import time
import numpy as nm
import numpy.linalg as nla
import scipy.sparse as sp
from sfepy.base.base import output, get_default, debug, Struct
from sfepy.base.log import get_logging_conf
from sfepy.solvers.solvers import make_get_conf, NonlinearSolver
from sfepy.solvers.nls import Newton, conv_test
from sfepy.linalg import compose_sparse
[docs]class SemismoothNewton(Newton):
r"""
The semi-smooth Newton method for solving problems of the following
structure:
.. math::
\begin{split}
& F(y) = 0 \\
& A(y) \ge 0 \;,\ B(y) \ge 0 \;,\ \langle A(y), B(y) \rangle = 0
\end{split}
The function :math:`F(y)` represents the smooth part of the problem.
Regular step: :math:`y \leftarrow y - J(y)^{-1} \Phi(y)`
Steepest descent step: :math:`y \leftarrow y - \beta J(y) \Phi(y)`
Notes
-----
Although fun_smooth_grad() computes the gradient of the smooth part only,
it should return the global matrix, where the non-smooth part is
uninitialized, but pre-allocated.
"""
name = 'nls.semismooth_newton'
_colors = {'regular' : 'g', 'steepest_descent' : 'k'}
@staticmethod
[docs] def process_conf(conf, kwargs):
"""
Missing items are set to default values.
Example configuration, all items::
solver_1 = {
'name' : 'semismooth_newton',
'kind' : 'nls.semismooth_newton',
'semismooth' : True,
'i_max' : 10,
'eps_a' : 1e-8,
'eps_r' : 1e-2,
'macheps' : 1e-16,
'lin_red' : 1e-2, # Linear system error < (eps_a * lin_red).
'ls_red_reg' : 0.1,
'ls_red_alt' : 0.01,
'ls_red_warp' : 0.001,
'ls_on' : 0.9,
'ls_min' : 1e-10,
'log' : {'plot' : 'convergence.png'},
}
"""
get = make_get_conf(conf, kwargs)
common = NonlinearSolver.process_conf(conf)
log = get_logging_conf(conf)
log = Struct(name='log_conf', **log)
is_any_log = (log.text is not None) or (log.plot is not None)
return Struct(semismooth=get('semismooth', True),
i_max=get('i_max', 1),
eps_a=get('eps_a', 1e-10),
eps_r=get('eps_r', 1.0),
macheps=get('macheps', nm.finfo(nm.float64).eps),
lin_red=get('lin_red', 1.0),
ls_red=get('ls_red', 0.1),
ls_red_warp=get('ls_red_warp', 0.001),
ls_on=get('ls_on', 0.99999),
ls_min=get('ls_min', 1e-5),
log=log,
is_any_log=is_any_log) + common
def __call__(self, vec_x0, conf=None, fun_smooth=None, fun_smooth_grad=None,
fun_a=None, fun_a_grad=None, fun_b=None, fun_b_grad=None,
lin_solver=None, status=None):
conf = get_default(conf, self.conf)
fun_smooth = get_default(fun_smooth, self.fun_smooth)
fun_smooth_grad = get_default(fun_smooth_grad, self.fun_smooth_grad)
fun_a = get_default(fun_a, self.fun_a)
fun_a_grad = get_default(fun_a_grad, self.fun_a_grad)
fun_b = get_default(fun_b, self.fun_b)
fun_b_grad = get_default(fun_b_grad, self.fun_b_grad)
lin_solver = get_default(lin_solver, self.lin_solver)
status = get_default(status, self.status)
time_stats = {}
vec_x = vec_x0.copy()
vec_x_last = vec_x0.copy()
vec_dx = None
if self.log is not None:
self.log.plot_vlines(color='r', linewidth=1.0)
err0 = -1.0
err_last = -1.0
it = 0
step_mode = 'regular'
r_last = None
reuse_matrix = False
while 1:
ls = 1.0
vec_dx0 = vec_dx;
i_ls = 0
while 1:
tt = time.clock()
try:
vec_smooth_r = fun_smooth(vec_x)
vec_a_r = fun_a(vec_x)
vec_b_r = fun_b(vec_x)
except ValueError:
vec_smooth_r = vec_semismooth_r = None
if (it == 0) or (ls < conf.ls_min):
output('giving up!')
raise
else:
ok = False
else:
if conf.semismooth:
# Semi-smooth equation.
vec_semismooth_r = nm.sqrt(vec_a_r**2.0 + vec_b_r**2.0) \
- (vec_a_r + vec_b_r)
else:
# Non-smooth equation (brute force).
vec_semismooth_r = nm.where(vec_a_r < vec_b_r,
vec_a_r, vec_b_r)
r_last = (vec_smooth_r, vec_a_r, vec_b_r, vec_semismooth_r)
ok = True
time_stats['rezidual'] = time.clock() - tt
if ok:
vec_r = nm.r_[vec_smooth_r, vec_semismooth_r]
try:
err = nla.norm(vec_r)
except:
output('infs or nans in the residual:', vec_semismooth_r)
output(nm.isfinite(vec_semismooth_r).all())
debug()
if self.log is not None:
self.log(err, it)
if it == 0:
err0 = err;
break
if err < (err_last * conf.ls_on):
step_mode = 'regular'
break
else:
output('%s step line search' % step_mode)
red = conf.ls_red[step_mode];
output('iter %d, (%.5e < %.5e) (new ls: %e)'\
% (it, err, err_last * conf.ls_on, red * ls))
else: # Failed to compute rezidual.
