def fget(self):
if(self._area is None or self._normals is None):
# Compute areas of cell faces
if(self.dim == 2):
xy = self.gridN
A, B = Utils.indexCube('AB', self.vnC+1, np.array([self.nNx, self.nCy]))
edge1 = xy[B, :] - xy[A, :]
normal1 = np.c_[edge1[:, 1], -edge1[:, 0]]
area1 = length2D(edge1)
A, D = Utils.indexCube('AD', self.vnC+1, np.array([self.nCx, self.nNy]))
# Note that we are doing A-D to make sure the normal points the right way.
# Think about it. Look at the picture. Normal points towards C iff you do this.
edge2 = xy[A, :] - xy[D, :]
normal2 = np.c_[edge2[:, 1], -edge2[:, 0]]
area2 = length2D(edge2)
self._area = np.r_[Utils.mkvc(area1), Utils.mkvc(area2)]
self._normals = [normalize2D(normal1), normalize2D(normal2)]
elif(self.dim == 3):
A, E, F, B = Utils.indexCube('AEFB', self.vnC+1, np.array([self.nNx, self.nCy, self.nCz]))
normal1, area1 = Utils.faceInfo(self.gridN, A, E, F, B, average=False, normalizeNormals=False)
A, D, H, E = Utils.indexCube('ADHE', self.vnC+1, np.array([self.nCx, self.nNy, self.nCz]))
normal2, area2 = Utils.faceInfo(self.gridN, A, D, H, E, average=False, normalizeNormals=False)
A, B, C, D = Utils.indexCube('ABCD', self.vnC+1, np.array([self.nCx, self.nCy, self.nNz]))
normal3, area3 = Utils.faceInfo(self.gridN, A, B, C, D, average=False, normalizeNormals=False)
self._area = np.r_[Utils.mkvc(area1), Utils.mkvc(area2), Utils.mkvc(area3)]
self._normals = [normal1, normal2, normal3]
return self._area
def test_regularizationMesh(self):
for i, mesh in enumerate(self.meshlist):
print('Testing {0:d}D'.format(mesh.dim))
# mapping = r.mapPair(mesh)
# reg = r(mesh, mapping=mapping)
# m = np.random.rand(mapping.nP)
if mesh.dim == 1:
indAct = Utils.mkvc(mesh.gridCC <= 0.8)
elif mesh.dim == 2:
indAct = (
Utils.mkvc(
mesh.gridCC[:,-1] <=
2*np.sin(2*np.pi*mesh.gridCC[:, 0]) + 0.5
)
)
elif mesh.dim == 3:
indAct = (
Utils.mkvc(
mesh.gridCC[:, -1] <=
2*np.sin(2*np.pi*mesh.gridCC[:, 0]) +
0.5 * 2*np.sin(2*np.pi*mesh.gridCC[:, 1]) + 0.5
)
)
regmesh = Regularization.RegularizationMesh(
mesh, indActive=indAct
)
assert (regmesh.vol == mesh.vol[indAct]).all()
def test_regularization_ActiveCells(self):
for R in dir(Regularization):
r = getattr(Regularization, R)
if not inspect.isclass(r):
continue
if not issubclass(r, Regularization.BaseRegularization):
continue
for i, mesh in enumerate(self.meshlist):
print('Testing Active Cells {0:d}D'.format((mesh.dim)))
if mesh.dim == 1:
indActive = Utils.mkvc(mesh.gridCC <= 0.8)
elif mesh.dim == 2:
indActive = Utils.mkvc(mesh.gridCC[:,-1] <= 2*np.sin(2*np.pi*mesh.gridCC[:,0])+0.5)
elif mesh.dim == 3:
indActive = Utils.mkvc(mesh.gridCC[:,-1] <= 2*np.sin(2*np.pi*mesh.gridCC[:,0])+0.5 * 2*np.sin(2*np.pi*mesh.gridCC[:,1])+0.5)
for indAct in [indActive, indActive.nonzero()[0]]: # test both bool and integers
reg = r(mesh, indActive=indAct)
m = np.random.rand(mesh.nC)[indAct]
reg.mref = np.ones_like(m)*np.mean(m)
print('Check: phi_m (mref) = {0:f}'.format(reg.eval(reg.mref)))
passed = reg.eval(reg.mref) < TOL
self.assertTrue(passed)
print('Check:', R)
passed = Tests.checkDerivative(lambda m : [reg.eval(m), reg.evalDeriv(m)], m, plotIt=False)
self.assertTrue(passed)
print('Check 2 Deriv:', R)
passed = Tests.checkDerivative(lambda m : [reg.evalDeriv(m), reg.eval2Deriv(m)], m, plotIt=False)
self.assertTrue(passed)
def set_ij_height(self):
"""
Compute (I, J) indicies to form sparse sensitivity matrix
This will be used in GlobalEM1DProblem when after sensitivity matrix
for each sounding is computed
"""
I = []
J = []
shift_for_J = 0
shift_for_I = 0
m = self.n_layer
for i in range(n_sounding):
n = self.nD_vec[i]
J_temp = np.tile(np.arange(m), (n, 1)) + shift_for_J
I_temp = (
np.tile(np.arange(n), (1, m)).reshape((n, m), order='F') +
shift_for_I
)
J.append(Utils.mkvc(J_temp))
I.append(Utils.mkvc(I_temp))
shift_for_J += m
shift_for_I = I_temp[-1, -1] + 1
J = np.hstack(J).astype(int)
I = np.hstack(I).astype(int)
return (I, J)
def Jvec(self, m, v, f=None):
if f is None:
f = self.fields(m)
self.curModel = m
Jv = self.dataPair(self.survey) #same size as the data
A = self.getA()
for src in self.survey.srcList:
u_src = f[src, self._solutionType] # solution vector
dA_dm_v = self.getADeriv(u_src, v)
dRHS_dm_v = self.getRHSDeriv(src, v)
du_dm_v = self.Ainv * ( - dA_dm_v + dRHS_dm_v )
for rx in src.rxList:
df_dmFun = getattr(f, '_{0!s}Deriv'.format(rx.projField), None)
df_dm_v = df_dmFun(src, du_dm_v, v, adjoint=False)
Jv[src, rx] = rx.evalDeriv(src, self.mesh, f, df_dm_v)
# Conductivity (d u / d log sigma)
if self._formulation is 'EB':
return -Utils.mkvc(Jv)
