class FastInv(BaseMatrix):
def __init__(self, mat, blocks1, blocks2):
super(FastInv, self).__init__()
self.parmat = mat
self.spmat = self.parmat if mpi_world.size == 1 else self.parmat.local_mat
self.inv1 = self.spmat.CreateBlockSmoother(blocks1)
self.inv2 = self.spmat.CreateBlockSmoother(
blocks2) if blocks2 is not None else None
self.res = self.spmat.CreateColVector()
self.ty = self.spmat.CreateColVector()
self.tmult = Timer("FastInv")
def Height(self):
return self.parmat.height
def Width(self):
return self.parmat.width
def CreateRowVector(self):
return self.spmat.CreateRowVector()
def CreateColVector(self):
return self.spmat.CreateColVector()
def MultAdd(self, scal, x, y):
self.ty[:] = 0
self.Mult(x, self.ty)
y.local_vec.data += scal * self.ty
def Mult(self, x, y):
self.tmult.Start()
y.local_vec.data = self.inv1 * x.local_vec
# self.res.data = x - self.spmat * y
# y.local_vec.data += self.inv2 * self.res
self.tmult.Stop()
def MultTransAdd(self, scal, x, y):
self.ty[:] = 0
self.MultTrans(x, self.ty)
y.local_vec.data += scal * self.ty
def MultTrans(self, x, y):
self.tmult.Start()
# y.local_vec.data = self.inv2.T * x.local_vec
# self.res.data = x - self.spmat.T * y
# y.local_vec.data += self.inv1.T * self.res
y.local_vec.data = self.inv1.T * x.local_vec
self.tmult.Stop()
def IsComplex(self):
return False
def solve_with_bramble_pasciak_cg(a_matrix, b_matrix, pre_a, pre_schur_complement, gfu, gfp, f, g, tolerance, max_steps):
sol = BlockVector([gfu, gfp])
bramble_pasciak_cg_timer = Timer("BramblePasciakCG")
bramble_pasciak_cg_timer.Start()
(solution, errors) = bramble_pasciak_cg(a_matrix, b_matrix, None, pre_a, pre_schur_complement, f, g, sol,
tolerance=tolerance, max_steps=max_steps)
bramble_pasciak_cg_timer.Stop()
print("Bramble Pasciak CG took",
bramble_pasciak_cg_timer.time, "seconds")
return (solution, errors, bramble_pasciak_cg_timer.time)
def orthonormalize(basis, tries=3):
orthonormalize_timer = Timer("orthonormalization")
orthonormalize_timer.Start()
for tries in range(tries):
for j in range(len(basis)):
for i in range(j):
basis[j].data -= InnerProduct(basis[i], basis[j]) / \
InnerProduct(basis[i], basis[i]) * basis[i]
basis[j].data = 1 / Norm(basis[j]) * basis[j]
orthonormalize_timer.Stop()
return basis
def solve_with_min_res(a, b, preA, preS, gfu, gfp, f, g, tolerance, max_steps):
K = BlockMatrix([[a, b.T], [b, None]])
C = BlockMatrix([[preA, None], [None, preS]])
rhs = BlockVector([f, g])
sol = BlockVector([gfu, gfp])
min_res_timer = Timer("MinRes")
min_res_timer.Start()
(solution, errors) = MinRes(mat=K, pre=C, rhs=rhs, sol=sol, initialize=False,
tol=tolerance, maxsteps=max_steps)
min_res_timer.Stop()
print("MinRes took", min_res_timer.time, "seconds")
return (solution, errors, min_res_timer.time)
def solve_hcurldiv(mesh, discretization, solver_factory):
V, S, Q = discretization(
mesh, velocity_dirichlet='wall|inlet|cyl', velocity_neumann='outlet')
X = FESpace([V, S, Q])
(u, sigma, p), (v, tau, q) = X.TnT()
n = specialcf.normal(mesh.dim)
a = BilinearForm(X, symmetric=True)
a += SymbolicBFI(InnerProduct(sigma, tau))
a += SymbolicBFI(div(sigma) * v + div(tau) * u)
a += SymbolicBFI(-(sigma * n) * n * (v * n) - (tau * n) *
n * (u * n), element_boundary=True)
a += SymbolicBFI(div(u) * q + div(v) * p)
a.Assemble()
f = LinearForm(X)
f += SymbolicLFI(CoefficientFunction((0, x - 0.5)) * v)
f.Assemble()
grid_function = GridFunction(X)
uin = CoefficientFunction((1.5 * 4 * y * (0.41 - y) / (0.41 * 0.41), 0))
grid_function.components[0].Set(uin, definedon=mesh.Boundaries("inlet"))
direct_timer = Timer("Direct Solver")
direct_timer.Start()
