def _select_step(self, shots, current_objective_value, gradient, iteration, objective_arguments, **kwargs):
"""Compute the Gauss-Newton update for a set of shots.
Gives the step s as a function of the gradient vector. Implemented as in p178 of Nocedal and Wright.
Parameters
----------
shots : list of pysit.Shot
List of Shots for which to compute.
grad : ndarray
Gradient vector.
iteration : int
Current time index.
"""
m0 = self.base_model
rhs = -1*gradient.asarray()
def matvec(x):
m1 = m0.perturbation(data=x)
return self.objective_function.apply_hessian(shots, self.base_model, x, **objective_arguments).data
A_shape = (len(rhs), len(rhs))
A = LinearOperator(shape=A_shape, matvec=matvec, dtype=gradient.dtype)
resid = []
# d, info = cg(A, rhs, maxiter=self.krylov_maxiter, residuals=resid)
d, info = gmres(A, rhs, maxiter=self.krylov_maxiter, residuals=resid)
d.shape = rhs.shape
direction = m0.perturbation(data=d)
if info < 0:
print "CG Failure"
if info == 0:
print "CG Converge"
if info > 0:
print "CG ran {0} iterations".format(info)
alpha0_kwargs = {'reset' : False}
if iteration == 0:
alpha0_kwargs = {'reset' : True}
alpha = self.select_alpha(shots, gradient, direction, objective_arguments,
current_objective_value=current_objective_value,
alpha0_kwargs=alpha0_kwargs, **kwargs)
self._print(' alpha {0}'.format(alpha))
self.store_history('alpha', iteration, alpha)
step = alpha * direction
return step
def smoother(A, x, b):
x[:] = (gmres(A,
b,
x0=x,
tol=tol,
maxiter=maxiter,
restrt=restrt,
M=M,
callback=callback,
residuals=residuals)[0]).reshape(x.shape)
def smoother(A, x, b):
x[:] = (
gmres(
A,
b,
x0=x,
tol=tol,
maxiter=maxiter,
restrt=restrt,
M=M,
callback=callback,
residuals=residuals)[0]).reshape(
x.shape)
def relax(A, x):
fn, kwargs = unpack_arg(prepostsmoother)
if fn == 'gauss_seidel':
gauss_seidel(A, x, np.zeros_like(x),
iterations=candidate_iters, sweep='symmetric')
elif fn == 'gauss_seidel_nr':
gauss_seidel_nr(A, x, np.zeros_like(x),
iterations=candidate_iters, sweep='symmetric')
elif fn == 'gauss_seidel_ne':
gauss_seidel_ne(A, x, np.zeros_like(x),
iterations=candidate_iters, sweep='symmetric')
elif fn == 'jacobi':
jacobi(A, x, np.zeros_like(x), iterations=1,
omega=1.0 / rho_D_inv_A(A))
elif fn == 'richardson':
polynomial(A, x, np.zeros_like(x), iterations=1,
coefficients=[1.0/approximate_spectral_radius(A)])
elif fn == 'gmres':
x[:] = (gmres(A, np.zeros_like(x), x0=x,
maxiter=candidate_iters)[0]).reshape(x.shape)
else:
raise TypeError('Unrecognized smoother')
Ub[ind1] = 1.-np.cos(np.pi*Y[ind1])
Ub[ind2] = 1.-np.cos(np.pi*Y[ind2])
rhs = - R.dot(Q.transpose().dot(Lgd).dot(Q).dot(Ub.reshape(-1,order='F')))
## Direct solve
t0 = time.time()
Lglu = sla.splu(Lg.tocsc())
usln1 = Lglu.solve(rhs)
t1 = time.time() - t0
print('SP LU: time=%f'%(t1))
## GMRES solve
res2 = []
t0 = time.time()
# usln2,info,niter = gmressolve(Lg,rhs,tol=1e-8)
usln2, info = gmres(Lg,rhs,x0=None,tol=1e-8,restrt=200, maxiter=5,residuals=res2)
t2 = time.time() - t0
print('GMRES: time=%f, iter=%d, (start, end) res=%e,%e'%(t2, len(res2), res2[0],res2[-1]))
## MG setup
mgschls = [('Divide',4),('Divide',2),('Subtract',2)]
mgschls = [('Divide',4)]
mgsmthl = ['RR','SC']
# mgsmthl = ['SC']
for sched in mgschls:
for smther in mgsmthl:
t0 = time.time()
semml = semml_md(p0,Ex,Ey,mxlv,initflg,X,Y,smoother=smther,schedule=sched,crs_solver='SPLU')
tmgsetup = time.time() - t0
print(semml, )
def general_setup_stage(ml, symmetry, candidate_iters, prepostsmoother, smooth,
eliminate_local, coarse_solver, work):
"""
Computes additional candidates and improvements
following Algorithm 4 in Brezina et al.