# Conductivity (d u / d log rho)
if self._formulation is 'HJ':
return Utils.mkvc(Jv)
def test_ana_forward(self):
survey = PF.BaseMag.BaseMagSurvey()
Inc = 45.0
Dec = 45.0
Btot = 51000
b0 = PF.MagAnalytics.IDTtoxyz(Inc, Dec, Btot)
survey.setBackgroundField(Inc, Dec, Btot)
xr = np.linspace(-300, 300, 41)
yr = np.linspace(-300, 300, 41)
X, Y = np.meshgrid(xr, yr)
Z = np.ones((xr.size, yr.size)) * 150
rxLoc = np.c_[Utils.mkvc(X), Utils.mkvc(Y), Utils.mkvc(Z)]
survey.rxLoc = rxLoc
self.prob.pair(survey)
u = self.prob.fields(self.chi)
B = u["B"]
bxa, bya, bza = PF.MagAnalytics.MagSphereAnaFunA(
rxLoc[:, 0], rxLoc[:, 1], rxLoc[:, 2], 100.0, 0.0, 0.0, 0.0, 0.01, b0, "secondary"
)
dpred = survey.projectFieldsAsVector(B)
err = np.linalg.norm(dpred - np.r_[bxa, bya, bza]) / np.linalg.norm(np.r_[bxa, bya, bza])
self.assertTrue(err < 0.08)
def Jtvec(self, m, v, f=None):
if f is None:
f = self.fields(m)
self.model = m
# Ensure v is a data object.
if not isinstance(v, self.dataPair):
v = self.dataPair(self.survey, v)
Jtv = np.zeros(m.size)
AT = self.getA()
for src in self.survey.srcList:
u_src = f[src, self._solutionType]
for rx in src.rxList:
PTv = rx.evalDeriv(src, self.mesh, f, v[src, rx], adjoint=True) # wrt f, need possibility wrt m
df_duTFun = getattr(f, '_{0!s}Deriv'.format(rx.projField), None)
df_duT, df_dmT = df_duTFun(src, None, PTv, adjoint=True)
ATinvdf_duT = self.Ainv * df_duT
dA_dmT = self.getADeriv(u_src, ATinvdf_duT, adjoint=True)
dRHS_dmT = self.getRHSDeriv(src, ATinvdf_duT, adjoint=True)
du_dmT = -dA_dmT + dRHS_dmT
Jtv += (df_dmT + du_dmT).astype(float)
# Conductivity ((d u / d log sigma).T)
if self._formulation == 'EB':
return -Utils.mkvc(Jtv)
# Conductivity ((d u / d log rho).T)
if self._formulation == 'HJ':
return Utils.mkvc(Jtv)
def forward(self, m, f=None):
if f is None:
f = self.fields(m)
self.curModel = m
Jv = self.dataPair(self.survey) #same size as the data
# A = self.getA()
JvAll = []
for tind in range(len(self.survey.times)):
#Pseudo-chareability
t = self.survey.times[tind]
v = self.DebyeTime(t)
for src in self.survey.srcList:
u_src = f[src, self._solutionType] # solution vector
dA_dm_v = self.getADeriv(u_src, v)
dRHS_dm_v = self.getRHSDeriv(src, v)
du_dm_v = self.Ainv * ( - dA_dm_v + dRHS_dm_v )
for rx in src.rxList:
timeindex = rx.getTimeP(self.survey.times)
if timeindex[tind]:
df_dmFun = getattr(f, '_{0!s}Deriv'.format(rx.projField), None)
df_dm_v = df_dmFun(src, du_dm_v, v, adjoint=False)
Jv[src, rx, t] = rx.evalDeriv(src, self.mesh, f, df_dm_v)
# Conductivity (d u / d log sigma)
if self._formulation is 'EB':
return -Utils.mkvc(Jv)
# Resistivity (d u / d log rho)
if self._formulation is 'HJ':
return Utils.mkvc(Jv)
def _fastInnerProduct(self, projType, prop=None, invProp=False, invMat=False):
"""
Fast version of getFaceInnerProduct.
This does not handle the case of a full tensor prop.
:param numpy.array prop: material property (tensor properties are possible) at each cell center (nC, (1, 3, or 6))
:param str projType: 'E' or 'F'
:param bool returnP: returns the projection matrices
:param bool invProp: inverts the material property
:param bool invMat: inverts the matrix
:rtype: scipy.sparse.csr_matrix
:return: M, the inner product matrix (nF, nF)
"""
assert projType in ["F", "E"], "projType must be 'F' for faces or 'E'" " for edges"
if prop is None:
prop = np.ones(self.nC)
if invProp:
prop = 1.0 / prop
if Utils.isScalar(prop):
prop = prop * np.ones(self.nC)
# number of elements we are averaging (equals dim for regular
# meshes, but for cyl, where we use symmetry, it is 1 for edge
# variables and 2 for face variables)
if self._meshType == "CYL":
n_elements = np.sum(getattr(self, "vn" + projType).nonzero())
else:
n_elements = self.dim
# Isotropic? or anisotropic?
if prop.size == self.nC:
Av = getattr(self, "ave" + projType + "2CC")
Vprop = self.vol * Utils.mkvc(prop)
M = n_elements * Utils.sdiag(Av.T * Vprop)
elif prop.size == self.nC * self.dim:
Av = getattr(self, "ave" + projType + "2CCV")
# if cyl, then only certain components are relevant due to symmetry
# for faces, x, z matters, for edges, y (which is theta) matters
if self._meshType == "CYL":
if projType == "E":
prop = prop[:, 1] # this is the action of a projection mat
elif projType == "F":
prop = prop[:, [0, 2]]
V = sp.kron(sp.identity(n_elements), Utils.sdiag(self.vol))
M = Utils.sdiag(Av.T * V * Utils.mkvc(prop))
else:
return None
if invMat:
return Utils.sdInv(M)
else:
return M
def getJ(self, sigma, emax, senseoption=True):
"""
Computing sensitivity for airbonre time domain EM IP inversion.