res = grid_function.vec.CreateVector()
res.data = f.vec - a.mat * grid_function.vec
inv = a.mat.Inverse(freedofs=X.FreeDofs(), inverse="umfpack")
grid_function.vec.data += inv * res
direct_timer.Stop()
velocity = CoefficientFunction(grid_function.components[0])
pressure = CoefficientFunction(grid_function.components[2])
Draw(velocity, mesh, "velocity")
Draw(pressure, mesh, "pressure")
ndofs = X.ndof
return (velocity, pressure, [], direct_timer.time, ndofs)
def bramble_pasciak_cg(a_matrix,
b_matrix,
c_matrix,
pre_a,
pre_schur_complement,
upper_rhs,
lower_rhs,
solution=None,
tolerance=1e-12,
max_steps=1000,
print_rates=True):
ev_timer = Timer("eigenvalues")
ev_timer.Start()
eigenvalues = EigenValues_Preconditioner(mat=a_matrix, pre=pre_a)
k = 1 / min(eigenvalues) + 1e-3
ev_timer.Stop()
print("scale factor: ", k)
print("condition number: ", max(eigenvalues) / min(eigenvalues))
scaled_pre_a = ScaledPreconditioner(k, a_matrix, pre_a)
original_matrix = BlockMatrix([[a_matrix, b_matrix.T],
[b_matrix, c_matrix]])
full_pre_a = BlockMatrix([[scaled_pre_a, None],
[None, IdentityMatrix(b_matrix.height)]])
full_b = BlockMatrix([[IdentityMatrix(a_matrix.width), None],
[b_matrix, -IdentityMatrix(b_matrix.height)]])
a_b_block_matrix = MatrixAB(a_matrix, b_matrix)
full_pre_schur_complement = BlockMatrix(
[[IdentityMatrix(a_matrix.width), None], [None, pre_schur_complement]])
rhs = BlockVector([upper_rhs, lower_rhs])
if not solution:
solution = rhs.CreateVector()
solution[:] = 0
residuum = rhs.CreateVector()
temp_1 = rhs.CreateVector()
full_preconditioned_residuum = rhs.CreateVector()
a_preconditioned_residuum = rhs.CreateVector()
temp_2 = rhs.CreateVector()
temp_2.data = rhs - original_matrix * solution
a_preconditioned_residuum.data = full_pre_a * temp_2
residuum.data = a_b_block_matrix * a_preconditioned_residuum \
- rhs + original_matrix * solution
temp_1.data = full_pre_schur_complement @ full_b * a_preconditioned_residuum
full_preconditioned_residuum.data = temp_1
current_residual_error_squared = InnerProduct(temp_1, residuum)
err0 = sqrt(abs(current_residual_error_squared))
errors = []
for iteration in range(max_steps):
iteration_timer = Timer("Bramble Pasciak CG Iteration " +
str(iteration))
iteration_timer.Start()
err = sqrt(abs(current_residual_error_squared))
if print_rates:
print("\rit =",
iteration,
"rel err =",
err / err0,
"abs err =",
err,
" " * 20,
end="")
errors.append(err / err0)
if err < tolerance * err0:
iteration_timer.Stop()
break
previous_residual_error_squared = current_residual_error_squared
temp_1.data = -original_matrix * full_preconditioned_residuum
temp_2.data = -full_pre_a * temp_1
temp_1.data += a_b_block_matrix * temp_2
alpha = previous_residual_error_squared / \
InnerProduct(full_preconditioned_residuum, temp_1)
solution.data += alpha * full_preconditioned_residuum
residuum.data += (-alpha) * temp_1
a_preconditioned_residuum.data += (-alpha) * temp_2
temp_1.data = full_pre_schur_complement @ full_b * a_preconditioned_residuum
current_residual_error_squared = InnerProduct(temp_1, residuum)
beta = current_residual_error_squared / previous_residual_error_squared
full_preconditioned_residuum *= beta
full_preconditioned_residuum.data += temp_1
iteration_timer.Stop()
else:
print("\nWarning: CG did not converge to TOL")
print("")
return (solution, errors)
def CG(mat,
rhs,
pre=None,
sol=None,
tol=1e-12,
maxsteps=100,
printrates=True,
initialize=True,
conjugate=False):
"""preconditioned conjugate gradient method
Parameters
----------
mat : Matrix
The left hand side of the equation to solve. The matrix has to be spd or hermitsch.
