def PINVIT(mata,
matm,
pre,
num=1,
maxit=20,
printrates=True,
GramSchmidt=False):
"""preconditioned inverse iteration"""
r = mata.CreateRowVector()
Av = mata.CreateRowVector()
Mv = mata.CreateRowVector()
uvecs = []
for i in range(num):
uvecs.append(mata.CreateRowVector())
vecs = []
for i in range(2 * num):
vecs.append(mata.CreateRowVector())
for v in uvecs:
r.FV().NumPy()[:] = random.rand(len(r.FV()))
v.data = pre * r
asmall = Matrix(2 * num, 2 * num)
msmall = Matrix(2 * num, 2 * num)
lams = num * [1]
for i in range(maxit):
for j in range(num):
vecs[j].data = uvecs[j]
r.data = mata * vecs[j] - lams[j] * matm * vecs[j]
vecs[num + j].data = pre * r
if GramSchmidt:
Orthogonalize(vecs, matm)
for j in range(2 * num):
Av.data = mata * vecs[j]
Mv.data = matm * vecs[j]
for k in range(2 * num):
asmall[j, k] = InnerProduct(Av, vecs[k])
msmall[j, k] = InnerProduct(Mv, vecs[k])
ev, evec = scipy.linalg.eigh(a=asmall, b=msmall)
lams[:] = ev[0:num]
if printrates:
print(i, ":", lams)
for j in range(num):
uvecs[j][:] = 0.0
for k in range(2 * num):
uvecs[j].data += float(evec[k, j]) * vecs[k]
return lams, uvecs
def Orthogonalize(vecs, mat):
mv = []
for i in range(len(vecs)):
for j in range(i):
vecs[i] -= InnerProduct(vecs[i], mv[j]) * vecs[j]
hv = mat.CreateRowVector()
hv.data = mat * vecs[i]
norm = sqrt(InnerProduct(vecs[i], hv))
vecs[i] *= 1 / norm
hv *= 1 / norm
mv.append(hv)
def pcg(A, B, b, x=None, tol=1.e-16, maxits=100, saveitfn=None):
"""Preconditioned Conjugate Gradient iterations for solving the
B-preconditioned A-linear system
B A x = B b
in the inv(B)-inner product.
"""
if x == None:
x = b.CreateVector() # if not given initial guess,
x[:] = 0.0 # set initial solution vector = 0
r = b.CreateVector() # r = residual
p = b.CreateVector() # p = search direction
Br = b.CreateVector() # for storing B*r and A*p
Ap = Br
r.data = b - A * x
Br.data = B * r
p.data = Br
rBr = [0, InnerProduct(r, Br)] # 2 consecutive residual B-norms
for it in range(maxits):
rBr[0] = rBr[1] # update residual's B-norm
Ap.data = A * p
pAp = InnerProduct(p, Ap)
alpha = rBr[0] / pAp # alpha = <r, B*r> / <p, A*p>
x.data += alpha * p # x = x + alpha * p
r.data += (-alpha) * Ap # r1 = r0 - alpha * A*p
Br.data = B * r
rBr[1] = InnerProduct(r, Br)
beta = rBr[1] / rBr[0] # beta = <r1, B*r1> / <r0, B*r0>
p.data = beta * p
p.data += Br # p = B*r1 + beta * p
if saveitfn != None:
saveitfn(x, it)
if abs(rBr[1].imag) > 1.e-8 or abs(pAp.imag) > 1.e-8:
print('\n*** rBr=', rBr, '\n*** pAp=', pAp)
print('*** System not Hermitian!')
else:
if rBr[0].real * rBr[1].real < 0:
print('\n*** Preconditioner indefinite!')
print('PCG%6d' % it, ': pAp=%12.9g %12.9g' % (pAp.real, rBr[1].real))
if abs(pAp) < tol or abs(rBr[1]) < tol:
break
return x
def Solve(self, maxit=100, maxerr=1e-11, dampfactor=1,
printing=False, callback=None, linesearch=False,
printenergy=False, print_wrong_direction=False):
numit = 0
err = 1.
