How to solve Ax=b when A is a Sparse Array ?

[crazyfireji]

Like the title, if A is a Sparse Array, we can solve it like: x=A\b, but it can't works in CUDA or gpu.

[amontoison]

Hi @crazyfireji!

If A is a sparse matrix on the GPU, you can use a Krylov solver depending on the properties and shape of A:

• x, stats = cg(A, b, atol=1e-10, rtol=0.0) if A is symmetric positive definite.
• x, stats = minres(A, b, atol=1e-10, rtol=0.0) if A is symmetric indefinite.
• x, stats = gmres(A, b, atol=1e-10, rtol=0.0) if A is square and unsymmetric.
• x, stats = lsmr(A, b, atol=1e-10, rtol=0.0) if A is rectangular.

I’ve detailed how to set this up for each GPU backend
--> here <--

[crazyfireji]

Hi @crazyfireji!

If A is a sparse matrix on the GPU, you can use a Krylov solver depending on the properties and shape of A:

• x, stats = cg(A, b, atol=1e-10, rtol=0.0) if A is symmetric positive definite.
• x, stats = minres(A, b, atol=1e-10, rtol=0.0) if A is symmetric indefinite.
• x, stats = gmres(A, b, atol=1e-10, rtol=0.0) if A is square and unsymmetric.
• x, stats = lsmr(A, b, atol=1e-10, rtol=0.0) if A is rectangular.

I’ve detailed how to set this up for each GPU backend --> here <--

but the effecience is very low when ,my A is very large（more than 200000*200000, is there any more effecient function can be recomanded?

[amontoison]

@crazyfireji Can you give more details please?
I don't know on which architectures you want to run your code (NVIDIA, AMD or Intel?) and what are the properties of A.

For large systems, you need a preconditioner to speed-up the convergence.
You have many exemples in the documentation.

If the system is "just" 200 000 x 200 000, I can also recommend CUDSS.jl but it will only work on NVIDIA GPUs and the system should not be too ill-conditioned:
https://github.com/exanauts/CUDSS.jl