def residualCheck(A, B, U, d):
"""
Test the l2 norm of the residual:
r[:,i] = d[i] B U[:,i] - A U[:,i]
"""
u = Vector()
Au = Vector()
Bu = Vector()
Binv_r = Vector()
A.init_vector(u, 1)
A.init_vector(Au, 0)
B.init_vector(Bu, 0)
B.init_vector(Binv_r, 0)
nvec = d.shape[0]
print("lambda", "||Au - lambdaBu||")
for i in range(0, nvec):
u.set_local(U[:, i])
A.mult(u, Au)
B.mult(u, Bu)
Au.axpy(-d[i], Bu)
print(d[i], Au.norm("l2"))
class CGSampler:
"""
This class implements the CG sampler algorithm to generate samples from N(0, A^-1).
REFERENCE:
Albert Parker and Colin Fox
Sampling Gaussian Distributions in Krylov Spaces with Conjugate Gradient
SIAM J SCI COMPUT, Vol 34, No. 3 pp. B312-B334
"""
def __init__(self):
"""
Construct the solver with default parameters
tolerance = 1e-4
print_level = 0
verbose = 0
"""
self.parameters = {}
self.parameters["tolerance"] = 1e-4
self.parameters["print_level"] = 0
self.parameters["verbose"] = 0
self.A = None
self.converged = False
self.iter = 0
self.b = Vector()
self.r = Vector()
self.p = Vector()
self.Ap = Vector()
def set_operator(self, A):
"""
Set the operator A, such that x ~ N(0, A^-1).
Note A is any object that provides the methods init_vector and mult.
"""
self.A = A
self.A.init_vector(self.r, 0)
self.A.init_vector(self.p, 0)
self.A.init_vector(self.Ap, 0)
self.A.init_vector(self.b, 0)
self.b.set_local(np.random.randn(self.b.array().shape[0]))
def sample(self, noise, s):
"""
Generate a sample s ~ N(0, A^-1).
noise is a numpy.array of i.i.d. normal variables used as input.
For a fixed realization of noise the algorithm is fully deterministic.
The size of noise determine the maximum number of CG iterations.
"""
s.zero()
self.iter = 0
self.converged = False
# r0 = b
self.r.zero()
self.r.axpy(1., self.b)
#p0 = r0
self.p.zero()
self.p.axpy(1., self.r)
self.A.mult(self.p, self.Ap)
d = self.p.inner(self.Ap)
tol2 = self.parameters["tolerance"] * self.parameters["tolerance"]
rnorm2_old = self.r.inner(self.r)
if self.parameters["verbose"] > 0:
print "initial residual = ", math.sqrt(rnorm2_old)
while (not self.converged) and (self.iter < noise.shape[0]):
gamma = rnorm2_old / d
s.axpy(noise[self.iter] / math.sqrt(d), self.p)
self.r.axpy(-gamma, self.Ap)
rnorm2 = self.r.inner(self.r)
beta = rnorm2 / rnorm2_old
# p_new = r + beta p
self.p *= beta
self.p.axpy(1., self.r)
self.A.mult(self.p, self.Ap)
d = self.p.inner(self.Ap)
rnorm2_old = rnorm2
if rnorm2 < tol2:
self.converged = True
else:
rnorm2_old = rnorm2
self.iter = self.iter + 1
if self.parameters["verbose"] > 0:
print "Final residual {0} after {1} iterations".format(
math.sqrt(rnorm2_old), self.iter)
class OasisFunction(Function):
"""Function with more or less efficient projection methods
of associated linear form.
The matvec option is provided for letting the right hand side
be computed through a fast matrix vector product. Both the matrix
and the Coefficient of the required vector must be provided.
method = "default"
Solve projection with regular linear algebra using solver_type
and preconditioner_type
method = "lumping"
Solve through lumping of mass matrix
"""
def __init__(self, form, Space, bcs=[],
name="x",
matvec=[None, None],
method="default",
solver_type="cg",
preconditioner_type="default"):
Function.__init__(self, Space, name=name)
self.form = form
self.method = method
self.bcs = bcs
self.matvec = matvec
self.trial = trial = TrialFunction(Space)
self.test = test = TestFunction(Space)
Mass = inner(trial, test) * dx()
self.bf = inner(form, test) * dx()
self.rhs = Vector(self.vector())
if method.lower() == "default":
self.A = A_cache[(Mass, tuple(bcs))]
self.sol = Solver_cache[(Mass, tuple(
bcs), solver_type, preconditioner_type)]
elif method.lower() == "lumping":
assert Space.ufl_element().degree() < 2
self.A = A_cache[(Mass, tuple(bcs))]
ones = Function(Space)
ones.vector()[:] = 1.
