def _ad_convert_type(self, value, options=None):
options = {} if options is None else options
riesz_representation = options.get("riesz_representation", "l2")
if riesz_representation == "l2":
return create_overloaded_object(
compat.function_from_vector(self.function_space(),
value,
cls=backend.Function))
elif riesz_representation == "L2":
ret = compat.create_function(self.function_space())
u = backend.TrialFunction(self.function_space())
v = backend.TestFunction(self.function_space())
M = backend.assemble(backend.inner(u, v) * backend.dx)
compat.linalg_solve(M, ret.vector(), value)
return ret
elif riesz_representation == "H1":
ret = compat.create_function(self.function_space())
u = backend.TrialFunction(self.function_space())
v = backend.TestFunction(self.function_space())
M = backend.assemble(
backend.inner(u, v) * backend.dx +
backend.inner(backend.grad(u), backend.grad(v)) * backend.dx)
compat.linalg_solve(M, ret.vector(), value)
return ret
elif callable(riesz_representation):
return riesz_representation(value)
else:
raise NotImplementedError("Unknown Riesz representation %s" %
riesz_representation)
def evaluate_hessian_component(self,
inputs,
hessian_inputs,
adj_inputs,
block_variable,
idx,
relevant_dependencies,
prepared=None):
form = prepared
hessian_input = hessian_inputs[0]
adj_input = adj_inputs[0]
c1 = block_variable.output
c1_rep = block_variable.saved_output
if isinstance(c1, backend.Function):
dc = backend.TestFunction(c1.function_space())
elif isinstance(c1, compat.ExpressionType):
mesh = form.ufl_domain().ufl_cargo()
W = c1._ad_function_space(mesh)
dc = backend.TestFunction(W)
elif isinstance(c1, backend.Constant):
mesh = compat.extract_mesh_from_form(form)
dc = backend.TestFunction(c1._ad_function_space(mesh))
elif isinstance(c1, compat.MeshType):
pass
else:
return None
if isinstance(c1, compat.MeshType):
X = backend.SpatialCoordinate(c1)
dform = backend.derivative(form, X)
else:
dform = backend.derivative(form, c1_rep, dc)
hessian_outputs = hessian_input * compat.assemble_adjoint_value(dform)
for other_idx, bv in relevant_dependencies:
c2_rep = bv.saved_output
tlm_input = bv.tlm_value
if tlm_input is None:
continue
if isinstance(c2_rep, compat.MeshType):
X = backend.SpatialCoordinate(c2_rep)
ddform = backend.derivative(dform, X, tlm_input)
else:
ddform = backend.derivative(dform, c2_rep, tlm_input)
hessian_outputs += adj_input * compat.assemble_adjoint_value(
ddform)
if isinstance(c1, compat.ExpressionType):
return [(hessian_outputs, W)]
else:
return hessian_outputs
def compute_gst(ic, final, nsv, ic_norm="mass", final_norm="mass", which=1):
'''This function computes the generalised stability analysis of a simulation.
Generalised stability theory computes the perturbations to a field (such as an
initial condition, forcing term, etc.) that /grow the most/ over the finite
time window of the simulation. For more details, see the mathematical documentation
on `the website <http://dolfin-adjoint.org>`_.
- :py:data:`ic` -- the input of the propagator
- :py:data:`final` -- the output of the propagator
- :py:data:`nsv` -- the number of optimal perturbations to compute
- :py:data:`ic_norm` -- a symmetric positive-definite bilinear form that defines the norm on the input space
- :py:data:`final_norm` -- a symmetric positive-definite bilinear form that defines the norm on the output space
- :py:data:`which` -- which singular vectors to compute. Use e.g. slepc4py.SLEPc.EPS.Which.LARGEST_REAL
You can supply :py:data:`"mass"` for :py:data:`ic_norm` and :py:data:`final_norm` to use the (default) mass matrices associated
with these spaces.
