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CG.py
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# -*- coding: utf-8 -*-
"""
Created on Tue Oct 25 12:54:55 2022
@author: uqalim8
"""
import torch
ZERO = 1e-8
def CG(A, b, tol = 1e-2, maxite = 100):
x0 = torch.zeros(b.shape[0], dtype = torch.float64)
rkm1 = b - Ax(A, x0)
pk = rkm1.clone()
xkm1 = x0
k = 0
while torch.norm(rkm1) > tol and k < maxite:
Apk = Ax(A, pk)
rk_1sq = torch.norm(rkm1)**2
alpha = rk_1sq / torch.dot(pk, Apk)
xk = xkm1 + alpha * pk
rk = rkm1 - alpha * Apk
beta = torch.norm(rk)**2 / rk_1sq
pk = rk + beta * pk
rkm1 = rk
xkm1 = xk
k += 1
return xkm1, k
def CGSteihaug(H, g, delta, tol, maxite):
z = torch.zeros_like(g)
# if torch.norm(g) < tol:
# return z, "||g||<tol", 1, 0
j = 0
d, r = -g.clone(), g.clone()
norm_g = torch.norm(g)
norm_r = norm_g
Bd = Ax(H, d)
norm_Hg = torch.norm(Bd)
norm_Hr = norm_Hg
while j <= maxite:
dBd = torch.dot(d, Bd)
j += 1
if dBd <= 0:
dz = torch.dot(d, z)
norm_d, norm_z = torch.norm(d), torch.norm(z)
numerator = - dz + torch.sqrt(dz**2 - norm_d**2 * (norm_z**2 - delta**2))
tau = numerator / norm_d**2
p = z + tau * d
m0_mk = - torch.dot(g, p) - torch.dot(p, Ax(H, p)) / 2
return p, norm_r / norm_g, norm_Hr / norm_Hg, "NC", m0_mk, j
norm_r = torch.dot(r, r)
alpha = norm_r / dBd
zp1 = z + alpha * d
if torch.norm(zp1) >= delta:
dz = torch.dot(d, z)
norm_d, norm_z = torch.norm(d), torch.norm(z)
numerator = - dz + torch.sqrt(dz**2 - norm_d**2 * (norm_z**2 - delta**2))
tau = numerator / norm_d**2
p = z + tau * d
m0_mk = - torch.dot(g, p) - torch.dot(p, Ax(H, p)) / 2
return p, norm_r / norm_g, norm_Hr / norm_Hg, "SOL,=", m0_mk, j
z = zp1
r = r + alpha * Bd
if torch.norm(r) < tol:
p = z
m0_mk = - torch.dot(g, p) - torch.dot(p, Ax(H, p)) / 2
return p, norm_r / norm_g, norm_Hr / norm_Hg, "SOL,<", m0_mk, j
norm_rp1 = torch.dot(r, r)
beta = norm_rp1 / norm_r
d = -r + beta * d
norm_r = norm_rp1
Bdkp1 = Ax(H, d)
norm_Hr = torch.norm(Bdkp1 - beta * Bd)
Bd = Bdkp1
p = z
m0_mk = - torch.dot(g, p) - torch.dot(p, Ax(H, p)) / 2
return p, norm_r / norm_g, norm_Hr / norm_Hg, "MAX,<", m0_mk, j
def CappedCG(H, b, zeta, epsilon, maxiter, M=0):
g = -b
y = torch.zeros_like(g)
kappa, tzeta, tau, T = para(M, epsilon, zeta)
tHy = y.clone()
tHY = y.reshape(-1, 1)
Y = y.reshape(-1, 1)
r = g
p = -g
tHp = Ax(H, p) + 2*epsilon*p
j = 1
ptHp = torch.dot(p, tHp)
norm_g = torch.norm(g)
norm_p = norm_g
rr = torch.dot(r, r)
dType = 'Sol'
relres = 1
if ptHp < epsilon*norm_p**2:
d = p
dType = 'NC'