"""
Meinv = self.mesh.getEdgeInnerProduct(invMat=True)
MeSig = self.mesh.getEdgeInnerProduct(sigma)
Asiginf = self.mesh.nodalGrad.T*MeSig*self.mesh.nodalGrad
MeDeriv = self.mesh.getEdgeInnerProductDeriv(np.ones(self.mesh.nC))
print ">>Factorizing ADC matrix"
if senseoption == True:
# ctx = DMumpsContext()
# if ctx.myid == 0:
# ctx.set_icntl(14, 60)
# ctx.set_centralized_sparse(Asiginf)
# ctx.run(job=4) # Factorization
Ainv = MumpsSolver(Asiginf)
ntx = self.survey.xyz_tx.shape[0]
J = np.zeros((ntx, self.mesh.nC))
J1_temp = np.zeros(self.mesh.nC)
J2_temp = np.zeros(self.mesh.nC)
print ">>Computing sensitivity"
for i in range (ntx):
stdout.write( (" \r%d/%d transmitter") % (i, ntx) )
stdout.flush()
S_temp = MeDeriv((emax[i,:]))*Utils.sdiag(sigma)
G_temp = BiotSavart.BiotSavartFun(self.mesh, self.survey.xyz_tx[i,:], component = 'z')
# Considering eIP term in jIP
if senseoption == True:
rhs = self.mesh.nodalGrad.T*MeSig*Meinv*self.mesh.aveE2CCV.T*G_temp.T
# if ctx.myid == 0:
# x = rhs.copy()
# ctx.set_rhs(x)
# ctx.run(job=3) # Solve
x = Ainv*rhs
J1_temp = Utils.mkvc((S_temp.T*self.mesh.nodalGrad*x).T)
J2_temp = Utils.mkvc(G_temp*self.mesh.aveE2CCV*Meinv*S_temp)
J[i,:] = J1_temp - J2_temp
# Only consider polarization current
else:
J2_temp = Utils.mkvc(G_temp*self.mesh.aveE2CCV*Meinv*S_temp)
J[i,:] = - J2_temp
stdout.write("\n")
if senseoption == True:
ctx.destroy()
self.J = J
self.sigma = sigma
self.emax = emax
def fget(self):
if self._gridFy is None:
N = self.r(self.gridN, 'N', 'N', 'M')
if self.dim == 2:
XY = [Utils.mkvc(0.5 * (n[:-1, :] + n[1:, :])) for n in N]
self._gridFy = np.c_[XY[0], XY[1]]
elif self.dim == 3:
XYZ = [Utils.mkvc(0.25 * (n[:-1, :, :-1] + n[:-1, :, 1:] + n[1:, :, :-1] + n[1:, :, 1:])) for n in N]
self._gridFy = np.c_[XYZ[0], XYZ[1], XYZ[2]]
return self._gridFy
def CondSphereAnalFunB(x, y, z, R, x0, y0, z0, sig1, sig2, E0, flag):
"""
test
Analytic function for Electro-Static problem. The set up here is
conductive sphere in whole-space.
* (x0,y0,z0)
* (x0, y0, z0 ): is the center location of sphere
* r: is the radius of the sphere
.. math::
\mathbf{E}_0 = E_0\hat{x}
"""
if (~np.size(x)==np.size(y)==np.size(z)):
print "Specify same size of x, y, z"
return
dim = x.shape
x = Utils.mkvc(x-x0)
y = Utils.mkvc(y-y0)
z = Utils.mkvc(z-z0)
ind = np.sqrt((x)**2+(y)**2+(z)**2 ) < R
r = Utils.mkvc(np.sqrt((x)**2+(y)**2+(z)**2 ))
Hx = np.zeros(x.size)
Hy = np.zeros(x.size)
Hz = np.zeros(x.size)
# Inside of the sphere
rf2 = 3*sig1/(sig2+2*sig1)
Hpy = -sig1*E0/2*z
Hpz = sig1*E0/2*y
if (flag == 'total'):
Hy[ind] = -3/2*sig2*E0*(rf2)*z[ind]
Hz[ind] = 3/2*sig2*E0*(rf2)*y[ind]
elif (flag == 'secondary'):
Hy[ind] = -3/2*sig2*E0*(rf2)*z[ind] - Hpy[ind]
Hz[ind] = 3/2*sig2*E0*(rf2)*y[ind] - Hpz[ind]
# Outside of the sphere
rf1 = (sig2-sig1)/(sig2+2*sig1)
if (flag == 'total'):
Hy[~ind] = sig1*(E0/r[~ind]**3*(R**3)*rf1*(-z[~ind]))+Hpy
Hz[~ind] = sig1*(E0/r[~ind]**3*(R**3)*rf1*( y[~ind]))+Hpz
elif (flag == 'secondary'):
Hy[~ind] = sig1*(E0/r[~ind]**3*(R**3)*rf1*(-z[~ind]))
Hz[~ind] = sig1*(E0/r[~ind]**3*(R**3)*rf1*( y[~ind]))
return np.reshape(mu_0*Hx, x.shape, order='F'), np.reshape(mu_0*Hy, x.shape, order='F'), np.reshape(mu_0*Hz, x.shape, order='F')
def getError(self):
#Test function
phi = lambda x: np.cos(0.5*np.pi*x)
j_fun = lambda x: -0.5*np.pi*np.sin(0.5*np.pi*x)
q_fun = lambda x: -0.25*(np.pi**2)*np.cos(0.5*np.pi*x)
xc_ana = phi(self.M.gridCC)
q_ana = q_fun(self.M.gridCC)
