def test_minres_deflation(self):
# Create sparse symmetric problem.
num_unknowns = 100
A = self._create_sparse_herm_indef_matrix(num_unknowns)
rhs = np.ones( (num_unknowns,1) )
x0 = np.zeros( (num_unknowns,1) )
# get projection
from scipy.sparse.linalg import eigs
num_vecs = 6
D, W = eigs(A, k=num_vecs, v0=np.ones((num_unknowns,1)))
# Uncomment next lines to perturb W
#W = W + 1.e-10*np.random.rand(W.shape[0], W.shape[1])
#from scipy.linalg import qr
#W, R = qr(W, mode='economic')
AW = nm._apply(A, W)
P, x0new = nm.get_projection( W, AW, rhs, x0 )
# Solve using MINRES.
tol = 1.0e-9
out = nm.minres( A, rhs, x0new, Mr=P, tol=tol, maxiter=num_unknowns-num_vecs, full_reortho=True, return_basis=True )
# TODO: move to new unit test
o = nm.get_p_ritz( W, AW, out['Vfull'], out['Hfull'] )
ritz_vals, ritz_vecs = nm.get_p_ritz( W, AW, out['Vfull'], out['Hfull'] )
# Make sure the method converged.
self.assertEqual(out['info'], 0)
# Check the residual.
res = rhs - A * out['xk']
self.assertAlmostEqual( np.linalg.norm(res)/np.linalg.norm(rhs), 0.0, delta=tol )
def test_get_projection(self):
N = 100
A = scipy.sparse.spdiags(range(1, N + 1), [0], N, N)
b = np.ones((N, 1))
x0 = np.ones((N, 1))
# 'last' 2 eigenvectors
W = np.zeros((N, 2))
W[-2, 0] = 1.
W[-1, 1] = 1
AW = A * W
P, x0new = nm.get_projection(W, AW, b, x0)
# Check A*(P*I) against exact A*P
AP = A * (P * np.eye(N))
AP_exact = scipy.sparse.spdiags(
list(range(1, N - 1)) + [0, 0], [0], N, N)
self.assertAlmostEqual(np.linalg.norm(AP - AP_exact), 0.0, delta=1e-14)
# Check x0new
x0new_exact = np.ones((N, 1))
x0new_exact[N - 2:N, 0] = [1. / (N - 1), 1. / N]
self.assertAlmostEqual(np.linalg.norm(x0new - x0new_exact),
0.0,
delta=1e-14)
def test_get_ritz(self):
N = 10
A = scipy.sparse.spdiags(range(1, N + 1), [0], N, N)
b = np.ones((N, 1))
x0 = np.ones((N, 1))
# 'last' 2 eigenvectors
W = np.zeros((N, 2))
W[-2, 0] = 1.
W[-1, 1] = 1
AW = A * W
# Get the projection
P, x0new = nm.get_projection(W, AW, b, x0)
# Run MINRES (we are only interested in the Lanczos basis and tridiag matrix)
out = nm.minres(A,
b,
x0new,
Mr=P,
tol=1e-14,
maxiter=11,
full_reortho=True,
return_basis=True)
# Get Ritz pairs
ritz_vals, ritz_vecs = nm.get_p_ritz(W, AW, out['Vfull'], out['Hfull'])
# Check Ritz pair residuals
#ritz_res_exact = A*ritz_vecs - np.dot(ritz_vecs,np.diag(ritz_vals))
#for i in range(0,len(ritz_vals)):
#norm_ritz_res_exact = np.linalg.norm(ritz_res_exact[:,i])
#self.assertAlmostEqual( abs(norm_ritz_res[i] - norm_ritz_res_exact), 0.0, delta=1e-13 )
# Check if Ritz values / vectors corresponding to W are still there ;)
order = np.argsort(ritz_vals)
# 'last' eigenvalue
self.assertAlmostEqual(abs(ritz_vals[order[-1]] - N), 0.0, delta=1e-13)
self.assertAlmostEqual(abs(ritz_vals[order[-2]] - (N - 1)),
0.0,
delta=1e-13)
# now the eigenvectors
self.assertAlmostEqual(abs(nm._ipstd(ritz_vecs[:, order[-1]], W[:,
1])),
1.0,
delta=1e-13)
self.assertAlmostEqual(abs(nm._ipstd(ritz_vecs[:, order[-2]], W[:,
0])),
1.0,
delta=1e-13)
def test_minres_deflation(self):