self.ML = self.A * ones.vector()
self.ML.set_local(1. / self.ML.array())
def assemble_rhs(self):
"""
Assemble right hand side (form*test*dx) in projection
"""
if not self.matvec[0] is None:
mat, func = self.matvec
self.rhs.zero()
self.rhs.axpy(1.0, mat * func.vector())
else:
assemble(self.bf, tensor=self.rhs)
def __call__(self, assemb_rhs=True):
"""
Compute the projection
"""
timer = Timer("Projecting {}".format(self.name()))
if assemb_rhs:
self.assemble_rhs()
for bc in self.bcs:
bc.apply(self.rhs)
if self.method.lower() == "default":
self.sol.solve(self.A, self.vector(), self.rhs)
else:
self.vector().zero()
self.vector().axpy(1.0, self.rhs * self.ML)
class OasisFunction(Function):
"""Function with more or less efficient projection methods
of associated linear form.
The matvec option is provided for letting the right hand side
be computed through a fast matrix vector product. Both the matrix
and the Coefficient of the required vector must be provided.
method = "default"
Solve projection with regular linear algebra using solver_type
and preconditioner_type
method = "lumping"
Solve through lumping of mass matrix
"""
def __init__(self,
form,
Space,
bcs=[],
name="x",
matvec=[None, None],
method="default",
solver_type="cg",
preconditioner_type="default"):
Function.__init__(self, Space, name=name)
self.form = form
self.method = method
self.bcs = bcs
self.matvec = matvec
self.trial = trial = TrialFunction(Space)
self.test = test = TestFunction(Space)
Mass = inner(trial, test) * dx()
self.bf = inner(form, test) * dx()
self.rhs = Vector(self.vector())
if method.lower() == "default":
self.A = A_cache[(Mass, tuple(bcs))]
self.sol = Solver_cache[(Mass, tuple(bcs), solver_type,
preconditioner_type)]
elif method.lower() == "lumping":
assert Space.ufl_element().degree() < 2
self.A = A_cache[(Mass, tuple(bcs))]
ones = Function(Space)
ones.vector()[:] = 1.
self.ML = self.A * ones.vector()
self.ML.set_local(1. / self.ML.array())
def assemble_rhs(self):
"""
Assemble right hand side (form*test*dx) in projection
"""
if not self.matvec[0] is None:
mat, func = self.matvec
self.rhs.zero()
self.rhs.axpy(1.0, mat * func.vector())
else:
assemble(self.bf, tensor=self.rhs)
def __call__(self, assemb_rhs=True):
"""
Compute the projection
"""
timer = Timer("Projecting {}".format(self.name()))
if assemb_rhs:
self.assemble_rhs()
for bc in self.bcs:
bc.apply(self.rhs)
if self.method.lower() == "default":
self.sol.solve(self.A, self.vector(), self.rhs)
else:
self.vector().zero()
self.vector().axpy(1.0, self.rhs * self.ML)
class CGSolverSteihaug:
"""
Solve the linear system A x = b using preconditioned conjugate gradient ( B preconditioner)
and the Steihaug stopping criterion:
- reason of termination 0: we reached the maximum number of iterations (no convergence)
- reason of termination 1: we reduced the residual up to the given tolerance (convergence)
- reason of termination 2: we reached a negative direction (premature termination due to not spd matrix)
The stopping criterion is based on either
- the absolute preconditioned residual norm check: || r^* ||_{B^{-1}} < atol
- the relative preconditioned residual norm check: || r^* ||_{B^{-1}}/|| r^0 ||_{B^{-1}} < rtol
where r^* = b - Ax^* is the residual at convergence and r^0 = b - Ax^0 is the initial residual.
The operator A is set using the method set_operator(A).
A must provide the following two methods:
- A.mult(x,y): y = A*x
- A.init_vector(x, dim): initialize the vector x so that it is compatible with the range (dim = 0) or
the domain (dim = 1) of A.
The preconditioner B is set using the method set_preconditioner(B).
B must provide the following method:
- B.solve(z,r): z is the action of the preconditioner B on the vector r
To solve the linear system A*x = b call self.solve(x,b).
Here x and b are assumed to be FEniCS::Vector objects.
The parameter attributes allows to set:
- rel_tolerance : the relative tolerance for the stopping criterion
- abs_tolerance : the absolute tolerance for the stopping criterion
- max_iter : the maximum number of iterations
- zero_initial_guess: if True we start with a 0 initial guess
if False we use the x as initial guess.