For example:
.. code-block:: python
gst = compute_gst("State", "State", nsv=10)
for i in range(gst.ncv): # number of converged vectors
(sigma, u, v) = gst.get_gst(i, return_vectors=True)
'''
ic_var = adjglobals.adj_variables[ic]
ic_var.c_object.timestep = 0
ic_var.c_object.iteration = 0
final_var = adjglobals.adj_variables[final]
if final_norm == "mass":
final_value = adjglobals.adjointer.get_variable_value(final_var).data
final_fnsp = final_value.function_space()
u = backend.TrialFunction(final_fnsp)
v = backend.TestFunction(final_fnsp)
final_mass = backend.inner(u, v) * backend.dx
final_norm = adjlinalg.Matrix(final_mass)
elif final_norm is not None:
final_norm = adjlinalg.Matrix(final_norm)
if ic_norm == "mass":
ic_value = adjglobals.adjointer.get_variable_value(ic_var).data
ic_fnsp = ic_value.function_space()
u = backend.TrialFunction(ic_fnsp)
v = backend.TestFunction(ic_fnsp)
ic_mass = backend.inner(u, v) * backend.dx
ic_norm = adjlinalg.Matrix(ic_mass)
elif ic_norm is not None:
ic_norm = adjlinalg.Matrix(ic_norm)
return adjglobals.adjointer.compute_gst(ic_var, ic_norm, final_var,
final_norm, nsv, which)
def __init__(self, form, control):
if not isinstance(control.control, backend.Function):
raise NotImplementedError("Only implemented for Function controls")
args = ufl.algorithms.extract_arguments(form)
if len(args) != 0:
raise ValueError("Must be a rank-zero form, i.e. a functional")
u = control.control
self.V = u.function_space()
# We want to make a copy of the control purely for use
# in the constraint, so that our writing it isn't
# bothering anyone else
self.u = backend_types.Function(self.V)
self.form = ufl.replace(form, {u: self.u})
self.trial = backend.TrialFunction(self.V)
self.dform = backend.derivative(self.form, self.u, self.trial)
if len(
ufl.algorithms.extract_arguments(
ufl.algorithms.expand_derivatives(self.dform))) == 0:
raise ValueError("Form must depend on control")
self.test = backend.TestFunction(self.V)
self.hess = ufl.algorithms.expand_derivatives(
backend.derivative(self.dform, self.u, self.test))
if len(ufl.algorithms.extract_arguments(self.hess)) == 0:
self.zero_hess = True
else:
self.zero_hess = False
def project_test(func):
if isinstance(func, backend.Function):
V = func.function_space()
u = backend.TrialFunction(V)
v = backend.TestFunction(V)
M = backend.assemble(backend.inner(u, v) * backend.dx)
proj = backend.Function(V)
backend.solve(M, proj.vector(), func.vector())
return proj
else:
return func
def __init__(self, v, V, output, bcs=[], *args, **kwargs):
mesh = kwargs.pop("mesh", None)
if mesh is None:
mesh = V.mesh()
dx = backend.dx(mesh)
w = backend.TestFunction(V)
Pv = backend.TrialFunction(V)
a = backend.inner(w, Pv) * dx
L = backend.inner(w, v) * dx
super(ProjectBlock, self).__init__(a == L, output, bcs, *args,
**kwargs)
def evaluate_adj_component(self,
inputs,
adj_inputs,
block_variable,
idx,
prepared=None):
form = prepared
adj_input = adj_inputs[0]
c = block_variable.output
c_rep = block_variable.saved_output
if isinstance(c, compat.ExpressionType):
# Create a FunctionSpace from self.form and Expression.
# And then make a TestFunction from this space.
mesh = self.form.ufl_domain().ufl_cargo()
V = c._ad_function_space(mesh)
dc = backend.TestFunction(V)
dform = backend.derivative(form, c_rep, dc)
output = compat.assemble_adjoint_value(dform)
return [[adj_input * output, V]]
elif isinstance(c, compat.MeshType):
X = backend.SpatialCoordinate(c_rep)
dform = backend.derivative(form, X)
output = compat.assemble_adjoint_value(dform)
return adj_input * output
if isinstance(c, backend.Function):
dc = backend.TestFunction(c.function_space())
elif isinstance(c, backend.Constant):
mesh = compat.extract_mesh_from_form(self.form)
dc = backend.TestFunction(c._ad_function_space(mesh))
dform = backend.derivative(form, c_rep, dc)
output = compat.assemble_adjoint_value(dform)
return adj_input * output
def derivative_action(self, dependencies, values, variable,
contraction_vector, hermitian):
idx = dependencies.index(variable)
# If you want to apply boundary conditions symmetrically in the adjoint
# -- and you often do --
# then we need to have a UFL representation of all the terms in the adjoint equation.
# However!
# Since UFL cannot represent the identity map,
# we need to find an f such that when
# assemble(inner(f, v)*dx)
# we get the contraction_vector.data back.
# This involves inverting a mass matrix.
if backend.parameters["adjoint"][
"symmetric_bcs"] and backend.__version__ <= '1.2.0':
backend.info_red(
"Warning: symmetric BC application requested but unavailable in dolfin <= 1.2.0."
