def getR2(y, y_fitted, chi=None):
"""
calculates the coefficient of determination R^2 for `y_fitted` as prediction for `y` over a region marked by chi>0 defined by
R^2=1 - S_res/S_tot
with S_res=int(chi*(y-y_fitted*1)**2, S_tot=int(chi*(y-m(y)*1)**2), m(y)=int(chi*y)/int(chi)
If R^2=1 then `y_fitted` is predicts `y` exactly. If R^2 then `y_fitted` does not make a better prediction than the mean.
:param y: target distribution
:type y: `esys.escript.Scalar`
:param y_fitted: fitted distribution
:type y_fitted: `esys.escript.Scalar`
:param chi: marker/weighting for region of interest
:type chi: `esys.escript.Scalar` or None
:rtype: `float`
"""
if chi is None:
chi = Scalar(1., Function(y_fitted.getFunctionSpace().getDomain()))
ybar = integrate(chi * y) / integrate(chi)
S_res = integrate(chi * (y - y_fitted)**2)
S_tot = integrate(chi * (y - ybar)**2)
if S_tot > 0:
R2 = 1 - S_res / S_tot
else:
if S_res > 0:
R2 = 0.
else:
R2 = 1.
return R2
def getDualProduct(self, m, r):
"""
returns the dual product of a gradient represented by X=r[1] and Y=r[0]
with a level set function m:
*Y_i*m_i + X_ij*m_{i,j}*
:type m: `Data`
:type r: `ArithmeticTuple`
:rtype: ``float``
"""
A = 0
if not r[0].isEmpty(): A += integrate(inner(r[0], m))
if not r[1].isEmpty(): A += integrate(inner(r[1], grad(m)))
return A
def getSourceScaling(self, u):
"""
returns the scaling factor s required to rescale source F to minimize defect ``|s * u- data|^2``
:param u: value of pressure solution (real and imaginary part)
:type u: ``escript.Data`` of shape (2,)
:rtype: `complex`
"""
uTu = escript.integrate(self.__weight * escript.length(u)**2)
uTar = escript.integrate(self.__weight * ( u[0]*self.__data[0]+u[1]*self.__data[1]) )
uTai = escript.integrate(self.__weight * ( u[0]*self.__data[1]-u[1]*self.__data[0]) )
if uTu > 0:
return complex(uTar/uTu, uTai/uTu)
else:
return complex(1.,0)
def getDualProduct(self, m, r):
"""
returns the dual product of a gradient represented by X=r[1] and Y=r[0]
with a level set function m:
*Y_i*m_i + X_ij*m_{i,j}*
:type m: `Data`
:type r: `ArithmeticTuple`
:rtype: ``float``
"""
A=0
if not r[0].isEmpty(): A+=integrate(inner(r[0], m))
if not r[1].isEmpty(): A+=integrate(inner(r[1], grad(m)))
return A
def getValue(self, m, grad_m):
"""
returns the value of the cost function J with respect to m.
This equation is specified in the inversion cookbook.
:rtype: ``float``
"""
if m != self.__pre_input:
raise RuntimeError("Attempt to change point using getValue")
# substituting cached values
m = self.__pre_input
grad_m = self.__pre_args
mu = self.__mu
mu_c = self.__mu_c
DIM = self.getDomain().getDim()
numLS = self.getNumLevelSets()
A = 0
if self.__w0 is not None:
r = inner(escript.integrate(m**2 * self.__w0), mu)
self.logger.debug("J_R[m^2] = %e" % r)
A += r
if self.__w1 is not None:
if numLS == 1:
r = escript.integrate(inner(grad_m**2, self.__w1)) * mu
self.logger.debug("J_R[grad(m)] = %e" % r)
A += r
else:
for k in range(numLS):
r = mu[k] * escript.integrate(
inner(grad_m[k, :]**2, self.__w1[k, :]))
self.logger.debug("J_R[grad(m)][%d] = %e" % (k, r))
A += r
if numLS > 1:
for k in range(numLS):
gk = grad_m[k, :]
len_gk = escript.length(gk)
for l in range(k):
gl = grad_m[l, :]
r = mu_c[l, k] * escript.integrate(self.__wc[l, k] * (
(len_gk * escript.length(gl))**2 - inner(gk, gl)**2))
self.logger.debug("J_R[cross][%d,%d] = %e" % (l, k, r))
A += r
return A / 2
def getDefect(self, rho, Hx, g_Hx):
"""
Returns the defect value.