- print_level : verbosity level:
-1 --> no output on screen
0 --> only final residual at convergence
or reason for not not convergence
"""
reason = [
"Maximum Number of Iterations Reached",
"Relative/Absolute residual less than tol",
"Reached a negative direction"
]
def __init__(self):
self.parameters = {}
self.parameters["rel_tolerance"] = 1e-9
self.parameters["abs_tolerance"] = 1e-12
self.parameters["max_iter"] = 1000
self.parameters["zero_initial_guess"] = True
self.parameters["print_level"] = 0
self.A = None
self.B = None
self.converged = False
self.iter = 0
self.reasonid = 0
self.final_norm = 0
self.r = Vector()
self.z = Vector()
self.d = Vector()
def set_operator(self, A):
"""
Set the operator A.
"""
self.A = A
self.A.init_vector(self.r, 0)
self.A.init_vector(self.z, 0)
self.A.init_vector(self.d, 0)
def set_preconditioner(self, B):
"""
Set the preconditioner B.
"""
self.B = B
def solve(self, x, b):
"""
Solve the linear system Ax = b
"""
self.iter = 0
self.converged = False
self.reasonid = 0
betanom = 0.0
alpha = 0.0
beta = 0.0
if self.parameters["zero_initial_guess"]:
self.r.zero()
self.r.axpy(1.0, b)
x.zero()
else:
self.A.mult(x, self.r)
self.r *= -1.0
self.r.axpy(1.0, b)
self.z.zero()
self.B.solve(self.z, self.r) #z = B^-1 r
self.d.zero()
self.d.axpy(1., self.z)
#d = z
nom0 = self.d.inner(self.r)
nom = nom0
if self.parameters["print_level"] == 1:
print(" Iterartion : ", 0, " (B r, r) = ", nom)
rtol2 = nom * self.parameters["rel_tolerance"] * self.parameters[
"rel_tolerance"]
atol2 = self.parameters["abs_tolerance"] * self.parameters[
"abs_tolerance"]
r0 = max(rtol2, atol2)
if nom <= r0:
self.converged = True
self.reasonid = 1
self.final_norm = math.sqrt(nom)
if (self.parameters["print_level"] >= 0):
print(self.reason[self.reasonid])
print("Converged in ", self.iter,
" iterations with final norm ", self.final_norm)
return
self.A.mult(self.d, self.z) #z = A d
den = self.z.inner(self.d)
if den <= 0.0:
self.converged = True
self.reasonid = 2
self.final_norm = math.sqrt(nom)
if (self.parameters["print_level"] >= 0):
print(self.reason[self.reasonid])
print("Converged in ", self.iter,
" iterations with final norm ", self.final_norm)
return
# start iteration
self.iter = 1
while True:
alpha = nom / den
x.axpy(alpha, self.d) # x = x + alpha d
self.r.axpy(-alpha, self.z) # r = r - alpha A d
self.B.solve(self.z, self.r) # z = B^-1 r
betanom = self.r.inner(self.z)
if self.parameters["print_level"] == 1:
print(" Iteration : ", self.iter, " (B r, r) = ", betanom)
if betanom < r0:
self.converged = True
self.reasonid = 1
self.final_norm = math.sqrt(betanom)
if (self.parameters["print_level"] >= 0):
print(self.reason[self.reasonid])
print("Converged in ", self.iter,
" iterations with final norm ", self.final_norm)
break
self.iter += 1
if self.iter > self.parameters["max_iter"]:
self.converged = False
self.reasonid = 0
self.final_norm = math.sqrt(betanom)
if (self.parameters["print_level"] >= 0):
print(self.reason[self.reasonid])
print("Not Converged. Final residual norm ",
self.final_norm)
break
beta = betanom / nom
self.d *= beta
self.d.axpy(1., self.z) #d = z + beta d
self.A.mult(self.d, self.z) # z = A d
den = self.d.inner(self.z)
if den <= 0.0:
self.converged = True
self.reasonid = 2
self.final_norm = math.sqrt(nom)
if (self.parameters["print_level"] >= 0):
print(self.reason[self.reasonid])
print("Converged in ", self.iter,
" iterations with final norm ", self.final_norm)
break
nom = betanom
class CGSolverSteihaug:
"""
Solve the linear system A x = b using preconditioned conjugate gradient ( B preconditioner)
and the Steihaug stopping criterion:
- reason of termination 0: we reached the maximum number of iterations (no convergence)
- reason of termination 1: we reduced the residual up to the given tolerance (convergence)
- reason of termination 2: we reached a negative direction (premature termination due to not spd matrix)
The operator A is set using the method set_operator(A).
