def gauss_newton_energy_norm(self, q):
r"""Compute the energy norm of a field w.r.t. the Gauss-Newton operator
The energy norm of a field :math:`q` w.r.t. the Gauss-Newton operator
:math:`H` can be computed using one fewer linear solve than if we were
to calculate the action of :math:`H\cdot q` on :math:`q`. This saves
computation when using the conjugate gradient method to solve for the
search direction.
"""
u, p = self.state, self.parameter
dE = derivative(self._E, u)
dR = derivative(self._R, p)
dF_du, dF_dp = self._dF_du, derivative(self._F, p)
v = firedrake.Function(u.function_space())
firedrake.solve(dF_du == action(dF_dp, q),
v,
self._bc,
solver_parameters=self._solver_params,
form_compiler_parameters=self._fc_params)
return self._assemble(
firedrake.energy_norm(derivative(dE, u), v) +
firedrake.energy_norm(derivative(dR, p), q))
def update_search_direction(self):
r"""Solve the Gauss-Newton system for the new search direction using
the preconditioned conjugate gradient method"""
p, q, dJ = self.parameter, self.search_direction, self.gradient
dR = derivative(self.regularization, self.parameter)
Q = q.function_space()
M = firedrake.TrialFunction(Q) * firedrake.TestFunction(Q) * dx + \
derivative(dR, p)
# Compute the preconditioned residual
z = firedrake.Function(Q)
firedrake.solve(M == -dJ,
z,
solver_parameters=self._solver_params,
form_compiler_parameters=self._fc_params)
# This variable is a search direction for a search direction, which
# is definitely not confusing at all.
s = z.copy(deepcopy=True)
q *= 0.0
old_cost = np.inf
while True:
z_mnorm = self._assemble(firedrake.energy_norm(M, z))
s_hnorm = self.gauss_newton_energy_norm(s)
α = z_mnorm / s_hnorm
δz = firedrake.Function(Q)
g = self.gauss_newton_mult(s)
firedrake.solve(M == g,
δz,
solver_parameters=self._solver_params,
form_compiler_parameters=self._fc_params)
q += α * s
z -= α * δz
β = self._assemble(firedrake.energy_norm(M, z)) / z_mnorm
s *= β
s += z
energy_norm = self.gauss_newton_energy_norm(q)
cost = 0.5 * energy_norm + self._assemble(action(dJ, q))
if (abs(old_cost - cost) /
(0.5 * energy_norm) < self._search_tolerance):
return
old_cost = cost
def step(self):
r"""Take one step of the conjugate gradient iteration"""
q = self.solution
s = self.search_direction
z = self.residual
δz = self._delta_residual
α = self.residual_energy / self.search_direction_energy
Gs = self.operator_product
dJ = self._rhs
delta_energy = α * (self._assemble(action(Gs, q)) +
0.5 * α * self.search_direction_energy)
self._energy += delta_energy
self._objective += delta_energy + α * self._assemble(action(dJ, s))
q.assign(q + Constant(α) * s)
z.assign(z - Constant(α) * δz)
M = self.preconditioner
residual_energy = self._assemble(firedrake.energy_norm(M, z))
β = residual_energy / self.residual_energy
s.assign(Constant(β) * s + z)
self.update_state()
self._residual_energy = residual_energy
Gs = self.operator_product
self._search_direction_energy = self._assemble(action(Gs, s))
self._iteration += 1
def reinit(self):
r"""Restart the solution to 0"""
self._iteration = 0
M = self._preconditioner
z = self.residual
s = self.search_direction
Gs = self.operator_product
Q = z.function_space()
self.solution.assign(firedrake.Function(Q))
self._residual_solver.solve()
s.assign(z)
self._residual_energy = self._assemble(firedrake.energy_norm(M, z))
self.update_state()
self._search_direction_energy = self._assemble(action(Gs, s))
self._energy = 0.
self._objective = 0.