rhs : Vector
The right hand side of the equation.
pre : Preconditioner
If provided the preconditioner is used.
sol : Vector
Start vector for CG method, if initialize is set False. Gets overwritten by the solution vector. If sol = None then a new vector is created.
tol : double
Tolerance of the residuum. CG stops if tolerance is reached.
maxsteps : int
Number of maximal steps for CG. If the maximal number is reached before the tolerance is reached CG stops.
printrates : bool
If set to True then the error of the iterations is displayed.
initialize : bool
If set to True then the initial guess for the CG method is set to zero. Otherwise the values of the vector sol, if provided, is used.
conjugate : bool
If set to True, then the complex inner product is used.
Returns
-------
(vector)
Solution vector of the CG method.
"""
timer = Timer("CG-Solver")
timer.Start()
u = sol if sol else rhs.CreateVector()
d = rhs.CreateVector()
w = rhs.CreateVector()
s = rhs.CreateVector()
if initialize: u[:] = 0.0
d.data = rhs - mat * u
w.data = pre * d if pre else d
err0 = sqrt(abs(InnerProduct(d, w)))
s.data = w
# wdn = InnerProduct (w,d)
wdn = w.InnerProduct(d, conjugate=conjugate)
if wdn == 0:
return u
for it in range(maxsteps):
w.data = mat * s
wd = wdn
# as_s = InnerProduct (s, w)
as_s = s.InnerProduct(w, conjugate=conjugate)
alpha = wd / as_s
u.data += alpha * s
d.data += (-alpha) * w
w.data = pre * d if pre else d
# wdn = InnerProduct (w, d)
wdn = w.InnerProduct(d, conjugate=conjugate)
beta = wdn / wd
s *= beta
s.data += w
err = sqrt(abs(wd))
if printrates:
print("it = ", it, " err = ", err)
if err < tol * err0: break
else:
print("Warning: CG did not converge to TOL")
timer.Stop()
return u
def solve(initial_temperature, end_time, time_step):
mesh = Mesh(unit_square.GenerateMesh(maxh=0.1))
Draw(mesh)
space = H1(mesh, order=10, dirichlet='bottom|right|top|left')
print(space)
Draw(initial_temperature, mesh, "initial_temperature")
trial_function = space.TrialFunction()
test_function = space.TestFunction()
diffusion_term = grad(trial_function) * grad(test_function)
diffusion = BilinearForm(space)
diffusion += diffusion_term * dx
mass_term = trial_function * test_function
mass = BilinearForm(space)
mass += mass_term * dx
heat = BilinearForm(space)
heat += mass_term * dx + time_step * diffusion_term * dx
source_term = 0 * test_function
source = LinearForm(space)
source += source_term * dx
diffusion.Assemble()
mass.Assemble()
heat.Assemble()
source.Assemble()
temperature = GridFunction(space, "temperature")
temperature.Set(initial_temperature)
for i in range(space.ndof):
if not space.FreeDofs()[i]:
temperature.vec[i] = 0
Draw(temperature)
residual = temperature.vec.CreateVector()
heat_inverse = heat.mat.Inverse(space.FreeDofs())
subspace_dimension = 5
dt = time_step / subspace_dimension
runge_kutta_weights = ImplicitRungeKuttaMethodWeights(10)
time = 0
print(f"time={time}")
with (TaskManager()):
while time < end_time:
time += time_step
print(f"time={time}")
timer = Timer("exponential integrators timer")
timer.Start()
subspace_basis = [temperature.vec.Copy()]
initial_condition_norm = Norm(temperature.vec)
subspace_basis_assembling_timer = Timer(
"subspace basis assembling")
