def solve(t_y, t_x, mask, t_y_weights=None, t_x_weights=None):
"""Solve for the image which best matches the target vertical gradients
t_y and horizontal gradients t_x, e.g. the one which minimizes sum of squares
of the residual
sum of (I[i,j] - I[i-1,j] - t_y[i,j])**2 + (I[i,j] - I[i,j-1] - t_x[i,j])**2
Only considers the target gradients lying entirely within the mask.
The first row of t_y and the first column of t_x are ignored. Optionally,
you may pass in an array with the weights corresponding to each target
gradient. The solution is unique up to a constant added to each of
the pixels. """
if t_y_weights is None:
t_y_weights = np.ones(t_y.shape)
if t_x_weights is None:
t_x_weights = np.ones(t_x.shape)
M, N = mask.shape
numbers = get_numbers(mask)
A = get_A(mask, t_y_weights, t_x_weights)
b = get_b(t_y, t_x, mask, t_y_weights, t_x_weights)
solver = pyamg.ruge_stuben_solver(A)
x = solver.solve(b)
I = np.zeros(mask.shape)
for i in range(M):
for j in range(N):
I[i,j] = x[numbers[i,j]]
return I
def test():
class Gamma0(Subdomain):
def is_inside(self, x):
return x[1] < 0.5
is_boundary_only = True
class Gamma1(Subdomain):
def is_inside(self, x):
return x[1] >= 0.5
is_boundary_only = True
class Poisson(object):
def apply(self, u):
return integrate(lambda x: -n_dot_grad(u(x)), dS) - integrate(
lambda x: 50 * sin(2 * pi * x[0]), dV
)
def dirichlet(self, u):
return [(lambda x: u(x) - 0.0, Gamma0()), (lambda x: u(x) - 1.0, Gamma1())]
# # Read the mesh from file
# mesh, _, _ = pyfvm.reader.read('circle.vtu')
# Create mesh using meshzoo
import meshzoo
vertices, cells = meshzoo.cube(0.0, 1.0, 0.0, 1.0, 0.0, 1.0, 30, 30, 30)
mesh = meshplex.MeshTetra(vertices, cells)
# vertices, cells = meshzoo.rectangle(0.0, 2.0, 0.0, 1.0, 401, 201)
# mesh = meshplex.MeshTri(vertices, cells)
print(len(vertices))
# import mshr
# import dolfin
# # h = 2.5e-3
# h = 1.e-1
# # cell_size = 2 * pi / num_boundary_points
# c = mshr.Circle(dolfin.Point(0., 0., 0.), 1, int(2*pi / h))
# # cell_size = 2 * bounding_box_radius / res
# m = mshr.generate_mesh(c, 2.0 / h)
# coords = m.coordinates()
# coords = numpy.c_[coords, numpy.zeros(len(coords))]
# cells = m.cells().copy()
# mesh = meshplex.MeshTri(coords, cells)
# # mesh = meshplex.lloyd_smoothing(mesh, 1.0e-4)
matrix, rhs = pyfvm.discretize_linear(Poisson(), mesh)
# ml = pyamg.smoothed_aggregation_solver(matrix)
ml = pyamg.ruge_stuben_solver(matrix)
u = ml.solve(rhs, tol=1e-10)
# from scipy.sparse import linalg
# u = linalg.spsolve(matrix, rhs)
mesh.write("out.vtk", point_data={"u": u})
return
def test_laplacian():
import pylab as pl
# Setup grid
nx, ny = 500, 500
d = 2*pi/nx
Lx, Ly = nx * d, ny * d
g = 0
# make grid
x = np.arange(-g, nx+g)*d
y = np.arange(-g, ny+g)*d
dx = x[1]-x[0]
dy = y[1]-y[0]
x, y = np.meshgrid(x, y, indexing='ij')
# build laplacian
A = build_laplacian_matrix(nx,ny,d=d)
# A = poisson((nx, ny), spacing=(dx, dy), format='csr')/d/d
# right hand side
f = np.sin(x)*np.cos(2*y)
f = np.random.rand(A.shape[0])
p_ex = np.sin(x)*np.cos(2*y)/(-1 - 4)
print("Timing information")
print("==================")
print("")
# with elapsed_timer() as elapsed:
# p_ap = la.spsolve(A, f.ravel())
# print("spsolve {0}".format(elapsed()))
with elapsed_timer() as elapsed:
p_cg = la.cg(A, f.ravel())[0]
print("cg {0}".format(elapsed()))
# with elapsed_timer() as elapsed:
# p_ap = la.gmres(A, f.ravel())
# print("gmres {0}".format(elapsed()))
ml = ruge_stuben_solver(A)
M = ml.aspreconditioner()
with elapsed_timer() as elapsed:
p_amg = la.cg(A, f.ravel(),M=M)
print("pyamg cg {0}".format(elapsed()))
pl.show()
def _solve_cg_mg(lap_sparse, B, tol):
if not amg_loaded:
print """the pyamg module (http://code.google.com/p/pyamg/)
must be installed to use the amg mode"""
raise ImportError
X = []
#lap_sparse = lap_sparse.tocsr()
ml = ruge_stuben_solver(lap_sparse)
M = ml.aspreconditioner(cycle='V')
for i in range(len(B)):
x0 = cg(lap_sparse, -B[i].todense(), tol=tol, M=M, maxiter=30)[0]
X.append(x0)
X = np.array(X)
X = np.argmax(X, axis=0)
return X
def _solve_cg_mg(lap_sparse, B, tol):
"""
solves lap_sparse X_i = B_i for each phase i, using the conjugate
gradient method with a multigrid preconditioner (ruge-stuben from
pyamg). For each pixel, the label i corresponding to the maximal
X_i is returned.
"""
X = []
ml = ruge_stuben_solver(lap_sparse)
M = ml.aspreconditioner(cycle='V')
for i in range(len(B)):
x0 = cg(lap_sparse, -B[i].todense(), tol=tol, M=M, maxiter=30)[0]
X.append(x0)
X = np.array(X)
X = np.argmax(X, axis=0)
return X
def _solve_cg_mg(lap_sparse, B, tol, return_full_prob=False):
"""
solves lap_sparse X_i = B_i for each phase i, using the conjugate
gradient method with a multigrid preconditioner (ruge-stuben from
pyamg). For each pixel, the label i corresponding to the maximal
X_i is returned.
"""
X = []
ml = ruge_stuben_solver(lap_sparse)
M = ml.aspreconditioner(cycle='V')
for i in range(len(B)):
x0 = cg(lap_sparse, -B[i].todense(), tol=tol, M=M, maxiter=30)[0]
X.append(x0)
if not return_full_prob:
X = np.array(X)
X = np.argmax(X, axis=0)
return X
def solver(A, b, **solve_time_kwargs):
kwargs.update(solve_time_kwargs)
import pyamg
ml = pyamg.ruge_stuben_solver(A)
kwargs['M'] = ml.aspreconditioner() # params to be developed
try:
sol, info, _ = krylov(A, b, **{'callback': callback, **kwargs})
except:
sol, info = krylov(A, b, **{'callback': callback, **kwargs})
if info > 0:
warnings.warn("Convergence not achieved!")
elif info == 0 and verbose:
print(f"{krylov.__name__} converged to "
+ f"tol={kwargs.get('tol', 'default')} and "
+ f"atol={kwargs.get('atol', 'default')}")
return sol
def solver(A, b):
num_iters = 0
def callback(xk):
nonlocal num_iters
num_iters += 1
ml = pyamg.ruge_stuben_solver(A)
ptmp = ml.solve(b, tol=1e-16, callback=callback)
# ptmp = scipy.sparse.linalg.spsolve(A,b,callback=callback)
atol = 1e-5
tol = 1e-12
# ptmp,_ = scipy.sparse.linalg.cg(A, b,p0[1:-1, 1:-1].ravel(),callback=callback,tol=tol)
# if max(tol*np.linalg.norm(b,2),atol) == atol:
# print('atol')
# else:
# print('tol')
return ptmp, num_iters
def precondieitoner(self):
tspace = self.tensorspace
vspace = self.vectorspace
tgdof = tspace.number_of_global_dofs()
gdim = tspace.geo_dimension()
bcs, ws = self.integrator.quadpts, self.integrator.weights
phi = tspace.basis(bcs)
# construct diag matrix D
if gdim == 2:
d = np.array([1, 1, 2])
elif gdim == 3:
d = np.array([1, 1, 1, 2, 2, 2])
D = np.einsum('i, ijkm, m, ijkm, j->jk', ws, phi, d, phi, self.measure)
tcell2dof = tspace.cell_to_dof()
self.D = np.bincount(tcell2dof.flat, weights=D.flat, minlength=tgdof)
# construct amg solver
A = stiff_matrix(self.cspace, self.integrator, self.measure)
isBdDof = self.cspace.boundary_dof()
bdIdx = np.zeros((A.shape[0], ), np.int)
bdIdx[isBdDof] = 1
Tbd = spdiags(bdIdx, 0, A.shape[0], A.shape[0])
T = spdiags(1-bdIdx, 0, A.shape[0], A.shape[0])
A = T@A@T + Tbd
self.ml = pyamg.ruge_stuben_solver(A) # 这里要求必须有网格内部节点
# Get interpolation matrix
NC = self.mesh.number_of_cells()
bc = self.vectorspace.dof.multiIndex/self.vectorspace.p
val = np.tile(bc, (NC, 1))
c2d0 = self.vectorspace.dof.cell2dof
c2d1 = self.cspace.cell_to_dof()
gdim = self.tensorspace.geo_dimension()
I = np.einsum('ij, k->ijk', c2d0, np.ones(gdim+1))
J = np.einsum('ik, j->ijk', c2d1, np.ones(len(bc)))
cgdof = self.cspace.number_of_global_dofs()
fgdof = self.vectorspace.number_of_global_dofs()/self.mesh.geo_dimension()
self.PI = csr_matrix((val.flat, (I.flat, J.flat)), shape=(fgdof, cgdof))
def get_new_points(mesh, tol=1.0e-10):
cells = mesh.cells["nodes"].T
row_idx = []
col_idx = []
val = []
a = numpy.ones(cells.shape[1], dtype=float)
for i in [[0, 1], [1, 2], [2, 0]]:
edges = cells[i]
row_idx += [edges[0], edges[1], edges[0], edges[1]]
col_idx += [edges[0], edges[1], edges[1], edges[0]]
val += [+a, +a, -a, -a]
row_idx = numpy.concatenate(row_idx)
col_idx = numpy.concatenate(col_idx)
val = numpy.concatenate(val)
n = mesh.node_coords.shape[0]
# Create CSR matrix for efficiency
matrix = scipy.sparse.coo_matrix((val, (row_idx, col_idx)), shape=(n, n))
matrix = matrix.tocsr()
# Apply Dirichlet conditions.
verts = numpy.where(mesh.is_boundary_node)[0]
# Set all Dirichlet rows to 0.
for i in verts:
matrix.data[matrix.indptr[i] : matrix.indptr[i + 1]] = 0.0
# Set the diagonal and RHS.
d = matrix.diagonal()
d[mesh.is_boundary_node] = 1.0
matrix.setdiag(d)
rhs = numpy.zeros((n, 2))
rhs[mesh.is_boundary_node] = mesh.node_coords[mesh.is_boundary_node]
# out = scipy.sparse.linalg.spsolve(matrix, rhs)
ml = pyamg.ruge_stuben_solver(matrix)