)
if backend.parameters["adjoint"][
"symmetric_bcs"] and backend.__version__ > '1.2.0':
V = contraction_vector.data.function_space()
v = backend.TestFunction(V)
if str(V) not in adjglobals.fsp_lu:
u = backend.TrialFunction(V)
A = backend.assemble(backend.inner(u, v) * backend.dx)
solver = "mumps" if "mumps" in backend.lu_solver_methods(
).keys() else "default"
lusolver = backend.LUSolver(A, solver)
lusolver.parameters["symmetric"] = True
lusolver.parameters["reuse_factorization"] = True
adjglobals.fsp_lu[str(V)] = lusolver
else:
lusolver = adjglobals.fsp_lu[str(V)]
riesz = backend.Function(V)
lusolver.solve(
riesz.vector(),
self.weights[idx] * contraction_vector.data.vector())
out = (backend.inner(riesz, v) * backend.dx)
else:
out = backend.Function(self.fn_space)
out.assign(self.weights[idx] * contraction_vector.data)
return adjlinalg.Vector(out)
def derivative_action(self, dependencies, values, variable,
contraction_vector, hermitian):
if not hermitian:
fn = Function_split(contraction_vector)[self.idx]
action = backend.inner(self.test, fn)
else:
bigtest = backend.TestFunction(self.function.function_space())
outfn = backend.Function(self.function.function_space())
# DOLFIN is a bit annoying when it comes to splits. Actually, it is very annoying.
# You can't do anything like
# outfn[idx].vector()[:] = values_I_want_to_assign_to_outfn[idx]
# or
# fn = outfn.split()[idx]; fn.vector()[:] = values_I_want_to_assign_to_outfn[idx]
# for whatever reason
assert False, "No idea how to assign to a subfunction yet .. "
assignment.dolfin_assign(outfn, contraction_vector.data)
action = backend.inner(bigtest, outfn) * backend.dx
return adjlinalg.Vector(action)
def perturbed_replay(parameter,
perturbation,
perturbation_scale,
observation,
perturbation_norm="mass",
observation_norm="mass",
callback=None,
forget=False):
r"""Perturb the forward run and compute
.. math::
\frac{
\left|\left| \delta \mathrm{observation} \right|\right|
}{
\left|\left| \delta \mathrm{input} \right| \right|
}
as a function of time.
:py:data:`parameter` -- an FunctionControl to say what variable should be
perturbed (e.g. FunctionControl('InitialConcentration'))
:py:data:`perturbation` -- a Function to give the perturbation direction (from a GST analysis, for example)
:py:data:`perturbation_norm` -- a bilinear Form which induces a norm on the space of perturbation inputs
:py:data:`perturbation_scale` -- how big the norm of the initial perturbation should be
:py:data:`observation` -- the variable to observe (e.g. 'Concentration')
:py:data:`observation_norm` -- a bilinear Form which induces a norm on the space of perturbation outputs
:py:data:`callback` -- a function f(var, perturbed, unperturbed) that the user can supply (e.g. to dump out variables during the perturbed replay)
"""
if not backend.parameters["adjoint"]["record_all"]:
info_red(
"Warning: your replay test will be much more effective with backend.parameters['adjoint']['record_all'] = True."
)
assert isinstance(parameter, controls.FunctionControl)
if perturbation_norm == "mass":
p_fnsp = perturbation.function_space()
u = backend.TrialFunction(p_fnsp)
v = backend.TestFunction(p_fnsp)
p_mass = backend.inner(u, v) * backend.dx
perturbation_norm = p_mass
if not isinstance(perturbation_norm, backend.GenericMatrix):
perturbation_norm = backend.assemble(perturbation_norm)
if not isinstance(observation_norm,
backend.GenericMatrix) and observation_norm != "mass":
observation_norm = backend.assemble(observation_norm)
def compute_norm(perturbation, norm):
# Need to compute <x, Ax> and then take its sqrt
# where x is perturbation, A is norm
try:
vec = perturbation.vector()
except:
vec = perturbation
Ax = vec.copy()
norm.mult(vec, Ax)
xAx = vec.inner(Ax)
return math.sqrt(xAx)
growths = []
for i in range(adjglobals.adjointer.equation_count):
(fwd_var, output) = adjglobals.adjointer.get_forward_solution(i)
if fwd_var == parameter.var: # we've hit the initial condition we want to perturb
current_norm = compute_norm(perturbation, perturbation_norm)
output.data.vector()[:] += (perturbation_scale /
current_norm) * perturbation.vector()
unperturbed = adjglobals.adjointer.get_variable_value(fwd_var).data
if fwd_var.name == observation: # we've hit something we want to observe
# Fetch the unperturbed result from the record
if observation_norm == "mass": # we can't do this earlier, because we don't have the observation function space yet
o_fnsp = output.data.function_space()
u = backend.TrialFunction(o_fnsp)
v = backend.TestFunction(o_fnsp)
o_mass = backend.inner(u, v) * backend.dx
observation_norm = backend.assemble(o_mass)
diff = output.data.vector() - unperturbed.vector()
growths.append(
compute_norm(diff, observation_norm) /
perturbation_scale) # <--- the action line
if callback is not None:
callback(fwd_var, output.data, unperturbed)
storage = libadjoint.MemoryStorage(output)
storage.set_compare(tol=None)
storage.set_overwrite(True)
out = adjglobals.adjointer.record_variable(fwd_var, storage)
if forget:
adjglobals.adjointer.forget_forward_equation(i)
# can happen if we initialised a nonlinear solve with a constant zero guess
if growths[0] == 0.0:
return growths[1:]
else:
return growths
def project_dolfin(v,
V=None,
bcs=None,
mesh=None,
solver_type="lu",
preconditioner_type="default",
form_compiler_parameters=None,
annotate=None,
name=None):
'''The project call performs an equation solve, and so it too must be annotated so that the
adjoint and tangent linear models may be constructed automatically by libadjoint.
