def makeVectors():
'''return two unit vectors that are randomly distributed on a sphere
with the condition that they are mutually perpendicular'''
p1, p2 = [[gauss(0, 1) for i in range(3)] for j in range(2)]
p3 = cross(p1, p2)
norm1 = norm(p1)
norm3 = norm(p3)
p1 = [x / norm1 for x in p1]
p3 = [x / norm3 for x in p3]
return p1, p3
def offset(self, o):
b = deepcopy(self)
pt0 = perp2(b.t0)
pt1 = perp2(b.t1)
c1 = proj(b.bp[0])
c2 = proj(b.bp[1])
c3 = proj(b.bp[2])
t2 = unit(c3 - c1)
pt2 = perp2(t2)
cc1 = unit(c2 - b.p0)
cc2 = unit(b.p1 - c2)
dp1 = dot(b.t0, cc1)
dp2 = dot(b.t1, cc2)
#t2 = unit(pt0 + refl(pt0,cc1))
#t3 = unit(pt1 + refl(pt1,cc2))
t2 = perp2(cc1)
t3 = perp2(cc2)
b.p0 = b.p0 + o * pt0
b.p1 = b.p1 + o * pt1
c2 = c2 + o * pt2
c1 = c1 + o / dp1 * t2
c3 = c3 + o / dp2 * t3
w1 = b.bp[0][2] #dot( b.t0,unit(c2 - b.p0))
w2 = b.bp[2][2] #dot( b.t1,unit(b.p1 - c2))
w1 = dot(b.t0, cc1)
w2 = dot(b.t1, cc2)
alpha = norm(c1 - c2)
beta = norm(c3 - c2)
b.r = alpha / beta
b.bp = (hom(c1, w1), hom(c2, 1), hom(c3, w2), alpha, beta)
b.h1 = [hom(b.p0, 1), b.bp[0], b.bp[1]]
b.h2 = [b.bp[1], b.bp[2], hom(b.p1, 1)]
return b
def __enter__(self):
self.f = self.func(self.X)
f = array([self.f]).T
self.J = self.Dfun(self.X)
self.A = self.J.T.dot(self.J)
self.g = -self.J.T.dot(f)
I = identity(len(self.A))
self.mu = self.tau*diag(self.A)*I
self.F0 = 0.5*f.T.dot(f)[0][0]
self.append((self.iter, self.X, norm(self.f)))
return self
def normalize_spfs(self):
"""Normalizes the spf vectors
"""
for i in range(self.nel):
ind = self.psistart[1,i]
for j in range(self.nmodes):
nspf = self.nspfs[i,j]
npbf = self.npbfs[j]
for k in range(nspf):
nrm = LA.norm(self.psi[ind:ind+npbf])
if abs(nrm) > 1.e-30:
self.psi[ind:ind+npbf] /= nrm
ind += npbf
def __enter__(self):
self.f = self.func(self.X)
f = array([self.f]).T
self.J = self.Dfun(self.X)
self.A = self.J.T.dot(self.J)
self.g = -self.J.T.dot(f)
self.mu = self.tau
self.F0 = 0.5*f.T.dot(f)[0][0]
X = self.X
if len(self.bounds)<>0:
X = type(X)([f(x) for x, f in zip(X, self.box_contraints_transformation)])
if len(self.scaling_of_variables)<>0:
X = [f.inverse(x) for x, f in zip(X, self.scaling_transformation)]
self.append((self.iter, X, norm(self.f)))
return self
def biarc_r_from_arcs(a1, a2):
alpha = norm(a1[0] - proj(a1[1]))
beta = norm(a1[3] - proj(a1[4]))
return alpha / beta
def check_qmr(self):
bx0 = self.x0.copy()
x, info = qmr(self.A, self.b, self.x0, callback=callback)
assert_array_equal(bx0, self.x0)
assert norm(dot(self.A, x) - self.b) < 5*self.tol
(1.0 / self[k][k] * (A[i][k] - s))
except ValueError:
raise LinAlgError('Matrix is not positive definite - Cholesky decomposition cannot be computed')
def solve(self, b):
n = len(b)
c = [0.0 for i in xrange(n)]
for i in xrange(n):
c[i] = (b[i] - sum( self[i][k] * c[k] for k in xrange(i) )) / self[i][i]
x = [0.0 for i in xrange(n)]
for i in xrange(n-1, -1, -1):
x[i] = (c[i] - sum( self[k][i] * x[k] for k in xrange(i+1, n) )) / self[i][i]
return x
if __name__ == '__main__':
A = array([[2, 1, 1, 3, 2],
[1, 2, 2, 1, 1],
[1, 2, 9, 1, 5],
[3, 1, 1, 7, 1],
[2, 1, 5, 1, 8]], dtype=float)
b = array([1, 1, 1, 1, 1], dtype=float)
L = cholesky( A )
X = L.solve(b)
from linalg import norm
from libarray import *
print 'Must be near 0: ', norm(A.dot(array([X]).T)- b)
def fmin(func, x0, Dfun=None, gtol=1.0e-06, xtol=1.0e-04, maxfev=1000, maxiter=50, epsfcn=1e-6, damping=(10e-3, 2.0), verbose=True):
with levmar(func, x0, Dfun=Dfun, gtol=gtol, xtol=xtol, maxfev=maxfev, maxiter=maxiter, epsfcn=epsfcn, damping=damping) as opt:
if verbose:
print 'Start Levenberg Marquart Optimizer...'
print '{step:>7}{x}{residual:>13}'.format(step='step', x='{:>13}'*len(x0), residual='residual').format(*('X[%d]'%i for i in xrange(len(x0))))
print '{step:>7}{x}{residual:>13.3e}'.format(step=0, x='{:>13.3e}'*len(x0), residual=norm(opt.f)).format(*x0)
while 1:
iter=0
try:
iter, X, residual = opt.next()
if verbose:
print '{step:>7}{x}{residual:>13.3e}'.format(step=iter, x='{:>13.3e}'*len(X), residual=residual).format(*X)
except StopIteration, exit_message:
if verbose:
print '---------'
print 'Optimization terminated successfully.'
print ' Exit Message: %s'%(exit_message)
print ' Iterations: %d'%(iter)
print ' Function evaluations: %d'%(opt.ifev)
break
def next(self):
isJnull = self.checkJacobian()
for i, f in enumerate(self.f):
if f == float('nan'):
raise LevMarError('The point %d is a Nan. The optimizer has been stopped.'%i)
# stopping criteria
if norm(self.g.T[0])<self.gtol:
raise StopIteration('Magnitude of gradient smaller than the \'gtol\' tolerance.')
# increment iter
self.iter += 1
if self.iter>self.maxiter:
raise StopIteration('Number of iterations exceeded \'maxiter\' or number of function evaluations exceeded \'maxfev\'.')
# save previous state
self.X0, self.A0, self.g0, self.f0 = self.X.copy(), self.A.copy(), self.g.copy(), self.f.copy(),
# compute dX
I = identity(len(self.A))
self.dX = array(solve(self.A + self.mu, self.g.T[0], verbose=False))
# stopping criteria
#if norm(self.dX)<self.xtol*(norm(self.X)+self.xtol):
# raise StopIteration()
if not(False in [a<b for a, b in zip(abs(self.dX), self.xtol*(abs(self.X)+self.xtol))]):
raise StopIteration('Change in x smaller than the \'xtol\' tolerance.')