# Create sparse symmetric problem.
num_unknowns = 100
A = self._create_sparse_herm_indef_matrix(num_unknowns)
rhs = np.ones((num_unknowns, 1))
x0 = np.zeros((num_unknowns, 1))
# get projection
from scipy.sparse.linalg import eigs
num_vecs = 6
D, W = eigs(A, k=num_vecs, v0=np.ones((num_unknowns, 1)))
# Uncomment next lines to perturb W
#W = W + 1.e-10*np.random.rand(W.shape[0], W.shape[1])
#from scipy.linalg import qr
#W, R = qr(W, mode='economic')
AW = nm._apply(A, W)
P, x0new = nm.get_projection(W, AW, rhs, x0)
# Solve using MINRES.
tol = 1.0e-9
out = nm.minres(A,
rhs,
x0new,
Mr=P,
tol=tol,
maxiter=num_unknowns - num_vecs,
full_reortho=True,
return_basis=True)
# TODO: move to new unit test
o = nm.get_p_ritz(W, AW, out['Vfull'], out['Hfull'])
ritz_vals, ritz_vecs = nm.get_p_ritz(W, AW, out['Vfull'], out['Hfull'])
# Make sure the method converged.
self.assertEqual(out['info'], 0)
# Check the residual.
res = rhs - A * out['xk']
self.assertAlmostEqual(np.linalg.norm(res) / np.linalg.norm(rhs),
0.0,
delta=tol)
def test_get_ritz(self):
N = 10
A = scipy.sparse.spdiags( range(1,N+1), [0], N, N)
b = np.ones( (N,1) )
x0 = np.ones( (N,1) )
# 'last' 2 eigenvectors
W = np.zeros( (N,2) )
W[-2,0]=1.
W[-1,1]=1
AW = A*W
# Get the projection
P, x0new = nm.get_projection(W, AW, b, x0)
# Run MINRES (we are only interested in the Lanczos basis and tridiag matrix)
out = nm.minres(A, b, x0new, Mr=P, tol=1e-14, maxiter=11,
full_reortho=True,
return_basis=True
)
# Get Ritz pairs
ritz_vals, ritz_vecs = nm.get_p_ritz(W, AW,
out['Vfull'],
out['Hfull']
)
# Check Ritz pair residuals
#ritz_res_exact = A*ritz_vecs - np.dot(ritz_vecs,np.diag(ritz_vals))
#for i in range(0,len(ritz_vals)):
#norm_ritz_res_exact = np.linalg.norm(ritz_res_exact[:,i])
#self.assertAlmostEqual( abs(norm_ritz_res[i] - norm_ritz_res_exact), 0.0, delta=1e-13 )
# Check if Ritz values / vectors corresponding to W are still there ;)
order = np.argsort(ritz_vals)
# 'last' eigenvalue
self.assertAlmostEqual( abs(ritz_vals[order[-1]] - N), 0.0, delta=1e-13 )
self.assertAlmostEqual( abs(ritz_vals[order[-2]] - (N-1)), 0.0, delta=1e-13 )
# now the eigenvectors
self.assertAlmostEqual( abs(nm._ipstd(ritz_vecs[:,order[-1]],W[:,1])), 1.0, delta=1e-13 )
self.assertAlmostEqual( abs(nm._ipstd(ritz_vecs[:,order[-2]],W[:,0])), 1.0, delta=1e-13 )
def test_get_projection(self):
N = 100
A = scipy.sparse.spdiags( range(1,N+1), [0], N, N)
b = np.ones( (N,1) )
x0 = np.ones( (N,1) )
# 'last' 2 eigenvectors
W = np.zeros( (N,2) )
W[-2,0]=1.