a, u, w, r,uh = self.a, self.u, self.w, self.r, self.uh
for it in range(maxit):
numit += 1
if printing:
print("Newton iteration ", it)
if printenergy:
print("Energy: ", a.Energy(u.vec))
a.AssembleLinearization(u.vec)
a.Apply(u.vec, r)
self._UpdateInverse()
if self.rhs is not None:
r.data -= self.rhs.vec
if a.condense:
r.data += a.harmonic_extension_trans * r
w.data = self.inv * r
w.data += a.harmonic_extension * w
w.data += a.inner_solve * r
else:
w.data = self.inv * r
err2 = InnerProduct(w,r)
if err2 < 0:
if print_wrong_direction:
print("wrong direction")
err = sqrt(abs(err2))
if printing:
print("err = ", err)
tau = min(1, numit*dampfactor)
if linesearch:
uh.data = u.vec - tau*w
energy = a.Energy(u.vec)
while a.Energy(uh) > energy+(max(1e-14*abs(energy),maxerr)) and tau > 1e-10:
tau *= 0.5
uh.data = u.vec - tau * w
if printing:
print ("tau = ", tau)
print ("energy uh = ", a.Energy(uh))
u.vec.data = uh
else:
u.vec.data -= tau * w
if callback is not None:
callback(it, err)
if abs(err) < maxerr: break
else:
print("Warning: Newton might not converge! Error = ", err)
return (-1,numit)
return (0,numit)
def PINVIT(mata,
matm,
pre,
num=1,
maxit=20,
printrates=True,
GramSchmidt=False):
"""preconditioned inverse iteration"""
r = mata.CreateRowVector()
uvecs = MultiVector(r, num)
vecs = MultiVector(r, 2 * num)
# hv = MultiVector(r, 2*num)
for v in vecs[0:num]:
v.SetRandom()
uvecs[:] = pre * vecs[0:num]
lams = Vector(num * [1])
for i in range(maxit):
vecs[0:num] = mata * uvecs - (matm * uvecs).Scale(lams)
vecs[num:2 * num] = pre * vecs[0:num]
vecs[0:num] = uvecs
vecs.Orthogonalize() # matm)
# hv[:] = mata * vecs
# asmall = InnerProduct (vecs, hv)
# hv[:] = matm * vecs
# msmall = InnerProduct (vecs, hv)
asmall = InnerProduct(vecs, mata * vecs)
msmall = InnerProduct(vecs, matm * vecs)
ev, evec = scipy.linalg.eigh(a=asmall, b=msmall)
lams = Vector(ev[0:num])
if printrates:
print(i, ":", list(lams))
uvecs[:] = vecs * Matrix(evec[:, 0:num])
return lams, uvecs
def CG(mat,
rhs,
pre=None,
sol=None,
tol=1e-12,
maxsteps=100,
printrates=True,
initialize=True,
conjugate=False):
"""preconditioned conjugate gradient method"""
u = sol if sol else rhs.CreateVector()
d = rhs.CreateVector()
w = rhs.CreateVector()
s = rhs.CreateVector()
if initialize: u[:] = 0.0
d.data = rhs - mat * u
w.data = pre * d if pre else d
err0 = sqrt(abs(InnerProduct(d, w)))
s.data = w
# wdn = InnerProduct (w,d)
wdn = w.InnerProduct(d, conjugate=conjugate)
if wdn == 0:
return u
for it in range(maxsteps):
w.data = mat * s
wd = wdn
# as_s = InnerProduct (s, w)
as_s = s.InnerProduct(w, conjugate=conjugate)
alpha = wd / as_s
u.data += alpha * s
d.data += (-alpha) * w
w.data = pre * d if pre else d
# wdn = InnerProduct (w, d)
wdn = w.InnerProduct(d, conjugate=conjugate)
beta = wdn / wd
s *= beta
s.data += w
err = sqrt(abs(wd))
if err < tol * err0: break
if printrates:
print("it = ", it, " err = ", err)
return u
def CG(mat, rhs, pre=None, sol=None, maxsteps = 100, printrates = True, initialize = True):