self.X = self.X + self.dX
# compute f using the new X
self.f = self.func(self.X)
f = array([self.f]).T
self.Fn = 0.5*f.T.dot(f)[0][0]
dX = array([self.dX]).T
self.dL = (0.5*(dX.T.dot(self.mu.dot(dX) + self.g)))[0][0]
self.dF = self.F0-self.Fn
if len(self.bounds):
isStepAcceptable = not(False in [min(bound)<=x<=max(bound) for x, bound in zip(self.X, self.bounds)])
else:
isStepAcceptable = True
# if step acceptable
if ((self.dF>0 and self.dL>0) or (self.dF<0 and self.dL<0)) and isStepAcceptable:
# compute jacobian, A and g
self.J = self.Dfun(self.X)
self.A = self.J.T.dot(self.J)
self.g = -self.J.T.dot(f)
# damp mu and update F0 parameter
self.mu = self.mu*max(0.333333333, (1.0-(2.0*(self.dF/self.dL)-1.0)**3)/2.0)
self.nu = self.damping[1]
self.F0 = self.Fn
else:
# restore
self.X, self.A, self.g, self.f = self.X0.copy(), self.A0.copy(), self.g0.copy(), self.f0.copy(),
# damp mu
self.mu = self.mu*self.nu
self.nu = 2.0*self.nu
# save and return X and residual
residual = norm(self.f)
self.append((self.iter, self.X, residual))
return self.iter, self.X, residual
except StopIteration, exit_message:
if verbose:
print '---------'
print 'Optimization terminated successfully.'
print ' Exit Message: %s'%(exit_message)
print ' Iterations: %d'%(iter)
print ' Function evaluations: %d'%(opt.ifev)
break
if verbose:
print ' Best step:'
if len(opt):
iter, X, residual = min(list(opt[i] for i in xrange(len(opt))), key=lambda step: step[2])
print '{step:>7}{x}{residual:>13.3e}'.format(step=iter, x='{:>13.3e}'*len(X), residual=residual).format(*X)
else:
iter = 0; residual = 0.0; X=x0
print '{step:>7}{x}{residual:>13}'.format(step=0, x='{:>13.3e}'*len(x0), residual=norm(opt.f)).format(*x0)
if opt.traceback:
print 'LevMarError:', opt.traceback[1]
if verbose:
print
return iter, X, residual
def fit(func, x0, xmeas, ymeas, gtol=1.0e-08, xtol=1.0e-06, maxfev=100, epsfcn=1e-8, damping=(10e-3, 2.0), verbose=True):
def f(X, xmeas, ymeas):
return array([(ymeas-func(x, *X))/ymeas for x in xmeas])
return fmin(f, x0, Dfun=None, gtol=gtol, xtol=xtol, maxfev=maxfev, epsfcn=epsfcn, damping=damping, verbose=verbose)
def makeRand():
'''return a single unit vector randomly distributed on a sphere'''
v1 = [gauss(0, 1) for i in range(3)]
return [x / norm(v1) for x in v1]
def makePerp(v1):
'''return a random unit vector on a circle perpendicular to v1'''
v2 = [gauss(0, 1) for i in range(3)]
v3 = cross(v1, v2)
return [x / norm(v3) for x in v3]
def main(argv):
# MPI init
comm = MPI.COMM_WORLD
rank = comm.Get_rank()
size = comm.Get_size()
# discretization object
data.discretization = data.Discretization()
discretization = data.discretization
# arguments processing
readcmdline(argv)
# remove old files generated by write_binary function
clean_directory(rank)
comm.Barrier()
# domain object (object per SubDomain)
data.domain = data.SubDomain(rank, size, discretization, comm)
domain = data.domain
# redefine variables for shortening the code
nx = domain.nx
ny = domain.ny
nt = discretization.nt
# iteration parameters
max_cg_iters = 200
max_newton_iters = 50
tolerance = 1.e-6
# message of beginning
if domain.rank == 0:
print(
"========================================================================"
)
print("Welcome to mini-stencil!")
print("Version :: Python 3 with MPI4Py: " + str(domain.size) +
" MPI ranks")
print("Mesh :: " + str(discretization.nx) + " * " +
str(discretization.ny) + " dx = " + str(discretization.dx))
print("Iteration :: " + "CG " + str(max_cg_iters) + ", Newton " +
str(max_newton_iters) + ", tolerance " + str(tolerance))
print(
"========================================================================"
)
# details of the cartesian division of the processors
if domain.rank == 0:
print("Cartesian division details for each processor (subdomain)\n")
domain.print()
if domain.rank == 0 and data.verbose_output:
print(
"========================================================================"
)
print("Verbose output\n")
