def get_norm_jac_iteration_matrix(m, nu, omega):
"""
Fuction to generate the norm of the iteration matrix solving the system
(nu*A^2 + I)*u = A (y_d + A^{-1}*f) = A*y_d - f
by using damped Jacobi
Parameters
----------
m : positive int
size the matrix should have (for d=2 the matrix size is m**2 x m**2),
regarding your desired discretization
nu : positive real number
regularization parameter of the optimal control problem
omega : real number between 0 and 1
weighing parameter of the damped Jacobi iteration
Returns
-------
Norm of the iteration matrix using damped Jacobi
"""
A = fd_laplace(m, 2).toarray()
n = m**2
A_2 = A.dot(A)
System = np.eye(n) + nu * A_2
D = np.diag(np.diag(System))
D_inv = omega * np.linalg.inv(D)
It_matrix = D_inv.dot((1 / omega) * D - System)
return np.linalg.norm(It_matrix, ord=2)
def test_jacobi(nu_start, m, omega, plot="nu"):
#y_d = 10*np.random.rand(m**2)
#f=np.random.rand(m**2)
u_sol = np.random.rand(m**2)
u_guess = np.random.rand(m**2)
maxIter = 10000
for nu in nu_start:
A = fd_laplace(m, 2)
A_2 = np.dot(A, A)
C = sparse.eye((m**2)) + nu * A_2
f = C.dot(u_sol)
#f = A.dot(y_d) - f
u_test, num_iters, res = solver_damped_jacobi(C, u_guess, omega, f,
maxIter)
x = np.arange(0, num_iters)
#u_test2 = sparse.linalg.spsolve(C, f)
#print("norm error = {}".format(np.linalg.norm(u_sol - u_test)))
if plot == "nu":
plt.semilogy(x, res, label="nu = {}".format(nu))
else:
plt.semilogy(x, res, label="m = {}".format(m))
def sparse_sol():
nu = 0.01
l = 4
m = 2**l - 1
A = fd_laplace(m, 2)
u_sol = np.random.rand(m**2)
A_2 = A.dot(A)
S_jacobi = sparse.eye((m**2)) + nu * A_2
f = S_jacobi.dot(u_sol)
sparse.linalg.spsolve(S_jacobi, f)
def plot_mulitgrid_jacobi():
omega = 0.5
plt.figure(figsize=(20, 20))
#plt.suptitle(r"Using multigrid to solve the system $(\ nu \cdot A^2+ I )v = f$ with dampeed jacobi with damping parameter $\omega =$ {} and comparing this to the convergence of the smoother as a stationary method".format(omega))
############# important nu1 = 2 is essentially one step since we break after nu-1 iterations
nu1 = 2
nu2 = 2
level = 3
j = 1
for i in range(0, 4):
nu = 0.01 * 10**i
for l in range(4, 7):
m = 2**l - 1
#h = 1/(m+1)
A = fd_laplace(m, 2)
u_sol = np.random.rand(m**2)
u_guess = np.zeros(m**2)
A_2 = A.dot(A)
S_jacobi = sparse.eye((m**2)) + nu * A_2
f = S_jacobi.dot(u_sol)
u_jac, res_jac, k_jac = multigrid_jacobi(S_jacobi,nu, f, u_guess, m, omega, \
nu1, nu2, level)
(u_jac_m, k_jac_m, res_jac_m) = solver_damped_jacobi(S_jacobi,
u_guess,
omega,
f,
maxIter=k_jac)
x_jac = np.arange(0, k_jac)
x_jac_m = np.arange(0, k_jac_m)
plt.subplot(4, 3, j)
plt.semilogy(x_jac,
res_jac,
"b-",
label="multigrid with {} levels".format(level))
plt.semilogy(x_jac_m, res_jac_m, "r-", label="damped jacobi")
plt.legend(loc="upper right")
plt.ylabel("normalized residual ")
plt.xlabel("iterations")
plt.title(r"$m =$ {0}, $\nu =$ {1}".format(m, nu))
print("norm differences = {}".format(np.linalg.norm(u_sol -
u_jac)))
j = j + 1
#plt.tight_layout()
plt.show()
def multi_sol():
omega = 0.5
nu1 = 3
nu2 = 3
nu = 0.01
level = 3
l = 4
m = 2**l - 1
A = fd_laplace(m, 2)
u_sol = np.random.rand(m**2)
u_guess = np.zeros((m**2))
A_2 = A.dot(A)
S_jacobi = sparse.eye((m**2)) + nu * A_2
f = S_jacobi.dot(u_sol)
multigrid_jacobi(nu, f, u_guess, m, omega, nu1, nu2, level)
def test_factored(nu_start, m, plot="nu"):
y_d = 10 * np.ones(m**2)
u_sol = np.random.rand(m**2)
u_guess = np.random.rand(m**2)
A = fd_laplace(m, d=2)
for nu in nu_start:
f = (-1) * (nu * A.dot(A.dot(u_sol)) + u_sol - A.dot(y_d))
u_test, info, num_iters, res = solver_poisson_factored_cg(
u_guess, nu, y_d, f, m)
x = np.arange(0, num_iters)
if plot == "nu":
plt.semilogy(x, res, label="nu = {}".format(nu))
else:
plt.semilogy(x, res, label="m = {}".format(m))
def test_stationary(nu_start, m, plot="nu", toler=1e-12):
y_d = 10 * np.ones(m**2)
u_sol = np.random.rand(m**2)
u_guess = np.random.rand(m**2)
A = fd_laplace(m, d=2)
for nu in nu_start:
f = (-1) * (nu * A.dot(A.dot(u_sol)) + u_sol - A.dot(y_d))
u_test, num_iters, res = solver_stationary(u_guess,
nu,
y_d,
f,
m,
tol=toler)
x = np.arange(0, num_iters)
if plot == "nu":
plt.semilogy(x, res, label="nu = {}".format(nu))
else:
plt.semilogy(x, res, label="m = {}".format(m))
@author: florianwolf
"""
import numpy as np
from scipy.sparse.linalg import LinearOperator
from ReducedModel import fd_laplace, condition_number_factored, condition_number_normal,\
multigrid_jacobi, get_system, multigrid_stat,solver_stationary_fixedRight,solver_damped_jacobi,\
fast_poisson, laplace_small_decomposition
from scipy import sparse
import matplotlib.pyplot as plt
import timeit
plt.close('all')
l = 6
m = 2**l - 1
A = fd_laplace(m, d=2)
for i in range(100):
x_sol = np.random.rand(m**2)
y = A.dot(x_sol)
lam, V = laplace_small_decomposition(m)
x_test = fast_poisson(V, V, lam, lam, y)
print("norm of the differences = {}".format(np.linalg.norm(x_sol -
x_test)))