W[-1,1]=1
AW = A*W
P, x0new = nm.get_projection(W, AW, b, x0)
# Check A*(P*I) against exact A*P
AP = A*(P*np.eye(N))
AP_exact = scipy.sparse.spdiags(list(range(1,N-1))+[0,0], [0], N, N)
self.assertAlmostEqual( np.linalg.norm(AP-AP_exact), 0.0, delta=1e-14 )
# Check x0new
x0new_exact = np.ones( (N,1) )
x0new_exact[N-2:N,0] = [1./(N-1),1./N]
self.assertAlmostEqual( np.linalg.norm(x0new-x0new_exact), 0.0, delta=1e-14 )
def _solve_system(modeleval, filename, timestep, mu, args):
# read the mesh
print('Reading current psi...',)
start = time.time()
mesh, point_data, field_data = voropy.read(filename,
timestep=timestep
)
total = time.time() - start
print('done (%gs).' % total)
num_coords = len( mesh.node_coords )
# --------------------------------------------------------------------------
# set psi at which to create the Jacobian
print('Creating initial guess and right-hand side...',)
start = time.time()
current_psi = (point_data['psi'][:,0] + 1j * point_data['psi'][:,1]).reshape(-1,1)
# Perturb a bit.
eps = 1.0e-10
perturbation = eps * (np.random.rand(num_coords) + 1j * np.random.rand(num_coords))
current_psi += perturbation.reshape(current_psi.shape)
if args.bordering:
phi0 = np.zeros((num_coords+1,1), dtype=complex)
# right hand side
x = np.empty((num_coords+1,1), dtype=complex)
x[0:num_coords] = current_psi
x[num_coords] = 0.0
else:
# create right hand side and initial guess
phi0 = np.zeros( (num_coords,1), dtype=complex )
x = current_psi
# right hand side
#rhs = np.ones( (num_coords,1), dtype=complex )
#rhs = np.empty( num_coords, dtype = complex )
#radius = np.random.rand( num_coords )
#arg = np.random.rand( num_coords ) * 2.0 * cmath.pi
#for k in range( num_coords ):
#rhs[ k ] = cmath.rect(radius[k], arg[k])
rhs = modeleval.compute_f(x=x, mu=mu, g=1.0)
end = time.time()
print('done. (%gs)' % (end - start))
print('||rhs|| = %g' % np.sqrt(modeleval.inner_product(rhs, rhs)))
# --------------------------------------------------------------------------
# create the linear operator
print('Getting Jacobian...',)
start_time = time.clock()
jacobian = modeleval.get_jacobian(x=x, mu=mu, g=1.0)
end_time = time.clock()
print('done. (%gs)' % (end_time - start_time))
# create precondictioner object
print('Getting preconditioner...',)
start_time = time.clock()
prec = modeleval.get_preconditioner_inverse(x=x, mu=mu, g=1.0)
end_time = time.clock()
print('done. (%gs)' % (end_time - start_time))
# --------------------------------------------------------------------------
# Get reference solution
#print 'Get reference solution (dim = %d)...' % (2*num_coords),
#start_time = time.clock()
#ref_sol, info, relresvec, errorvec = nm.minres_wrap( jacobian, rhs,
#x0 = phi0,
#tol = 1.0e-14,
#M = prec,
#inner_product = modeleval.inner_product,
#explicit_residual = True
#)
#end_time = time.clock()
#if info == 0:
#print 'success!',
#else:
#print 'no convergence.',
#print ' (', end_time - start_time, 's,', len(relresvec)-1 ,' iters).'
