Demo for using hifir4py

In this example, we show how to use hifir4py HIFIR preconditioner in its multilevel triangular solve and matrix-vector multiplication, which are the core operations in applying a preconditioner.

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

Factorize HIF

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(
    A,
    tau_L=0.01,
    tau_U=0.01,
    kappa=5.0,
    kappa_d=5.0,
    alpha_L=3.0,
    alpha_U=3.0,
)

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

Alternatively, one can use the following codes:

M = HIF()
params=Params()
params.tau = 1e-2  # equiv. to params["tau_L"]=params["tau_U"]=1e-2
params.kappa = 5.0
params.alpha = 3.0
M.factorize(A, params=params)

or

M = HIF()
M.factorize(A,
    tau_L=0.01,
    tau_U=0.01,
    kappa=5.0,
    kappa_d=5.0,
    alpha_L=3.0,
    alpha_U=3.0,
)

Apply HIF

We now consider applying M in triangular solve and matrix-vector multiplication two modes.

x = M.apply(b)
err = M.apply(x, op="M") - b
print("norm2(err)/norm2(b) =", np.linalg.norm(err)/np.linalg.norm(b))

norm2(err)/norm2(b) = 1.439076582138997e-17

# Tranpose
x = M.apply(b, op="SH")
err = M.apply(x, op="MH") - b
print("norm2(err)/norm2(b) =", np.linalg.norm(err)/np.linalg.norm(b))

norm2(err)/norm2(b) = 1.4514835137900503e-17