Demo for using hifir4py with its PIPIT solver

In this example, we show how to use hifir4py HIFIR-preconditioned PIPIT solver for seeking the pseudoinverse solution of a linear elasticity system with pure traction boundary conditions discretized by FEM. Note that this system has six-dimensional nullspace.

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

# load the MATFILE from scipy.io
f = loadmat("demo_inputs/LE_4.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 (2295, 2295), where the nnz is 83997

Now, we directly call pipit_hifir to seek for the pseudoinverse solution with default settings, i.e., default parameters in HIF, \(\text{rtol}=10^{-12}\) for the null-space residual tolerance, and \(\text{rtol}=10^{-6}\) for the least-squares solution (by HIF-preconditioned GMRES) relative residual tolerance; the restart and maxit are set to be 30 and 500, respectively.

x, ns, flag, stats = ksp.pipit_hifir(A, b, 6)  # 6 is the null-space dimension

HIF factorization finished in 0.1743s.

Starting computing left nullspace...
Finished left nullspace computation with total 46 GMRES iterations
and total 75 inner refinements in 0.463s.

Starting GMRES for least-squares solution...
Preconditioned provided as input.
Starting GMRES iterations...
Computed solution in 7 iterations and 0.02111s.

System is numerically symmetric; let vs=us.

We know analyze the accuracy of the null-space components. Note that since this system is symmetric (thus range-symmetric).

from scipy.sparse.linalg import norm as spnorm

vs = ns["vs"]
av = A.dot(vs)
av_res = np.asarray([np.linalg.norm(av[i]) for i in range(6)]) / spnorm(A)
print("relative residual of the six-dimensional nullspace are:", av_res)

relative residual of the six-dimensional nullspace are: [6.44680278e-13 7.25437491e-13 2.64731149e-14 6.97660024e-13
 6.79220534e-13 1.09590900e-14]