j_ana = j_fun(self.M.gridFx)
#TODO: Check where our boundary conditions are CCx or Nx
vecN = self.M.vectorNx
vecC = self.M.vectorCCx
phi_bc = phi(vecC[[0,-1]])
j_bc = j_fun(vecN[[0,-1]])
P, Pin, Pout = self.M.getBCProjWF([['dirichlet', 'neumann']])
Mc = self.M.getFaceInnerProduct()
McI = Utils.sdInv(self.M.getFaceInnerProduct())
V = Utils.sdiag(self.M.vol)
G = -Pin.T*Pin*self.M.faceDiv.T * V
D = self.M.faceDiv
j = McI*(G*xc_ana + P*phi_bc)
q = V*D*Pin.T*Pin*j + V*D*Pout.T*j_bc
# Rearrange if we know q to solve for x
A = V*D*Pin.T*Pin*McI*G
rhs = V*q_ana - V*D*Pin.T*Pin*McI*P*phi_bc - V*D*Pout.T*j_bc
# A = D*McI*G
# rhs = q_ana - D*McI*P*phi_bc
if self.myTest == 'j':
err = np.linalg.norm((Pin*j-Pin*j_ana), np.inf)
elif self.myTest == 'q':
err = np.linalg.norm((q-V*q_ana), np.inf)
elif self.myTest == 'xc':
#TODO: fix the null space
xc, info = sp.linalg.minres(A, rhs, tol = 1e-6)
err = np.linalg.norm((xc-xc_ana), np.inf)
if info > 0:
print('Solve does not work well')
print('ACCURACY', np.linalg.norm(Utils.mkvc(A*xc) - rhs))
elif self.myTest == 'xcJ':
#TODO: fix the null space
xc, info = sp.linalg.minres(A, rhs, tol = 1e-6)
j = McI*(G*xc + P*phi_bc)
err = np.linalg.norm((Pin*j-Pin*j_ana), np.inf)
if info > 0:
print('Solve does not work well')
print('ACCURACY', np.linalg.norm(Utils.mkvc(A*xc) - rhs))
return err
def test_SetGet(self):
F = self.F
nSrc = F.survey.nSrc
nT = F.survey.prob.nT + 1
e = np.random.rand(F.mesh.nE, nSrc, nT)
F[:, 'e'] = e
b = np.random.rand(F.mesh.nF, nSrc, nT)
F[:, 'b'] = b
self.assertTrue(np.all(F[:, 'e'] == e))
self.assertTrue(np.all(F[:, 'b'] == b))
F[:] = {'b': b, 'e': e}
self.assertTrue(np.all(F[:, 'e'] == e))
self.assertTrue(np.all(F[:, 'b'] == b))
for s in zero_types:
F[:, 'b'] = s
self.assertTrue(np.all(F[:, 'b'] == b*0))
b = np.random.rand(F.mesh.nF, 1, nT)
F[self.Src0, 'b'] = b
self.assertTrue(np.all(F[self.Src0, 'b'] == b[:, 0, :]))
b = np.random.rand(F.mesh.nF, 1, nT)
F[self.Src0, 'b', 0] = b[:, :, 0]
self.assertTrue(np.all(F[self.Src0, 'b', 0] == Utils.mkvc(b[:, 0, 0],
2)))
phi = np.random.rand(F.mesh.nC, 2, nT)
F[[self.Src0, self.Src1], 'phi'] = phi
self.assertTrue(np.all(F[[self.Src0, self.Src1], 'phi'] == phi))
fdict = F[:]
self.assertTrue(type(fdict) is dict)
self.assertTrue(sorted([k for k in fdict]) == ['b', 'e', 'phi'])
b = np.random.rand(F.mesh.nF, 2, nT)
F[[self.Src0, self.Src1], 'b'] = b
self.assertTrue(F[self.Src0]['b'].shape == (F.mesh.nF, nT))
self.assertTrue(F[self.Src0, 'b'].shape == (F.mesh.nF, nT))
self.assertTrue(np.all(F[self.Src0, 'b'] == b[:, 0, :]))
self.assertTrue(np.all(F[self.Src1, 'b'] == b[:, 1, :]))
self.assertTrue(np.all(F[self.Src0, 'b', 1] ==
Utils.mkvc(b[:, 0, 1], 2)))
self.assertTrue(np.all(F[self.Src1, 'b', 1] ==
Utils.mkvc(b[:, 1, 1], 2)))
self.assertTrue(np.all(F[self.Src0, 'b', 4] ==
Utils.mkvc(b[:, 0, 4], 2)))
self.assertTrue(np.all(F[self.Src1, 'b', 4] ==
Utils.mkvc(b[:, 1, 4], 2)))
b = np.random.rand(F.mesh.nF, 2, nT)
F[[self.Src0, self.Src1], 'b', 0] = b[:, :, 0]
def test_rotatePointsFromNormals(self):
np.random.seed(10)
v0 = np.random.rand(3)
v0*= 1./np.linalg.norm(v0)
np.random.seed(15)
v1 = np.random.rand(3)
v1*= 1./np.linalg.norm(v1)
v2 = Utils.mkvc(Utils.coordutils.rotatePointsFromNormals(Utils.mkvc(v0,2).T,v0,v1))
self.assertTrue(np.linalg.norm(v2-v1) < tol)
def MagSphereAnaFunA(x, y, z, R, xc, yc, zc, chi, Bo, flag):
"""Computing boundary condition using Congrous sphere method.