To disable the annotation of this function, just pass :py:data:`annotate=False`. This is useful in
cases where the solve is known to be irrelevant or diagnostic for the purposes of the adjoint
computation (such as projecting fields to other function spaces for the purposes of
visualisation).'''
to_annotate = utils.to_annotate(annotate)
if isinstance(v, backend.Expression) and (annotate is not True):
to_annotate = False
if isinstance(v, backend.Constant) and (annotate is not True):
to_annotate = False
out = backend.project(v=v,
V=V,
bcs=bcs,
mesh=mesh,
solver_type=solver_type,
preconditioner_type=preconditioner_type,
form_compiler_parameters=form_compiler_parameters)
out = utils.function_to_da_function(out)
if name is not None:
out.adj_name = name
out.rename(name, "a Function from dolfin-adjoint")
if to_annotate:
# reproduce the logic from project. This probably isn't future-safe, but anyway
if V is None:
V = backend.fem.projection._extract_function_space(v, mesh)
if mesh is None:
mesh = V.mesh()
# Define variational problem for projection
w = backend.TestFunction(V)
Pv = backend.TrialFunction(V)
a = backend.inner(w, Pv) * backend.dx(domain=mesh)
L = backend.inner(w, v) * backend.dx(domain=mesh)
solving.annotate(a == L,
out,
bcs,
solver_parameters={
"linear_solver": solver_type,
"preconditioner": preconditioner_type,
"symmetric": True
})
if backend.parameters["adjoint"]["record_all"]:
adjglobals.adjointer.record_variable(
adjglobals.adj_variables[out],
libadjoint.MemoryStorage(adjlinalg.Vector(out)))
return out
def annotate_split(bigfn, idx, smallfn, bcs):
fn_space = smallfn.function_space().collapse()
test = backend.TestFunction(fn_space)
trial = backend.TrialFunction(fn_space)
eq_lhs = backend.inner(test, trial) * backend.dx
key = "{}split{}{}{}{}".format(eq_lhs, smallfn, bigfn, idx,
random.random()).encode('utf8')
diag_name = "Split:%s:" % idx + hashlib.md5(key).hexdigest()
diag_deps = []
diag_block = libadjoint.Block(
diag_name,
dependencies=diag_deps,
test_hermitian=backend.parameters["adjoint"]["test_hermitian"],
test_derivative=backend.parameters["adjoint"]["test_derivative"])
solving.register_initial_conditions(
[(bigfn, adjglobals.adj_variables[bigfn])], linear=True, var=None)
var = adjglobals.adj_variables.next(smallfn)
frozen_expressions_dict = expressions.freeze_dict()
def diag_assembly_cb(dependencies, values, hermitian, coefficient,
context):
'''This callback must conform to the libadjoint Python block assembly
interface. It returns either the form or its transpose, depending on
the value of the logical hermitian.'''