if args.use_deflation:
W = 1j * x
AW = jacobian * W
P, x0new = nm.get_projection(W, AW, rhs, phi0,
inner_product = modeleval.inner_product
)
else:
#AW = np.zeros((len(current_psi),1), dtype=np.complex)
P = None
x0new = phi0
if args.krylov_method == 'cg':
lin_solve = nm.cg
elif args.krylov_method == 'minres':
lin_solve = nm.minres
#elif args.krylov_method == 'minresfo':
#lin_solve = nm.minres
#lin_solve_args.update({'full_reortho': True})
elif args.krylov_method == 'gmres':
lin_solve = nm.gmres
else:
raise ValueError('Unknown Krylov solver ''%s''.' % args.krylov_method)
print('Solving the system (len(x) = %d, bordering: %r)...' % (len(x), args.bordering),)
start_time = time.clock()
timer = False
out = lin_solve(jacobian, rhs,
x0new,
tol = args.tolerance,
Mr = P,
M = prec,
#maxiter = 2*num_coords,
maxiter = 500,
inner_product = modeleval.inner_product,
explicit_residual = True,
#timer=timer
#exact_solution = ref_sol
)
end_time = time.clock()
print('done. (%gs)' % (end_time - start_time))
print('(%d,%d)' % (2*num_coords, len(out['relresvec'])-1))
# compute actual residual
#res = rhs - jacobian * out['xk']
#print '||b-Ax|| = %g' % np.sqrt(modeleval.inner_product(res, res))
if timer:
# pretty-print timings
print(' '*22 + 'sum'.rjust(14) + 'mean'.rjust(14) + 'min'.rjust(14) + 'std dev'.rjust(14))
for key, item in out['times'].items():
print('\'%s\': %12g %12g %12g %12g'
% (key.ljust(20), item.sum(), item.mean(), item.min(), item.std()))
# Get the number of MG cycles.
# 'modeleval.num_cycles' contains the number of MG cycles executed
# for all AMG calls run.
# In nm.minres, two calls to the precondictioner are done prior to the
# actual iteration for the normalization of the residuals.
# With explicit_residual=True, *two* calls to the preconditioner are done
# in each iteration.
# What we would like to have here is the number of V-cycles done per loop
# when explicit_residual=False. Also, forget about the precondictioner
# calls for the initialization.
# Hence, cut of the first two and replace it by 0, and out of the
# remainder take every other one.
#nc = [0] + modeleval.tot_amg_cycles[2::2]
#nc_cumsum = np.cumsum(nc)
#pp.semilogy(nc_cumsum, out['relresvec'], color='0.0')
#pp.show()
#import matplotlib2tikz
#matplotlib2tikz.save('cycle10.tex')
#matplotlib2tikz.save('inf.tex')
return out['relresvec']
def _solve_system(modeleval, filename, timestep, mu, args):
# read the mesh
print(("Reading current psi...", ))
start = time.time()
mesh, point_data, field_data = meshplex.read(filename, timestep=timestep)
total = time.time() - start
print(("done (%gs)." % total))
num_coords = len(mesh.node_coords)
# set psi at which to create the Jacobian
print(("Creating initial guess and right-hand side...", ))
start = time.time()
current_psi = (point_data["psi"][:, 0] +
1j * point_data["psi"][:, 1]).reshape(-1, 1)
