def get_matrices_of_optimization_problem_bar(H_div, v, f_bar, invA, mesh, dim):
"""
:param H_div: funtional space for the flux function
:param v: approximate solution
:param f_bar: right-hand side function (modified RHS)
:param A: diffusion matrix
:param mesh: mesh
:param dim: geometrical problem dimension
:return DivDiv, PhiPhi, RhsDivPhi, RhsPhi: matrixes for the optimal reconstruction of the flux
"""
# Define variational problem
y = TrialFunction(H_div)
q = TestFunction(H_div)
# Define system of linear equation to find the majorant
# Phi_i, i = 1, ..., d are vector-valued basis of H_div
# \int_\Omega div(phi_i) div(phi_j) \dx
DivPhiDivPhi = assemble(inner(Div(y, dim), Div(q, dim)) * dx(domain=mesh))
#DivPhiDivPhi_sp = as_backend_type(assemble(inner(Div(y, dim), Div(q, dim)) * dx(domain=mesh))).mat()
#DivPhiDivPhi_sparray = csr_matrix(DivPhiDivPhi_sp.getValuesCSR()[::-1], shape=DivPhiDivPhi_sp.size)
# \int_\Omega A^{-1} phi_i \cdot phi_j \dx
PhiPhi = assemble(inner(invA * y, q) * dx(domain=mesh))
# Define vectors included into the RHS
RhsDivPhi = assemble(inner(-f_bar, Div(q, dim)) * dx(domain=mesh))
RhsPhi = assemble(inner(Grad(v, dim), q) * dx(domain=mesh))
#print "DivDiv = ", DivPhiDivPhi.array()
#print "PhiPhi = ", PhiPhi.array()
#print "RhsDivPhi = ", RhsDivPhi.array()
#print "RhsPhi = ", RhsPhi.array()
return DivPhiDivPhi, PhiPhi, RhsDivPhi, RhsPhi
def error_norm(u, ue, lmbd, A, invA, a, func_a, V, Ve, mesh, dim):
"""
:param u: approximate solution
:param ue: exact solution
:param lmbd: reaction function
:param A: diffusion operator
:param Ve: functional space of exact solution
:param mesh: mesh
:param dim: dimension of the problem
:return: error-norm between u and ue
"""
# Interpolate exact and approximate solution to the functional space of exact solution
u_ve = interpolate(u, Ve)
u_exact_ve = interpolate(ue, Ve)
e = abs(u_ve - u_exact_ve)
# Define variational form of the error
var_grad_e = inner(A * Grad(e, dim), Grad(e, dim))
delta = abs(lmbd - 0.5 * Div(func_a, dim))
var_delta_e = delta * inner(e, e)
var_lambda_e = lmbd * inner(e, e)
var_diva_e = (-0.5 * Div(func_a, dim)) * inner(e, e)
var_a_e = inner(invA * a * e, a * e)
# Assembling variational form of the error
grad_e = assemble(var_grad_e * dx(domain=mesh))
delta_e = assemble(var_delta_e * dx(domain=mesh))
lambda_e = assemble(var_lambda_e * dx(domain=mesh))
diva_e = assemble(var_diva_e * dx(domain=mesh))
a_e = assemble(var_a_e * dx(domain=mesh))
# Calculate L2 norm
l2_e = assemble(inner(e, e) * dx(domain=mesh))
# Calculate Linf norm
#e_func = project(e, Ve, form_compiler_parameters={'quadrature_degree': 4})
#linf_e = norm(e_func.vector(), 'linf')
# Calculate Linf based on the nodal values
u_exact_v = interpolate(ue, V)
linf_e = abs(u_exact_v.vector().array() - u.vector().array()).max()
print '%------------------------------------------------------------------------------------%'
print '% Error '
print '%------------------------------------------------------------------------------------%'
print "\| grad e \|^2_A = %8.2e" % grad_e
print "\| e \|^2 = %8.2e" % l2_e
print "\| e \|^2_inf = %8.2e" % linf_e