# solution matrices
# ny = ROW and nx = COLUMN (!! VERY IMPORTANT !!)
data.x_new = np.zeros((ny, nx), dtype=np.float64)
data.x_old = np.zeros((ny, nx), dtype=np.float64)
# boundary vectors
# initialized on 0 (dirichlet boundary condition)
data.bndN = np.zeros((1, nx), dtype=np.float64)
data.bndS = np.zeros((1, nx), dtype=np.float64)
data.bndE = np.zeros((1, ny), dtype=np.float64)
data.bndW = np.zeros((1, ny), dtype=np.float64)
# buffer vectors for exchanging boundary information
# they act as temporary dataholders for exchange
data.buffN = np.zeros((1, nx), dtype=np.float64)
data.buffS = np.zeros((1, nx), dtype=np.float64)
data.buffE = np.zeros((1, ny), dtype=np.float64)
data.buffW = np.zeros((1, ny), dtype=np.float64)
# Ax = b with b being 0 in the initial condition
b = np.zeros((ny, nx), dtype=np.float64)
# solution for conjugate gradient function
deltax = np.zeros((ny, nx), dtype=np.float64)
# convergence
if domain.rank == 0:
con = np.zeros((0),
dtype=np.float64) # convergence of the last iteration
res = np.empty((nt), dtype=np.float64) # residual of each timestep
if (data.custom_init):
for i in data.discretization.points:
x = i[0]
y = i[1]
s = i[2]
if (x >= (domain.startx) and x <= (domain.endx) and y >=
(domain.starty) and y <= (domain.endy)):
x -= (domain.startx)
y -= (domain.starty)
data.x_new[y, x] = s
else:
# Default initial condition (!! PROPORTIONAL TO THE GRID SIZE !!)
# A circle of concentration 0.1 centred at (xdim/4, ydim/4)
# with radius no larger than 1/8 of both xdim and ydim
# x lenght is always one while y lenght is variable depending on the dimensions
xc = 1.0 / 4.0
yc = (discretization.ny - 1) * discretization.dx / 4.0
# min ensure that radius will be not larger than 1/8 of xdim or ydim
radius = min(xc, yc) / 2.0
# Startx and starty begins artificially to count from 1 but it is actually 0
for j in range(domain.starty, domain.endy + 1):
# (j - 1) displace the circle slightly on the upper right of the domain
# instead of being in the corner
y = (j) * discretization.dx
for i in range(domain.startx, domain.endx + 1):
# (i - 1) displace the circle slightly on the upper right of the domain
# instead of being in the corner
x = (i) * discretization.dx
# Test if grid point is inside circle
# Actually the circle only lies in the (0, 0) domain
if ((x - xc) * (x - xc) + (y - yc) *
(y - yc)) < (radius * radius):
data.x_new[j - domain.starty][i - domain.startx] = 0.1
# Start time before solving the system
start_time = time.time()
# Main timeloop
for timestep in range(1, nt + 1):
# x_old = x_new
# get back the solution of previous timestep
# to compute the solution of the actual timestep
data.x_old[:] = data.x_new
# new residual at each time step
residual = 0.0
converged = False
# newton iterations used to solve a system of non linear equations
it = 0
for it in range(0, max_newton_iters):
# we get b in Ax = b by using the diffusion function
# b = f(x_n)
operators.diffusion(data.x_new, b, timestep, it)
# residual of newton methods
residual = linalg.norm(b)
# for plotting convergence of the newton methods
if domain.rank == 0 and timestep == nt:
con = np.append(con, residual)
if residual < tolerance:
converged = True
break
# solve linear system to get deltax
# Ax = b where A is J(x) and x deltax (we already have b)
# we have to calculate the Jacobian and then solve to get deltax
# J(x) * deltax = f(x)
cg_converged = False
cg_converged = linalg.cg_solver(deltax, b, max_cg_iters, tolerance,
cg_converged)
# Check that the CG solver converged
if not cg_converged:
break
# x_new = x_new - deltax
# deltax = f(x_n)/f'(x_n) = f(x_n) * J^-1(x)
# deltax got from cg_converged function
data.x_new -= deltax
# stats for printing results
data.iters_newton += it + 1
# Verbose mode (output stats at each Newton iteration)
if converged and data.verbose_output:
print("Step " + str(timestep) + " required " + str(it) +
" iterations for residual " + str(residual))
if not converged:
print("Step " + str(timestep) +
" ERROR : nonlinear iterations failed to converge")
break
# final residual for a series of newton iteration for each timestep
if domain.rank == 0:
res[timestep - 1] = residual
# total computational time for getting the solution
timespent = time.time() - start_time
# Avoid polluating printed output
time.sleep(0.2)
# final results printed
if (domain.rank == 0):
print(
"--------------------------------------------------------------------------------"
)
print("Simulation took " + str(timespent) + " seconds")
print(
str(data.iters_cg) +
" conjugate gradient iterations, at rate of " +
str(data.iters_cg / timespent) + " iters/second")
print(str(data.iters_newton) + " newton iterations")
print(str(data.flops_count) + " floating point operations")
print(
"--------------------------------------------------------------------------------"
)
print("Goodbye !")