# Perturb a bit.
eps = 1.0e-10
perturbation = eps * (np.random.rand(num_coords) +
1j * np.random.rand(num_coords))
current_psi += perturbation.reshape(current_psi.shape)
if args.bordering:
phi0 = np.zeros((num_coords + 1, 1), dtype=complex)
# right hand side
x = np.empty((num_coords + 1, 1), dtype=complex)
x[0:num_coords] = current_psi
x[num_coords] = 0.0
else:
# create right hand side and initial guess
phi0 = np.zeros((num_coords, 1), dtype=complex)
x = current_psi
# right hand side
# rhs = np.ones( (num_coords,1), dtype=complex )
# rhs = np.empty( num_coords, dtype = complex )
# radius = np.random.rand( num_coords )
# arg = np.random.rand( num_coords ) * 2.0 * cmath.pi
# for k in range( num_coords ):
# rhs[ k ] = cmath.rect(radius[k], arg[k])
rhs = modeleval.compute_f(x=x, mu=mu, g=1.0)
end = time.time()
print(("done. (%gs)" % (end - start)))
print(("||rhs|| = %g" % np.sqrt(modeleval.inner_product(rhs, rhs))))
# create the linear operator
print(("Getting Jacobian...", ))
start_time = time.clock()
jacobian = modeleval.get_jacobian(x=x, mu=mu, g=1.0)
end_time = time.clock()
print(("done. (%gs)" % (end_time - start_time)))
# create precondictioner object
print(("Getting preconditioner...", ))
start_time = time.clock()
prec = modeleval.get_preconditioner_inverse(x=x, mu=mu, g=1.0)
end_time = time.clock()
print(("done. (%gs)" % (end_time - start_time)))
# Get reference solution
# print 'Get reference solution (dim = %d)...' % (2*num_coords),
# start_time = time.clock()
# ref_sol, info, relresvec, errorvec = nm.minres_wrap( jacobian, rhs,
# x0 = phi0,
# tol = 1.0e-14,
# M = prec,
# inner_product = modeleval.inner_product,
# explicit_residual = True
# )
# end_time = time.clock()
# if info == 0:
# print 'success!',
# else:
# print 'no convergence.',
# print ' (', end_time - start_time, 's,', len(relresvec)-1 ,' iters).'
if args.use_deflation:
W = 1j * x
AW = jacobian * W
P, x0new = nm.get_projection(W,
AW,
rhs,
phi0,
inner_product=modeleval.inner_product)
else:
# AW = np.zeros((len(current_psi),1), dtype=np.complex)
P = None
x0new = phi0
if args.krylov_method == "cg":
lin_solve = nm.cg
elif args.krylov_method == "minres":
lin_solve = nm.minres
# elif args.krylov_method == 'minresfo':
# lin_solve = nm.minres
# lin_solve_args.update({'full_reortho': True})
elif args.krylov_method == "gmres":
lin_solve = nm.gmres
else:
raise ValueError("Unknown Krylov solver "
"%s"
"." % args.krylov_method)
print(("Solving the system (len(x) = %d, bordering: %r)..." %
(len(x), args.bordering), ))
start_time = time.clock()
timer = False
out = lin_solve(
jacobian,
rhs,
x0new,
tol=args.tolerance,
Mr=P,
M=prec,
# maxiter = 2*num_coords,
maxiter=500,
inner_product=modeleval.inner_product,
explicit_residual=True,
# timer=timer
# exact_solution = ref_sol
)
end_time = time.clock()
print(("done. (%gs)" % (end_time - start_time)))
print(("(%d,%d)" % (2 * num_coords, len(out["relresvec"]) - 1)))
# compute actual residual
# res = rhs - jacobian * out['xk']
# print '||b-Ax|| = %g' % np.sqrt(modeleval.inner_product(res, res))
if timer:
# pretty-print timings
print((" " * 22 + "sum".rjust(14) + "mean".rjust(14) +
"min".rjust(14) + "std dev".rjust(14)))
for key, item in list(out["times"].items()):
print(("'%s': %12g %12g %12g %12g" %
(key.ljust(20), item.sum(), item.mean(), item.min(),
item.std())))
# Get the number of MG cycles.
# 'modeleval.num_cycles' contains the number of MG cycles executed
# for all AMG calls run.
# In nm.minres, two calls to the precondictioner are done prior to the
# actual iteration for the normalization of the residuals.
# With explicit_residual=True, *two* calls to the preconditioner are done
# in each iteration.
# What we would like to have here is the number of V-cycles done per loop
# when explicit_residual=False. Also, forget about the precondictioner
# calls for the initialization.
# Hence, cut of the first two and replace it by 0, and out of the
# remainder take every other one.
# nc = [0] + modeleval.tot_amg_cycles[2::2]
# nc_cumsum = np.cumsum(nc)
# pp.semilogy(nc_cumsum, out['relresvec'], color='0.0')
# pp.show()
# import matplotlib2tikz
# matplotlib2tikz.save('cycle10.tex')
# matplotlib2tikz.save('inf.tex')
return out["relresvec"]
