def Mult(self,rhs,usol):
usol[:] = 0.
res = usol.CreateVector()
projres = usol.CreateVector()
proj = Projector(mask=self.CutVh.FreeDofs(),range=True)
fnew = rhs #f.vec.data
projres.data = proj*fnew #f.vec
normf = Norm(projres)
for it in range(self.maxit):
usol.data = self.MGpre*fnew
res.data = fnew - self.mats[-1]*usol
projres.data = proj*res
res_norm = Norm(projres) / normf
if self.printinfo:
print("it =", it+1, " ||res||_2 =", res_norm)
if res_norm < self.tol:
break
#udraw = GridFunction( self.CutVh )
#udraw.components[1].Set( solution[1], BND )
#udraw.vec.data += usol.vec
return usol
def __init__(self,
mat: BaseMatrix,
pre: Optional[Preconditioner] = None,
freedofs: Optional[BitArray] = None,
tol: float = None,
maxiter: int = 100,
atol: float = None,
callback: Optional[Callable[[int, float], None]] = None,
callback_sol: Optional[Callable[[BaseVector], None]] = None,
printrates: bool = False):
super().__init__()
if atol is None and tol is None:
tol = 1e-12
self.mat = mat
assert (freedofs is None) != (pre is None
) # either pre or freedofs must be given
self.pre = pre if pre else Projector(freedofs, True)
self.tol = tol
self.atol = atol
self.maxiter = maxiter
self.callback = callback
self.callback_sol = callback_sol
self.printrates = printrates
self.residuals = []
self.iterations = 0
def __init__(self,
mat: BaseMatrix,
pre: Optional[Preconditioner] = None,
freedofs: Optional[BitArray] = None,
conjugate: bool = False,
tol: float = 1e-12,
maxsteps: int = 100,
callback: Optional[Callable[[int, float], None]] = None,
printing=False,
abstol=None):
super().__init__()
self.mat = mat
assert (freedofs is None) != (pre is None
) # either pre or freedofs must be given
self.pre = pre if pre else Projector(freedofs, True)
self.conjugate = conjugate
self.tol = tol
self.abstol = abstol
self.maxsteps = maxsteps
self.callback = callback
self._tmp_vecs = [self.mat.CreateRowVector() for i in range(3)]
self.logger = logging.getLogger("CGSolver")
self.printing = printing
if printing:
self.handler = handler = logging.StreamHandler()
self.logger.addHandler(handler)
self.logger.setLevel(logging.INFO)
self.errors = []
self.iterations = 0
def BVP(bf,
lf,
gf,
pre=None,
maxsteps=200,
tol=1e-8,
print=True,
inverse="umfpack"):
"""
Solve a linear boundary value problem
"""
from ngsolve import Projector
from ngsolve.krylovspace import CG
r = lf.vec.CreateVector()
r.data = lf.vec
if bf.condense:
r.data += bf.harmonic_extension_trans * r
# zero local dofs
innerbits = gf.space.FreeDofs(False) & ~gf.space.FreeDofs(True)
Projector(innerbits, False).Project(gf.vec)
if pre:
CG(mat=bf.mat,
rhs=r,
pre=pre,
sol=gf.vec,
tol=tol,
maxsteps=maxsteps,
initialize=False,
printrates=print)
else:
inv = bf.mat.Inverse(gf.space.FreeDofs(bf.condense), inverse=inverse)
r.data -= bf.mat * gf.vec
gf.vec.data += inv * r
if bf.condense:
gf.vec.data += bf.harmonic_extension * gf.vec
gf.vec.data += bf.inner_solve * r
def PreconditionedRichardson(a,
rhs,
pre=None,
freedofs=None,
maxit=100,
tol=1e-8,
dampfactor=1.0,
printing=True):
""" Preconditioned Richardson Iteration
Parameters
----------
a : BilinearForm
The left hand side of the equation to solve
rhs : Vector
The right hand side of the equation.