This is designed for secondary field formulation.
>> Input
mesh: Mesh class
Bo: np.array([Box, Boy, Boz]): Primary magnetic flux
Chi: susceptibility at cell volume
.. math::
\\vec{B}(r) = \\frac{\mu_0}{4\pi}\\frac{m}{\| \\vec{r}-\\vec{r}_0\|^3}[3\hat{m}\cdot\hat{r}-\hat{m}]
"""
if (~np.size(x)==np.size(y)==np.size(z)):
print("Specify same size of x, y, z")
return
dim = x.shape
x = Utils.mkvc(x)
y = Utils.mkvc(y)
z = Utils.mkvc(z)
Bot = np.sqrt(sum(Bo**2))
mx = Bo[0]/Bot
my = Bo[1]/Bot
mz = Bo[2]/Bot
ind = np.sqrt((x-xc)**2+(y-yc)**2+(z-zc)**2 ) < R
Bx = np.zeros(x.size)
By = np.zeros(x.size)
Bz = np.zeros(x.size)
# Inside of the sphere
rf2 = 3/(chi+3)*(1+chi)
if (flag == 'total'):
Bx[ind] = Bo[0]*(rf2)
By[ind] = Bo[1]*(rf2)
Bz[ind] = Bo[2]*(rf2)
elif (flag == 'secondary'):
Bx[ind] = Bo[0]*(rf2)-Bo[0]
By[ind] = Bo[1]*(rf2)-Bo[1]
Bz[ind] = Bo[2]*(rf2)-Bo[2]
r = Utils.mkvc(np.sqrt((x-xc)**2+(y-yc)**2+(z-zc)**2 ))
V = 4*np.pi*R**3/3
mom = Bot/mu_0*chi/(1+chi/3)*V
const = mu_0/(4*np.pi)*mom
mdotr = (mx*(x[~ind]-xc)/r[~ind] + my*(y[~ind]-yc)/r[~ind] + mz*(z[~ind]-zc)/r[~ind])
Bx[~ind] = const/(r[~ind]**3)*(3*mdotr*(x[~ind]-xc)/r[~ind]-mx)
By[~ind] = const/(r[~ind]**3)*(3*mdotr*(y[~ind]-yc)/r[~ind]-my)
Bz[~ind] = const/(r[~ind]**3)*(3*mdotr*(z[~ind]-zc)/r[~ind]-mz)
return Bx, By, Bz
def getError(self):
#Test function
phi = lambda x: np.cos(np.pi*x)
j_fun = lambda x: -np.pi*np.sin(np.pi*x)
q_fun = lambda x: -(np.pi**2)*np.cos(np.pi*x)
xc_ana = phi(self.M.gridCC)
q_ana = q_fun(self.M.gridCC)
j_ana = j_fun(self.M.gridFx)
#TODO: Check where our boundary conditions are CCx or Nx
# vec = self.M.vectorNx
vec = self.M.vectorCCx
phi_bc = phi(vec[[0,-1]])
j_bc = j_fun(vec[[0,-1]])
P, Pin, Pout = self.M.getBCProjWF([['dirichlet', 'dirichlet']])
Mc = self.M.getFaceInnerProduct()
McI = Utils.sdInv(self.M.getFaceInnerProduct())
V = Utils.sdiag(self.M.vol)
G = -Pin.T*Pin*self.M.faceDiv.T * V
D = self.M.faceDiv
j = McI*(G*xc_ana + P*phi_bc)
q = V*D*Pin.T*Pin*j + V*D*Pout.T*j_bc
# Rearrange if we know q to solve for x
A = V*D*Pin.T*Pin*McI*G
rhs = V*q_ana - V*D*Pin.T*Pin*McI*P*phi_bc - V*D*Pout.T*j_bc
# A = D*McI*G
# rhs = q_ana - D*McI*P*phi_bc
if self.myTest == 'j':
err = np.linalg.norm((j-j_ana), np.inf)
elif self.myTest == 'q':
err = np.linalg.norm((q-V*q_ana), np.inf)
elif self.myTest == 'xc':
#TODO: fix the null space
solver = SolverCG(A, maxiter=1000)
xc = solver * (rhs)
print('ACCURACY', np.linalg.norm(Utils.mkvc(A*xc) - rhs))
err = np.linalg.norm((xc-xc_ana), np.inf)
elif self.myTest == 'xcJ':
#TODO: fix the null space
xc = Solver(A) * (rhs)
print(np.linalg.norm(Utils.mkvc(A*xc) - rhs))
j = McI*(G*xc + P*phi_bc)
err = np.linalg.norm((j-j_ana), np.inf)
return err
def run(plotIt=True, nx=5, ny=5):
# 2D mesh
mesh = Mesh.TensorMesh([nx, ny], x0='CC')
xtopo = mesh.vectorNx
# define a topographic surface
topo = 0.4*np.sin(xtopo*5)
# make it an array
Topo = np.hstack([Utils.mkvc(xtopo, 2), Utils.mkvc(topo, 2)])
# Compare the different options
indtopoCC_near = surface2ind_topo(mesh, Topo, gridLoc='CC', method='nearest')
indtopoN_near = surface2ind_topo(mesh, Topo, gridLoc='N', method='nearest')
indtopoCC_linear = surface2ind_topo(mesh, Topo, gridLoc='CC', method='linear')
indtopoN_linear = surface2ind_topo(mesh, Topo, gridLoc='N', method='linear')
indtopoCC_cubic = surface2ind_topo(mesh, Topo, gridLoc='CC', method='cubic')
indtopoN_cubic = surface2ind_topo(mesh, Topo, gridLoc='N', method='cubic')
if plotIt:
fig, ax = plt.subplots(2, 3, figsize=(9, 6))
ax = mkvc(ax)
xinterpolate = np.linspace(mesh.gridN[:, 0].min(), mesh.gridN[:, 0].max(),100)
listindex = [indtopoCC_near,indtopoN_near,indtopoCC_linear,indtopoN_linear,indtopoCC_cubic,indtopoN_cubic]