"""preconditioned conjugate gradient method"""
u = sol if sol else rhs.CreateVector()
d = rhs.CreateVector()
w = rhs.CreateVector()
s = rhs.CreateVector()
if initialize: u[:] = 0.0
d.data = rhs - mat * u
w.data = pre * d if pre else d
s.data = w
wdn = InnerProduct (w,d)
for it in range(maxsteps):
w.data = mat * s
wd = wdn
as_s = InnerProduct (s, w)
alpha = wd / as_s
u.data += alpha * s
d.data += (-alpha) * w
w.data = pre*d if pre else d
wdn = InnerProduct (w, d)
beta = wdn / wd
s *= beta
s.data += w
err = sqrt(wd)
if err < 1e-16: break
if printrates:
print ("it = ", it, " err = ", err)
return u
def pcg(mat, pre, rhs, maxits=100):
"""preconditioned conjugate gradient method"""
u = rhs.CreateVector()
d = rhs.CreateVector()
w = rhs.CreateVector()
s = rhs.CreateVector()
u[:] = 0.0
d.data = rhs - mat * u
w.data = pre * d
s.data = w
wdn = InnerProduct(w, d)
print("|u0| = ", wdn)
for it in range(maxits):
w.data = mat * s
wd = wdn
Ass = InnerProduct(s, w)
alpha = wd / Ass
u.data += alpha * s
d.data += (-alpha) * w
w.data = pre * d
wdn = InnerProduct(w, d)
beta = wdn / wd
s *= beta
s.data += w
err = sqrt(wd)
if err < 1e-16:
break
print("it = ", it, " err = ", err)
return u
def Arnoldi(mat, tol=1e-10, maxiter=200):
H = Matrix(maxiter, maxiter, complex=mat.is_complex)
H[:, :] = 0
v = mat.CreateVector(colvector=False)
abv = MultiVector(v, 0)
v.SetRandom()
v /= Norm(v)
for i in range(maxiter):
abv.Append(v)
v = (mat * v).Evaluate()
for j in range(i + 1):
H[j, i] = InnerProduct(v, abv[j])
v -= H[j, i] * abv[j]
if i + 1 < maxiter:
H[i + 1, i] = Norm(v)
v = 1 / Norm(v) * v
lam, ev = scipy.linalg.eig(H)
return Vector(lam), (abv * Matrix(ev)).Evaluate()
def QMR(mat,
rhs,
fdofs,
pre1=None,
pre2=None,
sol=None,
maxsteps=100,
printrates=True,
initialize=True,
ep=1.0,
tol=1e-7):
u = sol if sol else rhs.CreateVector()
r = rhs.CreateVector()
v = rhs.CreateVector()
v_tld = rhs.CreateVector()
w = rhs.CreateVector()
w_tld = rhs.CreateVector()
y = rhs.CreateVector()
y_tld = rhs.CreateVector()
z = rhs.CreateVector()
z_tld = rhs.CreateVector()
p = rhs.CreateVector()
p_tld = rhs.CreateVector()
q = rhs.CreateVector()
d = rhs.CreateVector()
s = rhs.CreateVector()
if (initialize): u[:] = 0.0
r.data = rhs - mat * u
v_tld.data = r
y.data = pre1 * v_tld if pre1 else v_tld
rho = InnerProduct(y, y)
rho = sqrt(rho)
w_tld.data = r
z.data = pre2.T * w_tld if pre2 else w_tld
xi = InnerProduct(z, z)
xi = sqrt(xi)
gamma = 1.0
eta = -1.0
theta = 0.0
for i in range(1, maxsteps + 1):
if (rho == 0.0):
print('Breakdown in rho')
return
if (xi == 0.0):
print('Breakdown in xi')
return
v.data = (1.0 / rho) * v_tld
y.data = (1.0 / rho) * y
w.data = (1.0 / xi) * w_tld
z.data = (1.0 / xi) * z
delta = InnerProduct(z, y)
if (delta == 0.0):
print('Breakdown in delta')
return
y_tld.data = pre2 * y if pre2 else y
z_tld.data = pre1.T * z if pre1 else z
if (i > 1):
p.data = (-xi * delta / ep) * p
p.data += y_tld
q.data = (-rho * delta / ep) * q
q.data += z_tld
else:
p.data = y_tld
q.data = z_tld
p_tld.data = mat * p
ep = InnerProduct(q, p_tld)