:param rho: a suggestion for resistivity
:type rho: ``Data`` of shape ()
:param Hx: magnetic field
:type Hx: ``Data`` of shape (2,)
:param g_Hx: gradient of magnetic field
:type g_Hx: ``Data`` of shape (2,2)
:rtype: ``float``
"""
x = g_Hx.getFunctionSpace().getX()
Hx = escript.interpolate(Hx, x.getFunctionSpace())
u0 = Hx[0]
u1 = Hx[1]
u01 = g_Hx[0, 1]
u11 = g_Hx[1, 1]
scale = rho / (u0**2 + u1**2)
Z = self._Z
A = escript.integrate(
self._weight * (Z[0]**2 + Z[1]**2 + scale *
(-2 * Z[0] * (u0 * u01 + u1 * u11) + 2 * Z[1] *
(u1 * u01 - u0 * u11) + rho * (u01**2 + u11**2))))
return A / 2
def getDefect(self, sigma, Ex, dExdz):
"""
Returns the defect value.
:param sigma: a suggestion for conductivity
:type sigma: ``Data`` of shape ()
:param Ex: electric field
:type Ex: ``Data`` of shape (2,)
:param dExdz: vertical derivative of electric field
:type dExdz: ``Data`` of shape (2,)
:rtype: ``float``
"""
x = dExdz.getFunctionSpace().getX()
Ex = escript.interpolate(Ex, x.getFunctionSpace())
u0 = Ex[0]
u1 = Ex[1]
u01 = dExdz[0]
u11 = dExdz[1]
scale = self._weight / (u01**2 + u11**2)
Z = self._Z
A = escript.integrate(
scale * ((Z[0]**2 + Z[1]**2) * (u01**2 + u11**2) + 2 * Z[1] *
(u0 * u11 - u01 * u1) - 2 * Z[0] *
(u0 * u01 + u11 * u1) + u0**2 + u1**2))
return A / 2
def rescaleWeights(self, scale=1., sigma_scale=1.):
"""
rescales the weights such that
:math: integrate( ( w omega**2 * sigma_scale * data * ((1/L_j)**2)**-1) +1 )/(data*omega**2 * ((1/L_j)**2)**-1) * sigma_scale )=scale
:param scale: scale of data weighting factors
:type scale: positive ``float``
:param sigma_scale: scale of 1/vp**2 velocity.
:type sigma_scale: ``Scalar``
"""
raise Warning("rescaleWeights is not tested yet.")
if not scale > 0:
raise ValueError("Value for scale must be positive.")
if not sigma_scale * omega**2 * d > 0:
raise ValueError(
"Rescaling of weights failed due to zero denominator.")
# copy back original weights before rescaling
#self.__weight=[1.*ow for ow in self.__origweight]
L2 = 1 / escript.length(1 / self.edge_length)**2
d = Lsup(escript.length(data))
A = escript.integrate(self.__weight *
(sigma_scale * omega**2 * d + 1) /
(sigma_scale * omega**2 * d))
if A > 0:
self.__weight *= 1. / A
if self.scaleF:
self.__data *= escript.sqrt(A)
else:
raise ValueError("Rescaling of weights failed.")
def getValue(self, m, grad_m):
"""
returns the value of the cost function J with respect to m.
This equation is specified in the inversion cookbook.