A must provide the following two methods:
- A.mult(x,y): y = A*x
- A.init_vector(x, dim): initialize the vector x so that it is compatible with the range (dim = 0) or
the domain (dim = 1) of A.
The preconditioner B is set using the method set_preconditioner(B).
B must provide the following method:
- B.solve(z,r): z is the action of the preconditioner B on the vector r
To solve the linear system A*x = b call solve(x,b). Here x and b are assumed
to be FEniCS::Vector objects
Maximum number of iterations, tolerances, verbosity level etc can be
set using the parameters attributes.
"""
reason = [
"Maximum Number of Iterations Reached",
"Relative/Absolute residual less than tol",
"Reached a negative direction"
]
def __init__(self):
self.parameters = {}
self.parameters["rel_tolerance"] = 1e-9
self.parameters["abs_tolerance"] = 1e-12
self.parameters["max_iter"] = 1000
self.parameters["zero_initial_guess"] = True
self.parameters["print_level"] = 0
self.A = None
self.B = None
self.converged = False
self.iter = 0
self.reasonid = 0
self.final_norm = 0
self.r = Vector()
self.z = Vector()
self.d = Vector()
if dolfin.__version__[2] == '3': self.vrs130 = True
else: self.vrs130 = False
def set_operator(self, A):
self.A = A
if self.vrs130:
self.r = self.A.init_vector130()
self.z = self.A.init_vector130()
self.d = self.A.init_vector130()
else:
self.A.init_vector(self.r, 0)
self.A.init_vector(self.z, 0)
self.A.init_vector(self.d, 0)
def set_preconditioner(self, B):
self.B = B
def solve(self, x, b):
self.iter = 0
self.converged = False
self.reasonid = 0
betanom = 0.0
alpha = 0.0
beta = 0.0
if self.parameters["zero_initial_guess"]:
self.r.zero()
self.r.axpy(1.0, b)
x.zero()
else:
self.A.mult(x, self.r)
self.r *= -1.0
self.r.axpy(1.0, b)
self.z.zero()
self.B.solve(self.z, self.r) #z = B^-1 r
self.d.zero()
self.d.axpy(1., self.z)
#d = z
nom0 = self.d.inner(self.r)
nom = nom0
if self.parameters["print_level"] == 1:
print " Iterartion : ", 0, " (B r, r) = ", nom
rtol2 = nom * self.parameters["rel_tolerance"] * self.parameters[
"rel_tolerance"]
atol2 = self.parameters["abs_tolerance"] * self.parameters[
"abs_tolerance"]
r0 = max(rtol2, atol2)
if nom <= r0:
self.converged = True
self.reasonid = 1
self.final_norm = math.sqrt(nom)
if (self.parameters["print_level"] >= 0):
print self.reason[self.reasonid]
print "Converged in ", self.iter, " iterations with final norm ", self.final_norm
return
self.A.mult(self.d, self.z) #z = A d
den = self.z.inner(self.d)
if den <= 0.0:
self.converged = True
self.reasonid = 2
self.final_norm = math.sqrt(nom)
if (self.parameters["print_level"] >= 0):
print self.reason[self.reasonid]
print "Converged in ", self.iter, " iterations with final norm ", self.final_norm
return
# start iteration
self.iter = 1
while True:
alpha = nom / den
x.axpy(alpha, self.d) # x = x + alpha d
self.r.axpy(-alpha, self.z) # r = r - alpha A d
self.B.solve(self.z, self.r) # z = B^-1 r
betanom = self.r.inner(self.z)
if self.parameters["print_level"] == 1:
print " Iteration : ", self.iter, " (B r, r) = ", betanom
if betanom < r0:
self.converged = True
self.reasonid = 1
self.final_norm = math.sqrt(betanom)
if (self.parameters["print_level"] >= 0):
print self.reason[self.reasonid]
print "Converged in ", self.iter, " iterations with final norm ", self.final_norm
break
self.iter += 1
if self.iter > self.parameters["max_iter"]:
self.converged = False
self.reasonid = 0
self.final_norm = math.sqrt(betanom)
if (self.parameters["print_level"] >= 0):
print self.reason[self.reasonid]
print "Not Converged. Final residual norm ", self.final_norm
break
beta = betanom / nom
self.d *= beta
self.d.axpy(1., self.z) #d = z + beta d
self.A.mult(self.d, self.z) # z = A d
den = self.d.inner(self.z)
if den <= 0.0:
self.converged = True
self.reasonid = 2
self.final_norm = math.sqrt(nom)
if (self.parameters["print_level"] >= 0):
print self.reason[self.reasonid]
print "Converged in ", self.iter, " iterations with final norm ", self.final_norm
break
nom = betanom
class CGSampler:
"""
This class implements the CG sampler algorithm to generate samples from :math:`\mathcal{N}(0, A^{-1})`.