if (ep == 0.0):
print('Breakdown in epsilon')
return
beta = ep / delta
if (beta == 0.0):
print('Breakdown in beta')
return
v_tld.data = p_tld - beta * v
y.data = pre1 * v_tld if pre1 else v_tld
rho_1 = rho
rho = InnerProduct(y, y)
rho = sqrt(rho)
w_tld.data = mat.T * q
w_tld.data -= beta * w
z.data = pre2.T * w_tld if pre2 else w_tld
xi = InnerProduct(z, z)
xi = sqrt(xi)
gamma_1 = gamma
theta_1 = theta
theta = rho / (gamma_1 * abs(beta))
gamma = 1.0 / sqrt(1.0 + theta * theta)
if (gamma == 0.0):
print('Breakdown in gamma')
return
eta = -eta * rho_1 * gamma * gamma / (beta * gamma_1 * gamma_1)
if (i > 1):
d.data = (theta_1 * theta_1 * gamma * gamma) * d
d.data += eta * p
s.data = (theta_1 * theta_1 * gamma * gamma) * s
s.data += eta * p_tld
else:
d.data = eta * p
s.data = eta * p_tld
u.data += d
r.data -= s
#Projected residuum: Better terminating condition necessary?
ResNorm = sqrt(np.dot(r.FV().NumPy()[fdofs], r.FV().NumPy()[fdofs]))
#ResNorm = sqrt(InnerProduct(r,r))
if (ResNorm <= tol):
break
if (printrates):
print("it = ", i, " err = ", ResNorm)
def MinRes(mat,
rhs,
pre=None,
sol=None,
maxsteps=100,
printrates=True,
initialize=True,
tol=1e-7):
u = sol if sol else rhs.CreateVector()
v_new = rhs.CreateVector()
v = rhs.CreateVector()
v_old = rhs.CreateVector()
w_new = rhs.CreateVector()
w = rhs.CreateVector()
w_old = rhs.CreateVector()
z_new = rhs.CreateVector()
z = rhs.CreateVector()
mz = rhs.CreateVector()
if (initialize):
u[:] = 0.0
v.data = rhs
else:
v.data = rhs - mat * u
z.data = pre * v if pre else v
#First Step
gamma = sqrt(InnerProduct(z, v))
gamma_new = 0
z.data = 1 / gamma * z
v.data = 1 / gamma * v
ResNorm = gamma
ResNorm_old = gamma
if (printrates):
print("it = ", 0, " err = ", ResNorm)
eta_old = gamma
c_old = 1
c = 1
s_new = 0
s = 0
s_old = 0
v_old[:] = 0.0
w_old[:] = 0.0
w[:] = 0.0
k = 1
while (k < maxsteps + 1 and ResNorm > tol):
mz.data = mat * z
delta = InnerProduct(mz, z)
v_new.data = mz - delta * v - gamma * v_old
z_new.data = pre * v_new if pre else v_new
gamma_new = sqrt(InnerProduct(z_new, v_new))
z_new *= 1 / gamma_new
v_new *= 1 / gamma_new
alpha0 = c * delta - c_old * s * gamma
alpha1 = sqrt(alpha0 * alpha0 + gamma_new * gamma_new) #**
alpha2 = s * delta + c_old * c * gamma
alpha3 = s_old * gamma
c_new = alpha0 / alpha1
s_new = gamma_new / alpha1
w_new.data = z - alpha3 * w_old - alpha2 * w
w_new.data = 1 / alpha1 * w_new
u.data += c_new * eta_old * w_new
eta = -s_new * eta_old
#update of residuum
ResNorm = abs(s_new) * ResNorm_old
if (printrates):
print("it = ", k, " err = ", ResNorm)
k += 1
# shift vectors by renaming
v_old, v, v_new = v, v_new, v_old
w_old, w, w_new = w, w_new, w_old
z, z_new = z_new, z
eta_old = eta
s_old = s
s = s_new
c_old = c
c = c_new
gamma = gamma_new
ResNorm_old = ResNorm
def NewtonMinimization(a,
u,
freedofs=None,
maxit=100,
maxerr=1e-11,
inverse="umfpack",
el_int=False,
dampfactor=1,
linesearch=False,
printing=True):
"""
Newton's method for solving non-linear problems of the form A(u)=0 involving energy integrators.
