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Fields calculation in Richards. Calculation of the diagonals of the Jacobian.

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This commit is contained in:
Rowan Cockett
2013-11-14 15:29:25 -08:00
parent a5032e07b1
commit 61e175cd12
4 changed files with 128 additions and 31 deletions
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SimPEG/forward/Richards.py
+105 -16
View File
@@ -1,14 +1,21 @@
from SimPEG.forward import Problem
import numpy as np
from SimPEG.utils import sdiag, mkvc, setKwargs
from SimPEG.inverse import NewtonRoot
class RichardsProblem(Problem):
"""docstring for RichardsProblem"""
timeStep = None
timeEnd = None
boundaryConditions = None
initialConditions = None
@property
def numIts(self):
"""The number of iterations in the time domain problem."""
return int(self.timeEnd/self.timeStep)
_method = 'mixed'
@property
@@ -51,7 +58,6 @@ class RichardsProblem(Problem):
assert value in ['mixed','head'], "method must be 'mixed' or 'head'."
self._method = value
_doNewton = False
@property
def doNewton(self):
"""Do a Newton iteration. If False, a Picard iteration will be completed."""
@@ -59,15 +65,82 @@ class RichardsProblem(Problem):
@doNewton.setter
def doNewton(self, value):
assert type(value) is bool, 'doNewton must be a boolean.'
self.rootFinder = NewtonRoot(doLS=value)
self._doNewton = value
@property
def dataType(self):
"""Choose how your data is collected, must be 'saturation' or 'pressureHead'."""
assert value in ['saturation','pressureHead'], "dataType must be 'saturation' or 'pressureHead'."
return self._dataType
@dataType.setter
def dataType(self, value):
self._dataType = value
def __init__(self, mesh, empirical):
def __init__(self, mesh, empirical, **kwargs):
Problem.__init__(self, mesh)
self.empirical = empirical
self.mesh.setCellGradBC('dirichlet')
self.doNewton = False # This also sets the rootFinder algorithm.
setKwargs(self, **kwargs)
def field(self, m):
self.empirical.setModel(m)
Hs = range(self.numIts+1)
Hs[0] = self.initialConditions
for ii in range(self.numIts):
hn = Hs[ii]
hn1 = self.rootFinder.root(lambda hn1: self.getResidual(hn,hn1), hn)
Hs[ii+1] = hn1
return Hs
def diagsJacobian(self, hn, hn1):
DIV = self.mesh.faceDiv
GRAD = self.mesh.cellGrad
BC = self.mesh.cellGradBC
AV = self.mesh.aveCC2F
Dz = self.mesh.faceDiv #TODO: fix this for more than one dimension.
bc = self.boundaryConditions
dt = self.timeStep
dT = self.empirical.moistureContentDeriv(hn)
dT1 = self.empirical.moistureContentDeriv(hn1)
K1 = self.empirical.hydraulicConductivity(hn1)
dK1 = self.empirical.hydraulicConductivityDeriv(hn1)
dKa1 = self.empirical.hydraulicConductivityModelDeriv(hn1)
# Compute part of the derivative of:
#
# DIV*diag(GRAD*hn1+BC*bc)*(AV*(1/K))^-1
DdiagGh1 = DIV*sdiag(GRAD*hn1+BC*bc)
diagAVk2_AVdiagK2 = sdiag((AV*(1./K1))**(-2)) * AV*sdiag(K1**(-2))
# The matrix that we are computing has the form:
#
# - - - - - -
# | Adiag | | h1 | | b1 |
# | Asub Adiag | | h2 | | b2 |
# | Asub Adiag | | h3 | = | b3 |
# | ... ... | | .. | | .. |
# | Asub Adiag | | hn | | bn |
# - - - - - -
Asub = (-1/dt)*dT
Adiag = (
(1/dt)*dT1
-DdiagGh1*diagAVk2_AVdiagK2*dK1
-DIV*sdiag(1./(AV*(1./K1)))*GRAD
-Dz*diagAVk2_AVdiagK2*dK1
)