print "\| (lmbd - 0.5 div a)^0.5 e \|^2 = %8.2e" % delta_e
print "\| lmbd^0.5 e \|^2 = %8.2e" % lambda_e
print "\| a e \|^2_{A^{-1}} = %8.2e" % a_e
'''
print "\| grad e \|^2_A + \| (lmbd - 0.5 div a)^0.5 e \|^2 = %8.2e" \
% (grad_e + delta_e)
print "\| grad e \|^2_A + \| lmbd^0.5 e \|^2 + \| a e \|^2_{A^{-1}} = %8.2e" \
% (grad_e + lambda_e + a_e)
'''
return grad_e, l2_e, linf_e, delta_e, lambda_e, a_e, var_grad_e, var_delta_e, var_lambda_e, var_a_e
def update_majorant_II_components(u, w, y, f, u0, mesh, dim, v_deg, W, Ve, T):
r_d = (y - Grad(u, dim) + Grad(w, dim))
r_f = (Div(y, dim) + f - D_t(u, dim) - D_t(w, dim))
L = D_t(u, dim) * w + inner(Grad(u, dim), Grad(w, dim)) - f * w
w_T, T_mesh = get_2d_slice_of_3d_function_on_Oz(mesh, w, T, dim, v_deg + 1)
w_0, O_mesh = get_2d_slice_of_3d_function_on_Oz(mesh, w, 0, dim, v_deg + 1)
u0_0, O_mesh = get_2d_slice_of_3d_function_on_Oz(mesh, interpolate(u0, Ve),
0, dim, v_deg) # varphi
u_0, O_mesh = get_2d_slice_of_3d_function_on_Oz(mesh, interpolate(u, Ve),
0, dim, v_deg)
r_0 = u0_0 - u_0
l = inner(r_0, r_0) - 2 * w_0 * r_0
w_T_form = inner(w_T, w_T)
m_d = assemble(inner(r_d, r_d) * dx(domain=mesh))
m_f = assemble(inner(r_f, r_f) * dx(domain=mesh))
m_L = assemble(
(D_t(u, dim) * w + inner(Grad(u, dim), Grad(w, dim)) - f * w) *
dx(domain=mesh))
m_T = assemble(w_T_form * dx(domain=T_mesh))
m_l = assemble(l * dx(domain=O_mesh))
return m_d, m_f, m_l, m_L, m_T
def get_matrices_of_optimization_problem_II(W, u, y, f, u0, T, mesh, dim,
v_deg):
# Define variational problem
w = TrialFunction(W)
mu = TestFunction(W)
#w_T, T_mesh = get_2d_slice_of_3d_function_on_Oz(mesh, w, T, dim, v_deg + 1)
#mu_T, T_mesh = get_2d_slice_of_3d_function_on_Oz(mesh, mu, T, dim, v_deg + 1)
#mu_0, O_mesh = get_2d_slice_of_3d_function_on_Oz(mesh, mu, 0, dim, v_deg + 1)
#u0_0, O_mesh = get_2d_slice_of_3d_function_on_Oz(mesh, interpolate(u0, W), 0, dim, v_deg + 1)
# Define system of linear equation to find the majorant
S = assemble(inner(Grad(w, dim), Grad(mu, dim)) * dx(domain=mesh))
K = assemble(inner(D_t(w, dim), D_t(mu, dim)) * dx(domain=mesh))
#F = assemble(inner(w_T, mu_T)) * dx(domain=T_mesh)
L = assemble(
(D_t(u, dim) * mu + inner(Grad(u, dim), Grad(mu, dim)) - f * mu) *
dx(domain=mesh))
z = assemble((inner(y - Grad(u, dim), Grad(mu, dim))) * dx(domain=mesh))
g = assemble(
(f - D_t(u, dim) + Div(y, dim)) * D_t(mu, dim) * dx(domain=mesh))
#I = assemble(u0_0 * mu_0 * dx(domain=O_mesh))
return S, K, L, z, g
def get_matrices_of_optimization_problem(H_div, u, f_bar, invA, mesh, dim):
# Define variational problem
y = TrialFunction(H_div)
q = TestFunction(H_div)
# Define system of linear equation to find the majorant
S = assemble(inner(Div(y, dim), Div(q, dim)) * dx(domain=mesh))
K = assemble(inner(invA * y, q) * dx(domain=mesh))
N = assemble(inner(y, q) * ds(neumann_bc_marker))
#z = assemble(inner(- (f - c_H * D_t(u, dim)), Div(q, dim)) * dx(domain=mesh))
z = assemble(inner(-f_bar, Div(q, dim)) * dx(domain=mesh))
g = assemble(inner(NablaGrad(u, dim), q) * dx(domain=mesh))
return S, K, z, g
def calculate_majorant_bar_mu_opt(u, y, beta, f_bar, A, invA, min_eig_A, lmbd,