return d, dType, j, ptHp, 1, 1
norm_Hp = torch.norm(tHp - 2*epsilon*p)
norm_Hb = norm_Hp
if norm_Hp > M*norm_p:
M = norm_Hp/norm_p
kappa, tzeta, tau, T = para(M, epsilon, zeta)
while j < maxiter:
alpha = rr/ptHp
y = y + alpha*p
#Y = torch.cat((Y, y.reshape(-1, 1)), 1) #record y
norm_y = torch.norm(y)
tHy = tHy + alpha*tHp
#tHY = torch.cat((tHY, tHy.reshape(-1, 1)), 1) # record tHy
norm_Hy = torch.norm(tHy - 2*epsilon*y)
r = r + alpha*tHp
rr_new = torch.dot(r, r)
beta = rr_new/rr
rr = rr_new
p = -r + beta*p #calculate Hr
norm_p = torch.norm(p)
tHp_new = Ax(H, p) + 2*epsilon*p #the only Hessian-vector product
j = j + 1
tHr = beta*tHp - tHp_new #calculate Hr
tHp = tHp_new
norm_Hp = torch.norm(tHp - 2*epsilon*p)
ptHp = torch.dot(p, tHp)
if norm_Hp> M*norm_p:
M = norm_Hp/norm_p
kappa, tzeta, tau, T = para(M, epsilon, zeta)
if norm_Hy > M*norm_y:
M = norm_Hy/norm_y
kappa, tzeta, tau, T = para(M, epsilon, zeta)
norm_r = torch.norm(r)
relres = norm_r/norm_g
# print(norm_r/norm_g, tzeta)
norm_Hr = torch.norm(tHr - 2*epsilon*r)
# print(norm_r, torch.norm(H(y) + g))
if norm_Hr> M*norm_r:
M = norm_Hr/norm_r
kappa, tzeta, tau, T = para(M, epsilon, zeta)
if torch.dot(y, tHy) < epsilon*norm_y**2:
d = y
dType = 'NC'
# print('y')
return d, dType, j, torch.dot(y, tHy), relres, norm_Hr / norm_Hb
elif norm_r < tzeta*norm_g:
# print('relres', relres)
d = y
return d, dType, j, 0, relres, norm_Hr / norm_Hb
elif torch.dot(p, tHp) < epsilon*norm_p**2:
d = p
dType = 'NC'
# print('p')
return d, dType, j, torch.dot(p, tHp), relres, norm_Hr / norm_Hb
elif norm_r > torch.sqrt(T*tau**j)*norm_g:
print('Uncomment tensors Y, tHY')
alpha_new = rr/ptHp
y_new = y + alpha_new*p
tHy_new = tHy + alpha_new*tHp
for i in range(j):
dy = y_new - Y[:, i]
dtHy = tHy_new - tHY[:, i]
if torch.dot(dy, dtHy) < epsilon*torch.norm(dy)**2:
d = dy
dType = 'NC'
print('dy')
return d, dType, j, torch.dot(dy, dtHy), relres, norm_Hr / norm_Hb
print('Maximum iteration exceeded!')
return y, dType, j, 0, relres, norm_Hr / norm_Hb
def para(M, epsilon, zeta):
# if torch.tensor(M):
# M = M.item()
kappa = (M + 2*epsilon)/epsilon
tzeta = zeta/3/kappa
# print('kappa', kappa)
sqk = torch.sqrt(torch.tensor(float(kappa)))
tau = sqk/(sqk + 1)
T = 4*kappa**4/(1 + torch.sqrt(tau))**2
return kappa, tzeta, tau, T
def Ax(A, x):
if callable(A):
Ax = A(x)
else:
Ax = A @ x
return Ax
# if __name__ == "__main__":
# A = np.random.rand(100, 100)
# A = A.T @ A
# b = A @ np.ones((100, 1))
# x, k, r = CG(A, b)
# print(np.linalg.norm(A @ x - b), k, r)