def BuildFilteredMatrix(A, neighborhoods, tol):
tab=Tab()
print('{}in BuildFilteredMatrix()'.format(tab))
timer = Timer('BuildFilteredMatrix')
timer.start()
# Expects A as csr
Af = A.copy()
for i in range(A.shape[0]):
N = neighborhoods[i]
start = Af.indptr[i]
end = Af.indptr[i+1]
for k in range(start, end):
j = Af.indices[k]
if i==j:
iPtr = k
break
for k in range(start, end):
j = Af.indices[k]
if j not in N:
Af.data[iPtr] -= Af.data[k]
Af.data[k]=0
timer.stop()
return Af
def makeProlongator(self, lev):
'''Make the prologator to go from level k to k+1.'''
tab = Tab()
print('{}making prolongator from level {} to {}'.format(tab,lev,lev+1))
(I_up, aggregates) = SA_coarsen(
self.matrix(lev+1), tol=self.tol, lvl=lev+1
)
return I_up
def reportFailure(self, iter, normR, normB):
'''If control.showFinal==True, print a message upon failure to converge.'''
tab = Tab()
if self._control.showFinal:
normRel = normR
if normR != 0:
normRel = normR / normB
print('%s%s solve FAILED: iters=%7d, ||r||/r0=%12.5g' %
(tab, self.name(), iter, normR / normB))
def reportSuccess(self, iter, normR, normB):
'''If control.showFinal==True, print a message upon convergence.'''
tab = Tab()
if self._control.showFinal:
normRel = normR
if normR != 0:
normRel = normR / normB
print('%s%s solve succeeded: iters=%7d, ||r||/r0=%12.5g' %
(tab, self.name(), iter, normRel))
def reportIter(self, iter, normR, normR0):
'''
If control.showIters==True, print information about the current iteration
at intervals specified by the control.interval parameter. Otherwise,
do nothing.
'''
tab = Tab()
if self._control.showIters and (iter % self._control.interval) == 0:
print('%s%s iter=%7d ||r||=%12.5g ||r||/r0=%12.5g' %
(tab, self.name(), iter, normR, normR / normR0))
def BuildTentativeProlongator(A, aggregates):
tab=Tab()
timer = Timer('BuildTentativeProlongator')
timer.start()
print('{}in BuildTentativeProlongator()'.format(tab))
P = sp.dok_matrix((A.shape[0], len(aggregates)))
for i in range(len(aggregates)):
for j in aggregates[i]:
P[j,i] = 1
timer.stop()
return P
def getNeighborhood(A, i, tol, a_diag):
tab = Tab()
#print('{}in getNeighborhood()'.format(tab))
N = {i}
a_ii = a_diag[i]
start = A.indptr[i]
end = A.indptr[i+1]
for k in range(start, end):
j = A.indices[k]
a_ij = A.data[k]
a_jj = a_diag[j]
if abs(a_ij) >= tol*np.sqrt(a_ii*a_jj):
N.add(j)
return N
def solve(self, A, b):
tab = Tab()
# Get size of matrix
n, nc = A.shape
# Make sure matrix is square
assert (n == nc)
# Make sure A and b are compatible
assert (n == len(b))
# Check for the trivial case b=0, x=0
normB = self.norm(b)
if normB == 0.0:
return self.handleConvergence(0, np.zeros_like(b), 0, 0)
# Create vectors for residual r and solution x
r = np.copy(b)
x = np.copy(b)
if self._cycleMgr == None or not self.matrixFrozen():
mlh = SmoothedAggregationMLHierarchy(A, numLevels=self.numLevels)
self._cycleMgr = VCycleManager(mlh,
nuPre=self.nuPre,
nuPost=self.nuPost,
smoother=self.smoother)
# Main loop
for k in range(self.maxiter()):
# Run a V-cycle
x = self._cycleMgr.runCycle(b, x)
# Compute residual
r = b - A * x
# Check for convergence
normR = self.norm(r)
self.reportIter(k, normR, normB)
if normR < self.tau() * normB:
return self.handleConvergence(k, x, normR, normB)
# If we're here, maxiter has been reached. This is normally a failure,
# but may be acceptable if failOnMaxiter is set to false.
return self.handleMaxiter(k, x, normR, normB)
def search(self, x0, normF0, newtStep, func):
tab = Tab()
t = 1.0
for k in range(self.maxsteps()):
x_k = x0 + t * newtStep
F_k = func.evalF(x_k)
normF_k = self.norm(F_k)
ratio = normF_k / normF0
self.report(k, t, ratio)
# Test for convergence of line search
if normF_k <= (1.0 - self.alpha() * t) * normF0:
return (True, x_k, F_k, normF_k)
# Shrink step
factor = 0.5 / ratio
if factor < self.low():
factor = self.low()