mesh, dim, C_FD):
"""
:param u: approximate solution
:param y: flux
:param beta: parameter minimizing majorant
:param f_bar: right-hand side function (modified RHS)
:param A: diffusion matrix
:param lambda_1: minimal eigenvalue of diffusion matrix
:param lmbd: reaction function
:param mesh: mesh
:param dim: geometrical problem dimension
:param C_FD: Freidrichs constant
:return maj: majorant value
m_d, m_f_one_minus_mu_opt: majorant components
beta: optimal parameter
var_m_d, var_m_f_w_opt: variational form of the majorant (to use them later in construction of error estimators)
"""
# Define optimal parameters
mu_opt = (C_FD**2) * (1.0 + beta) * lmbd / (beta * min_eig_A + (C_FD**2) *
(1.0 + beta) * lmbd)
#mu_opt = 1
w_opt = (C_FD**2) * (1.0 + beta) / (
beta * min_eig_A + (C_FD**2) * (1.0 + beta) * lmbd
) # this is the function in front of inner(r_f, r_f)
# Define residuals
r_d = y - A * NablaGrad(u, dim)
r_f = (Div(y, dim) + f_bar)
#r_d = y - eps * A * Grad(u, dim)
#r_f = Div(y, dim) + f_bar
# Define variational forms
var_m_d = inner(invA * r_d, r_d)
var_m_f_w_opt = w_opt * inner(r_f, r_f)
var_m_f_one_minus_mu_opt = ((1 - mu_opt)**2) * inner(
r_f, r_f) # only for the calculation of optimal beta
# Define majorant components
m_d = assemble(var_m_d * dx(domain=mesh))
m_f_w_opt = assemble(var_m_f_w_opt *
dx(domain=mesh)) # for calculating majorant
m_f_one_minus_mu_opt = assemble(var_m_f_one_minus_mu_opt *
dx(domain=mesh)) # fo calculating beta_opt
# Calculate majorant based on the parameter value
if m_f_w_opt <= DOLFIN_EPS:
maj = m_d
else:
if m_d <= DOLFIN_EPS:
maj = m_f_w_opt
else:
maj = (1.0 + beta) * m_d + m_f_w_opt
# Calculate the optimal value for beta parameter
beta = C_FD * sqrt(m_f_one_minus_mu_opt / m_d / min_eig_A)
return maj, m_d, m_f_one_minus_mu_opt, beta, var_m_d, var_m_f_w_opt
def get_indicators_CG0(e_form, norm_grad_e, V, V_star, f, u0, u0_boundary, u,
u_e, mesh, dim):
e_form = u_e - u
# Contruct the solution of the adjoint problem
z = solve_dual_problem(mesh, V_star, u0, u0_boundary, e_form, dim,
norm_grad_e)
Pz = project(z, V)
# Get parameters of the mesh
h = mesh.hmax()
n = FacetNormal(mesh)
# Construct the set of
DG0 = FunctionSpace(mesh, "DG", 0)
w = TestFunction(DG0)
r_h = jump(Grad(u, dim), n)
R_h = Div(Grad(u, dim), dim) + f
eta_var = w * (h * R_h)**2 * dx(
domain=mesh) + avg(w) * avg(h) * r_h**2 * dS(domain=mesh)
E_DWR_var = w * inner(R_h, z - Pz) * dx(
domain=mesh) - 0.5 * avg(w) * inner(r_h, avg(z - Pz)) * dS(domain=mesh)
J_e_var = w * J_energy_norm_error(e_form, Grad(e_form, dim), norm_grad_e,
dim) * dx(domain=mesh)
eta_DG0 = Function(DG0)
E_DWR_DG0 = Function(DG0)
J_e_DG0 = Function(DG0)
assemble(eta_var, tensor=eta_DG0.vector())
assemble(E_DWR_var, tensor=E_DWR_DG0.vector())
assemble(J_e_var, tensor=J_e_DG0.vector())
eta_distr = eta_DG0.vector().array()
E_DWR_distr = E_DWR_DG0.vector().array()
J_e_distr = J_e_DG0.vector().array()
print "eta_DG0 total:", numpy.sum(eta_distr)
print "E_DWR_DG0 total", numpy.sum(E_DWR_distr)
print "J_e_DG0 total", numpy.sum(J_e_distr)
return eta_distr, E_DWR_distr, J_e_distr