B = DdiagGh1*diagAVk2_AVdiagK2*dKa1 + Dz*diagAVk2_AVdiagK2*dKa1
def getResidual(self, hn, h):
"""
@@ -119,6 +192,9 @@ class Haverkamp(object):
def __init__(self, **kwargs):
setKwargs(self, **kwargs)
def setModel(self, m):
self.Ks = m
def moistureContent(self, h):
f = (self.alpha*(self.theta_s - self.theta_r )/
(self.alpha + abs(h)**self.beta) + self.theta_r)
@@ -141,14 +217,17 @@ class Haverkamp(object):
f[h >= 0] = np.exp(self.Ks)
return f
def hydraulicConductivityDeriv(self, h):
g = -(np.exp(self.Ks)*self.A*self.gamma*abs(h)**(self.gamma-1)*np.sign(h))/((self.A+abs(h)**self.gamma)**2)
g[h >= 0] = 0
g = sdiag(g)
def hydraulicConductivityModelDeriv(self, h):
#A
# dA = np.exp(self.Ks)/(self.A+abs(h)**self.gamma) - np.exp(self.Ks)*self.A/(self.A+abs(h)**self.gamma)**2;
#gamma
# dgamma = -(self.A*np.exp(self.Ks)*np.log(abs(h))*abs(h)**self.gamma)/(self.A + abs(h)**self.gamma)**2;
return sdiag(self.hydraulicConductivity(h)) # This assumes that the the model is Ks
def hydraulicConductivityDeriv(self, h):
g = -(np.exp(self.Ks)*self.A*self.gamma*abs(h)**(self.gamma-1)*np.sign(h))/((self.A+abs(h)**self.gamma)**2)
g[h >= 0] = 0
g = sdiag(g)
return g
@@ -180,6 +259,9 @@ class VanGenuchten(object):
def __init__(self, **kwargs):
setKwargs(self, **kwargs)
def setModel(self, m):
self.Ks = m
def moistureContent(self, h):
m = 1 - 1/self.n;
f = (( self.theta_s - self.theta_r )/
@@ -208,6 +290,15 @@ class VanGenuchten(object):
f[h >= 0] = np.exp(self.Ks)
return f
def hydraulicConductivityModelDeriv(self, h):
#alpha
# dA = I*h*n*np.exp(Ks)*abs(alpha*h)**(n - 1)*np.sign(alpha*h)*(1/n - 1)*((abs(alpha*h)**n + 1)**(1/n - 1))**(I - 1)*((1 - 1/((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1)))**(1 - 1/n) - 1)**2*(abs(alpha*h)**n + 1)**(1/n - 2) - (2*h*n*np.exp(Ks)*abs(alpha*h)**(n - 1)*np.sign(alpha*h)*(1/n - 1)*((abs(alpha*h)**n + 1)**(1/n - 1))**I*((1 - 1/((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1)))**(1 - 1/n) - 1)*(abs(alpha*h)**n + 1)**(1/n - 2))/(((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1) + 1)*(1 - 1/((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1)))**(1/n));
#n
# dn = 2*np.exp(Ks)*((np.log(1 - 1/((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1)))*(1 - 1/((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1)))**(1 - 1/n))/n**2 + ((1/n - 1)*(((np.log(abs(alpha*h)**n + 1)*(abs(alpha*h)**n + 1)**(1/n - 1))/n**2 - abs(alpha*h)**n*np.log(abs(alpha*h))*(1/n - 1)*(abs(alpha*h)**n + 1)**(1/n - 2))/((1/n - 1)*((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1) + 1)) - np.log((abs(alpha*h)**n + 1)**(1/n - 1))/(n**2*(1/n - 1)**2*((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1)))))/(1 - 1/((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1)))**(1/n))*((abs(alpha*h)**n + 1)**(1/n - 1))**I*((1 - 1/((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1)))**(1 - 1/n) - 1) - I*np.exp(Ks)*((np.log(abs(alpha*h)**n + 1)*(abs(alpha*h)**n + 1)**(1/n - 1))/n**2 - abs(alpha*h)**n*np.log(abs(alpha*h))*(1/n - 1)*(abs(alpha*h)**n + 1)**(1/n - 2))*((abs(alpha*h)**n + 1)**(1/n - 1))**(I - 1)*((1 - 1/((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1)))**(1 - 1/n) - 1)**2;
#I