listmethod = ['nearest','nearest', 'linear', 'linear', 'cubic', 'cubic']
for i in range(6):
mesh.plotGrid(ax=ax[i], nodes=True, centers=True)
mesh.plotImage(listindex[i], ax=ax[i], pcolorOpts = {"alpha":0.5, "cmap":plt.cm.gray})
ax[i].scatter(Topo[:,0], Topo[:,1], color = 'black', marker = 'o',s = 50)
ax[i].plot(
xinterpolate,
interp1d(Topo[:, 0], Topo[:, 1], kind=listmethod[i])(xinterpolate),
'--k',
linewidth=3
)
ax[i].xaxis.set_ticklabels([])
ax[i].yaxis.set_ticklabels([])
ax[i].set_aspect('equal')
ax[i].set_xlabel('')
ax[i].set_ylabel('')
ax[0].set_xlabel('Nearest Interpolation', fontsize=16)
ax[2].set_xlabel('Linear Interpolation', fontsize=16)
ax[4].set_xlabel('Cubic Interpolation', fontsize=16)
ax[0].set_ylabel('Cells Center \n based selection', fontsize=16)
ax[1].set_ylabel('Nodes \n based selection', fontsize=16)
plt.tight_layout()
def test_regularization_ActiveCells(self):
for R in dir(Regularization):
r = getattr(Regularization, R)
if not inspect.isclass(r):
continue
if not issubclass(r, ObjectiveFunction.BaseObjectiveFunction):
continue
if r.__name__ in IGNORE_ME:
continue
for i, mesh in enumerate(self.meshlist):
print('Testing Active Cells {0:d}D'.format((mesh.dim)))
if mesh.dim == 1:
indActive = Utils.mkvc(mesh.gridCC <= 0.8)
elif mesh.dim == 2:
indActive = Utils.mkvc(mesh.gridCC[:, -1] <= (
2*np.sin(2*np.pi*mesh.gridCC[:, 0])+0.5)
)
elif mesh.dim == 3:
indActive = Utils.mkvc(mesh.gridCC[:, -1] <= (
2 * np.sin(2*np.pi*mesh.gridCC[:, 0])+0.5 *
2 * np.sin(2*np.pi*mesh.gridCC[:, 1])+0.5)
)
if mesh.dim < 3 and r.__name__[-1] == 'z':
continue
if mesh.dim < 2 and r.__name__[-1] == 'y':
continue
for indAct in [indActive, indActive.nonzero()[0]]: # test both bool and integers
if indAct.dtype != bool:
nP = indAct.size
else:
nP = int(indAct.sum())
reg = r(
mesh, indActive=indAct, mapping=Maps.IdentityMap(nP=nP)
)
m = np.random.rand(mesh.nC)[indAct]
mref = np.ones_like(m)*np.mean(m)
reg.mref = mref
print(
'--- Checking {} ---\n'.format(
reg.__class__.__name__
)
)
passed = reg.test(m, eps=TOL)
self.assertTrue(passed)
def gridEy(self):
"""
Edge staggered grid in the y direction.
"""
if getattr(self, '_gridEy', None) is None:
N = self.r(self.gridN, 'N', 'N', 'M')
if self.dim == 2:
XY = [Utils.mkvc(0.5 * (n[:, :-1] + n[:, 1:])) for n in N]
self._gridEy = np.c_[XY[0], XY[1]]
elif self.dim == 3:
XYZ = [Utils.mkvc(0.5 * (n[:, :-1, :] + n[:, 1:, :])) for n in N]
self._gridEy = np.c_[XYZ[0], XYZ[1], XYZ[2]]
return self._gridEy
def test_data(self):
V = []
for src in self.D.survey.srcList:
for rx in src.rxList:
v = np.random.rand(rx.nD)
V += [v]
self.D[src, rx] = v
self.assertTrue(np.all(v == self.D[src, rx]))
V = np.concatenate(V)
self.assertTrue(np.all(V == Utils.mkvc(self.D)))
D2 = Survey.Data(self.D.survey, V)
self.assertTrue(np.all(Utils.mkvc(D2) == Utils.mkvc(self.D)))
def MagSphereAnaFun(x, y, z, R, x0, y0, z0, mu1, mu2, H0, flag='total'):
"""
test
Analytic function for Magnetics problem. The set up here is
magnetic sphere in whole-space assuming that the inducing field is oriented in the x-direction.
* (x0, y0, z0)
* (x0, y0, z0 ): is the center location of sphere
* r: is the radius of the sphere
.. math::
\mathbf{H}_0 = H_0\hat{x}
"""
if (~np.size(x)==np.size(y)==np.size(z)):
print("Specify same size of x, y, z")
return
dim = x.shape
x = Utils.mkvc(x)
y = Utils.mkvc(y)
z = Utils.mkvc(z)
ind = np.sqrt((x-x0)**2+(y-y0)**2+(z-z0)**2) < R
r = Utils.mkvc(np.sqrt((x-x0)**2+(y-y0)**2+(z-z0)**2))
Bx = np.zeros(x.size)
By = np.zeros(x.size)
Bz = np.zeros(x.size)
# Inside of the sphere
rf2 = 3*mu1/(mu2+2*mu1)
if flag is 'total' and any(ind):
Bx[ind] = mu2*H0*(rf2)
elif (flag == 'secondary'):
Bx[ind] = mu2*H0*(rf2)-mu1*H0
By[ind] = 0.
Bz[ind] = 0.