t = t * factor
# If here, the line step hasn't produced sufficient decrease.
return (False, x_k, F_k, normF_k)
def SmoothProlongator(Phat, A, Af, omega=(2/3)):
tab=Tab()
print('{}in SmoothProlongator()'.format(tab))
timer = Timer('SmoothProlongator')
timer.start()
smoothmat = omega*Af
d_A = A.diagonal()
for i in range(A.shape[0]):
start = smoothmat.indptr[i]
end = smoothmat.indptr[i+1]
for k in range(start, end):
j = smoothmat.indices[k]
smoothmat.data[k] /= d_A[i]
if i==j:
smoothmat.data[k] = 1 - smoothmat.data[k]
else:
smoothmat.data[k] = -smoothmat.data[k]
smoothed = smoothmat.dot(Phat)
timer.stop()
return smoothed
def SA_coarsen(A, tol=None, lvl=1):
tab=Tab()
print('{}in SA_coarsen()'.format(tab))
# lvl will only be used if tol is None
# If tol is None, use Vanek's suggestion
if tol is None:
tol = 0.08*(0.5)**(lvl-1)
# Build the aggregates
(aggregates, neighborhoods) = BuildAggregates(A, lvl=lvl)
# Build the tentative prolongator from the aggregates, A needed for it's dimensions
Phat = BuildTentativeProlongator(A, aggregates)
# Build the filtered matrix for the smoother
Af = BuildFilteredMatrix(A, neighborhoods, tol)
# Smooth the Prolongation Operator with weighted Jacobi using the filtered matrix
P = SmoothProlongator(Phat, A, Af)
return (P.tocsr(), aggregates)
def solve(self, A, b):
'''
Solve the system A*x=b for x.
* Input:
* A -- System matrix, can be a numpy 2D array or a scipy sparse matrix.
Must be SPD; this is not checked (doing so is too expensive).
* b -- RHS vector, must be a numpy 1D array compatible with A.
* Return:
* A SolveStatus object containing the solution estimate, a
success/failure flag, and convergence information.
'''
tab = Tab()
# Get size of matrix
n, nc = A.shape
# Make sure matrix is square
assert (n == nc)
# Make sure A and b are compatible
assert (n == len(b))
# Check for the trivial case b=0, x=0
normB = self.norm(b)
if normB == 0.0:
return self.handleConvergence(0, np.zeros_like(b), 0, 0)
# Form the preconditioner
print('prec frozen = ', self.precFrozen())
if self.precond == None or not self.precFrozen():
print('building prec')
self.precond = self.precondType().form(A)
# Initialize the step, residual, and solution vectors
r = np.copy(b)
p = self.precond.applyRight(r)
u = np.copy(p)
x = np.zeros_like(b)
uDotR = np.dot(u, r) # Should never be zero, since we've caught b=0
# Check anyway
if uDotR == 0.0:
return self.handleBreakdown(0, 'breakdown dot(u,r)==0')
# Preconditioned CG loop
for k in range(self.maxiter()):
# Compute A*p
Ap = mvmult(A, p)
pTAp = np.dot(p, Ap)
if pTAp == 0.0:
return self.handleBreakdown(k, 'breakdown dot(p, Ap)==0')
# calculate step length
alpha = uDotR / pTAp
# Make step and compute updated residual
x = x + alpha * p
r = r - alpha * Ap
u = self.precond.applyRight(r)
normR = self.norm(r)
self.reportIter(k, normR, normB)
# Check for convergence
if ((normR <= self.tau() * normB) or
((not self.failOnMaxiter()) and k == self.maxiter() - 1)):
return self.handleConvergence(k, x, normR, normB)
# Find next step direction
newUDotR = np.dot(u, r)
beta = newUDotR / uDotR
uDotR = newUDotR
p = u + beta * p
# If we're here, maxiter has been reached. This is normally a failure,
# but may be acceptable if failOnMaxiter is set to false.
return self.handleMaxiter(k, x, normR, normB)
def reportBreakdown(self, msg=''):
'''If control.showFinal==True, print a message upon breakdown.'''