# dI = np.exp(Ks)*np.log((abs(alpha*h)**n + 1)**(1/n - 1))*((abs(alpha*h)**n + 1)**(1/n - 1))**I*((1 - 1/((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1)))**(1 - 1/n) - 1)**2;
return sdiag(self.hydraulicConductivity(h)) # This assumes that the the model is Ks
def hydraulicConductivityDeriv(self, h):
alpha = self.alpha
I = self.I
@@ -218,14 +309,6 @@ class VanGenuchten(object):
g = I*alpha*n*np.exp(Ks)*abs(alpha*h)**(n - 1)*np.sign(alpha*h)*(1/n - 1)*((abs(alpha*h)**n + 1)**(1/n - 1))**(I - 1)*((1 - 1/((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1)))**(1 - 1/n) - 1)**2*(abs(alpha*h)**n + 1)**(1/n - 2) - (2*alpha*n*np.exp(Ks)*abs(alpha*h)**(n - 1)*np.sign(alpha*h)*(1/n - 1)*((abs(alpha*h)**n + 1)**(1/n - 1))**I*((1 - 1/((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1)))**(1 - 1/n) - 1)*(abs(alpha*h)**n + 1)**(1/n - 2))/(((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1) + 1)*(1 - 1/((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1)))**(1/n))
g[h >= 0] = 0
g = sdiag(g)
#alpha
# dA = I*h*n*np.exp(Ks)*abs(alpha*h)**(n - 1)*np.sign(alpha*h)*(1/n - 1)*((abs(alpha*h)**n + 1)**(1/n - 1))**(I - 1)*((1 - 1/((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1)))**(1 - 1/n) - 1)**2*(abs(alpha*h)**n + 1)**(1/n - 2) - (2*h*n*np.exp(Ks)*abs(alpha*h)**(n - 1)*np.sign(alpha*h)*(1/n - 1)*((abs(alpha*h)**n + 1)**(1/n - 1))**I*((1 - 1/((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1)))**(1 - 1/n) - 1)*(abs(alpha*h)**n + 1)**(1/n - 2))/(((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1) + 1)*(1 - 1/((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1)))**(1/n));
#n
# dn = 2*np.exp(Ks)*((np.log(1 - 1/((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1)))*(1 - 1/((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1)))**(1 - 1/n))/n**2 + ((1/n - 1)*(((np.log(abs(alpha*h)**n + 1)*(abs(alpha*h)**n + 1)**(1/n - 1))/n**2 - abs(alpha*h)**n*np.log(abs(alpha*h))*(1/n - 1)*(abs(alpha*h)**n + 1)**(1/n - 2))/((1/n - 1)*((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1) + 1)) - np.log((abs(alpha*h)**n + 1)**(1/n - 1))/(n**2*(1/n - 1)**2*((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1)))))/(1 - 1/((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1)))**(1/n))*((abs(alpha*h)**n + 1)**(1/n - 1))**I*((1 - 1/((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1)))**(1 - 1/n) - 1) - I*np.exp(Ks)*((np.log(abs(alpha*h)**n + 1)*(abs(alpha*h)**n + 1)**(1/n - 1))/n**2 - abs(alpha*h)**n*np.log(abs(alpha*h))*(1/n - 1)*(abs(alpha*h)**n + 1)**(1/n - 2))*((abs(alpha*h)**n + 1)**(1/n - 1))**(I - 1)*((1 - 1/((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1)))**(1 - 1/n) - 1)**2;
#I
# dI = np.exp(Ks)*np.log((abs(alpha*h)**n + 1)**(1/n - 1))*((abs(alpha*h)**n + 1)**(1/n - 1))**I*((1 - 1/((abs(alpha*h)**n + 1)**(1/n - 1))**(1/(1/n - 1)))**(1 - 1/n) - 1)**2;
return g
@@ -244,6 +327,12 @@ if __name__ == '__main__':
h = np.zeros(M.nC) + prob.boundaryConditions[0]
checkDerivative(lambda hn1: prob.getResidual(h,hn1), h)
numIts = 10
Hs = range(numIts+1)
Hs[0] = h
for ii in range(numIts):
hn = Hs[ii]
hn1 = NewtonRoot().root(lambda hn1: prob.getResidual(hn,hn1), hn)
Hs[ii+1] = hn1
M.plotImage(hn1,showIt=True)
SimPEG/inverse/Optimize.py
+13 -10
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@@ -663,12 +663,13 @@ class NewtonRoot(object):
"""
tol = 1.000e-06;
solveTol = 0; # Default direct solve.