subspace_basis_assembling_timer.Start()
for i in range(1, subspace_dimension):
residual.data = diffusion.mat * temperature.vec
temperature.vec.data -= dt * heat_inverse * residual
subspace_basis.append(temperature.vec.Copy())
subspace_basis = orthonormalize(subspace_basis)
subspace_basis_assembling_timer.Stop()
subspace_matrix_assembling_timer = Timer(
"subspace matrix assembling")
subspace_matrix_assembling_timer.Start()
subspace_diffusion = Matrix(subspace_dimension, subspace_dimension)
subspace_mass = Matrix(subspace_dimension, subspace_dimension)
for col in range(subspace_dimension):
residual.data = diffusion.mat * subspace_basis[col]
for row in range(subspace_dimension):
subspace_diffusion[row, col] = InnerProduct(
subspace_basis[row], residual)
residual.data = mass.mat * subspace_basis[col]
for row in range(subspace_dimension):
subspace_mass[row,
col] = InnerProduct(subspace_basis[row],
residual)
subspace_mass_inverse = Matrix(subspace_dimension,
subspace_dimension)
subspace_mass.Inverse(subspace_mass_inverse)
evolution_matrix = -subspace_mass_inverse * subspace_diffusion
subspace_matrix_assembling_timer.Stop()
large_timestep_timer = Timer("large timestep")
large_timestep_timer.Start()
subspace_temperature = Vector(subspace_dimension)
subspace_temperature[:] = 0
subspace_temperature[0] = initial_condition_norm
next_temperature = linear_implicit_runge_kutta_step(
runge_kutta_weights, evolution_matrix, subspace_temperature,
time_step)
temperature.vec[:] = 0
for i, basis_vector in enumerate(subspace_basis):
temperature.vec.data += next_temperature[i] * basis_vector
large_timestep_timer.Stop()
timer.Stop()
Redraw()
return (temperature, mesh, time)
def BramblePasciakCG(blfA,
blfB,
matC,
f,
g,
preA_unscaled,
preM,
sol=None,
tol=1e-6,
maxsteps=100,
printrates=True,
initialize=True,
rel_err=True):
"""preconditioned bramble pasciak conjugate gradient method
Parameters
----------
matA : Matrix
The left upper matrix of the saddle point matrix. The matrix has to be spd.
matB : Matrix
The B of the saddle point matrix. The matrix has to be spd.
matC : Matrix
The lower right of the saddle point matrix. The matrix has to be spd.
f : Vector
The first component right hand side of the equation.
g : Vector
The second component right hand side of the equation.
preA_unscaled :
Preconditioner of matA, preferedly BDDC.
preA_unscaled :
Preconditioner of matB, arbitrary (i.e. Jacobi).
sol : Vector
Start vector for CG method, if initialize is set False. Gets overwritten by the solution vector. If sol = None then a new vector is created.
tol : double
Tolerance of the residuum. BPCG stops if tolerance is reached.
maxsteps : int
Number of maximal steps for BPCG. If the maximal number is reached before the tolerance is reached CG stops.
printrates : bool
If set to True then the error of the iterations is displayed.
initialize : bool
If set to True then the initial guess for the BPCG method is set to zero. Otherwise the values of the vector sol, if provided, is used.
conjugate : bool
If set to True, then the complex inner product is used.
rel_err : bool
Whether to use the tolerance relative to the the initial error or not.
Returns
-------
(vector - call by reference)
Solution vector of the BPCG method.