tab = Tab()
if self._control.showFinal:
print('%s%s solve broke down: %s' % (tab, self.name(), msg))
def BuildAggregates(A, lvl=1, tol=None, phase=3):
tab=Tab()
tab1 = Tab()
print('{}in BuildAggregates()'.format(tab))
# If tol isn't specified, then use the default by Vanek
if tol is None:
tol = 0.08*(0.5)**(lvl-1)
timer0 = Timer('BuildAggregates init step')
timer0.start()
# Initialization
R = {i for i in range(A.shape[0])}
a_diag = A.diagonal()
neighborhoods = [getNeighborhood(A,i,tol,a_diag) for i in range(A.shape[0])]
aggregates = []
# isolated nodes aren't aggregated
for n in neighborhoods:
if len(n) == 1:
aggregates.append(n)
[elem] = n
R.remove(elem)
timer0.stop()
timer1 = Timer('BuildAggregates phase 1')
timer1.start()
# Phase 1
if phase > 0:
for i in range(A.shape[0]):
# If the neighborhood of i is completely in R, create an aggregate
# from the neighborhood
if i in R and neighborhoods[i].issubset(R):
aggregates.append(neighborhoods[i])
R -= neighborhoods[i]
timer1.stop()
timer2 = Timer('BuildAggregates phase 2')
timer2.start()
# Phase 2
if phase > 1:
# Copy the aggregates since we need to modify and check the
# original aggregates
timer_copy = Timer('agg copy')
timer_copy.start()
aggcopy = copy.deepcopy(aggregates)
timer_copy.stop()
# Loop through elements still in R
for i in range(A.shape[0]):
if i in R:
# We need the strongest connection
max_conn_strength = 0.0
agg_idx_of_max = -1
# Loop through all aggregates, looking to see if the current
# neighborhood intersections
for (j, agg) in enumerate(aggcopy):
#timer_disj = Timer('agg disjoint check')
#timer_disj.start()
isDisjoint_i_j = agg.isdisjoint(neighborhoods[i])
#timer_disj.stop()
if not isDisjoint_i_j:
#timer_loop = Timer('loop to find max strength')
#timer_loop.start()
for k in agg:
if abs(A[i,k]) > max_conn_strength:
max_conn_strength = abs(A[i,k])
agg_idx_of_max = j
#timer_loop.stop()
timer_insert = Timer('agg insertion')
timer_insert.start()
aggregates[agg_idx_of_max].add(i)
timer_insert.stop()
timer2.stop()
timer3 = Timer('BuildAggregates phase 2')
timer3.start()
# Phase 3
if phase > 2 and not R:
# Loop through elements still in R
for i in range(A.shape[0]):
if i in R:
aggregates.append(R.intersection(neighborhoods[i]))
R -= neighborhoods[i]
timer3.stop()
return (aggregates, neighborhoods)
def solve(self, func, xInit):
'''
'''
tab = Tab()
xCur = xInit.copy()
FCur = func.evalF(xCur)
newtStep = np.ones_like(xCur)
print('freeze prec for solver=', self.freezePrec)
freeze = PreconditionerFreeze(self.solver, self.freezePrec)
self.linesearch.setNorm(self.norm)
# Initial residual; to be used for relative residual tests
r0 = self.norm(FCur)
normFCur = r0
# Newton loop
for i in range(self.maxiter()):
# Report progress
self.reportIter(i, normFCur, r0)
# Check for convergence
if normFCur <= r0 * self.tau() + self.tau():
return self.handleConvergence(i, xCur, normFCur, r0)
# Evaluate Jacobian
J = func.evalJ(xCur)
# Set tolerance to be used in linear solve
if isinstance(self.solver, IterativeLinearSolver):
if self.fixLinTol: # Use fixed tolerance if desired (for testing)
tau_lin = self.minLinTol
else: # Adjust linear tol according to nonlinear residual.
# new linear tolerance is the larger of:
# (*) "fudge factor" times relative residual of nonlinear solve
# (*) a minimum linear tolerance.
# The minimum tolerance avoids pointlessly small tolerances
# for the linear solver
tau_lin = max(self.tolFudge * normFCur / r0,
self.minLinTol)
self.solver.setTolerance(tau_lin)
# Solve for the Newton step
tab.indent()
status = self.solver.solve(J, -FCur)
tab.unindent()
if not status.success():
return self.handleBreakdown(
i, 'solve for Newton step failed with msg={}'.format(
status.msg()))
p = status.soln()
# Do line search to find a step length giving sufficient decrease
tab.indent()
(success, xCur, FCur,
normFCur) = self.linesearch.search(xCur, normFCur, p, func)
tab.unindent()
if not success:
return self.handleBreakdown(i, msg='Line search failed')
# End of Newton loop. At this point, xCur, FCur, and normFCur have been
# updated to the new step.
# If here, we've reached the maximum number of iterations without
# convergence. Report failure to converge.
return self.handleMaxiter(self.maxiter(), xCur, normFCur, r0)
def report(self, k, t, ratio):
tab = Tab()
if self._report:
print('%sk=%4d t=%12.5g ||F_k||/||F_0||=%12.5g' %
(tab, k, t, ratio))