Reference:
`Albert Parker and Colin Fox Sampling Gaussian Distributions in Krylov Spaces with Conjugate Gradient
SIAM J SCI COMPUT, Vol 34, No. 3 pp. B312-B334`
"""
def __init__(self):
"""
Construct the solver with default parameters
:code:`tolerance = 1e-4`
:code:`print_level = 0`
:code:`verbose = 0`
"""
self.parameters = {}
self.parameters["tolerance"] = 1e-4
self.parameters["print_level"] = 0
self.parameters["verbose"] = 0
self.A = None
self.converged = False
self.iter = 0
self.b = Vector()
self.r = Vector()
self.p = Vector()
self.Ap = Vector()
def set_operator(self, A):
"""
Set the operator :code:`A`, such that :math:`x \sim \mathcal{N}(0, A^{-1})`.
.. note:: :code:`A` is any object that provides the methods :code:`init_vector()` and :code:`mult()`
"""
self.A = A
self.A.init_vector(self.r, 0)
self.A.init_vector(self.p, 0)
self.A.init_vector(self.Ap, 0)
self.A.init_vector(self.b, 0)
parRandom.normal(1., self.b)
def sample(self, noise, s):
"""
Generate a sample :math:`s ~ N(0, A^{-1})`.
:code:`noise` is a :code:`numpy.array` of i.i.d. normal variables used as input.
For a fixed realization of noise the algorithm is fully deterministic.
The size of noise determine the maximum number of CG iterations.
"""
s.zero()
self.iter = 0
self.converged = False
# r0 = b
self.r.zero()
self.r.axpy(1., self.b)
#p0 = r0
self.p.zero()
self.p.axpy(1., self.r)
self.A.mult(self.p, self.Ap)
d = self.p.inner(self.Ap)
tol2 = self.parameters["tolerance"] * self.parameters["tolerance"]
rnorm2_old = self.r.inner(self.r)
if self.parameters["verbose"] > 0:
print("initial residual = {0:g}".format(math.sqrt(rnorm2_old)))
while (not self.converged) and (self.iter < noise.shape[0]):
gamma = rnorm2_old / d
s.axpy(noise[self.iter] / math.sqrt(d), self.p)
self.r.axpy(-gamma, self.Ap)
rnorm2 = self.r.inner(self.r)
beta = rnorm2 / rnorm2_old
# p_new = r + beta p
self.p *= beta
self.p.axpy(1., self.r)
self.A.mult(self.p, self.Ap)
d = self.p.inner(self.Ap)
rnorm2_old = rnorm2
if rnorm2 < tol2:
self.converged = True
else:
rnorm2_old = rnorm2
self.iter = self.iter + 1
if self.parameters["verbose"] > 0:
print("Final residual {0} after {1} iterations".format(
math.sqrt(rnorm2_old), self.iter))
class CGSolverSteihaug:
"""
Solve the linear system A x = b using preconditioned conjugate gradient ( B preconditioner)
and the Steihaug stopping criterion:
- reason of termination 0: we reached the maximum number of iterations (no convergence)
- reason of termination 1: we reduced the residual up to the given tolerance (convergence)
- reason of termination 2: we reached a negative direction (premature termination due to not spd matrix)
- reason of termination 3: we reached the boundary of the trust region
The stopping criterion is based on either
- the absolute preconditioned residual norm check: || r^* ||_{B^{-1}} < atol
- the relative preconditioned residual norm check: || r^* ||_{B^{-1}}/|| r^0 ||_{B^{-1}} < rtol
where r^* = b - Ax^* is the residual at convergence and r^0 = b - Ax^0 is the initial residual.
The operator A is set using the method set_operator(A).
A must provide the following two methods:
- A.mult(x,y): y = A*x
- A.init_vector(x, dim): initialize the vector x so that it is compatible with the range (dim = 0) or
the domain (dim = 1) of A.
The preconditioner B is set using the method set_preconditioner(B).
B must provide the following method:
- B.solve(z,r): z is the action of the preconditioner B on the vector r
To solve the linear system A*x = b call self.solve(x,b).
Here x and b are assumed to be FEniCS::Vector objects.