Parameters
----------
a : BilinearForm
The BilinearForm of the non-linear variational problem. It does not have to be assembled.
u : GridFunction
The GridFunction where the solution is saved. The values are used as initial guess for Newton's method.
freedofs : BitArray
The FreeDofs on which the assembled matrix is inverted. If argument is 'None' then the FreeDofs of the underlying FESpace is used.
maxit : int
Number of maximal iteration for Newton. If the maximal number is reached before the maximal error Newton might no converge and a warning is displayed.
maxerr : float
The maximal error which Newton should reach before it stops. The error is computed by the square root of the inner product of the residuum and the correction.
inverse : string
A string of the sparse direct solver which should be solved for inverting the assembled Newton matrix.
el_int : bool
Flag if eliminate internal flag is used. If this is set to True then Newton uses static condensation to invert the matrix. If freedofs is not None, the user has to take care that the local dofs are set to zero in the freedofs array.
dampfactor : float
Set the damping factor for Newton's method. If dampfactor is 1 then no damping is done. If value is < 1 then the damping is done by the formula 'min(1,dampfactor*numit)' for the correction, where 'numit' denotes the Newton iteration.
linesearch : bool
If True then linesearch is used to guarantee that the energy decreases in every Newton iteration.
printing : bool
Set if Newton's method should print informations about the actual iteration like the error.
Returns
-------
(int, int)
List of two integers. The first one is 0 if Newton's method did converge, -1 otherwise. The second one gives the number of Newton iterations needed.
"""
w = u.vec.CreateVector()
r = u.vec.CreateVector()
uh = u.vec.CreateVector()
err = 1
numit = 0
inv = None
for it in range(maxit):
numit += 1
if printing:
print("Newton iteration ", it)
print("energy = ", a.Energy(u.vec))
a.Apply(u.vec, r)
a.AssembleLinearization(u.vec)
inv = a.mat.Inverse(freedofs if freedofs else u.space.FreeDofs(el_int),
inverse=inverse)
if el_int:
r.data += a.harmonic_extension_trans * r
w.data = inv * r
w.data += a.harmonic_extension * w
w.data += a.inner_solve * r
else:
w.data = inv * r
err = sqrt(abs(InnerProduct(w, r)))
if printing:
print("err = ", err)
energy = a.Energy(u.vec)
uh.data = u.vec - min(1, numit * dampfactor) * w
tau = min(1, numit * dampfactor)
if linesearch:
while a.Energy(uh) > energy + 1e-15:
tau *= 0.5
uh.data = u.vec - tau * w
if printing:
print("tau = ", tau)
print("energy uh = ", a.Energy(uh))
u.vec.data = uh
if abs(err) < maxerr: break
else:
print("Warning: Newton might not converge!")
return (-1, numit)
return (0, numit)
def QMR(mat,
rhs,
fdofs,
pre1=None,
pre2=None,
sol=None,
maxsteps=100,
printrates=True,
initialize=True,
ep=1.0,
tol=1e-7):
"""Quasi Minimal Residuum method
Parameters
----------
mat : Matrix
The left hand side of the equation to solve
rhs : Vector
The right hand side of the equation.