def majorant_distribution_DG0(mesh, f, lmbd, A, invA, u, e_form, y, beta, C_FD,
dim):
# Define the functional space used for distribution over cells
DG0 = FunctionSpace(mesh, "DG", 0)
w = TestFunction(DG0)
m_d_DG0 = Function(DG0)
m_df_DG0 = Function(DG0)
e_d_DG0 = Function(DG0)
# Define optimal parameters
w_opt = (C_FD**2) * (1 + beta) / (beta + (C_FD**2) * (1 + beta) * lmbd)
# Define the residuals of the majorant
r_d = y - A * Grad(u, dim)
r_f = (Div(y, dim) + f - lmbd * u)
# Define variational forms of dual component of majorant and the whole functional
m_d_var = w * sqrt(inner(invA * r_d, r_d)) * dx(domain=mesh)
m_df_var = w * sqrt((1.0 + beta) * inner(invA * r_d, r_d) +
w_opt * inner(r_f, r_f)) * dx(domain=mesh)
e_d_var = w * sqrt(inner(A * Grad(e_form, dim), Grad(
e_form, dim))) * dx(domain=mesh)
# Assemble the variation form and dumping obtained vector into function from DG0
assemble(m_d_var, tensor=m_d_DG0.vector())
assemble(m_df_var, tensor=m_df_DG0.vector())
assemble(e_d_var, tensor=e_d_DG0.vector())
m_d_distr = m_d_DG0.vector().array()
m_df_distr = m_df_DG0.vector().array()
e_d_distr = e_d_DG0.vector().array()
print "m_d_DG0 total", numpy.sum(m_d_distr)
print "m_df_DG0 total", numpy.sum(m_df_distr)
print "e_d_DG0 total", numpy.sum(e_d_distr)
return m_d_distr, m_df_distr, e_d_distr
def compare_error_indicators(mesh, V, V_star, f, u0, u0_boundary, u, u_e,
grad_u_e, y, beta, test_num, tag, dim):
z = solve_dual_problem(V_star, f, u0, u0_boundary, u, u_e)
Pz = project(z, V)
norm_grad_e = sqrt(
assemble(
inner(grad_u_e - Grad(u, dim), grad_u_e - Grad(u, dim)) *
dx(domain=mesh)))
h = mesh.hmax()
n = FacetNormal(mesh)
DG0 = FunctionSpace(mesh, "DG", 0)
w = TestFunction(DG0)
r_h = jump(Grad(u, dim), n)
R_h = div(Grad(u, dim)) + f
cell_num = mesh.num_cells()
e = abs(u_e - u)
r_d = abs(Grad(u, dim) - y)
r_f = abs(f + Div(y, dim))
eta_var = w * (h * R_h)**2 * dx + avg(w) * avg(h) * r_h**2 * dS
E_DWR_var = w * inner(R_h, z - Pz) * dx - 0.5 * avg(w) * inner(
r_h, avg(z - Pz)) * dS
J_e_var = w * J(w, e, norm_grad_e) * dx
m_d_var = w * sqrt(inner(r_d, r_d)) * dx
#C_FD = 1 / (sqrt(3) * DOLFIN_PI)
height = 1
width = 2
length = 2
C_FD = 1.0 / DOLFIN_PI / sqrt(1.0 / height**2 + 1.0 / width**2 +
1.0 / length**2)
m_df_var = w * sqrt((1 + beta) * inner(r_d, r_d) + C_FD**2 *
(1 + 1 / beta) * inner(r_f, r_f)) * dx
#M_e = w * (u - u_exact) * dx
eta_DG0 = Function(DG0)
E_DWR_DG0 = Function(DG0)
J_e_DG0 = Function(DG0)
m_d_DG0 = Function(DG0)
m_df_DG0 = Function(DG0)
assemble(eta_var, tensor=eta_DG0.vector())
assemble(E_DWR_var, tensor=E_DWR_DG0.vector())
assemble(J_e_var, tensor=J_e_DG0.vector())
assemble(m_d_var, tensor=m_d_DG0.vector())
assemble(m_df_var, tensor=m_df_DG0.vector())
eta_distr = eta_DG0.vector().array()
E_DWR_distr = E_DWR_DG0.vector().array()
J_e_distr = J_e_DG0.vector().array()
m_d_distr = m_d_DG0.vector().array()
m_df_distr = m_df_DG0.vector().array()
eta_DG0_total = numpy.sum(eta_distr)
E_DWR_DG0_total = numpy.sum(E_DWR_distr)
J_e_DG0_total = numpy.sum(J_e_distr)
m_d_DG0_total = numpy.sum(m_d_distr)
m_df_DG0_total = numpy.sum(m_df_distr)