Type: CGSolverSteihaug_ParameterList().showMe() for default parameters and their descriptions
"""
reason = ["Maximum Number of Iterations Reached",
"Relative/Absolute residual less than tol",
"Reached a negative direction",
"Reached trust region boundary"
]
def __init__(self, parameters=CGSolverSteihaug_ParameterList()):
self.parameters = parameters
self.A = None
self.B_solver = None
self.B_op = None
self.converged = False
self.iter = 0
self.reasonid = 0
self.final_norm = 0
self.TR_radius_2 = None
self.update_x = self.update_x_without_TR
self.r = Vector()
self.z = Vector()
self.d = Vector()
self.Bx = Vector()
def set_operator(self, A):
"""
Set the operator A.
"""
self.A = A
self.A.init_vector(self.r,0)
self.A.init_vector(self.z,0)
self.A.init_vector(self.d,0)
def set_preconditioner(self, B_solver):
"""
Set the preconditioner B.
"""
self.B_solver = B_solver
def set_TR(self,radius,B_op):
assert self.parameters["zero_initial_guess"]
self.TR_radius_2 = radius*radius
self.update_x = self.update_x_with_TR
self.B_op = B_op
self.B_op.init_vector(self.Bx,0)
def update_x_without_TR(self,x,alpha,d):
x.axpy(alpha,d)
return False
def update_x_with_TR(self,x,alpha,d):
x_bk = x.copy()
x.axpy(alpha,d)
self.Bx.zero()
self.B_op.mult(x, self.Bx)
x_Bnorm2 = self.Bx.inner(x)
if x_Bnorm2 < self.TR_radius_2:
return False
else:
# Move point to boundary of trust region
self.Bx.zero()
self.B_op.mult(x_bk, self.Bx)
x_Bnorm2 = self.Bx.inner(x_bk)
Bd = Vector()
self.B_op.init_vector(Bd,0)
Bd.zero()
self.B_op.mult(self.d,Bd)
d_Bnorm2 = Bd.inner(d)
d_Bx = Bd.inner(x_bk)
a_tau = alpha*alpha*d_Bnorm2
b_tau_half = alpha* d_Bx
c_tau = x_Bnorm2- self.TR_radius_2
# Solve quadratic for tau
tau = (-b_tau_half + math.sqrt(b_tau_half*b_tau_half - a_tau*c_tau))/a_tau
x.zero()
x.axpy(1,x_bk)
x.axpy(tau*alpha, d)
return True
def solve(self,x,b):
"""
Solve the linear system Ax = b
"""
self.iter = 0
self.converged = False
self.reasonid = 0
betanom = 0.0
alpha = 0.0
beta = 0.0
if self.parameters["zero_initial_guess"]:
self.r.zero()
self.r.axpy(1.0, b)
x.zero()
else:
assert self.TR_radius_2==None
self.A.mult(x,self.r)
self.r *= -1.0
self.r.axpy(1.0, b)
self.z.zero()
self.B_solver.solve(self.z,self.r) #z = B^-1 r
self.d.zero()
self.d.axpy(1.,self.z); #d = z
nom0 = self.d.inner(self.r)
nom = nom0
if self.parameters["print_level"] == 1:
print " Iterartion : ", 0, " (B r, r) = ", nom
rtol2 = nom * self.parameters["rel_tolerance"] * self.parameters["rel_tolerance"]
atol2 = self.parameters["abs_tolerance"] * self.parameters["abs_tolerance"]
r0 = max(rtol2, atol2)
if nom <= r0:
self.converged = True
self.reasonid = 1
self.final_norm = math.sqrt(nom)
if(self.parameters["print_level"] >= 0):
print self.reason[self.reasonid]
print "Converged in ", self.iter, " iterations with final norm ", self.final_norm
return
self.A.mult(self.d, self.z) #z = A d
den = self.z.inner(self.d)
if den <= 0.0:
self.converged = True
self.reasonid = 2
x.axpy(1., self.d)
self.r.axpy(-1., self.z)
self.B_solver.solve(self.z, self.r)
nom = self.r.inner(self.z)
self.final_norm = math.sqrt(nom)
if(self.parameters["print_level"] >= 0):
print self.reason[self.reasonid]
print "Converged in ", self.iter, " iterations with final norm ", self.final_norm
return
# start iteration
self.iter = 1
while True:
alpha = nom/den
TrustBool = self.update_x(x,alpha,self.d) # x = x + alpha d
if TrustBool == True:
self.converged = True
self.reasonid = 3
self.final_norm = math.sqrt(betanom)
if(self.parameters["print_level"] >= 0):
print self.reason[self.reasonid]
print "Converged in ", self.iter, " iterations with final norm ", self.final_norm
break
self.r.axpy(-alpha, self.z) # r = r - alpha A d
self.B_solver.solve(self.z, self.r) # z = B^-1 r
betanom = self.r.inner(self.z)
if self.parameters["print_level"] == 1:
print " Iteration : ", self.iter, " (B r, r) = ", betanom
if betanom < r0:
self.converged = True
self.reasonid = 1
self.final_norm = math.sqrt(betanom)
if(self.parameters["print_level"] >= 0):
print self.reason[self.reasonid]
print "Converged in ", self.iter, " iterations with final norm ", self.final_norm
break
self.iter += 1
if self.iter > self.parameters["max_iter"]:
self.converged = False
self.reasonid = 0
self.final_norm = math.sqrt(betanom)
if(self.parameters["print_level"] >= 0):
print self.reason[self.reasonid]
print "Not Converged. Final residual norm ", self.final_norm
break
beta = betanom/nom
self.d *= beta
self.d.axpy(1., self.z) #d = z + beta d
self.A.mult(self.d,self.z) # z = A d
den = self.d.inner(self.z)
if den <= 0.0:
self.converged = True
self.reasonid = 2
self.final_norm = math.sqrt(nom)
if(self.parameters["print_level"] >= 0):
print self.reason[self.reasonid]
print "Converged in ", self.iter, " iterations with final norm ", self.final_norm
break
nom = betanom
class CGSampler:
"""
This class implements the CG sampler algorithm to generate samples from :math:`\mathcal{N}(0, A^{-1})`.