pre : Preconditioner
If provided the preconditioner is used.
freedofs : BitArray
The FreeDofs on which the Richardson iteration acts. If argument is 'None' then the FreeDofs of the underlying FESpace is used.
maxit : int
Number of maximal iteration for Richardson iteration. If the maximal number is reached before the tolerance is reached a warning is displayed.
tol : double
Tolerance of the residuum. Richardson iteration stops if residuum < tolerance*initial_residuum is reached.
dampfactor : float
Set the damping factor for the Richardson iteration. If it is 1 then no damping is done. Values greater than 1 are allowed.
printing : bool
Set if Richardson iteration should print informations about the actual iteration like the residuum.
Returns
-------
(vector)
Solution vector of the Preconditioned Richardson iteration.
"""
u = rhs.CreateVector()
r = rhs.CreateVector()
u[:] = 0
projector = Projector(
freedofs if freedofs else a.space.FreeDofs(coupling=a.condense), False)
r.data = rhs # r.data = rhs - a.mat*u
r.data -= projector * r
it = 0
initial_res_norm = Norm(r)
if printing:
print("it =", it, " ||res||_2 =", initial_res_norm)
for it in range(1, maxit + 1):
u.data += dampfactor * (pre * r if pre else r)
r.data = rhs - a.mat * u
r.data -= projector * r
res_norm = Norm(r)
if printing:
print("it =", it, " ||res||_2 =", res_norm)
if res_norm < tol * initial_res_norm:
break
else:
print("Warning: Preconditioned Richardson did not converge to TOL")
return u
def Mult(self,rhs,usol):
proj = Projector(mask=self.fes.FreeDofs(),range=True)
usol[:] = 0.
#
projres = rhs.CreateVector()
projres.data = proj*rhs
normf = Norm(projres)
#
LinDof = self.linmgiter.getSpaceDim()
if( type(self.fes) == ngsolve.comp.CompoundFESpace or type(self.fes) == ngsolve.comp.FESpace):
numspaces = len( self.fes.components )
HOdof = [ self.fes.components[i].ndof for i in range(numspaces) ]
else:
HOdof = [ self.fes.ndof ]
#
print('lindof, hdof: ')
print(LinDof,HOdof)
oldres = Norm(projres)
for it in range(self.maxit):
# initial smoothing
self.blockjac.Smooth(usol, rhs, self.nu)
res = usol.CreateVector()
# perform multiplication more efficiently ...
# only 'upper' part (vertex dofs) needed
res.data = rhs -self.a.mat * usol
#eliminate boundary condition
projres.data = proj*res
mgrhs = self.linmgiter.createVec()
# copy data from two domains
# ... should be done in combination with vertexdofs ...
#mgrhs.Range(0, LinDof[0]).data = projres.Range(0, LinDof[0] )
#mgrhs.Range( LinDof[0], len(mgrhs) ).data = projres.Range(HOdof[0], HOdof[0] + LinDof[1] )
mgrhs.Range(0, LinDof[0]).data = projres.Range(0, LinDof[0] )
if( len(LinDof) > 1 ):
offsetc = 0
offsetf = 0
for i in range( len(LinDof) -1 ):
offsetc += LinDof[i]
offsetf += HOdof[i]
mgrhs.Range( offsetc, offsetc + LinDof[i+1] ).data = projres.Range(offsetf, offsetf + LinDof[i+1] )
# coarse grid correction: multigrid on linears
cup = mgrhs.CreateVector()
cup.data = self.linmgiter*mgrhs #maxit=1,printinfo=False)