fdofs : BitArray
BitArray of free degrees of freedoms.
pre1 : Preconditioner
First preconditioner if provided
pre2 : Preconditioner
Second preconditioner if provided
sol : Vector
Start vector for QMR method, if initialize is set False. Gets overwritten by the solution vector. If sol = None then a new vector is created.
maxsteps : int
Number of maximal steps for QMR. If the maximal number is reached before the tolerance is reached QMR stops.
printrates : bool
If set to True then the error of the iterations is displayed.
initialize : bool
If set to True then the initial guess for the QMR method is set to zero. Otherwise the values of the vector sol, if provided, is used.
ep : double
Start epsilon.
tol : double
Tolerance of the residuum. QMR stops if tolerance is reached.
Returns
-------
(vector)
Solution vector of the QMR method.
"""
u = sol if sol else rhs.CreateVector()
r = rhs.CreateVector()
v = rhs.CreateVector()
v_tld = rhs.CreateVector()
w = rhs.CreateVector()
w_tld = rhs.CreateVector()
y = rhs.CreateVector()
y_tld = rhs.CreateVector()
z = rhs.CreateVector()
z_tld = rhs.CreateVector()
p = rhs.CreateVector()
p_tld = rhs.CreateVector()
q = rhs.CreateVector()
d = rhs.CreateVector()
s = rhs.CreateVector()
if (initialize): u[:] = 0.0
r.data = rhs - mat * u
v_tld.data = r
y.data = pre1 * v_tld if pre1 else v_tld
rho = InnerProduct(y, y)
rho = sqrt(rho)
w_tld.data = r
z.data = pre2.T * w_tld if pre2 else w_tld
xi = InnerProduct(z, z)
xi = sqrt(xi)
gamma = 1.0
eta = -1.0
theta = 0.0
for i in range(1, maxsteps + 1):
if (rho == 0.0):
print('Breakdown in rho')
return
if (xi == 0.0):
print('Breakdown in xi')
return
v.data = (1.0 / rho) * v_tld
y.data = (1.0 / rho) * y
w.data = (1.0 / xi) * w_tld
z.data = (1.0 / xi) * z
delta = InnerProduct(z, y)
if (delta == 0.0):
print('Breakdown in delta')
return
y_tld.data = pre2 * y if pre2 else y
z_tld.data = pre1.T * z if pre1 else z
if (i > 1):
p.data = (-xi * delta / ep) * p
p.data += y_tld
q.data = (-rho * delta / ep) * q
q.data += z_tld
else:
p.data = y_tld
q.data = z_tld
p_tld.data = mat * p
ep = InnerProduct(q, p_tld)
if (ep == 0.0):
print('Breakdown in epsilon')
return
beta = ep / delta
if (beta == 0.0):
print('Breakdown in beta')
return
v_tld.data = p_tld - beta * v
y.data = pre1 * v_tld if pre1 else v_tld
rho_1 = rho
rho = InnerProduct(y, y)
rho = sqrt(rho)
w_tld.data = mat.T * q
w_tld.data -= beta * w
z.data = pre2.T * w_tld if pre2 else w_tld
xi = InnerProduct(z, z)
xi = sqrt(xi)
gamma_1 = gamma
theta_1 = theta
theta = rho / (gamma_1 * abs(beta))
gamma = 1.0 / sqrt(1.0 + theta * theta)
if (gamma == 0.0):
print('Breakdown in gamma')
return
eta = -eta * rho_1 * gamma * gamma / (beta * gamma_1 * gamma_1)
if (i > 1):
d.data = (theta_1 * theta_1 * gamma * gamma) * d
d.data += eta * p
s.data = (theta_1 * theta_1 * gamma * gamma) * s
s.data += eta * p_tld
else:
d.data = eta * p
s.data = eta * p_tld
u.data += d
r.data -= s
#Projected residuum: Better terminating condition necessary?
