How to set sparsity pattern and relaxation steps for ISAI preconditioner?

[learning-chip]

I saw the paper Incomplete Sparse Approximate Inverses for Parallel Preconditioning along with its implementation at gko::preconditioner::Isai (thanks for releasing for code!). I find the idea of ISAI very interesting and want to reproduce some results in the paper.

However, there doesn't seem a way to change the sparsity pattern (using L^2 pattern for example) and relaxation steps (Eq 15 of the paper) as did in the paper.

I think what's implemented in Ginkgo is referred as "SpMV SAI" in the paper, which does not use relaxation steps.

I'd like to produce something like this Table 4 in the paper that sweeps over ISAI parameters:

[learning-chip]

In the unit test reference/test/preconditioner/isai_kernels I found the call with_sparsity_power() that seems to set the sparsity pattern:

ginkgo/reference/test/preconditioner/isai_kernels.cpp

Lines 1144 to 1149 in 693bc4f

	const auto isai = TestFixture::LowerIsai::build()
	.with_sparsity_power(2)
	.on(this->exec)
	->generate(this->l_sparse);
	auto l_inv = isai->get_approximate_inverse();

As for the relaxation scheme, I could extract the inverse factor and manually type the formula (Eq 15.). Or is there a more efficient implementation available?

[fritzgoebel]

Hi, thanks for reaching out!
You seem to have already found the sparsity power parameter, that's great. For the relaxation steps, you can have a look at the example ir-ilu-preconditioned-solver. In line 120, there is a triangular solver defined as iterative refinement with block-Jacobi as the inner solver. If you replace the block-Jacobi with your ISAI factory, you should end up with what you are looking for. That would look something like

    auto isai_factory =
        gko::share(gko::preconditioner::LowerIsai<value_type, index_type>::build() 
                       .with_sparsity_power(2) 
                       .on(this->exec));

    auto trisolve_factory =
        gko::solver::Ir<value_type>::build()
            .with_solver(isai_factory)
            .with_criteria(
                gko::stop::Iteration::build().with_max_iters(20u).on(exec))
            .on(exec);

I hope this helps!

[learning-chip]

Thanks, composing gko::solver::Ir with ISAI is brilliant!