Reference:
`Albert Parker and Colin Fox Sampling Gaussian Distributions in Krylov Spaces with Conjugate Gradient
SIAM J SCI COMPUT, Vol 34, No. 3 pp. B312-B334`
"""
def __init__(self):
"""
Construct the solver with default parameters
:code:`tolerance = 1e-4`
:code:`print_level = 0`
:code:`verbose = 0`
"""
self.parameters = {}
self.parameters["tolerance"] = 1e-4
self.parameters["print_level"] = 0
self.parameters["verbose"] = 0
self.A = None
self.converged = False
self.iter = 0
self.b = Vector()
self.r = Vector()
self.p = Vector()
self.Ap = Vector()
def set_operator(self, A):
"""
Set the operator :code:`A`, such that :math:`x \sim \mathcal{N}(0, A^{-1})`.
.. note:: :code:`A` is any object that provides the methods :code:`init_vector()` and :code:`mult()`
"""
self.A = A
self.A.init_vector(self.r,0)
self.A.init_vector(self.p,0)
self.A.init_vector(self.Ap,0)
self.A.init_vector(self.b,0)
parRandom.normal(1., self.b)
def sample(self, noise, s):
"""
Generate a sample :math:`s ~ N(0, A^{-1})`.
:code:`noise` is a :code:`numpy.array` of i.i.d. normal variables used as input.
For a fixed realization of noise the algorithm is fully deterministic.
The size of noise determine the maximum number of CG iterations.
"""
s.zero()
self.iter = 0
self.converged = False
# r0 = b
self.r.zero()
self.r.axpy(1., self.b)
#p0 = r0
self.p.zero()
self.p.axpy(1., self.r)
self.A.mult(self.p, self.Ap)
d = self.p.inner(self.Ap)
tol2 = self.parameters["tolerance"]*self.parameters["tolerance"]
rnorm2_old = self.r.inner(self.r)
if self.parameters["verbose"] > 0:
print("initial residual = {0:g}".format( math.sqrt(rnorm2_old) ))
while (not self.converged) and (self.iter < noise.shape[0]):
gamma = rnorm2_old/d
s.axpy(noise[self.iter]/math.sqrt(d), self.p)
self.r.axpy(-gamma, self.Ap)
rnorm2 = self.r.inner(self.r)
beta = rnorm2/rnorm2_old
# p_new = r + beta p
self.p *= beta
self.p.axpy(1., self.r)
self.A.mult(self.p, self.Ap)
d = self.p.inner(self.Ap)
rnorm2_old = rnorm2
if rnorm2 < tol2:
self.converged = True
else:
rnorm2_old = rnorm2
self.iter = self.iter+1
if self.parameters["verbose"] > 0:
print("Final residual {0} after {1} iterations".format( math.sqrt(rnorm2_old), self.iter))
class CGSolverSteihaug:
"""
Solve the linear system A x = b using preconditioned conjugate gradient ( B preconditioner)
and the Steihaug stopping criterion:
- reason of termination 0: we reached the maximum number of iterations (no convergence)
- reason of termination 1: we reduced the residual up to the given tolerance (convergence)
- reason of termination 2: we reached a negative direction (premature termination due to not spd matrix)
The operator A is set using the method set_operator(A).