# update solution
# ... should be done in combination with vertexdofs ...
usol.Range(0, LinDof[0] ).data += cup.Range(0, LinDof[0])
if( len(LinDof) > 1 ):
offsetc = 0
offsetf = 0
for i in range( len(LinDof) -1 ):
offsetc += LinDof[i]
offsetf += HOdof[i]
usol.Range( offsetf, offsetf + LinDof[i+1] ).data += cup.Range( offsetc, offsetc + LinDof[i+1] )
'''
usol.Range(0, LinDof[0] ).data += cup.Range(0, LinDof[0])
usol.Range( HOdof[0], HOdof[0] + LinDof[1]).data += cup.Range( LinDof[0], len(mgrhs) )
'''
# compute residual for measurement
res.data = rhs - self.a.mat*usol
projres.data = proj*res
res_norm = Norm(projres)
print("tg-it =", it+1, "\t ||res||_2 = {0:.2E}".format(res_norm), "\t reduction: {0:.2f}".format(res_norm/oldres))
if res_norm < self.tol*normf:
break
oldres = res_norm
return usol
def GMRes(A,
b,
pre=None,
freedofs=None,
x=None,
maxsteps=100,
tol=1e-7,
innerproduct=None,
callback=None,
restart=None,
startiteration=0,
printrates=True):
""" Restarting preconditioned gmres solver for A*x=b. Minimizes the preconditioned
residuum pre*(b-A*x)
Parameters
----------
A : BaseMatrix
The left hand side of the linear system.
b : BaseVector
The right hand side of the linear system.
pre : BaseMatrix = None
The preconditioner for the system. If no preconditioner is given, the freedofs
of the system must be given.
freedofs : BitArray = None
Freedofs to solve on, only necessary if no preconditioner is given.
x : BaseVector = None
Startvector, if given it will be modified in the routine and returned. Will be created
if not given.
maxsteps : int = 100
Maximum iteration steps.
tol : float = 1e-7
Tolerance to be computed to. Gmres breaks if norm(pre*(b-A*x)) < tol.
innerproduct : function = None
Innerproduct to be used in iteration, all orthogonalizations/norms are computed with
respect to that inner product.
callback : function = None
If given, this function is called with the solution vector x in each step. Only for debugging
purposes, since it requires the overhead of computing x in every step.
restart : int = None
If given, gmres is restarted with the current solution x every 'restart' steps.
startiteration : int = 0
Internal value to count total number of iterations in restarted setup, no user input required
here.
printrates : bool = True
Print norm of preconditioned residual in each step.
"""
if not innerproduct:
innerproduct = lambda x, y: y.InnerProduct(x, conjugate=True)
norm = Norm
else:
norm = lambda x: sqrt(innerproduct(x, x).real)
is_complex = b.is_complex
if not pre:
assert freedofs is not None
pre = Projector(freedofs, True)
n = len(b)
m = maxsteps
if not x:
x = b.CreateVector()
x[:] = 0
if callback:
xstart = x.CreateVector()
xstart.data = x
else:
xstart = None
sn = Vector(m, is_complex)
cs = Vector(m, is_complex)
sn[:] = 0
cs[:] = 0
r = b.CreateVector()
tmp = b.CreateVector()
tmp.data = b - A * x
r.data = pre * tmp
Q = []
H = []
Q.append(b.CreateVector())
r_norm = norm(r)
if abs(r_norm) < tol:
return x
Q[0].data = 1. / r_norm * r
beta = Vector(m + 1, is_complex)
beta[:] = 0
beta[0] = r_norm
def arnoldi(A, Q, k):
q = b.CreateVector()
tmp.data = A * Q[k]
q.data = pre * tmp
h = Vector(m + 1, is_complex)
h[:] = 0
for i in range(k + 1):