# Keep an eye on multiple rhs-solves in pyamg,
# <https://github.com/pyamg/pyamg/issues/215>.
out = numpy.column_stack(
[ml.solve(rhs[:, 0], tol=tol), ml.solve(rhs[:, 1], tol=tol)]
)
return out[mesh.is_interior_node]
def _solve_linear_system(lap_sparse, B, tol, mode):
if mode is None:
mode = 'cg_j'
if mode == 'cg_mg' and not amg_loaded:
warn(
'"cg_mg" not available, it requires pyamg to be installed. '
'The "cg_j" mode will be used instead.',
stacklevel=2)
mode = 'cg_j'
if mode == 'bf':
X = spsolve(lap_sparse, B.toarray()).T
else:
maxiter = None
if mode == 'cg':
if UmfpackContext is None:
warn(
'"cg" mode may be slow because UMFPACK is not available. '
'Consider building Scipy with UMFPACK or use a '
'preconditioned version of CG ("cg_j" or "cg_mg" modes).',
stacklevel=2)
M = None
elif mode == 'cg_j':
M = sparse.diags(1.0 / lap_sparse.diagonal())
else:
# mode == 'cg_mg'
lap_sparse = lap_sparse.tocsr()
ml = ruge_stuben_solver(lap_sparse)
M = ml.aspreconditioner(cycle='V')
maxiter = 30
cg_out = [
cg(lap_sparse, B[:, i].toarray(), tol=tol, M=M, maxiter=maxiter)
for i in range(B.shape[1])
]
if np.any([info > 0 for _, info in cg_out]):
warn(
"Conjugate gradient convergence to tolerance not achieved. "
"Consider decreasing beta to improve system conditionning.",
stacklevel=2)
X = np.asarray([x for x, _ in cg_out])
return X
def picard(A, M, u0, tol=1e-12, atol=1e-12, ml=None):
if ml is None:
ml = pyamg.ruge_stuben_solver(A)
d0 = (u0 @ A @ u0) / (u0 @ M @ u0)
while True:
u1 = ml.solve(d0 * M @ u0, x0=u0, tol=1e-12, accel='cg').reshape(-1)
L0 = u1 @ M @ u1
L1 = u1 @ A @ u1
d1 = L1 / L0
if norm(u1 - u0, ord=1) / norm(u1, ord=1) < tol or np.abs(d1 -
d0) < atol:
u0 = u1 / np.max(np.abs(u1))
d0 = d1
break
else:
u0 = u1 / np.max(np.abs(u1))
d0 = d1
return u0, d0
def test(n=100):
omega = 1.0
iterations = 1
print("Creating A")
A = pyamg.gallery.poisson(
(n, n), format='csr') # 2D Poisson problem on nxn grid
print("Creating ref")
ml = pyamg.ruge_stuben_solver(A,
max_coarse=1,
max_levels=100,
presmoother=('jacobi', {
"withrho": True,
"omega": omega
}),
postsmoother=('jacobi', {
"withrho": True,
"omega": omega
})) # construct the multigrid hierarchy
mg = MultiGrid(A, omega=omega, iterations=iterations)
b = np.random.rand(A.shape[0]) # pick a random right hand side
x0 = np.random.rand(A.shape[0])
xP = x0.copy()
xV = x0.copy()
xVa = x0.copy()
xVa2 = x0.copy()
xW = x0.copy()
i = 0
print("residuals : ", n, i, np.linalg.norm(b - A * xP),
np.linalg.norm(b - A * xV), np.linalg.norm(b - A * xW))
while True:
i = i + 1
xP = ml.solve(b, tol=1e-9, maxiter=iterations, x0=xP)
mg.vcycle(xV, b)
mg.vcycle_alt(xVa, b)
mg.vcycle_alt2(xVa2, b)
mg.wcycle(xW, b)
print("residuals: ", n, i, np.linalg.norm(b - A * xP),
np.linalg.norm(b - A * xV), np.linalg.norm(b - A * xVa),
np.linalg.norm(b - A * xVa2), np.linalg.norm(b - A * xW))
if np.linalg.norm(b - A * xV) < 1e-3:
break
def solve1(a, L, uh, dirichlet=None, neuman=None, solver='cg'):
space = a.space
start = timer()
A = a.get_matrix()
b = L.get_vector()
end = timer()
print("Construct linear system time:", end - start)
if neuman is not None:
b += neuman.get_vector()
if dirichlet is not None:
AD, b = dirichlet.apply(A, b)
else:
AD = A
# print("The condtion number is: ", np.linalg.cond(AD.todense()))
if solver is 'cg':
start = timer()
D = AD.diagonal()
M = spdiags(1 / D, 0, AD.shape[0], AD.shape[1])
uh[:], info = cg(AD, b, tol=1e-14, M=M)
end = timer()
print(info)
elif solver is 'amg':
start = timer()
ml = pyamg.ruge_stuben_solver(AD)
uh[:] = ml.solve(b, tol=1e-12, accel='cg').reshape((-1, ))
end = timer()
print(ml)
elif solver is 'direct':
start = timer()
uh[:] = spsolve(AD, b)
end = timer()
else:
print("We don't support solver: " + solver)
print("Solve time:", end - start)
return A
def __init__(self, A, P, isBdDof):
"""
Notes
-----
这里的边界条件处理放到矩阵和向量的乘积运算当中, 所心不需要修改矩阵本身
"""
self.gdof = P.shape[0]
self.GD = A.shape[0] // self.gdof
self.A = A
self.isBdDof = isBdDof
# 处理预条件子的边界条件
bdIdx = np.zeros(P.shape[0], dtype=np.int_)
bdIdx[isBdDof] = 1
Tbd = spdiags(bdIdx, 0, P.shape[0], P.shape[0])
T = spdiags(1 - bdIdx, 0, P.shape[0], P.shape[0])
P = T @ P @ T + Tbd
self.ml = pyamg.ruge_stuben_solver(P)
def __init__(self, pde, n, tau, q):
self.pde = pde
self.mesh = pde.space_mesh(n)
self.timemesh, self.tau = self.pde.time_mesh(tau)
self.femspace = LagrangeFiniteElementSpace(self.mesh, 1)
self.uh0 = self.femspace.interpolation(pde.initdata)
self.uh1 = self.femspace.function()
self.area = self.mesh.entity_measure('cell')
self.integrator = self.mesh.integrator(q)
self.integralalg = IntegralAlg(self.integrator, self.mesh, self.area)
self.A, self.B, self.gradphi = grad_recovery_matrix(self.femspace)
self.M = doperator.mass_matrix(self.femspace, self.integrator, self.area)
self.K = self.get_stiff_matrix()
self.D = self.M + self.tau * self.K
self.ml = pyamg.ruge_stuben_solver(self.D)
print(self.ml)
self.current = 0
def mode_solver_initialisation(mode, solver_data, a, solver_tol):
solver_data['solver_timing'], solver_data['solver_tol'] = [], solver_tol
solver_data['is_solver_direct'] = []
id_n = sp.eye(a.shape[0], format='csr')
solver_data['a'] = a
solver_data['id_n'] = id_n
if mode == 'pyamg':
amg_hierarchy = pyamg.ruge_stuben_solver(a)
orig_hierarchy = copy_hierarchy(amg_hierarchy)
solver_data['amg_hierarchy'], solver_data['orig_hierarchy'] = amg_hierarchy, orig_hierarchy
elif mode == 'pyamgE':
B = np.ones((a.shape[0],1), dtype=a.dtype);
amg_hierarchy = pyamg.rootnode_solver(a, max_levels = 15, max_coarse = 300, coarse_solver = 'pinv', presmoother = ('block_gauss_seidel', {'sweep': 'symmetric', 'iterations': 1}), postsmoother = ('block_gauss_seidel', {'sweep': 'symmetric', 'iterations': 1}), BH = B.copy(),
strength = ('evolution', {'epsilon': 4.0, 'k': 2, 'proj_type': 'l2'}), aggregate ='standard', smooth = ('energy', {'weighting': 'local', 'krylov': 'gmres', 'degree': 1, 'maxiter': 2}), improve_candidates = [('block_gauss_seidel', {'sweep': 'symmetric', 'iterations': 4}), None, None, None, None, None, None, None, None, None, None, None, None, None, None])
orig_hierarchy = copy_hierarchy(amg_hierarchy)
solver_data['amg_hierarchy'], solver_data['orig_hierarchy'] = amg_hierarchy, orig_hierarchy
elif mode == 'splu':
solver_data['lu'] = sla.splu(a.tocsc())
return
def __init__(self, A, F, P, I, isBdDof):
"""
Notes
-----
求解高次拉格朗日有限元的快速算法
"""
self.gdof = len(isBdDof)
self.A = A # 矩阵 (gdof, gdof), 注意这里是没有处理 D 氏边界的矩阵
self.F = F # 右端 (gdof, ), 注意这里也没有处理 D 氏边界
self.I = I # 插值矩阵 (gdof, NN), 把线性元的解插值到 p 次解
self.isBdDof = isBdDof
# 获得磨光子
gdof = self.gdof
bdIdx = np.zeros(gdof, dtype=np.int_)
bdIdx[isBdDof] = 1 # 这里假定 A 的前 NN 个自由度是网格节点
Tbd = spdiags(bdIdx, 0, gdof, gdof)
T = spdiags(1-bdIdx, 0, gdof, gdof)
A = T@A@T + Tbd
self.L0 = tril(A).tocsr()
self.U0 = triu(A, k=1).tocsr()
self.U1 = self.L0.T.tocsr()
self.L1 = self.U0.T.tocsr()
# 处理预条件子的边界条件
NN = P.shape[0]
bdIdx = np.zeros(NN, dtype=np.int_)
bdIdx[isBdDof[:NN]] = 1 # 这里假定 A 的前 NN 个自由度是网格节点
Tbd = spdiags(bdIdx, 0, NN, NN)
T = spdiags(1-bdIdx, 0, NN, NN)
P = T@P@T + Tbd
self.ml = pyamg.ruge_stuben_solver(P) # P 的 D 氏边界条件用户先处理一下
def __init__(self, A, P, isBdDof):
"""
Notes
-----
A: [[A00, A01], [A10, A11]] (2*gdof, 2*gdof)
[[A00, A01, A02], [A10, A11, A12], [A20, A21, A22]] (3*gdof, 3*gdof)
P: 预条件子 (gdof, gdof)
这里的边界条件处理放到矩阵和向量的乘积运算当中, 所心不需要修改矩阵本身
"""
self.GD = len(A)
self.gdof = P.shape[0]
self.A = A
self.isBdDof = isBdDof
# 处理预条件子的边界条件
bdIdx = np.zeros(P.shape[0], dtype=np.int_)
bdIdx[isBdDof] = 1
Tbd = spdiags(bdIdx, 0, P.shape[0], P.shape[0])
T = spdiags(1 - bdIdx, 0, P.shape[0], P.shape[0])
P = T @ P @ T + Tbd
self.ml = pyamg.ruge_stuben_solver(P)
def solve(dmodel, uh, dirichlet=None, solver='cg'):
space = uh.space
start = timer()
A = dmodel.get_left_matrix()
b = dmodel.get_right_vector()
end = timer()
print("Construct linear system time:", end - start)
if dirichlet is not None:
AD, b = dirichlet.apply(A, b)
else:
AD = A
if solver is 'cg':
start = timer()
D = AD.diagonal()
M = spdiags(1/D, 0, AD.shape[0], AD.shape[1])
uh[:], info = cg(AD, b, tol=1e-14, M=M)
end = timer()
print(info)
elif solver is 'amg':
start = timer()
ml = pyamg.ruge_stuben_solver(AD)
uh[:] = ml.solve(b, tol=1e-12, accel='cg').reshape(-1)
end = timer()
print(ml)
elif solver is 'direct':
start = timer()
uh[:] = spsolve(AD, b)
end = timer()
else:
raise ValueError("We don't support solver `{}`! ".format(solver))
print("Solve time:", end-start)
return AD, b
def solve(self, uh, timeline):
''' piccard 迭代 C-N 方法 '''
from nonlinear_robin import nonlinear_robin
i = timeline.current
t1 = timeline.next_time_level()
dt = timeline.current_time_step_length()
F = self.pde.right_vector(uh, timeline)
# 网格数据
rho = self.mesh.meshdata['rho']
c = self.mesh.meshdata['c']
kappa = self.mesh.meshdata['kappa']
epsilon = self.mesh.meshdata['epsilon']
sigma = self.mesh.meshdata['sigma']
A = kappa * self.A
M = rho * c * self.M
b = self.apply_boundary_condition(A, F, uh, timeline)
e = 0.0000000001
error = 1
xi_new = self.space.function()
xi_new[:] = copy.deepcopy(uh[:, i])
while error > e:
xi_tmp = copy.deepcopy(xi_new[:])
R = self.nr.robin_bc(A,
xi_new,
lambda x, n: self.pde.robin(x, n, t1),
threshold=self.pde.is_robin_boundary)
r = M @ uh[:, i] + dt * b
R = M + dt * R
ml = pyamg.ruge_stuben_solver(R)
xi_new[:] = ml.solve(r, tol=1e-12, accel='cg').reshape(-1)
# xi_new[:] = spsolve(R, b).reshape(-1)
error = np.max(np.abs(xi_tmp - xi_new[:]))
uh[:, i + 1] = xi_new
def Asq_solve_sandwich(self, b, S, use_pyamg=False):
"""
Solves A @ S @ A.T @ x = b (where A is self.get_cycle_matrix()).
Parameters
----------
b : (Nf,) or (Nf, W) array
Right-hand side of system.
S : (Nj, Nj) sparse array
Sandwich matrix.
Returns
-------
x : shape of b
Solution of system.
"""
Sd = S.diagonal()
if S.nnz == np.count_nonzero(Sd):
if np.allclose(Sd[0], Sd):
return self.Asq_solve(b) / Sd[0]
if use_pyamg:
import pyamg
import pyamg.util.linalg
A = self.cycle_matrix @ S @ self.cycle_matrix.T
pyamg_cb = PyamgCallback(A, b)
ml = pyamg.ruge_stuben_solver(A, strength=('classical', {'theta': 0.2}))
try:
mg_out = ml.solve(b, tol=1e-8, maxiter=3, callback=pyamg_cb.cb)
except DivergenceError:
mg_out = pyamg_cb.x
if not pyamg_cb.has_diverged:
return mg_out
else:
return self.Asq_solve_sandwich(b, S, use_pyamg=False)
else:
AsqF = scipy.sparse.linalg.factorized(self.cycle_matrix @ S @ self.cycle_matrix.T)
return AsqF(b)
def generateQTBNQ(self):
self.QTBNQ = self.QT * self.BNQ
idx = list(self.QTBNQ.indices).index(0)
self.QTBNQ.data[idx] *= 2.0
self.ml = pyamg.ruge_stuben_solver(self.QTBNQ)
def apply_inverse(matrix, U, options=None):
"""Solve linear equation system.
Applies the inverse of `matrix` to the row vectors in `U`.
See :func:`sparse_options` for documentation of all possible options for
sparse matrices.
Parameters
----------
matrix
The |NumPy| matrix to invert.
U
2-dimensional |NumPy array| containing as row vectors
the right-hand sides of the linear equation systems to
solve.
options
|invert_options| to use. (See :func:`invert_options`.)
Returns
-------
|NumPy array| of the solution vectors.