(int)
Number of iterations the BCGP took
"""
class myAmatrix(BaseMatrix):
def __init__(self, blfA):
super(myAmatrix, self).__init__()
self.blfA = blfA
self.mat = (IdentityMatrix() - blfA.harmonic_extension_trans) @ (
blfA.mat + blfA.inner_matrix) @ (IdentityMatrix() -
blfA.harmonic_extension)
def Mult(self, x, y):
y.data = self.mat * x
def Height(self):
return self.blfA.mat.height
def Width(self):
return self.blfA.mat.width
def CreateColVector(self):
return self.blfA.mat.CreateColVector()
def CreateVector(self):
return self.blfA.mat.CreateColVector()
if blfA.condense:
matA = myAmatrix(blfA)
else:
matA = blfA.mat
matB = blfB.mat
timer_prep = Timer("BPCG-Preparation")
timer_prep.Start()
timer_prepev = Timer("BPCG-Preparation-EV")
timer_prepev.Start()
lams = EigenValues_Preconditioner(mat=matA, pre=preA_unscaled, tol=1e-3)
timer_prepev.Stop()
# print("min", min(lams), "max", max(lams))
k = 1. / min(lams) + 1e-3
print("condition", max(lams) / min(lams))
# print("scale factor", k)
preA = k * preA_unscaled
f_new = matA.CreateColVector()
# tmp0.data = preA * f
#tmp0 = harmonic_extension(f, blfA, preA)
tmp0 = f.CreateVector()
harmonic_extension(f, blfA, preA, result=tmp0)
f_new.data = matA * tmp0 - f
g_new = matB.CreateColVector()
g_new.data = matB * tmp0 - g
rhs = BlockVector([f_new, g_new])
u = sol if sol else rhs.CreateVector()
if initialize:
u[:] = 0.0
d = rhs.CreateVector()
w = rhs.CreateVector()
v = rhs.CreateVector()
z = rhs.CreateVector()
z_old = rhs.CreateVector()
s = rhs.CreateVector()
# MatOp = BP_Matrices(preA, matA, matB, preM)
# MatOp.update(u)
tmp0 = blfA.mat.CreateColVector()
#tmp1 = preA.CreateColVector()
tmp1 = blfA.mat.CreateColVector()
tmp2 = blfA.mat.CreateColVector()
tmp3 = matB.CreateColVector()
tmp4 = tmp1.CreateVector()
matA_s0 = blfA.mat.CreateColVector()
matB_s1 = matB.CreateRowVector()
tmp0.data = matA * u[0] + matB.T * u[1]
# tmp1.data = preA * tmp0
harmonic_extension(tmp0, blfA, preA, tmp1)
tmp2.data = matA * tmp1
tmp4.data = tmp1 - u[0]
tmp3.data = matB * tmp4
# d.data = rhs - MatOp.K * u
d[0].data = rhs[0] - (tmp2 - tmp0)
d[1].data = rhs[1] - tmp3
pr = rhs.CreateVector()
harmonic_extension(f, blfA, preA, pr[0])
# pr[0].data = preA * f
tmp5 = matB.CreateColVector()
tmp5.data = matB * pr[0] - g
pr[1].data = preM * tmp5
# w.data = pr - MatOp.Cinv_K * u
w[0].data = pr[0] - tmp1
w[1].data = pr[1] - preM * tmp3
wdn = InnerProduct(w, d)
err0 = sqrt(abs(wdn))
print("err0", err0)
s.data = w
if wdn == 0:
return u
timer_prep.Stop()
timer_its = Timer("BPCG-Iterations")
timer_its.Start()
matB_tranposed = matB.CreateTranspose()
for it in range(maxsteps):
if it == 0:
matA_s0.data = matA * s[0]
z[0].data = matA_s0 # A*w_0_0
else:
matA_s0.data = beta * matA_s0 + z_old[0] - alpha * tmp2
matB_s1.data = matB_tranposed * s[1]
tmp0.data = matA_s0 + matB_s1
# tmp1.data = preA * tmp0
harmonic_extension(f=tmp0, blfA=blfA, inverse=preA, result=tmp1)
tmp2.data = matA * tmp1
tmp4.data = tmp1 - s[0]
tmp3.data = matB * tmp4
z_old[0].data = z[0]
# w.data = MatOp.K * s
v[0].data = tmp2 - tmp0
v[1].data = tmp3
wd = wdn
as_s = InnerProduct(s, v)
# giving one (A,B) to the other side of the dot product
# as_s = InnerProduct(matA_s0, tmp1) - InnerProduct(s[0], tmp0) + InnerProduct(matB_s1, tmp4)
alpha = wd / as_s
u.data += alpha * s
d.data += (-alpha) * v
# w.data = w_old + (-alpha) * MatOp.Cinv_K * s
w[0].data = w[0] + (-alpha) * tmp1
w[1].data = w[1] + (-alpha) * preM * tmp3
wdn = InnerProduct(w, d)
beta = wdn / wd
z[0].data -= alpha * tmp2
s *= beta
s.data += w
err = sqrt(abs(wd))
if printrates:
print("it = ", it, " err = ", err, " " * 20)
if err < tol * (err0 if rel_err else 1):
break
else:
print("Warning: BPCG did not converge to TOL")
timer_its.Stop()
print("\n")
return (it, timer_its.time)