maxIter = 20;
stepDcr = 0.5;
maxLS = 30;
comments = False;
tol = 1.000e-06
solveTol = 0 # Default direct solve.
maxIter = 20
stepDcr = 0.5
maxLS = 30
comments = False
doLS = True
def __init__(self, **kwargs):
setKwargs(self, **kwargs)
@@ -688,14 +689,14 @@ class NewtonRoot(object):
# M = @(x) tril(J)\(diag(J).*(triu(J)\x));
# [dh, ~] = bicgstab(J,-r,O.solveTol,500,M);
muLS = 1
muLS = 1.
LScnt = 1
xt = x + dh
rt, Jt = fun(xt) # TODO: get rid of Jt
if self.comments: print '\tLinesearch:\n'
# Enter Linesearch
while True:
xt = x + muLS*dh
rt, Jt = fun(xt) # TODO: get rid of Jt
while True and self.doLS:
if self.comments:
print '\t\tResid: %e\n'%norm(rt)
if norm(rt) <= norm(r) or norm(rt) < self.tol:
@@ -708,6 +709,8 @@ class NewtonRoot(object):
print 'Newton Method: Line search break.'
root = NaN
return
xt = x + muLS*dh
rt, Jt = fun(xt) # TODO: get rid of Jt
x = xt
self._iter += 1
SimPEG/tests/TestUtils.py
+4 -4
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@@ -189,7 +189,7 @@ def Rosenbrock(x, return_g=True, return_H=True):
out += (H,)
return out if len(out) > 1 else out[0]
def checkDerivative(fctn, x0, num=7, plotIt=True, dx=None):
def checkDerivative(fctn, x0, num=7, plotIt=True, dx=None, expectedOrder=2, tolerance=0.9, eps=1e-10):
"""
Basic derivative check
@@ -201,6 +201,9 @@ def checkDerivative(fctn, x0, num=7, plotIt=True, dx=None):
:param int num: number of times to reduce step length, h
:param bool plotIt: if you would like to plot
:param numpy.array dx: step direction
:param int expectedOrder: The order that you expect the derivative to yield.
:param float tolerance: The tolerance on the expected order.
:param float eps: What is zero?
:rtype: bool
:return: did you pass the test?!
@@ -243,9 +246,6 @@ def checkDerivative(fctn, x0, num=7, plotIt=True, dx=None):
order1 = np.log10(E1[:-1]/E1[1:])
print "%d\t%1.2e\t%1.3e\t\t%1.3e\t\t%1.3f" % (i, t[i], E0[i], E1[i], np.nan if i == 0 else order1[i-1])
tolerance = 0.9
expectedOrder = 2
eps = 1e-10
order0 = order0[E0[1:] > eps]
order1 = order1[E1[1:] > eps]
belowTol = order1.size == 0 and order0.size > 0
SimPEG/tests/test_Richards.py
+6 -1
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@@ -69,9 +69,14 @@ class RichardsTests(unittest.TestCase):
passed = checkDerivative(wrapper, np.random.randn(n), plotIt=False)
self.assertTrue(passed,True)
def test_Richards_getResidual(self):
def test_Richards_getResidual_Newton(self):
self.prob.doNewton = True
checkDerivative(lambda hn1: self.prob.getResidual(self.h0,hn1), self.h0, plotIt=False)
def test_Richards_getResidual_Picard(self):
self.prob.doNewton = False
checkDerivative(lambda hn1: self.prob.getResidual(self.h0,hn1), self.h0, plotIt=False, expectedOrder=1)
if __name__ == '__main__':