"""
default_options = invert_options(matrix)
if options is None:
options = default_options.values()[0]
elif isinstance(options, str):
if options == 'least_squares':
for k, v in default_options.iteritems():
if k.startswith('least_squares'):
options = v
break
assert not isinstance(options, str)
else:
options = default_options[options]
else:
assert 'type' in options and options['type'] in default_options \
and options.viewkeys() <= default_options[options['type']].viewkeys()
user_options = options
options = default_options[user_options['type']]
options.update(user_options)
R = np.empty((len(U), matrix.shape[1]))
if options['type'] == 'solve':
for i, UU in enumerate(U):
try:
R[i] = np.linalg.solve(matrix, UU)
except np.linalg.LinAlgError as e:
raise InversionError('{}: {}'.format(str(type(e)), str(e)))
elif options['type'] == 'least_squares_lstsq':
for i, UU in enumerate(U):
try:
R[i], _, _, _ = np.linalg.lstsq(matrix, UU, rcond=options['rcond'])
except np.linalg.LinAlgError as e:
raise InversionError('{}: {}'.format(str(type(e)), str(e)))
elif options['type'] == 'bicgstab':
for i, UU in enumerate(U):
R[i], info = bicgstab(matrix, UU, tol=options['tol'], maxiter=options['maxiter'])
if info != 0:
if info > 0:
raise InversionError('bicgstab failed to converge after {} iterations'.format(info))
else:
raise InversionError('bicgstab failed with error code {} (illegal input or breakdown)'.
format(info))
elif options['type'] == 'bicgstab_spilu':
ilu = spilu(matrix, drop_tol=options['spilu_drop_tol'], fill_factor=options['spilu_fill_factor'],
drop_rule=options['spilu_drop_rule'], permc_spec=options['spilu_permc_spec'])
precond = LinearOperator(matrix.shape, ilu.solve)
for i, UU in enumerate(U):
R[i], info = bicgstab(matrix, UU, tol=options['tol'], maxiter=options['maxiter'], M=precond)
if info != 0:
if info > 0:
raise InversionError('bicgstab failed to converge after {} iterations'.format(info))
else:
raise InversionError('bicgstab failed with error code {} (illegal input or breakdown)'.
format(info))
elif options['type'] == 'spsolve':
for i, UU in enumerate(U):
R[i] = spsolve(matrix, UU, permc_spec=options['permc_spec'])
elif options['type'] == 'lgmres':
for i, UU in enumerate(U):
R[i], info = lgmres(matrix, UU.copy(i),
tol=options['tol'],
maxiter=options['maxiter'],
inner_m=options['inner_m'],
outer_k=options['outer_k'])
if info > 0:
raise InversionError('lgmres failed to converge after {} iterations'.format(info))
assert info == 0
elif options['type'] == 'least_squares_lsmr':
for i, UU in enumerate(U):
R[i], info, itn, _, _, _, _, _ = lsmr(matrix, UU.copy(i),
damp=options['damp'],
atol=options['atol'],
btol=options['btol'],
conlim=options['conlim'],
maxiter=options['maxiter'],
show=options['show'])
assert 0 <= info <= 7
if info == 7:
raise InversionError('lsmr failed to converge after {} iterations'.format(itn))
elif options['type'] == 'least_squares_lsqr':
for i, UU in enumerate(U):
R[i], info, itn, _, _, _, _, _, _, _ = lsqr(matrix, UU.copy(i),
damp=options['damp'],
atol=options['atol'],
btol=options['btol'],
conlim=options['conlim'],
iter_lim=options['iter_lim'],
show=options['show'])
assert 0 <= info <= 7
if info == 7:
raise InversionError('lsmr failed to converge after {} iterations'.format(itn))
elif options['type'] == 'pyamg':
if len(U) > 0:
U_iter = iter(enumerate(U))
R[0], ml = pyamg.solve(matrix, next(U_iter)[1],
tol=options['tol'],
maxiter=options['maxiter'],
return_solver=True)
for i, UU in U_iter:
R[i] = pyamg.solve(matrix, UU,
tol=options['tol'],
maxiter=options['maxiter'],
existing_solver=ml)
elif options['type'] == 'pyamg-rs':
ml = pyamg.ruge_stuben_solver(matrix,
strength=options['strength'],
CF=options['CF'],
presmoother=options['presmoother'],
postsmoother=options['postsmoother'],
max_levels=options['max_levels'],
max_coarse=options['max_coarse'],
coarse_solver=options['coarse_solver'])
for i, UU in enumerate(U):
R[i] = ml.solve(UU,
tol=options['tol'],
maxiter=options['maxiter'],
cycle=options['cycle'],
accel=options['accel'])
elif options['type'] == 'pyamg-sa':
ml = pyamg.smoothed_aggregation_solver(matrix,
symmetry=options['symmetry'],
strength=options['strength'],
aggregate=options['aggregate'],
smooth=options['smooth'],
presmoother=options['presmoother'],
postsmoother=options['postsmoother'],
improve_candidates=options['improve_candidates'],
max_levels=options['max_levels'],
max_coarse=options['max_coarse'],
diagonal_dominance=options['diagonal_dominance'])
for i, UU in enumerate(U):
R[i] = ml.solve(UU,
tol=options['tol'],
maxiter=options['maxiter'],
cycle=options['cycle'],
accel=options['accel'])
elif options['type'].startswith('generic') or options['type'].startswith('least_squares_generic'):
logger = getLogger('pymor.la.numpysolvers.apply_inverse')
logger.warn('You have selected a (potentially slow) generic solver for a NumPy matrix operator!')
from pymor.operators.basic import NumpyMatrixOperator
from pymor.la import NumpyVectorArray
return genericsolvers.apply_inverse(NumpyMatrixOperator(matrix),
NumpyVectorArray(U, copy=False),
options=options).data
else:
raise ValueError('Unknown solver type')
return R
import pyamg
import numpy as np
import scipy.sparse
n = 100
omega = 1.2
A = pyamg.gallery.poisson((n, n),
format='csr') # 2D Poisson problem on 500x500 grid
ml = pyamg.ruge_stuben_solver(A,
max_coarse=1,
max_levels=100,
presmoother=('jacobi', {
"withrho": True,
"omega": omega
}),
postsmoother=('jacobi', {
"withrho": True,
"omega": omega
})) # construct the multigrid hierarchy
print(ml) # print hierarchy information
b = np.random.rand(A.shape[0]) # pick a random right hand side
x = ml.solve(b, maxiter=2) # solve Ax=b to a tolerance of 1e-8
print("residual: ",
np.linalg.norm(b - A * x)) # compute norm of residual vector
def jacobi(A, x, b, omega):
Ad = A.diagonal()
Atmp = scipy.sparse.lil_matrix(A.shape)
def search(self, itr=False):
import numpy as np
if self.amg:
if self.itr<0:
self.itr+=1
from pyamg import ruge_stuben_solver
from scipy.sparse import diags
tempB = diags([self.B.tolist()] , [0])
tempy = self.y
#self.C = (self.D * (self.I + self.B) ) -self.A
self.C = (self.D * (self.I + tempB) ) -self.A
#self.H = self.D * (tempB * self.y.T)
self.H = self.D * (tempB * tempy)
#self.H = self.H.todense()
self.H = np.asarray(self.H)
ml = ruge_stuben_solver(self.C)
self.ml = ml
if itr:
solution = ml.solve(self.H, tol=0.5e-4, return_residuals=True)
f = np.squeeze(solution[0])
x = solution[1]
return (f, x)
else:
f = np.squeeze(ml.solve(self.H, tol=0.5e-4))
self.out = f
return f
else:
self.ml = self.create_solver()
self.out = self.ml.solve(self.H, tol=0.5e-4,x0=self.out,return_residuals=False)
return self.out
else:
from scipy.sparse.linalg import spsolve
self.C = (self.D * (self.I + self.B) ) -self.A
self.H = self.D * np.dot(np.diag(self.B) , self.y.T)
f = np.squeeze(spsolve(self.C, self.H))
return f
def apply_inverse(self, U, ind=None, mu=None, options=None):
default_options = self.invert_options
if options is None:
options = default_options.values()[0]
elif isinstance(options, str):
options = default_options[options]
else:
assert 'type' in options and options['type'] in default_options \
and options.viewkeys() <= default_options[options['type']].viewkeys()
user_options = options
options = default_options[user_options['type']]
options.update(user_options)
assert isinstance(U, NumpyVectorArray)
assert self.dim_range == U.dim
U = U._array[:U._len] if ind is None else U._array[ind]
if U.shape[1] == 0:
return NumpyVectorArray(U)
R = np.empty((len(U), self.dim_source))
if self.sparse:
if options['type'] == 'bicgstab':
for i, UU in enumerate(U):
R[i], info = bicgstab(self._matrix, UU, tol=options['tol'], maxiter=options['maxiter'])
if info != 0:
if info > 0:
raise InversionError('bicgstab failed to converge after {} iterations'.format(info))
else:
raise InversionError('bicgstab failed with error code {} (illegal input or breakdown)'.
format(info))
elif options['type'] == 'bicgstab-spilu':
ilu = spilu(self._matrix, drop_tol=options['spilu_drop_tol'], fill_factor=options['spilu_fill_factor'],
drop_rule=options['spilu_drop_rule'], permc_spec=options['spilu_permc_spec'])
precond = LinearOperator(self._matrix.shape, ilu.solve)
for i, UU in enumerate(U):
R[i], info = bicgstab(self._matrix, UU, tol=options['tol'], maxiter=options['maxiter'], M=precond)
if info != 0:
if info > 0:
raise InversionError('bicgstab failed to converge after {} iterations'.format(info))
else:
raise InversionError('bicgstab failed with error code {} (illegal input or breakdown)'.
format(info))
elif options['type'] == 'spsolve':
for i, UU in enumerate(U):
R[i] = spsolve(self._matrix, UU, permc_spec=options['permc_spec'])
elif options['type'] == 'pyamg':
if len(U) > 0:
U_iter = iter(enumerate(U))
R[0], ml = pyamg.solve(self._matrix, next(U_iter)[1],
tol=options['tol'],
maxiter=options['maxiter'],
return_solver=True)
for i, UU in U_iter:
R[i] = pyamg.solve(self._matrix, UU,
tol=options['tol'],
maxiter=options['maxiter'],
existing_solver=ml)
elif options['type'] == 'pyamg-rs':
ml = pyamg.ruge_stuben_solver(self._matrix,
strength=options['strength'],
CF=options['CF'],
presmoother=options['presmoother'],
postsmoother=options['postsmoother'],
max_levels=options['max_levels'],
max_coarse=options['max_coarse'],
coarse_solver=options['coarse_solver'])
for i, UU in enumerate(U):
R[i] = ml.solve(UU,
tol=options['tol'],
maxiter=options['maxiter'],
cycle=options['cycle'],
accel=options['accel'])
elif options['type'] == 'pyamg-sa':
ml = pyamg.smoothed_aggregation_solver(self._matrix,
symmetry=options['symmetry'],
strength=options['strength'],
aggregate=options['aggregate'],
smooth=options['smooth'],
presmoother=options['presmoother'],
postsmoother=options['postsmoother'],
improve_candidates=options['improve_candidates'],
max_levels=options['max_levels'],
max_coarse=options['max_coarse'],
diagonal_dominance=options['diagonal_dominance'])
for i, UU in enumerate(U):
R[i] = ml.solve(UU,
tol=options['tol'],
maxiter=options['maxiter'],
cycle=options['cycle'],
accel=options['accel'])
else:
raise ValueError('Unknown solver type')
else:
for i, UU in enumerate(U):
try:
R[i] = np.linalg.solve(self._matrix, UU)
except np.linalg.LinAlgError as e:
raise InversionError('{}: {}'.format(str(type(e)), str(e)))
return NumpyVectorArray(R)
def __init__(self, A):
super().__init__(A)
ml = pyamg.ruge_stuben_solver(A.tocsr(), strength=('classical'))
self.gamg = ml.aspreconditioner()
def alg_3_2(self, maxit=None):
"""
1. 自适应求解 -\Delta u = 1。
1. 在最细网格上求最小特征值和特征向量。
refine maxit， picard: 1
"""
print("算法 3.2")
if maxit is None:
maxit = self.maxit
start = timer()
if self.step == 0:
idx = []
else:
idx = list(range(0, self.maxit, self.step)) + [self.maxit - 1]
mesh = self.pde.init_mesh(n=self.numrefine)
GD = mesh.geo_dimension()
if (self.step > 0) and (0 in idx):
NN = mesh.number_of_nodes()
fig = plt.figure()
fig.set_facecolor('white')
if GD == 2:
axes = fig.gca()
else:
axes = Axes3D(fig)
mesh.add_plot(axes, cellcolor='w')
fig.savefig(self.resultdir + 'mesh_3_2_0_' + str(NN) + '.pdf')
plt.close()
self.savemesh(mesh,
self.resultdir + 'mesh_3_2_0_' + str(NN) + '.mat')
final = 0
integrator = mesh.integrator(self.q)
for i in range(maxit):
space = LagrangeFiniteElementSpace(mesh, 1)
gdof = space.number_of_global_dofs()
A = self.get_stiff_matrix(space)
M = self.get_mass_matrix(space)
b = M @ np.ones(gdof)
isFreeDof = ~(space.boundary_dof())
A = A[isFreeDof, :][:, isFreeDof].tocsr()
M = M[isFreeDof, :][:, isFreeDof].tocsr()
uh = space.function()
if self.matlab is False:
uh[isFreeDof] = self.psolve(A, b[isFreeDof], M)
else:
uh[isFreeDof] = self.msolve(A, b[isFreeDof])
eta = self.residual_estimate(uh)
markedCell = mark(eta, self.theta)
IM = mesh.bisect(markedCell, returnim=True)
NN = mesh.number_of_nodes()
print(i + 1, "refine :", NN)
if (self.step > 0) and (i in idx):
fig = plt.figure()
fig.set_facecolor('white')
if GD == 2:
axes = fig.gca()
else:
axes = Axes3D(fig)
mesh.add_plot(axes, cellcolor='w')
fig.savefig(self.resultdir + 'mesh_3_2_' + str(i + 1) + '_' +
str(NN) + '.pdf')
plt.close()
self.savemesh(
mesh, self.resultdir + 'mesh_3_2_' + str(i + 1) + '_' +
str(NN) + '.mat')
final = i + 1
if NN > self.maxdof:
break
if self.step > 0:
NN = mesh.number_of_nodes()
fig = plt.figure()
fig.set_facecolor('white')
if GD == 2:
axes = fig.gca()
else:
axes = Axes3D(fig)
mesh.add_plot(axes, cellcolor='w')
fig.savefig(self.resultdir + 'mesh_3_2_' + str(final) + '_' +
str(NN) + '.pdf')
plt.close()
self.savemesh(
mesh, self.resultdir + 'mesh_3_2_' + str(final) + '_' +
str(NN) + '.mat')
space = LagrangeFiniteElementSpace(mesh, 1)
gdof = space.number_of_global_dofs()
A = self.get_stiff_matrix(space)
M = self.get_mass_matrix(space)
isFreeDof = ~(space.boundary_dof())
A = A[isFreeDof, :][:, isFreeDof].tocsr()
M = M[isFreeDof, :][:, isFreeDof].tocsr()
if self.multieigs is True:
self.A = A
self.M = M
self.ml = pyamg.ruge_stuben_solver(self.A)
self.eigs()
else:
uh = IM @ uh
if self.matlab is False:
uh[isFreeDof], d = self.eig(A, M)
else:
uh[isFreeDof], d = self.meigs(A, M)
print("smallest eigns:", d)
return space.function(array=uh)
end = timer()
print("with time: ", end - start)
def perform_convergence_tests(
problem, exact_sol, get_mesh, rng, verbose=False
):
n = len(rng)
H = numpy.empty(n)
error_norm_1 = numpy.empty(n)
order_1 = numpy.empty(n-1)
error_norm_inf = numpy.empty(n)
order_inf = numpy.empty(n-1)
if verbose:
print(79 * '-')
print('k' + 5*' ' + 'num verts' + 4*' ' + 'max edge length' + 4*' ' +
'||error||_1' + 8*' ' + '||error||_inf'
)
print(38*' ' + '(order)' + 12*' ' + '(order)')
print(79 * '-')