assert coefficient == 1
expressions.update_expressions(frozen_expressions_dict)
value_coeffs = [v.data for v in values]
eq_l = eq_lhs
if hermitian:
adjoint_bcs = [
utils.homogenize(bc)
for bc in bcs if isinstance(bc, backend.DirichletBC)
] + [bc for bc in bcs if not isinstance(bc, backend.DirichletBC)]
if len(adjoint_bcs) == 0: adjoint_bcs = None
return (adjlinalg.Matrix(backend.adjoint(eq_l), bcs=adjoint_bcs),
adjlinalg.Vector(None, fn_space=fn_space))
else:
return (adjlinalg.Matrix(eq_l, bcs=bcs),
adjlinalg.Vector(None, fn_space=fn_space))
diag_block.assemble = diag_assembly_cb
rhs = SplitRHS(test, bigfn, idx)
eqn = libadjoint.Equation(var, blocks=[diag_block], targets=[var], rhs=rhs)
cs = adjglobals.adjointer.register_equation(eqn)
solving.do_checkpoint(cs, var, rhs)
if backend.parameters["adjoint"]["fussy_replay"]:
mass = eq_lhs
smallfn_massed = backend.Function(fn_space)
backend.solve(mass == backend.action(mass, smallfn), smallfn_massed)
assert False, "No idea how to assign to a subfunction yet .. "
#assignment.dolfin_assign(bigfn, smallfn_massed)
if backend.parameters["adjoint"]["record_all"]:
smallfn_record = backend.Function(fn_space)
assignment.dolfin_assign(smallfn_record, smallfn)
adjglobals.adjointer.record_variable(
var, libadjoint.MemoryStorage(adjlinalg.Vector(smallfn_record)))
def evaluate_hessian_component(self,
inputs,
hessian_inputs,
adj_inputs,
block_variable,
idx,
relevant_dependencies,
prepared=None):
c = block_variable.output
if c == self.func and not self.linear:
return None
adj_sol2 = prepared["adj_sol2"]
adj_sol2_bdy = prepared["adj_sol2_bdy"]
F_form = prepared["form"]
adj_sol = prepared["adj_sol"]
fwd_block_variable = self.get_outputs()[0]
tlm_output = fwd_block_variable.tlm_value
c_rep = block_variable.saved_output
# If m = DirichletBC then d^2F(u,m)/dm^2 = 0 and d^2F(u,m)/dudm = 0,
# so we only have the term dF(u,m)/dm * adj_sol2
if isinstance(c, backend.DirichletBC):
tmp_bc = compat.create_bc(c,
value=extract_subfunction(
adj_sol2_bdy, c.function_space()))
return [tmp_bc]
if isinstance(c_rep, backend.Constant):
mesh = compat.extract_mesh_from_form(F_form)
W = c._ad_function_space(mesh)
elif isinstance(c, compat.ExpressionType):
mesh = F_form.ufl_domain().ufl_cargo()
W = c._ad_function_space(mesh)
elif isinstance(c, compat.MeshType):
X = backend.SpatialCoordinate(c)
element = X.ufl_domain().ufl_coordinate_element()
W = backend.FunctionSpace(c, element)
else:
W = c.function_space()
dc = backend.TestFunction(W)
form_adj = backend.action(F_form, adj_sol)
form_adj2 = backend.action(F_form, adj_sol2)
if isinstance(c, compat.MeshType):
dFdm_adj = backend.derivative(form_adj, X, dc)
dFdm_adj2 = backend.derivative(form_adj2, X, dc)
else:
dFdm_adj = backend.derivative(form_adj, c_rep, dc)
dFdm_adj2 = backend.derivative(form_adj2, c_rep, dc)
# TODO: Old comment claims this might break on split. Confirm if true or not.
d2Fdudm = ufl.algorithms.expand_derivatives(
backend.derivative(dFdm_adj, fwd_block_variable.saved_output,
tlm_output))
hessian_output = 0
# We need to add terms from every other dependency
# i.e. the terms d^2F/dm_1dm_2
for _, bv in relevant_dependencies:
c2 = bv.output
c2_rep = bv.saved_output
if isinstance(c2, backend.DirichletBC):
continue
tlm_input = bv.tlm_value
if tlm_input is None:
continue
if c2 == self.func and not self.linear:
continue
# TODO: If tlm_input is a Sum, this crashes in some instances?
if isinstance(c2_rep, compat.MeshType):
X = backend.SpatialCoordinate(c2_rep)
d2Fdm2 = ufl.algorithms.expand_derivatives(
backend.derivative(dFdm_adj, X, tlm_input))
else:
d2Fdm2 = ufl.algorithms.expand_derivatives(
backend.derivative(dFdm_adj, c2_rep, tlm_input))
if d2Fdm2.empty():
continue
hessian_output -= compat.assemble_adjoint_value(d2Fdm2)
if not d2Fdudm.empty():
# FIXME: This can be empty in the multimesh case, ask sebastian
hessian_output -= compat.assemble_adjoint_value(d2Fdudm)
hessian_output -= compat.assemble_adjoint_value(dFdm_adj2)
if isinstance(c, compat.ExpressionType):
return [(hessian_output, W)]
else:
return hessian_output