# Outside of the sphere
rf1 = (mu2-mu1)/(mu2+2*mu1)
if (flag == 'total'):
Bx[~ind] = mu1*(H0+H0/r[~ind]**5*(R**3)*rf1*(2*(x[~ind]-x0)**2-(y[~ind]-y0)**2-(z[~ind]-z0)**2))
elif (flag == 'secondary'):
Bx[~ind] = mu1*(H0/r[~ind]**5*(R**3)*rf1*(2*(x[~ind]-x0)**2-(y[~ind]-y0)**2-(z[~ind]-z0)**2))
By[~ind] = mu1*(H0/r[~ind]**5*(R**3)*rf1*(3*(x[~ind]-x0)*(y[~ind]-y0)))
Bz[~ind] = mu1*(H0/r[~ind]**5*(R**3)*rf1*(3*(x[~ind]-x0)*(z[~ind]-z0)))
return np.reshape(Bx, x.shape, order='F'), np.reshape(By, x.shape, order='F'), np.reshape(Bz, x.shape, order='F')
def setUp(self):
# Define sphere parameters
self.rad = 2.
self.rho = 0.1
# Define a mesh
cs = 0.2
hxind = [(cs, 21)]
hyind = [(cs, 21)]
hzind = [(cs, 21)]
mesh = Mesh.TensorMesh([hxind, hyind, hzind], 'CCC')
# Get cells inside the sphere
sph_ind = PF.MagAnalytics.spheremodel(mesh, 0., 0., 0., self.rad)
# Adjust density for volume difference
Vratio = (4./3.*np.pi*self.rad**3.) / (np.sum(sph_ind)*cs**3.)
model = np.ones(mesh.nC)*self.rho*Vratio
self.model = model[sph_ind]
# Create reduced identity map for Linear Pproblem
idenMap = Maps.IdentityMap(nP=int(sum(sph_ind)))
# Create plane of observations
xr = np.linspace(-20, 20, 21)
yr = np.linspace(-20, 20, 21)
X, Y = np.meshgrid(xr, yr)
# Move obs plane 2 radius away from sphere
Z = np.ones((xr.size, yr.size))*2.*self.rad
self.locXyz = np.c_[Utils.mkvc(X), Utils.mkvc(Y), Utils.mkvc(Z)]
rxLoc = PF.BaseGrav.RxObs(self.locXyz)
srcField = PF.BaseGrav.SrcField([rxLoc])
self.survey = PF.BaseGrav.LinearSurvey(srcField)
self.prob_x = PF.Gravity.GravityIntegral(mesh, rhoMap=idenMap,
actInd=sph_ind,
forwardOnly=True,
rx_type='x')
self.prob_y = PF.Gravity.GravityIntegral(mesh, rhoMap=idenMap,
actInd=sph_ind,
forwardOnly=True,
rx_type='y')
self.prob_z = PF.Gravity.GravityIntegral(mesh, rhoMap=idenMap,
actInd=sph_ind,
forwardOnly=True,
rx_type='z')
def vol(self):
"""Construct cell volumes of the 3D model as 1d array."""
if getattr(self, '_vol', None) is None:
vh = self.h
# Compute cell volumes
if self.dim == 1:
self._vol = Utils.mkvc(vh[0])
elif self.dim == 2:
# Cell sizes in each direction
self._vol = Utils.mkvc(np.outer(vh[0], vh[1]))
elif self.dim == 3:
# Cell sizes in each direction
self._vol = Utils.mkvc(np.outer(Utils.mkvc(np.outer(vh[0], vh[1])), vh[2]))
return self._vol
def test_rotationMatrixFromNormals(self):
np.random.seed(0)
v0 = np.random.rand(3)
v0 *= 1./np.linalg.norm(v0)
np.random.seed(5)
v1 = np.random.rand(3)
v1 *= 1./np.linalg.norm(v1)
Rf = Utils.coordutils.rotationMatrixFromNormals(v0,v1)
Ri = Utils.coordutils.rotationMatrixFromNormals(v1,v0)
self.assertTrue(np.linalg.norm(Utils.mkvc(Rf.dot(v0) - v1)) < tol)
self.assertTrue(np.linalg.norm(Utils.mkvc(Ri.dot(v1) - v0)) < tol)
def CondSphereAnalFunJ(x, y, z, R, x0, y0, z0, sig1, sig2, E0, flag):
"""
test
Analytic function for Electro-Static problem. The set up here is
conductive sphere in whole-space.
* (x0,y0,z0)
* (x0, y0, z0 ): is the center location of sphere
* r: is the radius of the sphere
.. math::
\mathbf{E}_0 = E_0\hat{x}
"""
if (~np.size(x)==np.size(y)==np.size(z)):
print "Specify same size of x, y, z"
return
dim = x.shape
x = Utils.mkvc(x-x0)
y = Utils.mkvc(y-y0)
z = Utils.mkvc(z-z0)
ind = np.sqrt((x)**2+(y)**2+(z)**2 ) < R
r = Utils.mkvc(np.sqrt((x)**2+(y)**2+(z)**2 ))
Jx = np.zeros(x.size)
Jy = np.zeros(x.size)
Jz = np.zeros(x.size)
# Inside of the sphere
rf2 = 3*sig1/(sig2+2*sig1)
if (flag == 'total'):
Jx[ind] = sig2*E0*(rf2)
elif (flag == 'secondary'):
Jx[ind] = sig2*E0*(rf2)-sig1*E0
Jy[ind] = 0.
Jz[ind] = 0.