# Add "zero" to all entities. This later gets translated into
# np.zeros with the appropriate length, making sure that scalar
# terms in the lambda expression correctly return np.arrays.
zero = sympy.Symbol('zero')
x = sympy.DeferredVector('x')
# See <http://docs.sympy.org/dev/modules/utilities/lambdify.html>.
array2array = [{'ImmutableMatrix': numpy.array}, 'numpy']
exact_eval = sympy.lambdify((x, zero), exact_sol(x), modules=array2array)
for k in rng:
mesh = get_mesh(k)
H[k] = max(mesh.edge_lengths)
linear_system = pyfvm.discretize(problem, mesh)
# x = linalg.spsolve(linear_system.matrix, linear_system.rhs)
ml = pyamg.ruge_stuben_solver(linear_system.matrix)
x = ml.solve(linear_system.rhs, tol=1e-10)
zero = numpy.zeros(len(mesh.node_coords))
error = x - exact_eval(mesh.node_coords.T, zero)
# import meshio
# meshio.write(
# 'sol%d.vtu' % k,
# mesh.node_coords, {'triangle': mesh.cells['nodes']},
# point_data={'x': x, 'error': error},
# )
error_norm_1[k] = numpy.sum(abs(mesh.control_volumes * error))
error_norm_inf[k] = max(abs(error))
# numerical orders of convergence
if k > 0:
order_1[k-1] = \
numpy.log(error_norm_1[k-1] / error_norm_1[k]) / \
numpy.log(H[k-1] / H[k])
order_inf[k-1] = \
numpy.log(error_norm_inf[k-1] / error_norm_inf[k]) / \
numpy.log(H[k-1] / H[k])
if verbose:
print
print((38*' ' + '%0.5f' + 12*' ' + '%0.5f') %
(order_1[k-1], order_inf[k-1])
)
print
if verbose:
num_nodes = len(mesh.control_volumes)
print('%2d %5.3e %0.10e %0.10e %0.10e' %
(k, num_nodes, H[k], error_norm_1[k], error_norm_inf[k])
)
return H, error_norm_1, error_norm_inf, order_1, order_inf
t = time.time()
res = []
x = ml.solve(b, residuals=res, tol=1e-12, accel='cg')
print(res)
print("residual: ", numpy.linalg.norm(b - A * x))
elapsed = time.time() - t
timescale = numpy.linspace(0, elapsed, num=len(res))
plt.semilogy(timescale, res, marker='o', label='SA')
print('solve time = ', elapsed)
# Classical -------------------
print("****** Ruge Stuben solver ******")
ml = pyamg.ruge_stuben_solver(A, max_coarse=100)
print(ml)
n = 0
def residual(w):
global n
res = A * w - b
write_meshio('file%d.xdmf' % n, mesh, res)
n += 1
print(numpy.linalg.norm(res))
t = time.time()
res = []
# Illustrates the selection of Coarse-Fine (CF)
# splittings in Classical AMG.
import numpy
from scipy.io import loadmat
from pyamg import ruge_stuben_solver
from pyamg.gallery import load_example
data = loadmat('square.mat') #load_example('airfoil')
A = data['A'] # matrix
V = data['vertices'][:A.shape[0]] # vertices of each variable
E = numpy.vstack((A.tocoo().row, A.tocoo().col)).T # edges of the matrix graph
# Use Ruge-Stuben Splitting Algorithm (use 'keep' in order to retain the splitting)
mls = ruge_stuben_solver(A, max_levels=2, max_coarse=1, CF='RS', keep=True)
print mls
# The CF splitting, 1 == C-node and 0 == F-node
splitting = mls.levels[0].splitting
C_nodes = splitting == 1
F_nodes = splitting == 0
from draw import lineplot
from pylab import figure, axis, scatter, show
figure(figsize=(6, 6))
axis('equal')
lineplot(V, E)
scatter(V[:, 0][C_nodes], V[:, 1][C_nodes], c='r',
s=100.0) #plot C-nodes in red
for i in range(maxit):
space = LagrangeFiniteElementSpace(mesh, p=p)
NDof[i] = space.number_of_global_dofs()
uh = space.function()
A = space.stiff_matrix()
F = space.source_vector(pde.source)
bc = RobinBC(space, pde.robin)
A, F = bc.apply(A, F)
#uh[:] = spsolve(A, F).reshape(-1)
ml = pyamg.ruge_stuben_solver(A)
uh[:] = ml.solve(F, tol=1e-12, accel='cg').reshape(-1)
errorMatrix[0, i] = space.integralalg.L2_error(pde.solution, uh)
errorMatrix[1, i] = space.integralalg.L2_error(pde.gradient, uh.grad_value)
if i < maxit - 1:
mesh.uniform_refine()
if d == 2:
fig = plt.figure()
axes = fig.gca(projection='3d')
uh.add_plot(axes, cmap='rainbow')
elif d == 3:
print('The 3d function plot is not been implemented!')
def apply_inverse(op,
V,
options=None,
least_squares=False,
check_finite=True,
default_solver='pyamg_solve'):
"""Solve linear equation system.
Applies the inverse of `op` to the vectors in `rhs` using PyAMG.
Parameters
----------
op
The linear, non-parametric |Operator| to invert.
rhs
|VectorArray| of right-hand sides for the equation system.
options
The |solver_options| to use (see :func:`solver_options`).
least_squares
Must be `False`.
check_finite
Test if solution only contains finite values.
default_solver
Default solver to use (pyamg_solve, pyamg_rs, pyamg_sa).
Returns
-------
|VectorArray| of the solution vectors.
"""
assert V in op.range
if least_squares:
raise NotImplementedError
if isinstance(op, NumpyMatrixOperator):
matrix = op.matrix
else:
from pymor.algorithms.to_matrix import to_matrix
matrix = to_matrix(op)
options = _parse_options(options, solver_options(), default_solver,
None, least_squares)
V = V.to_numpy()
promoted_type = np.promote_types(matrix.dtype, V.dtype)
R = np.empty((len(V), matrix.shape[1]), dtype=promoted_type)
if options['type'] == 'pyamg_solve':
if len(V) > 0:
V_iter = iter(enumerate(V))
R[0], ml = pyamg.solve(matrix,
next(V_iter)[1],
tol=options['tol'],
maxiter=options['maxiter'],
return_solver=True)
for i, VV in V_iter:
R[i] = pyamg.solve(matrix,
VV,
tol=options['tol'],
maxiter=options['maxiter'],
existing_solver=ml)
elif options['type'] == 'pyamg_rs':
ml = pyamg.ruge_stuben_solver(
matrix,
strength=options['strength'],
CF=options['CF'],
presmoother=options['presmoother'],
postsmoother=options['postsmoother'],
max_levels=options['max_levels'],
max_coarse=options['max_coarse'],
coarse_solver=options['coarse_solver'])
for i, VV in enumerate(V):
R[i] = ml.solve(VV,
tol=options['tol'],
maxiter=options['maxiter'],
cycle=options['cycle'],
accel=options['accel'])
elif options['type'] == 'pyamg_sa':
ml = pyamg.smoothed_aggregation_solver(
matrix,
symmetry=options['symmetry'],
strength=options['strength'],
aggregate=options['aggregate'],
smooth=options['smooth'],
presmoother=options['presmoother'],
postsmoother=options['postsmoother'],
improve_candidates=options['improve_candidates'],
max_levels=options['max_levels'],
max_coarse=options['max_coarse'],
diagonal_dominance=options['diagonal_dominance'])
for i, VV in enumerate(V):
R[i] = ml.solve(VV,
tol=options['tol'],
maxiter=options['maxiter'],
cycle=options['cycle'],
accel=options['accel'])
else:
raise ValueError('Unknown solver type')
if check_finite:
if not np.isfinite(np.sum(R)):
raise InversionError('Result contains non-finite values')
return op.source.from_numpy(R)
def alg_3_4(self, maxit=None):
"""
1. 最粗网格上求解最小特征特征值问题，得到最小特征值 d_H 和特征向量 u_H
2. 自适应求解 - \Delta u_h = u_H
* u_H 插值到下一层网格上做为新 u_H
3. 最新网格层上求出的 uh 做为一个基函数，加入到最粗网格的有限元空间中，
求解最小特征值问题。
自适应 maxit， picard:2
"""
print("算法 3.4")
if maxit is None:
maxit = self.maxit
start = timer()
if self.step == 0:
idx = []
else:
idx = list(range(0, self.maxit, self.step)) + [self.maxit - 1]
mesh = self.pde.init_mesh(n=self.numrefine)
integrator = mesh.integrator(self.q)
# 1. 粗网格上求解最小特征值问题
space = LagrangeFiniteElementSpace(mesh, 1)
AH = self.get_stiff_matrix(space)
MH = self.get_mass_matrix(space)
isFreeHDof = ~(space.boundary_dof())
gdof = space.number_of_global_dofs()
print("initial mesh :", gdof)
uH = np.zeros(gdof, dtype=np.float)
A = AH[isFreeHDof, :][:, isFreeHDof].tocsr()
M = MH[isFreeHDof, :][:, isFreeHDof].tocsr()
if self.matlab is False:
uH[isFreeHDof], d = self.eig(A, M)
else:
uH[isFreeHDof], d = self.meigs(A, M)
uh = space.function()
uh[:] = uH
GD = mesh.geo_dimension()
if (self.step > 0) and (0 in idx):
NN = mesh.number_of_nodes()
fig = plt.figure()
fig.set_facecolor('white')
if GD == 2:
axes = fig.gca()
else:
axes = Axes3D(fig)
mesh.add_plot(axes, cellcolor='w')
fig.savefig(self.resultdir + 'mesh_3_4_0_' + str(NN) + '.pdf')
plt.close()
self.savemesh(mesh,
self.resultdir + 'mesh_3_4_0_' + str(NN) + '.mat')
# 2. 以 u_H 为右端项自适应求解 -\Deta u = u_H
I = eye(gdof)
final = 0
for i in range(maxit):
eta = self.residual_estimate(uh)
markedCell = mark(eta, self.theta)
IM = mesh.bisect(markedCell, returnim=True)
print(i + 1, "refine :", mesh.number_of_nodes())
if (self.step > 0) and (i in idx):
NN = mesh.number_of_nodes()
fig = plt.figure()
fig.set_facecolor('white')
if GD == 2:
axes = fig.gca()
else:
axes = Axes3D(fig)
mesh.add_plot(axes, cellcolor='w')
fig.savefig(self.resultdir + 'mesh_3_4_' + str(i + 1) + '_' +
str(NN) + '.pdf')
plt.close()
self.savemesh(
mesh, self.resultdir + 'mesh_3_4_' + str(i + 1) + '_' +
str(NN) + '.mat')
I = IM @ I
uH = IM @ uH
space = LagrangeFiniteElementSpace(mesh, 1)
gdof = space.number_of_global_dofs()
A = self.get_stiff_matrix(space)
M = self.get_mass_matrix(space)
isFreeDof = ~(space.boundary_dof())
b = M @ uH
uh = space.function()
if self.matlab is False:
uh[isFreeDof] = self.psolve(
A[isFreeDof, :][:, isFreeDof].tocsr(), b[isFreeDof],
M[isFreeDof, :][:, isFreeDof].tocsr())
else:
uh[isFreeDof] = self.msolve(
A[isFreeDof, :][:, isFreeDof].tocsr(), b[isFreeDof])
final = i + 1
if gdof > self.maxdof:
break
if self.step > 0:
NN = mesh.number_of_nodes()
fig = plt.figure()
fig.set_facecolor('white')
if GD == 2:
axes = fig.gca()
else:
axes = Axes3D(fig)
mesh.add_plot(axes, cellcolor='w')
fig.savefig(self.resultdir + 'mesh_3_4_' + str(final) + '_' +
str(NN) + '.pdf')
plt.close()
self.savemesh(
mesh, self.resultdir + 'mesh_3_4_' + str(final) + '_' +
str(NN) + '.mat')
# 3. 把 uh 加入粗网格空间, 组装刚度和质量矩阵
w0 = uh @ A
w1 = w0 @ uh
w2 = w0 @ I
AA = bmat([[AH, w2.reshape(-1, 1)], [w2, w1]], format='csr')
w0 = uh @ M
w1 = w0 @ uh
w2 = w0 @ I
MM = bmat([[MH, w2.reshape(-1, 1)], [w2, w1]], format='csr')
isFreeDof = np.r_[isFreeHDof, True]
u = np.zeros(len(isFreeDof))
## 求解特征值
A = AA[isFreeDof, :][:, isFreeDof].tocsr()
M = MM[isFreeDof, :][:, isFreeDof].tocsr()
if self.multieigs is True:
self.A = A
self.M = M
self.ml = pyamg.ruge_stuben_solver(self.A)
self.eigs()
else:
if self.matlab is False:
u[isFreeDof], d = self.eig(A, M)
else:
u[isFreeDof], d = self.meigs(A, M)
print("smallest eigns:", d)
uh *= u[-1]
uh += I @ u[:-1]
uh /= np.max(np.abs(uh))
uh = space.function(array=uh)
return uh
end = timer()
print("with time: ", end - start)
def alg_3_3(self, maxit=None):
"""
1. 最粗网格上求解最小特征特征值问题，得到最小特征值 d_H 和特征向量 u_H
2. 自适应求解 - \Delta u_h = u_H
* u_H 插值到下一层网格上做为新 u_H
3. 在最细网格上求解一次最小特征值问题
自适应 maxit, picard:2
"""
print("算法 3.3")
if maxit is None:
maxit = self.maxit
start = timer()
if self.step == 0:
idx = []
else:
idx = list(range(0, self.maxit, self.step))
mesh = self.pde.init_mesh(n=self.numrefine)
# 1. 粗网格上求解最小特征值问题
space = LagrangeFiniteElementSpace(mesh, 1)
AH = self.get_stiff_matrix(space)
MH = self.get_mass_matrix(space)
isFreeHDof = ~(space.boundary_dof())
gdof = space.number_of_global_dofs()
uH = np.zeros(gdof, dtype=mesh.ftype)
print("initial mesh :", gdof)
A = AH[isFreeHDof, :][:, isFreeHDof].tocsr()
M = MH[isFreeHDof, :][:, isFreeHDof].tocsr()
if self.matlab is False:
uH[isFreeHDof], d = self.eig(A, M)
else:
uH[isFreeHDof], d = self.meigs(A, M)
uh = space.function()
uh[:] = uH
GD = mesh.geo_dimension()
if (self.step > 0) and (0 in idx):
NN = mesh.number_of_nodes()
fig = plt.figure()
fig.set_facecolor('white')
if GD == 2:
axes = fig.gca()
else:
axes = Axes3D(fig)
mesh.add_plot(axes, cellcolor='w')
fig.savefig(self.resultdir + 'mesh_3_3_0_' + str(NN) + '.pdf')
plt.close()
self.savemesh(mesh,
self.resultdir + 'mesh_3_3_0_' + str(NN) + '.mat')
# 2. 以 u_H 为右端项自适应求解 -\Deta u = u_H
I = eye(gdof)
final = 0
for i in range(maxit - 1):
eta = self.residual_estimate(uh)
markedCell = mark(eta, self.theta)
IM = mesh.bisect(markedCell, returnim=True)
NN = mesh.number_of_nodes()
print(i + 1, "refine : ", NN)
if (self.step > 0) and (i in idx):
NN = mesh.number_of_nodes()
fig = plt.figure()
fig.set_facecolor('white')