ResNorm = sqrt(np.dot(r.FV().NumPy()[fdofs], r.FV().NumPy()[fdofs]))
#ResNorm = sqrt(InnerProduct(r,r))
if (printrates):
print("it = ", i, " err = ", ResNorm)
if (ResNorm <= tol):
break
else:
print("Warning: QMR did not converge to TOL")
return u
def CG(mat,
rhs,
pre=None,
sol=None,
tol=1e-12,
maxsteps=100,
printrates=True,
initialize=True,
conjugate=False):
"""preconditioned conjugate gradient method
Parameters
----------
mat : Matrix
The left hand side of the equation to solve. The matrix has to be spd or hermitsch.
rhs : Vector
The right hand side of the equation.
pre : Preconditioner
If provided the preconditioner is used.
sol : Vector
Start vector for CG method, if initialize is set False. Gets overwritten by the solution vector. If sol = None then a new vector is created.
tol : double
Tolerance of the residuum. CG stops if tolerance is reached.
maxsteps : int
Number of maximal steps for CG. If the maximal number is reached before the tolerance is reached CG stops.
printrates : bool
If set to True then the error of the iterations is displayed.
initialize : bool
If set to True then the initial guess for the CG method is set to zero. Otherwise the values of the vector sol, if provided, is used.
conjugate : bool
If set to True, then the complex inner product is used.
Returns
-------
(vector)
Solution vector of the CG method.
"""
u = sol if sol else rhs.CreateVector()
d = rhs.CreateVector()
w = rhs.CreateVector()
s = rhs.CreateVector()
if initialize: u[:] = 0.0
d.data = rhs - mat * u
w.data = pre * d if pre else d
err0 = sqrt(abs(InnerProduct(d, w)))
s.data = w
# wdn = InnerProduct (w,d)
wdn = w.InnerProduct(d, conjugate=conjugate)
if wdn == 0:
return u
for it in range(maxsteps):
w.data = mat * s
wd = wdn
# as_s = InnerProduct (s, w)
as_s = s.InnerProduct(w, conjugate=conjugate)
alpha = wd / as_s
u.data += alpha * s
d.data += (-alpha) * w
w.data = pre * d if pre else d
# wdn = InnerProduct (w, d)
wdn = w.InnerProduct(d, conjugate=conjugate)
beta = wdn / wd
s *= beta
s.data += w
err = sqrt(abs(wd))
if printrates:
print("it = ", it, " err = ", err)
if err < tol * err0: break
else:
print("Warning: CG did not converge to TOL")
return u
def MinRes(mat,
rhs,
pre=None,
sol=None,
maxsteps=100,
printrates=True,
initialize=True,
tol=1e-7):
"""Minimal Residuum method
Parameters
----------
mat : Matrix
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.
sol : Vector
Start vector for MinRes method, if initialize is set False. Gets overwritten by the solution vector. If sol = None then a new vector is created.
maxsteps : int
Number of maximal steps for MinRes. If the maximal number is reached before the tolerance is reached MinRes stops.
printrates : bool
If set to True then the error of the iterations is displayed.
initialize : bool
If set to True then the initial guess for the MinRes method is set to zero. Otherwise the values of the vector sol, if prevented, is used.
tol : double
Tolerance of the residuum. MinRes stops if tolerance is reached.
Returns
-------
(vector)
Solution vector of the MinRes method.