red = conf.ls_red_warp;
output('rezidual computation failed for iter %d'
' (new ls: %e)!' % (it, red * ls))
if ls < conf.ls_min:
if step_mode == 'regular':
output('restore previous state')
vec_x = vec_x_last.copy()
vec_smooth_r, vec_a_r, vec_b_r, vec_semismooth_r = r_last
err = err_last
reuse_matrix = True
step_mode = 'steepest_descent'
else:
output('linesearch failed, continuing anyway')
break
ls *= red;
vec_dx = ls * vec_dx0;
vec_x = vec_x_last.copy() - vec_dx
i_ls += 1
# End residual loop.
output('%s step' % step_mode)
if self.log is not None:
self.log.plot_vlines([1],
color=self._colors[step_mode],
linewidth=0.5)
err_last = err;
vec_x_last = vec_x.copy()
condition = conv_test(conf, it, err, err0)
if condition >= 0:
break
tt = time.clock()
if not reuse_matrix:
mtx_jac = self.compute_jacobian(vec_x, fun_smooth_grad,
fun_a_grad, fun_b_grad,
vec_smooth_r,
vec_a_r, vec_b_r)
else:
reuse_matrix = False
time_stats['matrix'] = time.clock() - tt
tt = time.clock()
if step_mode == 'regular':
vec_dx = lin_solver(vec_r, mtx=mtx_jac)
vec_e = mtx_jac * vec_dx - vec_r
lerr = nla.norm(vec_e)
if lerr > (conf.eps_a * conf.lin_red):
output('linear system not solved! (err = %e)' % lerr)
output('switching to steepest descent step')
step_mode = 'steepest_descent'
vec_dx = mtx_jac.T * vec_r
else:
vec_dx = mtx_jac.T * vec_r
time_stats['solve'] = time.clock() - tt
for kv in time_stats.iteritems():
output('%10s: %7.2f [s]' % kv)
vec_x -= vec_dx
it += 1
if status is not None:
status['time_stats'] = time_stats
status['err0'] = err0
status['err'] = err
status['condition'] = condition
if conf.log.plot is not None:
if self.log is not None:
self.log(save_figure=conf.log.plot)
return vec_x
[docs] def compute_jacobian(self, vec_x, fun_smooth_grad, fun_a_grad, fun_b_grad,
vec_smooth_r, vec_a_r, vec_b_r):
conf = self.conf
mtx_s = fun_smooth_grad(vec_x)
mtx_a = fun_a_grad(vec_x)
mtx_b = fun_b_grad(vec_x)
n_s = vec_smooth_r.shape[0]
n_ns = vec_a_r.shape[0]
if conf.semismooth:
aa = nm.abs(vec_a_r)
ab = nm.abs(vec_b_r)
iz = nm.where((aa < (conf.macheps * max(aa.max(), 1.0)))
& (ab < (conf.macheps * max(ab.max(), 1.0))))[0]
inz = nm.setdiff1d(nm.arange(n_ns), iz)
output('non_active/active: %d/%d' % (len(inz), len(iz)))
mul_a = nm.empty_like(vec_a_r)
mul_b = nm.empty_like(mul_a)
# Non-active part of the jacobian.
if len(inz) > 0:
a_r_nz = vec_a_r[inz]
b_r_nz = vec_b_r[inz]
sqrt_ab = nm.sqrt(a_r_nz**2.0 + b_r_nz**2.0)
mul_a[inz] = (a_r_nz / sqrt_ab) - 1.0
mul_b[inz] = (b_r_nz / sqrt_ab) - 1.0
# Active part of the jacobian.
if len(iz) > 0:
vec_z = nm.zeros_like(vec_x)
vec_z[n_s+iz] = 1.0
mtx_a_z = mtx_a[iz]
mtx_b_z = mtx_b[iz]
sqrt_ab = nm.empty((iz.shape[0],), dtype=vec_a_r.dtype)
for ir in range(len(iz)):
row_a_z = mtx_a_z[ir]
row_b_z = mtx_b_z[ir]
sqrt_ab[ir] = nm.sqrt((row_a_z * row_a_z.T).todense()
+ (row_b_z * row_b_z.T).todense())
mul_a[iz] = ((mtx_a_z * vec_z) / sqrt_ab) - 1.0
mul_b[iz] = ((mtx_b_z * vec_z) / sqrt_ab) - 1.0
else:
iz = nm.where(vec_a_r > vec_b_r)[0]
mul_a = nm.zeros_like(vec_a_r)
mul_b = nm.ones_like(mul_a)
mul_a[iz] = 1.0
mul_b[iz] = 0.0
mtx_ns = sp.spdiags(mul_a, 0, n_ns, n_ns) * mtx_a \
+ sp.spdiags(mul_b, 0, n_ns, n_ns) * mtx_b
mtx_jac = compose_sparse([[mtx_s], [mtx_ns]]).tocsr()
mtx_jac.sort_indices()
return mtx_jac