if GD == 2:
axes = fig.gca()
else:
axes = Axes3D(fig)
mesh.add_plot(axes, cellcolor='w')
fig.savefig(self.resultdir + 'mesh_3_3_' + str(i + 1) + '_' +
str(NN) + '.pdf')
plt.close()
self.savemesh(
mesh, self.resultdir + 'mesh_3_3_' + str(i + 1) + '_' +
str(NN) + '.mat')
final = i + 1
if NN > self.maxdof:
break
I = IM @ I
uH = IM @ uH
space = LagrangeFiniteElementSpace(mesh, 1)
gdof = space.number_of_global_dofs()
A = self.get_stiff_matrix(space)
M = self.get_mass_matrix(space)
isFreeDof = ~(space.boundary_dof())
b = M @ uH
uh = space.function()
if self.matlab is False:
uh[isFreeDof] = self.psolve(
A[isFreeDof, :][:, isFreeDof].tocsr(), b[isFreeDof],
M[isFreeDof, :][:, isFreeDof].tocsr())
else:
uh[isFreeDof] = self.msolve(
A[isFreeDof, :][:, isFreeDof].tocsr(), b[isFreeDof])
# 3. 在最细网格上求解一次最小特征值问题
if self.step > 0:
NN = mesh.number_of_nodes()
fig = plt.figure()
fig.set_facecolor('white')
if GD == 2:
axes = fig.gca()
else:
axes = Axes3D(fig)
mesh.add_plot(axes, cellcolor='w')
fig.savefig(self.resultdir + 'mesh_3_3_' + str(final) + '_' +
str(NN) + '.pdf')
plt.close()
self.savemesh(
mesh, self.resultdir + 'mesh_3_3_' + str(final) + '_' +
str(NN) + '.mat')
space = LagrangeFiniteElementSpace(mesh, 1)
gdof = space.number_of_global_dofs()
A = self.get_stiff_matrix(space)
M = self.get_mass_matrix(space)
isFreeDof = ~(space.boundary_dof())
uh = space.function(array=uh)
A = A[isFreeDof, :][:, isFreeDof].tocsr()
M = M[isFreeDof, :][:, isFreeDof].tocsr()
uh = space.function()
if self.multieigs is True:
self.A = A
self.M = M
self.ml = pyamg.ruge_stuben_solver(self.A)
self.eigs()
else:
if self.matlab is False:
uh[isFreeDof], d = self.eig(A, M)
else:
uh[isFreeDof], d = self.meigs(A, M)
print("smallest eigns:", d)
return uh
end = timer()
print("with time: ", end - start)
def apply_inverse(op, V, options=None, least_squares=False, check_finite=True, default_solver='pyamg_solve'):
"""Solve linear equation system.
Applies the inverse of `op` to the vectors in `rhs` using PyAMG.
Parameters
----------
op
The linear, non-parametric |Operator| to invert.
rhs
|VectorArray| of right-hand sides for the equation system.
options
The |solver_options| to use (see :func:`solver_options`).
check_finite
Test if solution only containes finite values.
default_solver
Default solver to use (pyamg_solve, pyamg_rs, pyamg_sa).
Returns
-------
|VectorArray| of the solution vectors.
"""
assert V in op.range
if isinstance(op, NumpyMatrixOperator):
matrix = op._matrix
else:
from pymor.algorithms.to_matrix import to_matrix
matrix = to_matrix(op)
options = _parse_options(options, solver_options(), default_solver, None, least_squares)
V = V.data
promoted_type = np.promote_types(matrix.dtype, V.dtype)
R = np.empty((len(V), matrix.shape[1]), dtype=promoted_type)
if options['type'] == 'pyamg_solve':
if len(V) > 0:
V_iter = iter(enumerate(V))
R[0], ml = pyamg.solve(matrix, next(V_iter)[1],
tol=options['tol'],
maxiter=options['maxiter'],
return_solver=True)
for i, VV in V_iter:
R[i] = pyamg.solve(matrix, VV,
tol=options['tol'],
maxiter=options['maxiter'],
existing_solver=ml)
elif options['type'] == 'pyamg_rs':
ml = pyamg.ruge_stuben_solver(matrix,
strength=options['strength'],
CF=options['CF'],
presmoother=options['presmoother'],
postsmoother=options['postsmoother'],
max_levels=options['max_levels'],
max_coarse=options['max_coarse'],
coarse_solver=options['coarse_solver'])
for i, VV in enumerate(V):
R[i] = ml.solve(VV,
tol=options['tol'],
maxiter=options['maxiter'],
cycle=options['cycle'],
accel=options['accel'])
elif options['type'] == 'pyamg_sa':
ml = pyamg.smoothed_aggregation_solver(matrix,
symmetry=options['symmetry'],
strength=options['strength'],
aggregate=options['aggregate'],
smooth=options['smooth'],
presmoother=options['presmoother'],
postsmoother=options['postsmoother'],
improve_candidates=options['improve_candidates'],
max_levels=options['max_levels'],
max_coarse=options['max_coarse'],
diagonal_dominance=options['diagonal_dominance'])
for i, VV in enumerate(V):
R[i] = ml.solve(VV,
tol=options['tol'],
maxiter=options['maxiter'],
cycle=options['cycle'],
accel=options['accel'])
else:
raise ValueError('Unknown solver type')
if check_finite:
if not np.isfinite(np.sum(R)):
raise InversionError('Result contains non-finite values')
return op.source.from_data(R)
# Illustrates the selection of Coarse-Fine (CF)
# splittings in Classical AMG.
import numpy
from pyamg import ruge_stuben_solver
from scipy.io import loadmat
from scipy.io import savemat
data = loadmat('Mat/NcutData.mat')
Acsc = data['NcutMatrix'] # matrix
Acsr = Acsc.tocsr()
mls = ruge_stuben_solver(Acsr,
max_levels=6,
max_coarse=150,
CF='RS',
keep=True)
#mls = ruge_stuben_solver(Acsr, keep=True)
'''
Parameters
#----------
A : csr_matrix
Square matrix in CSR format
strength : ['symmetric', 'classical', 'evolution', None]
Method used to determine the strength of connection between unknowns
of the linear system. Method-specific parameters may be passed in
using a tuple, e.g. strength=('symmetric',{'theta' : 0.25 }). If
strength=None, all nonzero entries of the matrix are considered strong.
CF : {string} : default 'RS'
Method used for coarse grid selection (C/F splitting)
Supported methods are RS, PMIS, PMISc, CLJP, and CLJPc
presmoother : {string or dict}
Method used for presmoothing at each level. Method-specific parameters
def apply_inverse(matrix, U, options=None):
"""Solve linear equation system.
Applies the inverse of `matrix` to the row vectors in `U`.
See :func:`sparse_options` for documentation of all possible options for
sparse matrices.
This method is called by :meth:`pymor.core.NumpyMatrixOperator.apply_inverse`
and usually should not be used directly.
Parameters
----------
matrix
The |NumPy| matrix to invert.
U
2-dimensional |NumPy array| containing as row vectors
the right-hand sides of the linear equation systems to
solve.
options
|invert_options| to use. (See :func:`invert_options`.)
Returns
-------
|NumPy array| of the solution vectors.
"""
default_options = invert_options(matrix)
if options is None:
options = default_options.values()[0]
elif isinstance(options, str):
if options == 'least_squares':
for k, v in default_options.iteritems():
if k.startswith('least_squares'):
options = v
break
assert not isinstance(options, str)
else:
options = default_options[options]
else:
assert 'type' in options and options['type'] in default_options \
and options.viewkeys() <= default_options[options['type']].viewkeys()
user_options = options
options = default_options[user_options['type']]
options.update(user_options)
R = np.empty((len(U), matrix.shape[1]))
if options['type'] == 'solve':
for i, UU in enumerate(U):
try:
R[i] = np.linalg.solve(matrix, UU)
except np.linalg.LinAlgError as e:
raise InversionError('{}: {}'.format(str(type(e)), str(e)))
elif options['type'] == 'least_squares_lstsq':
for i, UU in enumerate(U):
try:
R[i], _, _, _ = np.linalg.lstsq(matrix,
UU,
rcond=options['rcond'])
except np.linalg.LinAlgError as e:
raise InversionError('{}: {}'.format(str(type(e)), str(e)))
elif options['type'] == 'bicgstab':
for i, UU in enumerate(U):
R[i], info = bicgstab(matrix,
UU,
tol=options['tol'],
maxiter=options['maxiter'])
if info != 0:
if info > 0:
raise InversionError(
'bicgstab failed to converge after {} iterations'.
format(info))
else:
raise InversionError(
'bicgstab failed with error code {} (illegal input or breakdown)'
.format(info))
elif options['type'] == 'bicgstab_spilu':
ilu = spilu(matrix,
drop_tol=options['spilu_drop_tol'],
fill_factor=options['spilu_fill_factor'],
drop_rule=options['spilu_drop_rule'],
permc_spec=options['spilu_permc_spec'])
precond = LinearOperator(matrix.shape, ilu.solve)
for i, UU in enumerate(U):
R[i], info = bicgstab(matrix,
UU,
tol=options['tol'],
maxiter=options['maxiter'],
M=precond)
if info != 0:
if info > 0:
raise InversionError(
'bicgstab failed to converge after {} iterations'.
format(info))
else:
raise InversionError(
'bicgstab failed with error code {} (illegal input or breakdown)'
.format(info))
elif options['type'] == 'spsolve':
if scipy.version.version >= '0.14':
if hasattr(matrix, 'factorization'):
R = matrix.factorization.solve(U.T).T
elif options['keep_factorization']:
matrix.factorization = splu(matrix,
permc_spec=options['permc_spec'])
R = matrix.factorization.solve(U.T).T
else:
R = spsolve(matrix, U.T, permc_spec=options['permc_spec']).T
else:
if hasattr(matrix, 'factorization'):
for i, UU in enumerate(U):
R[i] = matrix.factorization.solve(UU)
elif options['keep_factorization']:
matrix.factorization = splu(matrix,
permc_spec=options['permc_spec'])
for i, UU in enumerate(U):
R[i] = matrix.factorization.solve(UU)
elif len(U) > 1:
factorization = splu(matrix, permc_spec=options['permc_spec'])
for i, UU in enumerate(U):
R[i] = factorization.solve(UU)
else:
R = spsolve(matrix, U.T,
permc_spec=options['permc_spec']).reshape((1, -1))
elif options['type'] == 'lgmres':
for i, UU in enumerate(U):
R[i], info = lgmres(matrix,
UU.copy(i),
tol=options['tol'],
maxiter=options['maxiter'],
inner_m=options['inner_m'],
outer_k=options['outer_k'])
if info > 0:
raise InversionError(
'lgmres failed to converge after {} iterations'.format(
info))
assert info == 0
elif options['type'] == 'least_squares_lsmr':
for i, UU in enumerate(U):
R[i], info, itn, _, _, _, _, _ = lsmr(matrix,
UU.copy(i),
damp=options['damp'],
atol=options['atol'],
btol=options['btol'],
conlim=options['conlim'],
maxiter=options['maxiter'],
show=options['show'])
assert 0 <= info <= 7
if info == 7:
raise InversionError(
'lsmr failed to converge after {} iterations'.format(itn))
elif options['type'] == 'least_squares_lsqr':
for i, UU in enumerate(U):
R[i], info, itn, _, _, _, _, _, _, _ = lsqr(
matrix,
UU.copy(i),
damp=options['damp'],
atol=options['atol'],
btol=options['btol'],
conlim=options['conlim'],
iter_lim=options['iter_lim'],
show=options['show'])
assert 0 <= info <= 7
if info == 7:
raise InversionError(
'lsmr failed to converge after {} iterations'.format(itn))
elif options['type'] == 'pyamg':
if len(U) > 0:
U_iter = iter(enumerate(U))
R[0], ml = pyamg.solve(matrix,
next(U_iter)[1],
tol=options['tol'],
maxiter=options['maxiter'],
return_solver=True)
for i, UU in U_iter:
R[i] = pyamg.solve(matrix,
UU,
tol=options['tol'],
maxiter=options['maxiter'],
existing_solver=ml)
elif options['type'] == 'pyamg-rs':
ml = pyamg.ruge_stuben_solver(matrix,
strength=options['strength'],
CF=options['CF'],
presmoother=options['presmoother'],
postsmoother=options['postsmoother'],
max_levels=options['max_levels'],
max_coarse=options['max_coarse'],
coarse_solver=options['coarse_solver'])
for i, UU in enumerate(U):
R[i] = ml.solve(UU,
tol=options['tol'],
maxiter=options['maxiter'],
cycle=options['cycle'],
accel=options['accel'])
elif options['type'] == 'pyamg-sa':
ml = pyamg.smoothed_aggregation_solver(
matrix,
symmetry=options['symmetry'],
strength=options['strength'],
aggregate=options['aggregate'],
smooth=options['smooth'],
presmoother=options['presmoother'],
postsmoother=options['postsmoother'],
improve_candidates=options['improve_candidates'],
max_levels=options['max_levels'],
max_coarse=options['max_coarse'],
diagonal_dominance=options['diagonal_dominance'])
for i, UU in enumerate(U):
R[i] = ml.solve(UU,
tol=options['tol'],
maxiter=options['maxiter'],
cycle=options['cycle'],
accel=options['accel'])
elif options['type'].startswith('generic') or options['type'].startswith(
'least_squares_generic'):
logger = getLogger('pymor.la.numpysolvers.apply_inverse')
logger.warn(
'You have selected a (potentially slow) generic solver for a NumPy matrix operator!'
