带 ILU 预调节器的一般最小 RESidual (GMRES)

Question

我正在尝试在我编写的这段 GMRES 代码中实现 ILU 预调节器（为了求解线性系统 Ax = b。我正在尝试使用一个简单的 25x25 维三对角 SPD 矩阵。如您所见我正在用 spilu 方法计算预条件子。代码是运行没有错误，但解决方案显然是错误的，因为在代码末尾，我打印了 b 的范数和乘积的范数A*x. 他们几乎不一样.. 代码运行在没有预处理器的情况下很好，并且对同一矩阵进行 13 次迭代收敛。

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)))

真的希望有人能解决！！

Answer 1

也许为时已晚，但供将来参考：

你更新x的时候忘记乘以conditioner了:

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

见here

通过该修复，算法在 1 次迭代中收敛。

其他评论：

你可以直接做：M2 = sp.sparse.linalg.LinearOperator((matrixSize,matrixSize),M.solve)
最后，比较Ax和b，最好打印差异（残差），因为你会得到更精确的结果：print(sp.linalg.norm(b - np.dot(A.todense(),x)))

带 ILU 预调节器的一般最小 RESidual (GMRES)

General Minimum RESidual (GMRES) with ILU preconditioner

python

optimization

scipy

preconditions

convergence