A must provide the following two methods:
- A.mult(x,y): y = A*x
- A.init_vector(x, dim): initialize the vector x so that it is compatible with the range (dim = 0) or
the domain (dim = 1) of A.
The preconditioner B is set using the method set_preconditioner(B).
B must provide the following method:
- B.solve(z,r): z is the action of the preconditioner B on the vector r
To solve the linear system A*x = b call solve(x,b). Here x and b are assumed
to be FEniCS::Vector objects
Maximum number of iterations, tolerances, verbosity level etc can be
set using the parameters attributes.
"""
reason = ["Maximum Number of Iterations Reached",
"Relative/Absolute residual less than tol",
"Reached a negative direction"
]
def __init__(self):
self.parameters = {}
self.parameters["rel_tolerance"] = 1e-9
self.parameters["abs_tolerance"] = 1e-12
self.parameters["max_iter"] = 1000
self.parameters["zero_initial_guess"] = True
self.parameters["print_level"] = 0
self.A = None
self.B = None
self.converged = False
self.iter = 0
self.reasonid = 0
self.final_norm = 0
self.r = Vector()
self.z = Vector()
self.d = Vector()
if dolfin.__version__[2] == '5': self.vrs150 = True
else: self.vrs150 = False
def set_operator(self, A):
self.A = A
if self.vrs150:
self.A.init_vector(self.r,0)
self.A.init_vector(self.z,0)
self.A.init_vector(self.d,0)
else:
self.r = self.A.init_vector130()
self.z = self.A.init_vector130()
self.d = self.A.init_vector130()
def set_preconditioner(self, B):
self.B = B
def solve(self,x,b):
self.iter = 0
self.converged = False
self.reasonid = 0
betanom = 0.0
alpha = 0.0
beta = 0.0
if self.parameters["zero_initial_guess"]:
self.r.zero()
self.r.axpy(1.0, b)
x.zero()
else:
self.A.mult(x,self.r)
self.r *= -1.0
self.r.axpy(1.0, b)
self.z.zero()
self.B.solve(self.z,self.r) #z = B^-1 r
self.d.zero()
self.d.axpy(1.,self.z); #d = z
nom0 = self.d.inner(self.r)
nom = nom0
if self.parameters["print_level"] == 1:
print " Iterartion : ", 0, " (B r, r) = ", nom
rtol2 = nom * self.parameters["rel_tolerance"] * self.parameters["rel_tolerance"]
atol2 = self.parameters["abs_tolerance"] * self.parameters["abs_tolerance"]
r0 = max(rtol2, atol2)
if nom <= r0:
self.converged = True
self.reasonid = 1
self.final_norm = math.sqrt(nom)
if(self.parameters["print_level"] >= 0):
print self.reason[self.reasonid]
print "Converged in ", self.iter, " iterations with final norm ", self.final_norm
return
self.A.mult(self.d, self.z) #z = A d
den = self.z.inner(self.d)
if den <= 0.0:
self.converged = True
self.reasonid = 2
self.final_norm = math.sqrt(nom)
if(self.parameters["print_level"] >= 0):
print self.reason[self.reasonid]
print "Converged in ", self.iter, " iterations with final norm ", self.final_norm
return
# start iteration
self.iter = 1
while True:
alpha = nom/den
x.axpy(alpha,self.d) # x = x + alpha d
self.r.axpy(-alpha, self.z) # r = r - alpha A d
self.B.solve(self.z, self.r) # z = B^-1 r
betanom = self.r.inner(self.z)
if self.parameters["print_level"] == 1:
print " Iteration : ", self.iter, " (B r, r) = ", betanom
if betanom < r0:
self.converged = True
self.reasonid = 1
self.final_norm = math.sqrt(betanom)
if(self.parameters["print_level"] >= 0):
print self.reason[self.reasonid]
print "Converged in ", self.iter, " iterations with final norm ", self.final_norm
break
self.iter += 1
if self.iter > self.parameters["max_iter"]:
self.converged = False
self.reasonid = 0
self.final_norm = math.sqrt(betanom)
if(self.parameters["print_level"] >= 0):
print self.reason[self.reasonid]
print "Not Converged. Final residual norm ", self.final_norm
break
beta = betanom/nom
self.d *= beta
self.d.axpy(1., self.z) #d = z + beta d
self.A.mult(self.d,self.z) # z = A d
den = self.d.inner(self.z)
if den <= 0.0:
self.converged = True
self.reasonid = 2
self.final_norm = math.sqrt(nom)
if(self.parameters["print_level"] >= 0):
print self.reason[self.reasonid]
print "Converged in ", self.iter, " iterations with final norm ", self.final_norm
break
nom = betanom