)
from pymor.operators.numpy import NumpyMatrixOperator
from pymor.la.numpyvectorarray import NumpyVectorArray
return genericsolvers.apply_inverse(NumpyMatrixOperator(matrix),
NumpyVectorArray(U, copy=False),
options=options).data
else:
raise ValueError('Unknown solver type')
return R
def _solve(self, A=None, b=None, x0=None):
r"""
Sends the A and b matrices to the specified solver, and solves for *x*
given the boundary conditions, and source terms based on the present
value of *x*. This method does NOT iterate to solve for non-linear
source terms or march time steps.
Parameters
----------
A : sparse matrix
The coefficient matrix in sparse format. If not specified, then
it uses the ``A`` matrix attached to the object.
b : ND-array
The RHS matrix in any format. If not specified, then it uses
the ``b`` matrix attached to the object.
x0 : ND-array
The initial guess for the solution of Ax = b
Notes
-----
The solver used here is specified in the ``settings`` attribute of the
algorithm.
"""
# Fetch A and b from self if not given, and throw error if not found
A = self.A if A is None else A
b = self.b if b is None else b
if A is None or b is None:
raise Exception('The A matrix or the b vector not yet built.')
A = A.tocsr()
x0 = np.zeros_like(b) if x0 is None else x0
# Check if A and b are well-defined
self._check_for_nans()
# Raise error if solver_family not available
if self.settings["solver_family"] not in ["scipy", "petsc", "pyamg"]:
raise Exception(f"{self.settings['solver_family']} not available.")
# Set tolerance for iterative solvers
tol = self.settings["solver_tol"]
max_it = self.settings["solver_maxiter"]
atol = self._get_atol()
rtol = self._get_rtol(x0=x0)
# Check if A is symmetric
is_sym = op.utils.is_symmetric(self.A)
if self.settings['solver_type'] == 'cg' and not is_sym:
raise Exception('CG solver only works on symmetric matrices.')
# SciPy
if self.settings['solver_family'] == 'scipy':
# Umfpack by default uses its 32-bit build -> memory overflow
try:
import scikits.umfpack
A.indices = A.indices.astype(np.int64)
A.indptr = A.indptr.astype(np.int64)
except ModuleNotFoundError:
pass
solver = getattr(scipy.sparse.linalg, self.settings['solver_type'])
iterative = [
'bicg', 'bicgstab', 'cg', 'cgs', 'gmres', 'lgmres', 'minres',
'gcrotmk', 'qmr'
]
if solver.__name__ in iterative:
x, exit_code = solver(A=A,
b=b,
atol=atol,
tol=tol,
maxiter=max_it,
x0=x0)
if exit_code > 0:
raise Exception(
f'Solver did not converge, exit code: {exit_code}')
else:
x = solver(A=A, b=b)
# PETSc
if self.settings['solver_family'] == 'petsc':
# Check if petsc is available
try:
import petsc4py
from openpnm.utils.petsc import PETScSparseLinearSolver as SLS
except Exception:
raise ModuleNotFoundError('PETSc is not installed.')
temp = {
"type": self.settings["solver_type"],
"preconditioner": self.settings["solver_preconditioner"]
}
ls = SLS(A=A, b=b, settings=temp)
x = ls.solve(x0=x0, atol=atol, rtol=rtol, max_it=max_it)
# PyAMG
if self.settings['solver_family'] == 'pyamg':
# Check if PyAMG is available
try:
import pyamg
except Exception:
raise ModuleNotFoundError('PyAMG is not installed.')
ml = pyamg.ruge_stuben_solver(A)
x = ml.solve(b=b, tol=rtol, maxiter=max_it)
return x
"""
Test the convergence of a 100x100 anisotropic diffusion equation
"""
import numpy as np
import pyamg
n = 100
nx = n
ny = n
# Rotated Anisotropic Diffusion
stencil = pyamg.gallery.diffusion_stencil_2d(
type='FE', epsilon=0.001, theta=np.pi / 3)
A = pyamg.gallery.stencil_grid(stencil, (nx, ny), format='csr')
S = pyamg.strength.classical_strength_of_connection(A, 0.0)
np.random.seed(625)
x = np.random.rand(A.shape[0])
b = A * np.random.rand(A.shape[0])
ml = pyamg.ruge_stuben_solver(A, max_coarse=10)
resvec = []
x = ml.solve(b, x0=x, maxiter=20, tol=1e-14, residuals=resvec)
for i, r in enumerate(resvec):
print("residual at iteration {0:2}: {1:^6.2e}".format(i, r))
def _solve(self, A=None, b=None):
r"""
Sends the A and b matrices to the specified solver, and solves for *x*
given the boundary conditions, and source terms based on the present
value of *x*. This method does NOT iterate to solve for non-linear
source terms or march time steps.
Parameters
----------
A : sparse matrix
The coefficient matrix in sparse format. If not specified, then
it uses the ``A`` matrix attached to the object.
b : ND-array
The RHS matrix in any format. If not specified, then it uses
the ``b`` matrix attached to the object.
Notes
-----
The solver used here is specified in the ``settings`` attribute of the
algorithm.
"""
# Fetch A and b from self if not given, and throw error if they've not
# been calculated
if A is None:
A = self.A
if A is None:
raise Exception('The A matrix has not been built yet')
if b is None:
b = self.b
if b is None:
raise Exception('The b matrix has not been built yet')
A = A.tocsr()
# Default behavior -> use Scipy's default solver (spsolve)
if self.settings['solver'] == 'pyamg':
self.settings['solver_family'] = 'pyamg'
if self.settings['solver'] == 'petsc':
self.settings['solver_family'] = 'petsc'
# Set tolerance for iterative solvers
rtol = self.settings['solver_rtol']
# Reference for residual's normalization
ref = np.sum(np.absolute(self.A.diagonal())) or 1
atol = ref * rtol
# SciPy
if self.settings['solver_family'] == 'scipy':
if importlib.util.find_spec('scikit-umfpack'):
A.indices = A.indices.astype(np.int64)
A.indptr = A.indptr.astype(np.int64)
iterative = ['bicg', 'bicgstab', 'cg', 'cgs', 'gmres', 'lgmres',
'minres', 'gcrotmk', 'qmr']
solver = getattr(sprs.linalg, self.settings['solver_type'])
if self.settings['solver_type'] in iterative:
x, exit_code = solver(A=A, b=b, atol=atol, tol=rtol,
maxiter=self.settings['solver_maxiter'])
if exit_code > 0:
raise Exception('SciPy solver did not converge! '
+ 'Exit code: ' + str(exit_code))
else:
x = solver(A=A, b=b)
return x
# PETSc
if self.settings['solver_family'] == 'petsc':
# Check if petsc is available
if importlib.util.find_spec('petsc4py'):
from openpnm.utils.petsc import PETScSparseLinearSolver as SLS
else:
raise Exception('PETSc is not installed.')
# Define the petsc linear system converting the scipy objects
ls = SLS(A=A, b=b)
sets = self.settings
sets = {k: v for k, v in sets.items() if k.startswith('solver_')}
sets = {k.split('solver_')[1]: v for k, v in sets.items()}
ls.settings.update(sets)
x = SLS.solve(ls)
del(ls)
return x
# PyAMG
if self.settings['solver_family'] == 'pyamg':
if importlib.util.find_spec('pyamg'):
import pyamg
else:
raise Exception('pyamg is not installed.')
ml = pyamg.ruge_stuben_solver(A)
x = ml.solve(b=b, tol=1e-10)
return x
def _apply_inverse(matrix, V, options=None):
"""Solve linear equation system.
Applies the inverse of `matrix` to the row vectors in `V`.
See :func:`dense_options` for documentation of all possible options for
sparse matrices.
See :func:`sparse_options` for documentation of all possible options for
sparse matrices.
This method is called by :meth:`pymor.core.NumpyMatrixOperator.apply_inverse`
and usually should not be used directly.
Parameters
----------
matrix
The |NumPy| matrix to invert.
V
2-dimensional |NumPy array| containing as row vectors
the right-hand sides of the linear equation systems to
solve.
options
The solver options to use. (See :func:`_options`.)
Returns
-------
|NumPy array| of the solution vectors.
"""
default_options = _options(matrix)
if options is None:
options = default_options.values()[0]
elif isinstance(options, str):
if options == 'least_squares':
for k, v in default_options.iteritems():
if k.startswith('least_squares'):
options = v
break
assert not isinstance(options, str)
else:
options = default_options[options]
else:
assert 'type' in options and options['type'] in default_options \
and options.viewkeys() <= default_options[options['type']].viewkeys()
user_options = options
options = default_options[user_options['type']]
options.update(user_options)
promoted_type = np.promote_types(matrix.dtype, V.dtype)
R = np.empty((len(V), matrix.shape[1]), dtype=promoted_type)
if options['type'] == 'solve':
for i, VV in enumerate(V):
try:
R[i] = np.linalg.solve(matrix, VV)
except np.linalg.LinAlgError as e:
raise InversionError('{}: {}'.format(str(type(e)), str(e)))
elif options['type'] == 'least_squares_lstsq':
for i, VV in enumerate(V):
try:
R[i], _, _, _ = np.linalg.lstsq(matrix, VV, rcond=options['rcond'])
except np.linalg.LinAlgError as e:
raise InversionError('{}: {}'.format(str(type(e)), str(e)))
elif options['type'] == 'bicgstab':
for i, VV in enumerate(V):
R[i], info = bicgstab(matrix, VV, tol=options['tol'], maxiter=options['maxiter'])
if info != 0:
if info > 0:
raise InversionError('bicgstab failed to converge after {} iterations'.format(info))
else:
raise InversionError('bicgstab failed with error code {} (illegal input or breakdown)'.
format(info))
elif options['type'] == 'bicgstab_spilu':
ilu = spilu(matrix, drop_tol=options['spilu_drop_tol'], fill_factor=options['spilu_fill_factor'],
drop_rule=options['spilu_drop_rule'], permc_spec=options['spilu_permc_spec'])
precond = LinearOperator(matrix.shape, ilu.solve)
for i, VV in enumerate(V):
R[i], info = bicgstab(matrix, VV, tol=options['tol'], maxiter=options['maxiter'], M=precond)
if info != 0:
if info > 0:
raise InversionError('bicgstab failed to converge after {} iterations'.format(info))
else:
raise InversionError('bicgstab failed with error code {} (illegal input or breakdown)'.