Parameters
----------
candidate_iters
number of test relaxation iterations
epsilon
minimum acceptable relaxation convergence factor
References
----------
.. [1] Brezina, Falgout, MacLachlan, Manteuffel, McCormick, and Ruge
"Adaptive Smoothed Aggregation (alphaSA) Multigrid"
SIAM Review Volume 47, Issue 2 (2005)
http://www.cs.umn.edu/~maclach/research/aSA2.pdf
"""
def make_bridge(T):
M, N = T.shape
K = T.blocksize[0]
bnnz = T.indptr[-1]
# the K+1 represents the new dof introduced by the new candidate. the
# bridge 'T' ignores this new dof and just maps zeros there
data = np.zeros((bnnz, K + 1, K), dtype=T.dtype)
data[:, :-1, :] = T.data
return bsr_matrix((data, T.indices, T.indptr),
shape=((K + 1) * int(M / K), N))
def expand_candidates(B_old, nodesize):
# insert a new dof that is always zero, to create NullDim+1 dofs per
# node in B
NullDim = B_old.shape[1]
nnodes = int(B_old.shape[0] / nodesize)
Bnew = np.zeros((nnodes, nodesize + 1, NullDim), dtype=B_old.dtype)
Bnew[:, :-1, :] = B_old.reshape(nnodes, nodesize, NullDim)
return Bnew.reshape(-1, NullDim)
levels = ml.levels
x = sp.rand(levels[0].A.shape[0], 1)
if levels[0].A.dtype.name.startswith('complex'):
x = x + 1.0j * sp.rand(levels[0].A.shape[0], 1)
b = np.zeros_like(x)
x = ml.solve(b,
x0=x,
tol=float(np.finfo(np.float).tiny),
maxiter=candidate_iters)
work[:] += ml.operator_complexity(
) * ml.levels[0].A.nnz * candidate_iters * 2
T0 = levels[0].T.copy()
# TEST FOR CONVERGENCE HERE
for i in range(len(ml.levels) - 2):
# alpha-SA paper does local elimination here, but after talking
# to Marian, its not clear that this helps things
# fn, kwargs = unpack_arg(eliminate_local)
# if fn == True:
# eliminate_local_candidates(x,levels[i].AggOp,levels[i].A,
# levels[i].T, **kwargs)
# add candidate to B
B = np.hstack((levels[i].B, x.reshape(-1, 1)))
# construct Ptent
T, R = fit_candidates(levels[i].AggOp, B)
levels[i].T = T
x = R[:, -1].reshape(-1, 1)
# smooth P
fn, kwargs = unpack_arg(smooth[i])
if fn == 'jacobi':
levels[i].P = jacobi_prolongation_smoother(levels[i].A, T,
levels[i].C, R,
**kwargs)
elif fn == 'richardson':
levels[i].P = richardson_prolongation_smoother(
levels[i].A, T, **kwargs)
elif fn == 'energy':
levels[i].P = energy_prolongation_smoother(levels[i].A, T,
levels[i].C, R, None,
(False, {}), **kwargs)
x = R[:, -1].reshape(-1, 1)
elif fn is None:
levels[i].P = T
else:
raise ValueError('unrecognized prolongation smoother method %s' %
str(fn))
# construct R
if symmetry == 'symmetric': # R should reflect A's structure
levels[i].R = levels[i].P.T.asformat(levels[i].P.format)
elif symmetry == 'hermitian':
levels[i].R = levels[i].P.H.asformat(levels[i].P.format)
# construct coarse A
levels[i + 1].A = levels[i].R * levels[i].A * levels[i].P
# construct bridging P
T_bridge = make_bridge(levels[i + 1].T)
R_bridge = levels[i + 2].B
