General Minimum RESidual (GMRES) with ILU preconditioner

问题

I'm trying to implement the ILU preconditioner in this GMRES code I wrote (in order to solve the linear sistem Ax = b. I'm trying with an easy tridiagonal SPD matrix of dimension 25x25. As you can see I'm calculating the preconditioner with spilu method. The code is running without error, but the solution is clearly wrong since, at the end of the code, I'm printing the norm of b and the norm of the product A*x. They are not nearly the same.. The code Run fine without preconditioner and converge with 13 iteration for the same matrix.

This is the code I followed

import numpy as np
import scipy as sp
import matplotlib.pyplot as plt

'Size controller'

matrixSize =25

'Building a tri-diagonal matrix'

def Atridiag(val_0, val_sup, val_inf, mSize):
    cen     = np.ones((1, mSize))*val_0
    sup     = np.ones((1, mSize-1))*val_sup
    inf     = np.ones((1, mSize-1))*val_inf
    diag_cen  = np.diagflat(cen, 0)
    diag_sup  = np.diagflat(sup, 1)
    diag_inf  = np.diagflat(inf, -1)
    return diag_cen + diag_sup + diag_inf

A = Atridiag(2, -1, -1, matrixSize)

A = sp.sparse.csc_matrix (A)

'Plot matrix sparsity'

plt.clf()
plt.spy(A, marker ='.', markersize=2)
plt.show()


'random b and x0 vectors'

b = np.matrix(np.ones((matrixSize, 1)))
x = np.matrix(np.ones((matrixSize, 1)))

'Incomplete LU'

M = sp.sparse.linalg.dsolve.spilu(A)
M1 = lambda x: M.solve(x)
M2=sp.sparse.linalg.LinearOperator((matrixSize,matrixSize),M1)


'Initial Data'

nmax_iter = 30
rstart = 2
tol = 1e-7
e = np.zeros((nmax_iter + 1, 1))
rr = 1

'Starting GMRES'

for rs in range (0, rstart+1):

    'first check on residual'

    if rr < tol :
        break

    else:

        r0 = (b - A.dot(x))
        betha = np.linalg.norm(r0)
        e[0] = betha
        H = np.zeros((nmax_iter + 1, nmax_iter))
        V = np.zeros((matrixSize, nmax_iter+1))
        V[:, 0:1] = r0/betha

    for k in range (1, nmax_iter+1):

        'Appling the Preconditioner'

        t = A.dot(V[:, k-1])
        V[:, k] = M2.matvec(t)

        'Ortogonalizzazione GS'

        for j in range (k):
            H[j, k-1] = np.dot(V[:, k].T, V[:, j])
            V[:, k] = V[:, k] - (np.dot(H[j, k-1], V[:, j]))

        H[k, k-1] = np.linalg.norm(V[:, k])
        V[:, k] = V[:, k] / H[k, k-1] 

        'QR Decomposition'

        n=k
        Q = np.zeros((n+1, n))
        R = np.zeros((n, n))
        R[0, 0] = np.linalg.norm(H[0:n+2, 0])
        Q[:, 0] = H[0:n+1, 0] / R[0,0]
        for j in range (0, n+1):
            t = H[0:n+1, j-1]
            for i in range (0, j-1):
                R[i, j-1] = np.dot(Q[:, i], t)
                t = t - np.dot(R[i, j-1], Q[:, i])
            R[j-1, j-1] = np.linalg.norm(t)
            Q[:, j-1] = t / R[j-1, j-1]

        g = np.dot(Q.T, e[0:k+1]) 

        Z = np.dot(np.linalg.inv(R), g)

        Res = e[0:n] - np.dot(H[0:n, 0:n], Z[0:n])
        rr = np.linalg.norm(Res)

        'second check on residual'

        if rr < tol:
            break

    'Updating the solution'    

    x = x + np.dot(V[:, 0:k], Z)



print(sp.linalg.norm(b))
print(sp.linalg.norm(np.dot(A.todense(),x)))

Really Hope somebody can figure it out!!

回答1:

Maybe it's too late, but for future reference :

You forgot to multiply by the conditioner when updating x :

x = x + M2.dot(np.dot(V[:, 0:k], Z)    # M2.matvec() works the same

See here

With that fix, the algorithm converges in 1 iteration.

---

Other comments:

• You can directly do : M2 = sp.sparse.linalg.LinearOperator((matrixSize,matrixSize),M.solve)
• At the end, to compare Ax and b, it's better to print the difference (residual) because you will get a much more precise result: print(sp.linalg.norm(b - np.dot(A.todense(),x)))

来源：https://stackoverflow.com/questions/42233966/general-minimum-residual-gmres-with-ilu-preconditioner