"""
u = sol if sol else rhs.CreateVector()
v_new = rhs.CreateVector()
v = rhs.CreateVector()
v_old = rhs.CreateVector()
w_new = rhs.CreateVector()
w = rhs.CreateVector()
w_old = rhs.CreateVector()
z_new = rhs.CreateVector()
z = rhs.CreateVector()
mz = rhs.CreateVector()
if (initialize):
u[:] = 0.0
v.data = rhs
else:
v.data = rhs - mat * u
z.data = pre * v if pre else v
#First Step
gamma = sqrt(InnerProduct(z, v))
gamma_new = 0
z.data = 1 / gamma * z
v.data = 1 / gamma * v
ResNorm = gamma
ResNorm_old = gamma
if (printrates):
print("it = ", 0, " err = ", ResNorm)
eta_old = gamma
c_old = 1
c = 1
s_new = 0
s = 0
s_old = 0
v_old[:] = 0.0
w_old[:] = 0.0
w[:] = 0.0
k = 1
while (k < maxsteps + 1 and ResNorm > tol):
mz.data = mat * z
delta = InnerProduct(mz, z)
v_new.data = mz - delta * v - gamma * v_old
z_new.data = pre * v_new if pre else v_new
gamma_new = sqrt(InnerProduct(z_new, v_new))
z_new *= 1 / gamma_new
v_new *= 1 / gamma_new
alpha0 = c * delta - c_old * s * gamma
alpha1 = sqrt(alpha0 * alpha0 + gamma_new * gamma_new) #**
alpha2 = s * delta + c_old * c * gamma
alpha3 = s_old * gamma
c_new = alpha0 / alpha1
s_new = gamma_new / alpha1
w_new.data = z - alpha3 * w_old - alpha2 * w
w_new.data = 1 / alpha1 * w_new
u.data += c_new * eta_old * w_new
eta = -s_new * eta_old
#update of residuum
ResNorm = abs(s_new) * ResNorm_old
if (printrates):
print("it = ", k, " err = ", ResNorm)
k += 1
# shift vectors by renaming
v_old, v, v_new = v, v_new, v_old
w_old, w, w_new = w, w_new, w_old
z, z_new = z_new, z
eta_old = eta
s_old = s
s = s_new
c_old = c
c = c_new
gamma = gamma_new
ResNorm_old = ResNorm
else:
print("Warning: MinRes did not converge to TOL")
return u
def Newton(a,
u,
freedofs=None,
maxit=100,
maxerr=1e-11,
inverse="umfpack",
dampfactor=1,
printing=True):
"""
Newton's method for solving non-linear problems of the form A(u)=0.
Parameters
----------
a : BilinearForm
The BilinearForm of the non-linear variational problem. It does not have to be assembled.
u : GridFunction
The GridFunction where the solution is saved. The values are used as initial guess for Newton's method.
freedofs : BitArray
The FreeDofs on which the assembled matrix is inverted. If argument is 'None' then the FreeDofs of the underlying FESpace is used.
maxit : int
Number of maximal iteration for Newton. If the maximal number is reached before the maximal error Newton might no converge and a warning is displayed.
maxerr : float
The maximal error which Newton should reach before it stops. The error is computed by the square root of the inner product of the residuum and the correction.
inverse : string
A string of the sparse direct solver which should be solved for inverting the assembled Newton matrix.
dampfactor : float
Set the damping factor for Newton's method. If dampfactor is 1 then no damping is done. If value is < 1 then the damping is done by the formula 'min(1,dampfactor*numit)' for the correction, where 'numit' denotes the Newton iteration.
printing : bool
Set if Newton's method should print informations about the actual iteration like the error.
Returns
-------
(int, int)
List of two integers. The first one is 0 if Newton's method did converge, -1 otherwise. The second one gives the number of Newton iterations needed.
"""
w = u.vec.CreateVector()
r = u.vec.CreateVector()
err = 1
numit = 0
inv = None
for it in range(maxit):
numit += 1
if printing:
print("Newton iteration ", it)
a.Apply(u.vec, r)
a.AssembleLinearization(u.vec)
inv = a.mat.Inverse(
freedofs if freedofs else u.space.FreeDofs(a.condense),
inverse=inverse)
if a.condense:
r.data += a.harmonic_extension_trans * r
w.data = inv * r
w.data += a.harmonic_extension * w
w.data += a.inner_solve * r
else:
w.data = inv * r
err = sqrt(abs(InnerProduct(w, r)))
if printing:
print("err = ", err)
u.vec.data -= min(1, numit * dampfactor) * w
if abs(err) < maxerr: break
else:
print("Warning: Newton might not converge!")
return (-1, numit)
return (0, numit)