# smooth bridging P
fn, kwargs = unpack_arg(smooth[i + 1])
if fn == 'jacobi':
levels[i + 1].P = jacobi_prolongation_smoother(
levels[i + 1].A, T_bridge, levels[i + 1].C, R_bridge, **kwargs)
elif fn == 'richardson':
levels[i + 1].P = richardson_prolongation_smoother(
levels[i + 1].A, T_bridge, **kwargs)
elif fn == 'energy':
levels[i + 1].P = energy_prolongation_smoother(
levels[i + 1].A, T_bridge, levels[i + 1].C, R_bridge, None,
(False, {}), **kwargs)
elif fn is None:
levels[i + 1].P = T_bridge
else:
raise ValueError('unrecognized prolongation smoother method %s' %
str(fn))
# construct the "bridging" R
if symmetry == 'symmetric': # R should reflect A's structure
levels[i + 1].R = levels[i + 1].P.T.asformat(levels[i +
1].P.format)
elif symmetry == 'hermitian':
levels[i + 1].R = levels[i + 1].P.H.asformat(levels[i +
1].P.format)
# run solver on candidate
solver = multilevel_solver(levels[i + 1:], coarse_solver=coarse_solver)
change_smoothers(solver,
presmoother=prepostsmoother,
postsmoother=prepostsmoother)
x = solver.solve(np.zeros_like(x),
x0=x,
tol=float(np.finfo(np.float).tiny),
maxiter=candidate_iters)
work[:] += 2 * solver.operator_complexity() * solver.levels[0].A.nnz *\
candidate_iters*2
# update values on next level
levels[i + 1].B = R[:, :-1].copy()
levels[i + 1].T = T_bridge
# note that we only use the x from the second coarsest level
fn, kwargs = unpack_arg(prepostsmoother)
for lvl in reversed(levels[:-2]):
x = lvl.P * x
work[:] += lvl.A.nnz * candidate_iters * 2
if fn == 'gauss_seidel':
# only relax at nonzeros, so as not to mess up any locally dropped
# candidates
indices = np.ravel(x).nonzero()[0]
gauss_seidel_indexed(lvl.A,
x,
np.zeros_like(x),
indices,
iterations=candidate_iters,
sweep='symmetric')
elif fn == 'gauss_seidel_ne':
gauss_seidel_ne(lvl.A,
x,
np.zeros_like(x),
iterations=candidate_iters,
sweep='symmetric')
elif fn == 'gauss_seidel_nr':
gauss_seidel_nr(lvl.A,
x,
np.zeros_like(x),
iterations=candidate_iters,
sweep='symmetric')
elif fn == 'jacobi':
jacobi(lvl.A,
x,
np.zeros_like(x),
iterations=1,
omega=1.0 / rho_D_inv_A(lvl.A))
elif fn == 'richardson':
polynomial(lvl.A,
x,
np.zeros_like(x),
iterations=1,
coefficients=[1.0 / approximate_spectral_radius(lvl.A)])
elif fn == 'gmres':
x[:] = (gmres(lvl.A,
np.zeros_like(x),
x0=x,
maxiter=candidate_iters)[0]).reshape(x.shape)
else:
raise TypeError('Unrecognized smoother')
# x will be dense again, so we have to drop locally again
elim, elim_kwargs = unpack_arg(eliminate_local)
if elim is True:
x = x / norm(x, 'inf')
eliminate_local_candidates(x, levels[0].AggOp, levels[0].A, T0,
**elim_kwargs)
return x.reshape(-1, 1)
def general_setup_stage(ml, symmetry, candidate_iters, prepostsmoother,
smooth, eliminate_local, coarse_solver, work):
"""Compute additional candidates and improvements following Algorithm 4 in Brezina et al.