h[i] = innerproduct(Q[i], q)
q.data += (-1) * h[i] * Q[i]
h[k + 1] = norm(q)
if abs(h[k + 1]) < 1e-12:
return h, None
q *= 1. / h[k + 1].real
return h, q
def givens_rotation(v1, v2):
if v2 == 0:
return 1, 0
elif v1 == 0:
return 0, v2 / abs(v2)
else:
t = sqrt((v1.conjugate() * v1 + v2.conjugate() * v2).real)
cs = abs(v1) / t
sn = v1 / abs(v1) * v2.conjugate() / t
return cs, sn
def apply_givens_rotation(h, cs, sn, k):
for i in range(k):
temp = cs[i] * h[i] + sn[i] * h[i + 1]
h[i + 1] = -sn[i].conjugate() * h[i] + cs[i].conjugate() * h[i + 1]
h[i] = temp
cs[k], sn[k] = givens_rotation(h[k], h[k + 1])
h[k] = cs[k] * h[k] + sn[k] * h[k + 1]
h[k + 1] = 0
def calcSolution(k):
mat = Matrix(k + 1, k + 1, is_complex)
for i in range(k + 1):
mat[:, i] = H[i][:k + 1]
rs = Vector(k + 1, is_complex)
rs[:] = beta[:k + 1]
y = mat.I * rs
if xstart:
x.data = xstart
for i in range(k + 1):
x.data += y[i] * Q[i]
for k in range(m):
startiteration += 1
h, q = arnoldi(A, Q, k)
H.append(h)
if q is None:
break
Q.append(q)
apply_givens_rotation(h, cs, sn, k)
beta[k + 1] = -sn[k].conjugate() * beta[k]
beta[k] = cs[k] * beta[k]
error = abs(beta[k + 1])
if printrates:
print("Step", startiteration, ", error = ", error)
if callback:
calcSolution(k)
callback(x)
if error < tol:
break
if restart and k + 1 == restart and not (restart == maxsteps):
calcSolution(k)
del Q
return GMRes(A,
b,
freedofs=freedofs,
pre=pre,
x=x,
maxsteps=maxsteps - restart,
callback=callback,
tol=tol,
innerproduct=innerproduct,
restart=restart,
startiteration=startiteration,
printrates=printrates)
calcSolution(k)
return x
def BVP(bf, lf, gf, \
pre=None, pre_flags={}, \
solver=None, solver_flags={},
maxsteps=200, tol=1e-8, print=True, inverse="umfpack",
needsassembling=True):
"""
Solve a linear boundary value problem A(u,v) = f(v)
Parameters
----------
bf : BilinearForm
provides the matrix.
lf : LinearForm
provides the right hand side.
gf : GridFunction
provides the solution vector
pre : Basematrix or class or string = None
used if an iterative solver is used
can be one of
* a preconditioner object
* a preconditioner class
* a preconditioner class name
pre_flags : dictionary = { }
flags used to create preconditioner
"""
from ngsolve import Projector, Preconditioner
from ngsolve.krylovspace import CG
if isinstance(pre, type):
pre = pre(bf, **pre_flags)
if not needsassembling:
pre.Update()
if isinstance(pre, str):
pre = Preconditioner(bf, pre, **pre_flags)
if not needsassembling:
pre.Update()
if needsassembling:
bf.Assemble()
lf.Assemble()
r = lf.vec.CreateVector()
r.data = lf.vec
if bf.condense:
r.data += bf.harmonic_extension_trans * r
# zero local dofs
innerbits = gf.space.FreeDofs(False) & ~gf.space.FreeDofs(True)
Projector(innerbits, False).Project(gf.vec)
if solver:
inv = solver(mat=bf.mat, pre=pre, **solver_flags)
r.data -= bf.mat * gf.vec
gf.vec.data += inv * r
elif pre:
CG(mat=bf.mat,
rhs=r,
pre=pre,
sol=gf.vec,
tol=tol,
maxsteps=maxsteps,
initialize=False,
printrates=print)
else:
inv = bf.mat.Inverse(gf.space.FreeDofs(bf.condense), inverse=inverse)
r.data -= bf.mat * gf.vec
gf.vec.data += inv * r
if bf.condense:
gf.vec.data += bf.harmonic_extension * gf.vec
gf.vec.data += bf.inner_solve * r