# write final solution to BIN file for visualization
# wld way used in the old c version
# wept and modified for showing how to write in one file with MPI
if data.printed_output:
filename = write_binary("output", data.x_old, domain, discretization)
read_binary(filename, domain, discretization)
# interactive visualization of the final solution
# all the subdomains solutions are merged into one matrix using MPI
if data.interactive_output or data.printed_matrix:
# merged results from all subdomains
result = merge_solution(discretization, domain)
# print final matrix result in txt file
if data.printed_matrix and domain.rank == 0:
print("output.txt generated in current directory !")
np.savetxt("./output.txt", result)
# generate interactive visualization
if data.interactive_output and domain.rank == 0:
print(
"(Close the python interactive windows to stop the program.)")
plot_solution(result)
plot_convergence(con)
def next(self):
isJnull = self.checkJacobian()
if 0 in isJnull:
print LevMarWarning('The Jacobian is rank deficient.')
if nan in self.f:
raise LevMarError('One point of \'f\' is a Nan. The optimizer has been stopped.')
# stopping criteria
if norm(self.g)<self.gtol:
raise StopIteration('Magnitude of gradient smaller than the \'gtol\' tolerance.')
# increment iter
self.iter += 1
if self.iter>self.maxiter:
raise StopIteration('Number of iterations exceeded \'maxiter\' or number of function evaluations exceeded \'maxfev\'.')
# save previous state
self.X0 = self.X.copy()
self.A0 = self.A.copy()
self.g0 = self.g.copy()
self.f0 = self.f.copy()
# compute dX
I = identity(len(self.A))
A = self.A
mu = self.mu
g = self.g
def geth(A, g, mu):
mA = max(abs(A.flatten()))
for i in xrange(5):
try:
I = identity(len(A))
L = chol(A + mu*diag(A)*I)
h = L.solve(g)
return array(h), mu
except LinAlgError:
mu = max(10.0*mu, 1e-15*mA)
raise LevMarError('The matrix A + mu*diag(A) is not positive. The optimizer has been stopped.')
#self.dX = array(solve(A + mu*diag(A)*I, g.T[0]))
self.dX, self.mu = geth(A, g.T[0], mu)
# stopping criteria
if norm(self.dX)<self.xtol*(norm(self.X)+self.xtol):
raise StopIteration('Change in x smaller than the \'xtol\' tolerance.')