Parameters
----------
candidate_iters
number of test relaxation iterations
epsilon
minimum acceptable relaxation convergence factor
References
----------
.. [1] Brezina, Falgout, MacLachlan, Manteuffel, McCormick, and Ruge
"Adaptive Smoothed Aggregation (alphaSA) Multigrid"
SIAM Review Volume 47, Issue 2 (2005)
http://www.cs.umn.edu/~maclach/research/aSA2.pdf
"""
def make_bridge(T):
M, N = T.shape
K = T.blocksize[0]
bnnz = T.indptr[-1]
# the K+1 represents the new dof introduced by the new candidate. the
# bridge 'T' ignores this new dof and just maps zeros there
data = np.zeros((bnnz, K+1, K), dtype=T.dtype)
data[:, :-1, :] = T.data
return bsr_matrix((data, T.indices, T.indptr),
shape=((K + 1) * int(M / K), N))
def expand_candidates(B_old, nodesize):
# insert a new dof that is always zero, to create NullDim+1 dofs per
# node in B
NullDim = B_old.shape[1]
nnodes = int(B_old.shape[0] / nodesize)
Bnew = np.zeros((nnodes, nodesize+1, NullDim), dtype=B_old.dtype)
Bnew[:, :-1, :] = B_old.reshape(nnodes, nodesize, NullDim)
return Bnew.reshape(-1, NullDim)
levels = ml.levels
x = sp.rand(levels[0].A.shape[0], 1)
if levels[0].A.dtype.name.startswith('complex'):
x = x + 1.0j*sp.rand(levels[0].A.shape[0], 1)
b = np.zeros_like(x)
x = ml.solve(b, x0=x, tol=float(np.finfo(np.float).tiny),
maxiter=candidate_iters)
work[:] += ml.operator_complexity()*ml.levels[0].A.nnz*candidate_iters*2
T0 = levels[0].T.copy()
# TEST FOR CONVERGENCE HERE
for i in range(len(ml.levels) - 2):
# alpha-SA paper does local elimination here, but after talking
# to Marian, its not clear that this helps things
# fn, kwargs = unpack_arg(eliminate_local)
# if fn == True:
# eliminate_local_candidates(x,levels[i].AggOp,levels[i].A,
# levels[i].T, **kwargs)
# add candidate to B
B = np.hstack((levels[i].B, x.reshape(-1, 1)))
# construct Ptent
T, R = fit_candidates(levels[i].AggOp, B)
levels[i].T = T
x = R[:, -1].reshape(-1, 1)
# smooth P
fn, kwargs = unpack_arg(smooth[i])
if fn == 'jacobi':
levels[i].P = jacobi_prolongation_smoother(levels[i].A, T,
levels[i].C, R,
**kwargs)
elif fn == 'richardson':
levels[i].P = richardson_prolongation_smoother(levels[i].A, T,
**kwargs)
elif fn == 'energy':
levels[i].P = energy_prolongation_smoother(levels[i].A, T,
levels[i].C, R, None,
(False, {}), **kwargs)
x = R[:, -1].reshape(-1, 1)
elif fn is None:
levels[i].P = T
else:
raise ValueError('unrecognized prolongation smoother method %s' %
str(fn))
# construct R
if symmetry == 'symmetric': # R should reflect A's structure
levels[i].R = levels[i].P.T.asformat(levels[i].P.format)
elif symmetry == 'hermitian':
levels[i].R = levels[i].P.H.asformat(levels[i].P.format)
# construct coarse A
levels[i+1].A = levels[i].R * levels[i].A * levels[i].P
# construct bridging P
T_bridge = make_bridge(levels[i+1].T)
R_bridge = levels[i+2].B
# smooth bridging P
fn, kwargs = unpack_arg(smooth[i+1])
if fn == 'jacobi':
levels[i+1].P = jacobi_prolongation_smoother(levels[i+1].A,
T_bridge,
levels[i+1].C,
R_bridge, **kwargs)
elif fn == 'richardson':
levels[i+1].P = richardson_prolongation_smoother(levels[i+1].A,
T_bridge,
**kwargs)
elif fn == 'energy':
levels[i+1].P = energy_prolongation_smoother(levels[i+1].A,
T_bridge,
levels[i+1].C,
R_bridge, None,
(False, {}), **kwargs)
elif fn is None:
levels[i+1].P = T_bridge
else:
raise ValueError('unrecognized prolongation smoother method %s' %
str(fn))
# construct the "bridging" R
if symmetry == 'symmetric': # R should reflect A's structure
levels[i+1].R = levels[i+1].P.T.asformat(levels[i+1].P.format)
elif symmetry == 'hermitian':
levels[i+1].R = levels[i+1].P.H.asformat(levels[i+1].P.format)
# run solver on candidate
solver = multilevel_solver(levels[i+1:], coarse_solver=coarse_solver)
change_smoothers(solver, presmoother=prepostsmoother,
postsmoother=prepostsmoother)
