Demo for using hifir4py with built-in gmres_hif

In this example, we show how to use hifir4py HIFIR preconditioner coupling with the built-in gmres_hif solver. The example system is a saddle-point formulation of 3D Stokes equation with Taylor-Hood elements.

import numpy as np
from scipy.io import loadmat
from hifir4py import *

# load the MATFILE from scipy.io
f = loadmat("demo_inputs/data.mat")
A = f["A"]
b = f["b"].reshape(-1)

Let’s show some basic information of the system, including shape, nnz, and leading block symmetry

# A is scipy.sparse.csr_matrix
print("The system shape is {}, where the nnz is {}".format(A.shape, A.nnz))

The system shape is (2990, 2990), where the nnz is 44632

Now, let’s build the preconditioenr \(\boldsymbol{M}\) with more aggressive options, i.e. droptol for L and U factors is 1e-2, condest for L, U, and D is 5, and \(\alpha\) for L and U is 3.

M = HIF()
params = Params()
params.tau = 0.01
params.kappa = 5.0
params.alpha = 3.0
M.factorize(A, params=params)

With the preconditioenr successfully been built, let’s print out some basic information

print("M levels are {}, with nnz {}".format(M.levels, M.nnz))

M levels are 2, with nnz 114848

Now, we solve with the built-in flexible GMRES solver in SciPy. Notice that the GMRES in SciPy is left-preconditioned, which is not recommended.

x, flag, stats = ksp.gmres_hif(A, b, M=M)

Preconditioned provided as input.
Starting GMRES iterations...
Computed solution in 3 iterations and 0.07072s.

print("solver done, flag={}, res={}".format(flag, np.linalg.norm(b-A.dot(x))/np.linalg.norm(b)))

solver done, flag=0, res=8.254808859114319e-07