:rtype: ``float``
"""
if m != self.__pre_input:
raise RuntimeError("Attempt to change point using getValue")
# substituting cached values
m = self.__pre_input
grad_m = self.__pre_args
mu = self.__mu
mu_c = self.__mu_c
DIM = self.getDomain().getDim()
numLS = self.getNumLevelSets()
A = 0
if self.__w0 is not None:
r = inner(integrate(m ** 2 * self.__w0), mu)
self.logger.debug("J_R[m^2] = %e" % r)
A += r
if self.__w1 is not None:
if numLS == 1:
r = integrate(inner(grad_m ** 2, self.__w1)) * mu
self.logger.debug("J_R[grad(m)] = %e" % r)
A += r
else:
for k in range(numLS):
r = mu[k] * integrate(inner(grad_m[k, :] ** 2, self.__w1[k, :]))
self.logger.debug("J_R[grad(m)][%d] = %e" % (k, r))
A += r
if numLS > 1:
for k in range(numLS):
gk = grad_m[k, :]
len_gk = length(gk)
for l in range(k):
gl = grad_m[l, :]
r = mu_c[l, k] * integrate(self.__wc[l, k] * ((len_gk * length(gl)) ** 2 - inner(gk, gl) ** 2))
self.logger.debug("J_R[cross][%d,%d] = %e" % (l, k, r))
A += r
return A / 2
def getNorm(self, m):
"""
returns the norm of ``m``.
:param m: level set function
:type m: `Data`
:rtype: ``float``
"""
return sqrt(integrate(length(m)**2) / self.__vol_d)
def getNorm(self, m):
"""
returns the norm of ``m``.
:param m: level set function
:type m: `Data`
:rtype: ``float``
"""
return sqrt(integrate(length(m) ** 2) / self.__vol_d)
def getValue(self, m, grad_m):
"""
returns the value of the cost function J with respect to m.
This equation is specified in the inversion cookbook.
:rtype: ``float``
"""
mu = self.__mu
mu_c = self.__mu_c
DIM = self.getDomain().getDim()
numLS = self.getNumLevelSets()
A = 0
if self.__w0 is not None:
r = inner(integrate(m**2 * self.__w0), mu)
self.logger.debug("J_R[m^2] = %e" % r)
A += r
if self.__w1 is not None:
if numLS == 1:
r = integrate(inner(grad_m**2, self.__w1)) * mu
self.logger.debug("J_R[grad(m)] = %e" % r)
A += r
else:
for k in range(numLS):
r = mu[k] * integrate(
inner(grad_m[k, :]**2, self.__w1[k, :]))
self.logger.debug("J_R[grad(m)][%d] = %e" % (k, r))
A += r
if numLS > 1:
for k in range(numLS):
gk = grad_m[k, :]
len_gk = length(gk)
for l in range(k):
gl = grad_m[l, :]
r = mu_c[l, k] * integrate(self.__wc[l, k] * (
(len_gk * length(gl))**2 - inner(gk, gl)**2))
self.logger.debug("J_R[cross][%d,%d] = %e" % (l, k, r))
A += r
return A / 2
def getArguments(self, sigma):
"""
Returns precomputed values shared by `getDefect()` and `getGradient()`.
:param sigma: a suggestion for complex 1/V**2
:type sigma: ``escript.Data`` of shape (2,)
:return: solution, uTar, uTai, uTu
:rtype: ``escript.Data`` of shape (2,), 3 x `float`
"""
pde=self.setUpPDE()
D=pde.getCoefficient('D')
D[0,0]=-self.__omega**2 * sigma[0]
D[0,1]= self.__omega**2 * sigma[1]
D[1,0]=-self.__omega**2 * sigma[1]
D[1,1]=-self.__omega**2 * sigma[0]
pde.setValue(D=D, Y=self.__F, y=self.__f, y_dirac=self.__f_dirac)
u=pde.getSolution()
uTar=escript.integrate(self.__weight * ( u[0]*self.__data[0]+u[1]*self.__data[1]) )
uTai=escript.integrate(self.__weight * ( u[0]*self.__data[1]-u[1]*self.__data[0]) )
uTu = escript.integrate( self.__weight * escript.length(u)**2 )
return u, uTar, uTai, uTu
def getDefect(self, sigma, u, uTar, uTai, uTu):
"""
Returns the defect value.