x = solver.solve(np.zeros_like(x), x0=x,
tol=float(np.finfo(np.float).tiny),
maxiter=candidate_iters)
work[:] += 2 * solver.operator_complexity() * solver.levels[0].A.nnz *\
candidate_iters*2
# update values on next level
levels[i+1].B = R[:, :-1].copy()
levels[i+1].T = T_bridge
# note that we only use the x from the second coarsest level
fn, kwargs = unpack_arg(prepostsmoother)
for lvl in reversed(levels[:-2]):
x = lvl.P * x
work[:] += lvl.A.nnz*candidate_iters*2
if fn == 'gauss_seidel':
# only relax at nonzeros, so as not to mess up any locally dropped
# candidates
indices = np.ravel(x).nonzero()[0]
gauss_seidel_indexed(lvl.A, x, np.zeros_like(x), indices,
iterations=candidate_iters, sweep='symmetric')
elif fn == 'gauss_seidel_ne':
gauss_seidel_ne(lvl.A, x, np.zeros_like(x),
iterations=candidate_iters, sweep='symmetric')
elif fn == 'gauss_seidel_nr':
gauss_seidel_nr(lvl.A, x, np.zeros_like(x),
iterations=candidate_iters, sweep='symmetric')
elif fn == 'jacobi':
jacobi(lvl.A, x, np.zeros_like(x), iterations=1,
omega=1.0 / rho_D_inv_A(lvl.A))
elif fn == 'richardson':
polynomial(lvl.A, x, np.zeros_like(x), iterations=1,
coefficients=[1.0/approximate_spectral_radius(lvl.A)])
elif fn == 'gmres':
x[:] = (gmres(lvl.A, np.zeros_like(x), x0=x,
maxiter=candidate_iters)[0]).reshape(x.shape)
else:
raise TypeError('Unrecognized smoother')
# x will be dense again, so we have to drop locally again
elim, elim_kwargs = unpack_arg(eliminate_local)
if elim is True:
x = x/norm(x, 'inf')
eliminate_local_candidates(x, levels[0].AggOp, levels[0].A, T0,
**elim_kwargs)
return x.reshape(-1, 1)
def _select_step(self, shots, current_objective_value, gradient, iteration,
objective_arguments, **kwargs):
"""Compute the Gauss-Newton update for a set of shots.
Gives the step s as a function of the gradient vector. Implemented as in p178 of Nocedal and Wright.
Parameters
----------
shots : list of pysit.Shot
List of Shots for which to compute.
grad : ndarray
Gradient vector.
iteration : int
Current time index.
"""
m0 = self.base_model
rhs = -1 * gradient.asarray()
def matvec(x):
m1 = m0.perturbation(data=x)
return self.objective_function.apply_hessian(
shots, self.base_model, x, **objective_arguments).data
A_shape = (len(rhs), len(rhs))
A = LinearOperator(shape=A_shape, matvec=matvec, dtype=gradient.dtype)
resid = []
# d, info = cg(A, rhs, maxiter=self.krylov_maxiter, residuals=resid)
d, info = gmres(A, rhs, maxiter=self.krylov_maxiter, residuals=resid)
d.shape = rhs.shape
direction = m0.perturbation(data=d)
if info < 0:
print "CG Failure"
if info == 0:
print "CG Converge"
if info > 0:
print "CG ran {0} iterations".format(info)
alpha0_kwargs = {'reset': False}
if iteration == 0:
alpha0_kwargs = {'reset': True}
alpha = self.select_alpha(
shots,
gradient,
direction,
objective_arguments,
current_objective_value=current_objective_value,
alpha0_kwargs=alpha0_kwargs,
**kwargs)
self._print(' alpha {0}'.format(alpha))
self.store_history('alpha', iteration, alpha)
step = alpha * direction
return step
bb = np.r_[b.array(), np.zeros(len(points))]
# Preconditioner diag(ML(A), I)
ml = smoothed_aggregation_solver(A_)
PA = ml.aspreconditioner()
def Pmat_vec(x):
yA = PA.matvec(x[:V.dim()])
# Identity on part of x from point sources
return np.r_[yA, x[V.dim():]]
P = LinearOperator(shape=AA.shape, matvec=Pmat_vec)
residuals = []
# Run the iterations
x, failed = gmres(AA, bb, M=P,
tol=1e-10, maxiter=1000, residuals=residuals)
uh = Function(V)
uh.vector()[:] = x[:V.dim()]
# Solve
# xx = spsolve(AA, bb)
# Get the coefs of !multipliers
# U = xx[:V.dim()]
# uh = Function(V)
# uh.vector()[:] = U
print len(residuals)-1, residuals[-1]
plot(uh)
interactive()
def _select_step(self, shots, current_objective_value, gradient, iteration,
objective_arguments, **kwargs):
"""Compute the update for a set of shots.