self.X = self.X + self.dX
# compute f using the new X
self.f = self.func(self.X)
f = array([self.f]).T
dX = array([self.dX]).T
self.Fn = 0.5*f.T.dot(f)[0][0]
# Using the state A, mu, g, h and considering a Taylor expansion model L, the trust region dL = L(h)-L(0) is calculated.
# then dF/dL is the gain of the trust region. dL is positive
self.dL = 0.5*dX.T.dot((self.mu*diag(self.A)*I).dot(dX) + self.g)[0][0]
self.dF = self.F0-self.Fn
# if step acceptable, dF is sometimes null due to machine accuracy
if self.dF>0 and self.dL>0 :
# compute jacobian, A and g
self.J = self.Dfun(self.X)
self.A = self.J.T.dot(self.J)
self.g = -self.J.T.dot(f)
# damp mu and update F0 parameter
self.mu = self.mu*max(1.0/3.0, (1.0-(2.0*(self.dF/self.dL)-1.0)**3)/2.0)
self.nu = self.damping[1]
self.F0 = self.Fn
if norm(self.g0-self.g) < self.gtol:
raise StopIteration('Change in g smaller than the \'gtol\' tolerance.')
else:
# restore
self.X = self.X0.copy()
self.A = self.A0.copy()
self.g = self.g0.copy()
self.f = self.f0.copy()
# damp mu
self.mu = self.mu*self.nu
self.nu = 2.0*self.nu
# save and return X and residual
residual = norm(self.f)
X = self.X
if len(self.bounds)<>0:
X = type(X)([f(x) for x, f in zip(X, self.box_contraints_transformation)])
if len(self.scaling_of_variables)<>0:
X = [f.inverse(x) for x, f in zip(X, self.scaling_transformation)]
self.append((self.iter, X, residual))
return self.iter, X, residual
def proc_orient(line, inFile, fragDict):
'''
This is called when a line containing '!!orient' is read from inFile.
It continues reading inFile until reaching the line '!!oend'. It parses the
arguments on the !!orient line, adds lines for the dummy atoms used to
orient the fragment, and then stores the rest of the lines in the !!orient
block. If the centroid coordinates were not given, it calculates them from
the atoms it reads. It returns a template with placeholders for the dummy
atoms and a Namespace containing the !!orient arguments.
'''
# Process the !!orient line. Find the correct FRAG block from fragDict and
# add dummy atoms to it. Store this modified FRAG block.
orientArgs = parse_instructions(line)
fragLines = make_frag(fragDict[orientArgs.frag])
dummyAtoms = [
'afix 99{}\n'.format(orientArgs.afix),
'REM !! To remove the dummy atoms while preserving the rest of\n'
'REM !! the model, simply refine this model in ShelXL and delete\n'
'REM !! the lines from here through "PERP 1 ..." from the \n'
'REM !! resulting .res file.\n'
'part -1 10.0 !!ornt\n', 'cent 1 cent_xyz 10 10.02 !!ornt\n',
'pivt 1 pivt_xyz 10 10.02 !!ornt\n',
'perp 1 perp_xyz 10 10.02 !!ornt\n'
]
# Read and process lines until !!oend. Atoms are saved to atomMatchList and
# their coordinates are changed to 0 0 0. All other lines are unchanged.
# All lines are written to the list orientLines.
atomMatchList, orientLines = [], []
while True:
atomMatch = atomRE.match(line)
if atomMatch and not line[0].upper() in shelxWords:
atomMatchList.append(atomMatch)
line = atomMatch.group(1) + ' 0 0 0 11 {}\n'.format(
orientArgs.uiso)
if line.lower().startswith('!!oend'):
orientLines.append('afix 0\n')
break
orientLines.append(line)
line = inFile.readline()
# If the centroid was given on the !!orient line, then we name it here;
# otherwise calculate it from the atoms within the orient...oend block.
# If the centroid is on 0,0,0 then we nudge it slightly so that ShelXL
# treats it as a real position.
if orientArgs.x is not None:
cent = [orientArgs.x, orientArgs.y, orientArgs.z]
else:
xyzArray = [[0, 0, 0] for i in range(len(atomMatchList))]
for i in range(len(atomMatchList)):
xyzArray[i] = [
float(atomMatchList[i].group(j)) for j in range(2, 5)
]
cent = centroid(xyzArray)
if norm(cent) < 0.001: cent = vec_sum(cent, [.001, .001, .001])
orientArgs.x, orientArgs.y, orientArgs.z = cent[0], cent[1], cent[2]
orientArgs.cent = cent
outLines = fragLines + dummyAtoms + orientLines
return outLines, orientArgs