original Scipy CPU implementation to see the speedup—you should see anywhere from 10x to 100x depending on your GPU.

内容的提问来源于stack exchange，提问作者rkp