format(info))
elif options['type'] == 'spsolve':
try:
# maybe remove unusable factorization:
if hasattr(matrix, 'factorization'):
fdtype = matrix.factorizationdtype
if not np.can_cast(V.dtype, fdtype, casting='safe'):
del matrix.factorization
if map(int, scipy.version.version.split('.')) >= [0, 14, 0]:
if hasattr(matrix, 'factorization'):
# we may use a complex factorization of a real matrix to
# apply it to a real vector. In that case, we downcast
# the result here, removing the imaginary part,
# which should be zero.
R = matrix.factorization.solve(V.T).T.astype(promoted_type, copy=False)
elif options['keep_factorization']:
# the matrix is always converted to the promoted type.
# if matrix.dtype == promoted_type, this is a no_op
matrix.factorization = splu(matrix_astype_nocopy(matrix, promoted_type), permc_spec=options['permc_spec'])
matrix.factorizationdtype = promoted_type
R = matrix.factorization.solve(V.T).T
else:
# the matrix is always converted to the promoted type.
# if matrix.dtype == promoted_type, this is a no_op
R = spsolve(matrix_astype_nocopy(matrix, promoted_type), V.T, permc_spec=options['permc_spec']).T
else:
# see if-part for documentation
if hasattr(matrix, 'factorization'):
for i, VV in enumerate(V):
R[i] = matrix.factorization.solve(VV).astype(promoted_type, copy=False)
elif options['keep_factorization']:
matrix.factorization = splu(matrix_astype_nocopy(matrix, promoted_type), permc_spec=options['permc_spec'])
matrix.factorizationdtype = promoted_type
for i, VV in enumerate(V):
R[i] = matrix.factorization.solve(VV)
elif len(V) > 1:
factorization = splu(matrix_astype_nocopy(matrix, promoted_type), permc_spec=options['permc_spec'])
for i, VV in enumerate(V):
R[i] = factorization.solve(VV)
else:
R = spsolve(matrix_astype_nocopy(matrix, promoted_type), V.T, permc_spec=options['permc_spec']).reshape((1, -1))
except RuntimeError as e:
raise InversionError(e)
elif options['type'] == 'lgmres':
for i, VV in enumerate(V):
R[i], info = lgmres(matrix, VV.copy(i),
tol=options['tol'],
maxiter=options['maxiter'],
inner_m=options['inner_m'],
outer_k=options['outer_k'])
if info > 0:
raise InversionError('lgmres failed to converge after {} iterations'.format(info))
assert info == 0
elif options['type'] == 'least_squares_lsmr':
for i, VV in enumerate(V):
R[i], info, itn, _, _, _, _, _ = lsmr(matrix, VV.copy(i),
damp=options['damp'],
atol=options['atol'],
btol=options['btol'],
conlim=options['conlim'],
maxiter=options['maxiter'],
show=options['show'])
assert 0 <= info <= 7
if info == 7:
raise InversionError('lsmr failed to converge after {} iterations'.format(itn))
elif options['type'] == 'least_squares_lsqr':
for i, VV in enumerate(V):
R[i], info, itn, _, _, _, _, _, _, _ = lsqr(matrix, VV.copy(i),
damp=options['damp'],
atol=options['atol'],
btol=options['btol'],
conlim=options['conlim'],
iter_lim=options['iter_lim'],
show=options['show'])
assert 0 <= info <= 7
if info == 7:
raise InversionError('lsmr failed to converge after {} iterations'.format(itn))
elif options['type'] == 'pyamg':
if len(V) > 0:
V_iter = iter(enumerate(V))
R[0], ml = pyamg.solve(matrix, next(V_iter)[1],
tol=options['tol'],
maxiter=options['maxiter'],
return_solver=True)
for i, VV in V_iter:
R[i] = pyamg.solve(matrix, VV,
tol=options['tol'],
maxiter=options['maxiter'],
existing_solver=ml)
elif options['type'] == 'pyamg-rs':
ml = pyamg.ruge_stuben_solver(matrix,
strength=options['strength'],
CF=options['CF'],
presmoother=options['presmoother'],
postsmoother=options['postsmoother'],
max_levels=options['max_levels'],
max_coarse=options['max_coarse'],
coarse_solver=options['coarse_solver'])
for i, VV in enumerate(V):
R[i] = ml.solve(VV,
tol=options['tol'],
maxiter=options['maxiter'],
cycle=options['cycle'],
accel=options['accel'])
elif options['type'] == 'pyamg-sa':
ml = pyamg.smoothed_aggregation_solver(matrix,
symmetry=options['symmetry'],
strength=options['strength'],
aggregate=options['aggregate'],
smooth=options['smooth'],
presmoother=options['presmoother'],
postsmoother=options['postsmoother'],
improve_candidates=options['improve_candidates'],
max_levels=options['max_levels'],
max_coarse=options['max_coarse'],
diagonal_dominance=options['diagonal_dominance'])
for i, VV in enumerate(V):
R[i] = ml.solve(VV,
tol=options['tol'],
maxiter=options['maxiter'],
cycle=options['cycle'],
accel=options['accel'])
elif options['type'].startswith('generic') or options['type'].startswith('least_squares_generic'):
logger = getLogger('pymor.operators.numpy._apply_inverse')
logger.warn('You have selected a (potentially slow) generic solver for a NumPy matrix operator!')
from pymor.operators.numpy import NumpyMatrixOperator
from pymor.vectorarrays.numpy import NumpyVectorArray
return genericsolvers.apply_inverse(NumpyMatrixOperator(matrix),
NumpyVectorArray(V, copy=False),
options=options).data
else:
raise ValueError('Unknown solver type')
return R
def testgrid(parameters):
exec ','.join(parameters) + ', = parameters.values()' # unpack the parameters into the local namespace
description = '-' + description
dt = 24.0*7 #hours per time step
Nsteps = 1 #number of steps
Nx = problemscale #number of cells, x-dimension
Ny = problemscale
Nz = problemscale
ndims = 3
N = Nx*Ny*Nz #number of cells, total
size=N
if verbose:
print "N is %i." % N
#h in feet
hx=16.0 #length, x-dimension
hy=16.0
hz=10.0
#perm in md, conversion factor for hours
Perm =ones([3,Nz,Ny,Nx]).astype(np.float64)*100.0*.0002637 #mD is millidarcy=m^2/1000 #permeability
Kinv = 1.0/Perm
phi = .20 #unitless #porosity
ct = 10.0E-06 #1/psi #conductivity?
#B=1.0
alpha = hx*hy*hz*phi*ct/dt #ft^3/hours/psi #???
gamma = .433 #psi/ft #specific weight
Phi =ones([Nz,Ny,Nx]).astype(np.float64)*2500.0 #Flow Potential
Z=zeros([Nz,Ny,Nx]).astype(np.float64)
for k in range(Nz):
Z[k,:,:]+=k*hz # 3D field of all 0's in the bottom slice (z=0), adding 10 for each slice up (all 10s in slice where z=1, all 20s in slice where z=2, etc.)
p = Phi-gamma*Z #potential
Phi = Phi.flatten() #Phi is now just a row vector of 2500.0's, either Nx or Ny long. Really? Not Nx*Ny*Nz?
# We're switching from using a 3-tuple as an index to each cell to having a scalar between 0 and (N-1) as an index to each cell.
TX = zeros([Nz,Ny,Nx+1]).astype(np.float64)#zeros([Nx+1,Ny,Nz]) #a 3D scalar field: x-components of transmissibility vectors
TY = zeros([Nz,Ny+1,Nx]).astype(np.float64)#zeros([Nx,Ny+1,Nz])
TZ = zeros([Nz+1,Ny,Nx]).astype(np.float64)#zeros([Nx,Ny,Nz+1])
# (for Nx=3,Ny=4,Nz=5):
TX[:,:,1:-1] = (2*hy*hz/hx)/(Kinv[0,:,:,0:-1]+Kinv[0,:,:,1: ]) # 2x4x5
TY[:,1:-1,:] = (2*hx*hz/hy)/(Kinv [1,:,0:-1,:]+Kinv[1,:,1:,:]) # 3x3x5 Why are these not all 3x4x5?
TZ[1:-1,:,:] = (2*hx*hy/hz)/(Kinv[2,0:-1,:,:]+Kinv[2,1:,:,:]) # 3x4x4
x1 = TX[:,:,0:Nx].flatten()
x2 = TX[:,:,1:].flatten()
y1 = TY[:,0:Ny,:].flatten()
y2 = TY[:,1:,:].flatten()
z1 = TZ[0:Nz,:,:].flatten()
z2 = TZ[1:,:,:].flatten()
q = zeros([Nz,Ny,Nx]).astype(np.float64)
q[:,4,4]=10.0*5.61/24.0
Iw = (Nx*Ny-1)/2
q = q.flatten()
main_diag=x1+x2+y1+y2+z1+z2#+q2 #(when q2==1, sum is zero and vice versa)
#DiagVecs = vstack((-z2,-y2,-x2,main_diag,-x1,-y1,-z1))
DiagVecs = vstack((-z2,-y2,-x2, -x1,-y1,-z1))
#DiagIndx = array([-Nx*Ny,-Nx,-1,0,1,Nx,Nx*Ny])
DiagIndx = array([-Nx*Ny,-Nx,-1,1, Nx,Nx*Ny])
Phi_w = zeros(Nsteps+1).astype(np.float64)
Phi_w[0] = Phi[Iw]
Phi_c = zeros(Nsteps+1).astype(np.float64)
Phi_c[0] = Phi[-1]
Phi_cl = zeros(Nsteps+1).astype(np.float64)
Phi_cl[0] = Phi[0]
level=0
#whoami()
#global N
#global ndims
A = sparse.spdiags(DiagVecs,DiagIndx,N/(2**ndims)**level,N/(2**ndims)**level,format='csr')
DM = sparse.spdiags(main_diag,array([0]),N/(2**ndims)**level,N/(2**ndims)**level,format='csr')
D0 = sparse.spdiags(ones(N/(2**ndims)**level)*alpha,array([0]),N/(2**ndims)**level,N/(2**ndims)**level,format='csr')
A = sparse.spdiags(DiagVecs,DiagIndx,N/(2**ndims)**level,N/(2**ndims)**level,format='csr')
DM = sparse.spdiags(main_diag,array([0]),N/(2**ndims)**level,N/(2**ndims)**level,format='csr')
D0 = sparse.spdiags(ones(N/(2**ndims)**level)*alpha,array([0]),N/(2**ndims)**level,N/(2**ndims)**level,format='csr')
LHS = (A+D0+DM).toarray().astype(np.float64)
if verbose:
print 'LHS shape is:'
print LHS.shape
# Main Timestep Loop
info_dict = {'test_grid':True}
for k in range(Nsteps):
Qa = (-A.astype(np.float64)*Phi.astype(np.float64)).astype(np.float64)
Qa-= DM*Phi
if verbose: print 'solving with %s...' % solver
##CHOOSE THY SOLVER:
if solver == 'pyamg': ##PyAMG: Ruge-Steuben
#make_sparsity_graph('graphs/pyamg_LHS.png','pyamg LHS',A+D0+DM)
ml = pyamg.ruge_stuben_solver(A+D0+DM,max_levels=gridlevels,presmoother=('gauss_seidel',{'iterations':iterations}))
u = ml.solve(q+Qa,cycle='F',tol=1e-06, accel='cg')
info_dict['solverstring']='PyAMG-%icells' % N
info_dict['cycle']=0
info_dict['norm']=np.linalg.norm(q+Qa-np.dot(LHS,u))
if solver == 'pyamg-linagg': ##PyAMG; linear aggregation:
#make_sparsity_graph('graphs/pyamg_LHS.png','pyamg LHS',A+D0+DM)
ml=pyamg.smoothed_aggregation_solver(A+D0+DM)#,mat_flag= 'symmetric',strength= 'symmetric')
u = ml.solve(q+Qa,cycle='F',tol=1e-06, accel='cg')
info_dict['solverstring']='PyAMG-%icells' % N
info_dict['cycle']=0
info_dict['norm']=np.linalg.norm(q+Qa-np.dot(LHS,u))
elif solver == 'mmg': ##Our Multi-Multi-Grid
#parameters['verbose']=False #uncomment to disable verbosity within the OpenMG code.
if graph_pressure: parameters['cycles']=1
(u,info_dict) = mg_solve(LHS,q+Qa,parameters) # info_dict here contains both cycle and norm
if graph_pressure: parameters['cycles']=2
if graph_pressure: (utwo,info_dict) = mg_solve(LHS,q+Qa,parameters) # info_dict here contains both cycle and norm
if graph_pressure: parameters['cycles']=3
if graph_pressure: (uthree,info_dict) = mg_solve(LHS,q+Qa,parameters) # info_dict here contains both cycle and norm
info_dict['solverstring'] = '%igrid-%iiter-%icells%sthreshold-%icycles' % (gridlevels,iterations,N,threshold,cycles)
elif solver == 'gs': ##Our Gauss-Seidel iterative solver
(u,gs_iterations) = iterative_solve_to_threshold(LHS, q+Qa, np.zeros((q.size,)), threshold,verbose=verbose)
info_dict['solverstring']='GaussSeidel-%iiter-%icells' % (gs_iterations,N)
info_dict['cycle']=0
info_dict['norm']=np.linalg.norm(q+Qa-np.dot(LHS,u))
elif solver == 'gs_xiter': ##Our Gauss-Seidel iterative solver, with a specified number of iterations
#def iterative_solve(A,b,x,iterations,verbose=False):
u = iterative_solve(LHS, q+Qa, np.zeros((q.size,)), iterations,verbose)
info_dict={}
gs_iterations = iterations
info_dict['solverstring']='GaussSeidel-%iiter-%icells' % (gs_iterations,N)
info_dict['cycle']=0
info_dict['norm']=np.linalg.norm(q+Qa-np.dot(LHS,u))
else: ##Numpy's (pretty good) direct solver.
solver = 'direct'
u = np.linalg.solve(LHS,q+Qa)
info_dict['solverstring']= 'direct-%icells' % N
info_dict['cycle']=0
info_dict['norm']=np.linalg.norm(q+Qa-np.dot(LHS,u))
if graph_pressure: oldPhi = Phi.copy()
Phi += u
Phi_w[k+1]=Phi[Iw]
Phi_cl[k+1]=Phi[0]
Phi_c[k+1]=Phi[-1]
info_dict['solverstring'] += description
if graph_pressure:
u_correct = np.linalg.solve(LHS,q+Qa)
Phi_correct = oldPhi + u_correct
Phi_graph_correct = reshape(Phi_correct,[Nz,Ny,Nx])
Phi_two = oldPhi + utwo
Phi_two_graph=reshape(Phi_two,[Nz,Ny,Nx])
Phi_three = oldPhi + uthree
Phi_three_graph=reshape(Phi_three,[Nz,Ny,Nx])
Phi_graph=reshape(Phi,[Nz,Ny,Nx])
filename='graphs/USS_output-%s.png' % (info_dict['solverstring'],)
import visualization as viz
viz.make_multiple_3d_graphs([Phi_graph_correct.T,Phi_graph.T,Phi_two_graph.T,Phi_three_graph.T],filename)
del [Phi,Phi_c,Phi_cl,Phi_graph,Phi_w,u,Qa,DM,D0,LHS,A]
else:
del [Phi,Phi_c,Phi_cl,Phi_w,u,Qa,DM,D0,LHS,A]
if verbose: print "completed solverstring is", info_dict['solverstring']
return info_dict
def test_block_diagonalize_P():
"""Check if eigenvalues of block matrices are the same as eigenvalues of original block-circulant matrix"""
init_octave(1)
num_unknowns_per_block = 64
root_num_blocks = 3
A, _ = generate_A_delaunay_block_periodic_lognormal(
num_unknowns_per_block, root_num_blocks)
# A = A + 0.1 * scipy.sparse.diags(np.ones(num_unknowns_per_block * root_num_blocks**2))
orig_solver = pyamg.ruge_stuben_solver(A,
max_levels=2,
max_coarse=1,
CF='CLJP',
keep=True)
orig_splitting = orig_solver.levels[0].splitting
block_splitting = list(chunks(orig_splitting, num_unknowns_per_block))
common_block_splitting = most_frequent_splitting(block_splitting)
repeated_splitting = np.tile(common_block_splitting, root_num_blocks**2)
# we recompute the Ruge-Stuben prolongation matrix with the modified splitting, and the original strength
# matrix. We assume the strength matrix is block-circulant (because A is block-circulant)
C = orig_solver.levels[0].C
P = direct_interpolation(A, C, repeated_splitting)
P = tf.convert_to_tensor(P.toarray(), dtype=tf.float64)
P_blocks = block_diagonalize_P(P, root_num_blocks,
repeated_splitting.nonzero())
P_blocks = P_blocks.numpy()
A_c = P.numpy().T @ A.toarray() @ P.numpy()
# double_W, double_W_conj_t = create_double_W(num_unknowns_per_block * root_num_blocks, root_num_blocks)
# A_c_full_block_diag = double_W_conj_t @ A_c_full @ double_W
# P_full_block_diag = double_W_conj_t @ full_P @ double_W
# A_block_diag = double_W_conj_t @ A.toarray() @ double_W
# A_c_full_block_diag_2 = P_full_block_diag.conj().T @ A_block_diag @ P_full_block_diag
tf_A = tf.cast(tf.stack([A.toarray()]), tf.complex128)
A_blocks = block_diagonalize_A_fast(
tf_A, root_num_blocks,
True).numpy()[0][1:] # ignore the first zero mode block
def relaxation_matrices(As, w=0.8):
I = np.eye(As[0].shape[0])
res = [I - w * np.diag(1 / (np.diag(A))) @ A for A in As]