:param sigma: a suggestion for complex 1/V**2
:type sigma: ``escript.Data`` of shape (2,)
:param u: a u vector
:type u: ``escript.Data`` of shape (2,)
:param uTar: equals `integrate( w * (data[0]*u[0]+data[1]*u[1]))`
:type uTar: `float`
:param uTai: equals `integrate( w * (data[1]*u[0]-data[0]*u[1]))`
:type uTa: `float`
:param uTu: equals `integrate( w * (u,u))`
:type uTu: `float`
:rtype: ``float``
"""
# assuming integrate(w * length(data)**2) =1
if self.scaleF and abs(uTu) >0:
A = 1.-(uTar**2 + uTai**2)/uTu
else:
A = escript.integrate(self.__weight*escript.length(self.__data)**2)- 2 * uTar + uTu
return A/2
def RegionalCalculation(reg_mask):
"""
Calculates the "regional" from the entire FEILDS model excluding the
selected region and outputs gravity at the specified altitude...
see above for the "residual"
"""
# read in a gravity data grid to define data computation space
G_DATA = os.path.join(DATADIR,'Final_BouguerTC_UC15K_qrtdeg.nc')
FS=ReducedFunction(dom)
nValues=[NX, NY, 1]
first = [0, 0, cell_at_altitude]
multiplier = [1, 1, 1]
reverse = [0, 0, 0]
byteorder = BYTEORDER_NATIVE
gdata = readBinaryGrid(G_DATA, FS, shape=(),
fill=-999999, byteOrder=byteorder,
dataType=DATATYPE_FLOAT32, first=first, numValues=nValues,
multiplier=multiplier, reverse=reverse)
print("Grid successfully read")
# get the masking and units sorted out for the data-space
g_mask = whereNonZero(gdata+999999)
gdata=gdata*g_mask * GRAV_UNITS
# if people choose to have air in their region we exclude it from the
# specified gravity calculation region
if h_top < 0.:
reg_mask = reg_mask+mask_air
live_model = initial_model* whereNonPositive(reg_mask)
dead_model = initial_model* wherePositive(reg_mask)
if UseMean is True:
# calculate the mean density within the selected region
BackgroundDensity = integrate(dead_model)/integrate(wherePositive(reg_mask))
print("Density mean for selected region equals = %s"%BackgroundDensity)
live_model = live_model + BackgroundDensity * wherePositive(reg_mask)
# create mapping
rho_mapping = DensityMapping(dom, rho0=live_model)
# invert sign of gravity field to account for escript's coordinate system
gdata = -GRAV_UNITS * gdata
# turn the scalars into vectors (vertical direction)
d=kronecker(DIM)[DIM-1]
w=safeDiv(1., g_mask)
gravity_model=GravityModel(dom, w*d, gdata*d, fixPotentialAtBottom=False, coordinates=COORDINATES)
gravity_model.rescaleWeights(rho_scale=rho_mapping.getTypicalDerivative())
phi,_ = gravity_model.getArguments(live_model)
g_init = -gravity_model.getCoordinateTransformation().getGradient(phi)
g_init = interpolate(g_init, gdata.getFunctionSpace())
print("Computed gravity: %s"%(g_init[2]))
fn=os.path.join(OUTPUTDIR,'regional-gravity')
if SiloOutput is True:
saveSilo(fn, density=live_model, gravity_init=g_init, g_initz=-g_init[2], gravitymask=g_mask, modelmask=reg_mask)
print('SILO file written with the following fields: density (kg/m^3), gravity vector (m/s^2), gz (m/s^2), gravitymask, modelmask')