print "eta_DG0 total:", eta_DG0_total
print "E_DWR_DG0 total", E_DWR_DG0_total
print "J_e_DG0 total", J_e_DG0_total
print "m_d_DG0 total", m_d_DG0_total
print "m_df_DG0 total", m_df_DG0_total
def error_norm(u, ue, A, lmbd, a, f, c_H, v_degree, mesh, T, dim, V, Ve):
"""
:param u: approximate solution
:param ue: exact solution
:param A:
:param lmbd:
:param lambd:
:param Ve: functional space of exact solution
:return: L2 error-norm between u and ue
"""
u_ve = interpolate(u, Ve)
u_exact_ve = interpolate(ue, Ve)
e = abs(u_ve - u_exact_ve)
#e = (u_ve - u_exact_ve)
res = Div(A * NablaGrad(u, dim), dim) \
+ f \
- c_H * D_t(u, dim) \
- lmbd * u \
- inner(a, NablaGrad(u, dim))
var_e = inner(e, e)
var_delta_e = inner((lmbd - 0.5 * Div(a, dim)) * e, e)
var_lambda_e = lmbd * inner(e, e)
var_grad_e = inner(A * NablaGrad(e, dim), NablaGrad(e, dim))
var_e_t = inner(c_H * D_t(e, dim), c_H * D_t(e, dim))
var_laplas_e = inner(Div(A * NablaGrad(e, dim), dim),
Div(A * NablaGrad(e, dim), dim))
var_e_id = inner(res, res)
val_e = assemble(var_e * dx(domain=mesh))
val_delta_e = assemble(var_delta_e * dx(domain=mesh))
val_lmbd_e = assemble(var_lambda_e * dx(domain=mesh))
val_grad_e = assemble(var_grad_e * dx(domain=mesh))
val_e_t = assemble(var_e_t * dx(domain=mesh))
val_laplas_e = assemble(var_laplas_e * dx(domain=mesh))
val_e_id = assemble(var_e_id * dx(domain=mesh))
u_T, mesh_T = get_2d_slice_of_3d_function_on_Oz(mesh, u_ve, T, dim,
v_degree)
ue_T, mesh_T = get_2d_slice_of_3d_function_on_Oz(mesh, u_exact_ve, T, dim,
v_degree)
var_e_T = abs(ue_T - u_T)
val_e_T = assemble(inner(var_e_T, var_e_T) * dx(domain=mesh_T))
u_0, mesh_0 = get_2d_slice_of_3d_function_on_Oz(mesh, u_ve, 0.0, dim,
v_degree)
ue_0, mesh_0 = get_2d_slice_of_3d_function_on_Oz(mesh, u_exact_ve, 0.0,
dim, v_degree)
var_e_0 = abs(ue_0 - u_0)
val_e_0 = assemble(inner(A * var_e_0, var_e_0) * dx(domain=mesh_0))
print '%------------------------------------------------------------------------------------%'
print '% Error '
print '%------------------------------------------------------------------------------------%\n'
print "\| grad_x e \|^2_A = %8.2e" % val_grad_e
print "\| e \|^2 = %8.2e" % val_e
print "\| (lmbd - 0.5 div a)^0.5 e \|^2 = %8.2e" % val_delta_e
print "\| lmbd^0.5 e \|^2 = %8.2e\n" % val_lmbd_e
print "\| laplas e \|^2 = %8.2e" % val_laplas_e
print "\| e_t \|^2 = %8.2e" % val_e_t
print "\| r_v \|^2 = %8.2e" % val_e_id
print "\| e \|^2_T = %8.2e" % val_e_T
print "\| e \|^2_0 = %8.2e\n" % val_e_0
#print "\| grad_x e \|^2_T = %8.2e" % val_grad_e_T
#print "\| grad_x e \|^2_0 = %8.2e\n" % val_grad_e_0
print '%------------------------------------------------------------------------------------%'
print '% Error identity '
print '%------------------------------------------------------------------------------------%\n'
print "id = \| e \|^2_T + \| r_v \|^2 = %8.2e" % (val_e_T +
val_e_id)
print "\| laplas e \|^2 + \| e_t \|^2 + \| e \|^2_0 = %8.2e\n" % (
val_e_0 + val_laplas_e + val_e_t + val_delta_e)
return e, val_e, val_grad_e, val_delta_e, val_e_T, val_laplas_e + val_e_t, val_e_id, \
var_e, var_delta_e, var_grad_e