# computes the iteration matrix of the relaxation, here Gauss-Seidel is used.
# This function is called on each block seperately.
# num_As = len(As)
# grid_sizes = [A.shape[0] for A in As]
# Bs = [A.copy() for A in As]
# for B, grid_size in zip(Bs, grid_sizes):
# B[np.tril_indices(grid_size, 0)[0], np.tril_indices(grid_size, 0)[1]] = 0. # B is the upper part of A
# res = []
# for i in tqdm(range(num_As)): # range(A.shape[0] // batch_size):
# res.append(scipy.linalg.solve_triangular(a=As[i],
# b=-Bs[i],
# lower=True, unit_diagonal=False,
# overwrite_b=False, debug=None, check_finite=True).astype(
# np.float64))
return res
S = relaxation_matrices([A.toarray()])[0]
S_blocks = relaxation_matrices(A_blocks)
# A_c = P.numpy().T @ A.toarray() @ P.numpy()
A_c_blocks = P_blocks.transpose([0, 2, 1]).conj() @ A_blocks @ P_blocks
A = A.toarray()
C = np.eye(A.shape[0]) - P.numpy() @ np.linalg.inv(A_c) @ P.numpy().T @ A
M = S @ C @ S
I = np.eye(A_blocks[0].shape[0])
C_blocks = [
I - P_block @ np.linalg.inv(A_c_block) @ P_block.conj().T @ A_block
for (P_block, A_c_block,
A_block) in zip(P_blocks, A_c_blocks, A_blocks)
]
M_blocks = [
S_block @ C_block @ S_block
for (S_block, C_block) in zip(S_blocks, C_blocks)
]
# # extract only elements that correspond to coarse nodes
# A_c_blocks = A_c_blocks[:, common_block_splitting.nonzero()[0][:, None], common_block_splitting.nonzero()[0]]
A_c_block_eigs = np.sort(np.linalg.eigvals(A_c_blocks).flatten())
A_c_eigs = np.sort(np.linalg.eigvals(A_c))
C_block_eigs = np.sort(np.linalg.eigvals(C_blocks).flatten())
C_eigs = np.sort(np.linalg.eigvals(C))
M_block_eigs = np.sort(np.linalg.eigvals(M_blocks).flatten())
M_eigs = np.sort(np.linalg.eigvals(M))
pass
# Illustrates the selection of Coarse-Fine (CF)
# splittings in Classical AMG.
import numpy
from scipy.io import loadmat
from pyamg import ruge_stuben_solver
from pyamg.gallery import load_example
data = loadmat('square.mat') #load_example('airfoil')
A = data['A'] # matrix
V = data['vertices'][:A.shape[0]] # vertices of each variable
E = numpy.vstack((A.tocoo().row,A.tocoo().col)).T # edges of the matrix graph
# Use Ruge-Stuben Splitting Algorithm (use 'keep' in order to retain the splitting)
mls = ruge_stuben_solver(A, max_levels=2, max_coarse=1, CF='RS',keep=True)
print mls
# The CF splitting, 1 == C-node and 0 == F-node
splitting = mls.levels[0].splitting
C_nodes = splitting == 1
F_nodes = splitting == 0
from draw import lineplot
from pylab import figure, axis, scatter, show
figure(figsize=(6,6))
axis('equal')
lineplot(V, E)
scatter(V[:,0][C_nodes], V[:,1][C_nodes], c='r', s=100.0) #plot C-nodes in red
scatter(V[:,0][F_nodes], V[:,1][F_nodes], c='b', s=100.0) #plot F-nodes in blue
def _apply_inverse(matrix, V, options=None):
"""Solve linear equation system.
Applies the inverse of `matrix` to the row vectors in `V`.
See :func:`dense_options` for documentation of all possible options for
sparse matrices.
See :func:`sparse_options` for documentation of all possible options for
sparse matrices.
This method is called by :meth:`pymor.core.NumpyMatrixOperator.apply_inverse`
and usually should not be used directly.
Parameters
----------
matrix
The |NumPy| matrix to invert.
V
2-dimensional |NumPy array| containing as row vectors
the right-hand sides of the linear equation systems to
solve.
options
The solver options to use. (See :func:`_options`.)
Returns
-------
|NumPy array| of the solution vectors.
"""
default_options = _options(matrix)
if options is None:
options = default_options.values()[0]
elif isinstance(options, str):
if options == "least_squares":
for k, v in default_options.iteritems():
if k.startswith("least_squares"):
options = v
break
assert not isinstance(options, str)
else:
options = default_options[options]
else:
assert (
"type" in options
and options["type"] in default_options
and options.viewkeys() <= default_options[options["type"]].viewkeys()
)
user_options = options
options = default_options[user_options["type"]]
options.update(user_options)
R = np.empty((len(V), matrix.shape[1]), dtype=np.promote_types(matrix.dtype, V.dtype))
if options["type"] == "solve":
for i, VV in enumerate(V):
try:
R[i] = np.linalg.solve(matrix, VV)
except np.linalg.LinAlgError as e:
raise InversionError("{}: {}".format(str(type(e)), str(e)))
elif options["type"] == "least_squares_lstsq":
for i, VV in enumerate(V):
try:
R[i], _, _, _ = np.linalg.lstsq(matrix, VV, rcond=options["rcond"])
except np.linalg.LinAlgError as e:
raise InversionError("{}: {}".format(str(type(e)), str(e)))
elif options["type"] == "bicgstab":
for i, VV in enumerate(V):
R[i], info = bicgstab(matrix, VV, tol=options["tol"], maxiter=options["maxiter"])
if info != 0:
if info > 0:
raise InversionError("bicgstab failed to converge after {} iterations".format(info))
else:
raise InversionError("bicgstab failed with error code {} (illegal input or breakdown)".format(info))
elif options["type"] == "bicgstab_spilu":
ilu = spilu(
matrix,
drop_tol=options["spilu_drop_tol"],
fill_factor=options["spilu_fill_factor"],
drop_rule=options["spilu_drop_rule"],
permc_spec=options["spilu_permc_spec"],
)
precond = LinearOperator(matrix.shape, ilu.solve)
for i, VV in enumerate(V):
R[i], info = bicgstab(matrix, VV, tol=options["tol"], maxiter=options["maxiter"], M=precond)
if info != 0:
if info > 0:
raise InversionError("bicgstab failed to converge after {} iterations".format(info))
else:
raise InversionError("bicgstab failed with error code {} (illegal input or breakdown)".format(info))
elif options["type"] == "spsolve":
try:
if scipy.version.version >= "0.14":
if hasattr(matrix, "factorization"):
R = matrix.factorization.solve(V.T).T
elif options["keep_factorization"]:
matrix.factorization = splu(matrix, permc_spec=options["permc_spec"])
R = matrix.factorization.solve(V.T).T
else:
R = spsolve(matrix, V.T, permc_spec=options["permc_spec"]).T
else:
if hasattr(matrix, "factorization"):
for i, VV in enumerate(V):
R[i] = matrix.factorization.solve(VV)
elif options["keep_factorization"]:
matrix.factorization = splu(matrix, permc_spec=options["permc_spec"])
for i, VV in enumerate(V):
R[i] = matrix.factorization.solve(VV)
elif len(V) > 1:
factorization = splu(matrix, permc_spec=options["permc_spec"])
for i, VV in enumerate(V):
R[i] = factorization.solve(VV)
else:
R = spsolve(matrix, V.T, permc_spec=options["permc_spec"]).reshape((1, -1))
except RuntimeError as e:
raise InversionError(e)
elif options["type"] == "lgmres":
for i, VV in enumerate(V):
R[i], info = lgmres(
matrix,
VV.copy(i),
tol=options["tol"],
maxiter=options["maxiter"],
inner_m=options["inner_m"],
outer_k=options["outer_k"],
)
if info > 0:
raise InversionError("lgmres failed to converge after {} iterations".format(info))
assert info == 0
elif options["type"] == "least_squares_lsmr":
for i, VV in enumerate(V):
R[i], info, itn, _, _, _, _, _ = lsmr(
matrix,
VV.copy(i),
damp=options["damp"],
atol=options["atol"],
btol=options["btol"],
conlim=options["conlim"],
maxiter=options["maxiter"],
show=options["show"],
)
assert 0 <= info <= 7
if info == 7:
raise InversionError("lsmr failed to converge after {} iterations".format(itn))
elif options["type"] == "least_squares_lsqr":
for i, VV in enumerate(V):
R[i], info, itn, _, _, _, _, _, _, _ = lsqr(
matrix,
VV.copy(i),
damp=options["damp"],
atol=options["atol"],
btol=options["btol"],
conlim=options["conlim"],
iter_lim=options["iter_lim"],
show=options["show"],
)
assert 0 <= info <= 7
if info == 7:
raise InversionError("lsmr failed to converge after {} iterations".format(itn))
elif options["type"] == "pyamg":
if len(V) > 0:
V_iter = iter(enumerate(V))
R[0], ml = pyamg.solve(
matrix, next(V_iter)[1], tol=options["tol"], maxiter=options["maxiter"], return_solver=True
)
for i, VV in V_iter:
R[i] = pyamg.solve(matrix, VV, tol=options["tol"], maxiter=options["maxiter"], existing_solver=ml)
elif options["type"] == "pyamg-rs":
ml = pyamg.ruge_stuben_solver(
matrix,
strength=options["strength"],
CF=options["CF"],
presmoother=options["presmoother"],
postsmoother=options["postsmoother"],
max_levels=options["max_levels"],
max_coarse=options["max_coarse"],
coarse_solver=options["coarse_solver"],
)
for i, VV in enumerate(V):
R[i] = ml.solve(
VV, tol=options["tol"], maxiter=options["maxiter"], cycle=options["cycle"], accel=options["accel"]
)
elif options["type"] == "pyamg-sa":
ml = pyamg.smoothed_aggregation_solver(
matrix,
symmetry=options["symmetry"],
strength=options["strength"],
aggregate=options["aggregate"],
smooth=options["smooth"],
presmoother=options["presmoother"],
postsmoother=options["postsmoother"],
improve_candidates=options["improve_candidates"],
max_levels=options["max_levels"],
max_coarse=options["max_coarse"],
diagonal_dominance=options["diagonal_dominance"],
)
for i, VV in enumerate(V):
R[i] = ml.solve(
VV, tol=options["tol"], maxiter=options["maxiter"], cycle=options["cycle"], accel=options["accel"]
)
elif options["type"].startswith("generic") or options["type"].startswith("least_squares_generic"):
logger = getLogger("pymor.operators.numpy._apply_inverse")
logger.warn("You have selected a (potentially slow) generic solver for a NumPy matrix operator!")
from pymor.operators.numpy import NumpyMatrixOperator
from pymor.vectorarrays.numpy import NumpyVectorArray
return genericsolvers.apply_inverse(
NumpyMatrixOperator(matrix), NumpyVectorArray(V, copy=False), options=options
).data
else:
raise ValueError("Unknown solver type")
return R
# 25, 25, 25
# )
# mesh = pyfvm.meshTetra.meshTetra(vertices, cells)
# vertices, cells = meshzoo.rectangle.create_mesh(
# 0.0, 2.0,
# 0.0, 1.0,
# 401, 201,
# zigzag=True
# )
# mesh = pyfvm.meshTri.meshTri(vertices, cells)
import mshr
import dolfin
h = 2.5e-2
# cell_size = 2 * pi / num_boundary_points
c = mshr.Circle(dolfin.Point(0., 0., 0.), 1, int(2*pi / h))
# cell_size = 2 * bounding_box_radius / res
m = mshr.generate_mesh(c, 2.0 / h)
coords = m.coordinates()
coords = numpy.c_[coords, numpy.zeros(len(coords))]
mesh = pyfvm.meshTri.meshTri(coords, m.cells())
linear_system = pyfvm.discretize(Poisson(), mesh)
ml = pyamg.ruge_stuben_solver(linear_system.matrix)
x = ml.solve(linear_system.rhs, tol=1e-10)
# from scipy.sparse import linalg
# x = linalg.spsolve(linear_system.matrix, linear_system.rhs)
mesh.write('out.vtu', point_data={'x': x})
def generateQTBNQ(self): self.QTBNQ = self.QT*self.BNQ idx = list(self.QTBNQ.indices).index(0) self.QTBNQ.data[idx] *=2.0 self.ml = pyamg.ruge_stuben_solver(self.QTBNQ)