Gives the step s as a function of the gradient vector. Implemented as in p178 of Nocedal and Wright.
Parameters
----------
shots : list of pysit.Shot
List of Shots for which to compute.
grad : ndarray
Gradient vector.
i : int
Current time index.
objective_arguments['frequencies'] gives a list with the frequencies
nrealizations: should be passed on through kwargs
"""
if type(self.objective_function) != FrequencyLeastSquares:
raise NotImplementedError(
"Stochastic Gauss Newton is only implemented for a frequency least squares objective function"
)
rhs = -1 * gradient.asarray()
stochastic_hessian = self._get_stoch_hessian(
objective_arguments['frequencies'], shots, **kwargs)
#stochastic_hessian = stochastic_hessian / 900.0
m = self.solver.mesh
dx = m.x.delta
dz = m.z.delta
#Need to multiply stochastic hessian by (dx*dz) to get values at same order of magnitude as you would get using adjoint method to calculate H_appr
#This will decrease entries in the result of 'd'
#Difference probably due to difference in definition norm. volume integral vs sum etc...
#If I don't multiply, the step I get a 'step' with entries that seem to be too large. After 10 reductions of alpha, norm is still decreasing each time alpha is reduced.
#final result does not seem to change since value of alpha will adapt accordingly with different order of magnitude of 'd'
#stochastic_hessian = stochastic_hessian * float(dx*dz)
resid = []
if self.use_diag_preconditioner:
diagonal = stochastic_hessian.diagonal()
idiagonal = 1. / diagonal
Pinv = dia_matrix((idiagonal, 0), shape=stochastic_hessian.shape)
d, info = gmres(stochastic_hessian,
rhs,
maxiter=self.krylov_maxiter,
residuals=resid,
M=Pinv)
else:
d, info = gmres(stochastic_hessian,
rhs,
maxiter=self.krylov_maxiter,
residuals=resid)
d.shape = rhs.shape
direction = self.solver.model_parameters.perturbation()
direction = direction.without_padding()
direction.data = d
# d, info = cg(A, rhs, maxiter=self.krylov_maxiter, residuals=resid)
#direction = mo.perturbation(data=d)
if info < 0:
print "CG Failure"
if info == 0:
print "CG Converge"
if info > 0:
print "CG ran {0} iterations".format(info)
alpha0_kwargs = {
'reset': False
} #COPIED FROM LBFGS THINK ABOUT THIS LATER
if iteration == 0:
alpha0_kwargs = {'reset': True}
alpha = self.select_alpha(
shots,
gradient,
direction,
objective_arguments,
current_objective_value=current_objective_value,
alpha0_kwargs=alpha0_kwargs,
**kwargs)
self._print(' alpha {0}'.format(alpha))
self.store_history('alpha', iteration, alpha)
step = alpha * direction
if kwargs.has_key('printfigs'):
if kwargs['printfigs']: #if it is True
import numpy as np
import matplotlib.pyplot as plt
nx = m.x.n
nz = m.z.n
gradientplot = np.reshape(gradient.data, (nz, nx), 'F')
directionplot = np.reshape(direction.data, (nz, nx), 'F')
stepplot = np.reshape(step.data, (nz, nz), 'F')
plt.ion()
f1 = plt.figure(1)
plt.imshow(gradientplot, interpolation='nearest')
plt.colorbar()
plt.title('gradient')
f2 = plt.figure(2)
plt.imshow(directionplot, interpolation='nearest')
plt.colorbar()
plt.title('Direction after applying Hessian')
f3 = plt.figure(3)
plt.imshow(stepplot, interpolation='nearest')
plt.colorbar()
plt.title('Step after applying Hessian and doing linesearch')
f4 = plt.figure(4)
plt.imshow(stochastic_hessian.todense(),
interpolation='nearest')
plt.colorbar()
plt.title("incomplete stochastic hessian")
wait = raw_input("PRESS ENTER TO CONTINUE.")
plt.close(f1)
plt.close(f2)
plt.close(f3)
plt.close(f4)
return step