# Outside of the sphere
rf1 = (sig2-sig1)/(sig2+2*sig1)
if (flag == 'total'):
Jx[~ind] = sig1*(E0+E0/r[~ind]**5*(R**3)*rf1*(2*x[~ind]**2-y[~ind]**2-z[~ind]**2))
elif (flag == 'secondary'):
Jx[~ind] = sig1*(E0/r[~ind]**5*(R**3)*rf1*(2*x[~ind]**2-y[~ind]**2-z[~ind]**2))
Jy[~ind] = sig1*(E0/r[~ind]**5*(R**3)*rf1*(3*x[~ind]*y[~ind]))
Jz[~ind] = sig1*(E0/r[~ind]**5*(R**3)*rf1*(3*x[~ind]*z[~ind]))
return np.reshape(Jx, x.shape, order='F'), np.reshape(Jy, x.shape, order='F'), np.reshape(Jz, x.shape, order='F')
def hzAnalyticDipoleF(r, freq, sigma, secondary=True, mu=mu_0):
"""
4.56 in Ward and Hohmann
.. plot::
import matplotlib.pyplot as plt
from SimPEG import EM
freq = np.logspace(-1, 6, 61)
test = EM.Analytics.FDEM.hzAnalyticDipoleF(100, freq, 0.001, secondary=False)
plt.loglog(freq, abs(test.real))
plt.loglog(freq, abs(test.imag))
plt.title('Response at $r$=100m')
plt.xlabel('Frequency')
plt.ylabel('Response')
plt.legend(('real','imag'))
plt.show()
"""
r = np.abs(r)
k = np.sqrt(-1j*2.*np.pi*freq*mu*sigma)
m = 1
front = m / (2. * np.pi * (k**2) * (r**5) )
back = 9 - ( 9 + 9j * k * r - 4 * (k**2) * (r**2) - 1j * (k**3) * (r**3)) * np.exp(-1j*k*r)
hz = front*back
if secondary:
hp =-1/(4*np.pi*r**3)
hz = hz-hp
if hz.ndim == 1:
hz = Utils.mkvc(hz,2)
return hz
def Jvec(self, m, v, u=None):
"""
Sensitivity times a vector.
:param numpy.array m: inversion model (nP,)
:param numpy.array v: vector which we take sensitivity product with (nP,)
:param SimPEG.EM.FDEM.Fields u: fields object
:rtype numpy.array:
:return: Jv (ndata,)
"""
if u is None:
u = self.fields(m)
self.curModel = m
Jv = self.dataPair(self.survey)
for freq in self.survey.freqs:
A = self.getA(freq)
Ainv = self.Solver(A, **self.solverOpts)
for src in self.survey.getSrcByFreq(freq):
u_src = u[src, self._solutionType]
dA_dm_v = self.getADeriv(freq, u_src, v)
dRHS_dm_v = self.getRHSDeriv(freq, src, v)
du_dm_v = Ainv * ( - dA_dm_v + dRHS_dm_v )
for rx in src.rxList:
df_dmFun = getattr(u, '_%sDeriv'%rx.projField, None)
df_dm_v = df_dmFun(src, du_dm_v, v, adjoint=False)
Jv[src, rx] = rx.evalDeriv(src, self.mesh, u, df_dm_v)
Ainv.clean()
return Utils.mkvc(Jv)
def Wd(self):
"""getWd(survey)
The data weighting matrix.
The default is based on the norm of the data plus a noise floor.
:rtype: scipy.sparse.csr_matrix
:return: Wd
"""
if getattr(self, '_Wd', None) is None:
survey = self.survey
if getattr(survey,'std', None) is None:
print('SimPEG.DataMisfit.l2_DataMisfit assigning default std of 5%')
survey.std = 0.05
if getattr(survey, 'eps', None) is None:
print('SimPEG.DataMisfit.l2_DataMisfit assigning default eps of 1e-5 * ||dobs||')
survey.eps = np.linalg.norm(Utils.mkvc(survey.dobs),2)*1e-5
self._Wd = Utils.sdiag(1/(abs(survey.dobs)*survey.std+survey.eps))
return self._Wd
def Jvec(self, m, v, f=None):
self.model = m
# When sensitivity matrix J is stored
if self.storeJ:
J = self.getJ(m, f=f)
Jv = Utils.mkvc(np.dot(J, v))
return self.sign * Jv
else:
if f is None:
f = self.fields(m)
Jv = []
for src in self.survey.srcList:
u_src = f[src, self._solutionType] # solution vector
dA_dm_v = self.getADeriv(u_src.flatten(), v, adjoint=False)
dRHS_dm_v = self.getRHSDeriv(src, v)
du_dm_v = self.Ainv * ( - dA_dm_v + dRHS_dm_v )
for rx in src.rxList:
df_dmFun = getattr(f, '_{0!s}Deriv'.format(rx.projField), None)
df_dm_v = df_dmFun(src, du_dm_v, v, adjoint=False)
Jv.append(rx.evalDeriv(src, self.mesh, f, df_dm_v))
# Conductivity (d u / d log sigma) - EB form
# Resistivity (d u / d log rho) - HJ form
return self.sign*np.hstack(Jv)
def test_rotateMatrixFromNormals(self):
np.random.seed(20)
n0 = np.random.rand(3)
n0 *= 1. / np.linalg.norm(n0)
np.random.seed(25)
n1 = np.random.rand(3)
n1 *= 1. / np.linalg.norm(n1)
np.random.seed(30)
scale = np.random.rand(100, 1)
XYZ0 = scale * n0
XYZ1 = scale * n1
XYZ2 = Utils.coordutils.rotatePointsFromNormals(XYZ0, n0, n1)
self.assertTrue(
np.linalg.norm(Utils.mkvc(XYZ1) - Utils.mkvc(XYZ2)) /
np.linalg.norm(Utils.mkvc(XYZ1)) < tol)
def gridEz(self):
"""
Edge staggered grid in the z direction.
"""
if getattr(self, '_gridEz', None) is None and self.dim == 3:
N = self.r(self.gridN, 'N', 'N', 'M')
XYZ = [Utils.mkvc(0.5 * (n[:, :, :-1] + n[:, :, 1:])) for n in N]
self._gridEz = np.c_[XYZ[0], XYZ[1], XYZ[2]]
return self._gridEz