# to compare calculated data against input dataset.
# Not used by default but should work if the input dataset is correct
#gslice = g_init[2]*wherePositive(g_mask)
#g_dash = integrate(gslice)/integrate(wherePositive(g_mask))
#gdataslice = gdata*wherePositive(g_mask)
#gdata_dash = integrate(gdataslice)/integrate(wherePositive(g_mask))
#misfit=(gdataslice-gdata_dash)-(gslice-g_dash)
saveDataCSV(fn+".csv", mask=g_mask, gz=-g_init[2], Long=datacoords[0], Lat=datacoords[1], h=datacoords[2])
print('CSV file written with the following fields: Longitude (degrees) Latitude (degrees), h (100km), gz (m/s^2)')
def __init__(self,
domain,
numLevelSets=1,
w0=None,
w1=None,
wc=None,
location_of_set_m=escript.Data(),
useDiagonalHessianApproximation=False,
tol=1e-8,
coordinates=None,
scale=None,
scale_c=None):
"""
initialization.
:param domain: domain
:type domain: `Domain`
:param numLevelSets: number of level sets
:type numLevelSets: ``int``
:param w0: weighting factor for the m**2 term. If not set zero is assumed.
:type w0: ``Scalar`` if ``numLevelSets`` == 1 or `Data` object of shape
(``numLevelSets`` ,) if ``numLevelSets`` > 1
:param w1: weighting factor for the grad(m_i) terms. If not set zero is assumed
:type w1: ``Vector`` if ``numLevelSets`` == 1 or `Data` object of shape
(``numLevelSets`` , DIM) if ``numLevelSets`` > 1
:param wc: weighting factor for the cross gradient terms. If not set
zero is assumed. Used for the case if ``numLevelSets`` > 1
only. Only values ``wc[l,k]`` in the lower triangle (l<k)
are used.
:type wc: `Data` object of shape (``numLevelSets`` , ``numLevelSets``)
:param location_of_set_m: marks location of zero values of the level
set function ``m`` by a positive entry.
:type location_of_set_m: ``Scalar`` if ``numLevelSets`` == 1 or `Data`
object of shape (``numLevelSets`` ,) if ``numLevelSets`` > 1
:param useDiagonalHessianApproximation: if True cross gradient terms
between level set components are ignored when calculating
approximations of the inverse of the Hessian Operator.
This can speed-up the calculation of the inverse but may
lead to an increase of the number of iteration steps in the
inversion.
:type useDiagonalHessianApproximation: ``bool``
:param tol: tolerance when solving the PDE for the inverse of the
Hessian Operator
:type tol: positive ``float``
:param coordinates: defines coordinate system to be used
:type coordinates: ReferenceSystem` or `SpatialCoordinateTransformation`
:param scale: weighting factor for level set function variation terms.
If not set one is used.
:type scale: ``Scalar`` if ``numLevelSets`` == 1 or `Data` object of
shape (``numLevelSets`` ,) if ``numLevelSets`` > 1
:param scale_c: scale for the cross gradient terms. If not set
one is assumed. Used for the case if ``numLevelSets`` > 1
only. Only values ``scale_c[l,k]`` in the lower triangle
(l<k) are used.
:type scale_c: `Data` object of shape (``numLevelSets``,``numLevelSets``)
"""
if w0 is None and w1 is None:
raise ValueError("Values for w0 or for w1 must be given.")
if wc is None and numLevelSets > 1:
raise ValueError("Values for wc must be given.")
self.__pre_input = None
self.__pre_args = None
self.logger = logging.getLogger('inv.%s' % self.__class__.__name__)
self.__domain = domain
DIM = self.__domain.getDim()
self.__numLevelSets = numLevelSets
self.__trafo = makeTransformation(domain, coordinates)
self.__pde = linearPDEs.LinearPDE(self.__domain,
numEquations=self.__numLevelSets,
numSolutions=self.__numLevelSets)
self.__pde.getSolverOptions().setTolerance(tol)
self.__pde.setSymmetryOn()
self.__pde.setValue(
A=self.__pde.createCoefficient('A'),
D=self.__pde.createCoefficient('D'),
)
try:
self.__pde.setValue(q=location_of_set_m)
except linearPDEs.IllegalCoefficientValue:
raise ValueError(
"Unable to set location of fixed level set function.")
# =========== check the shape of the scales: ========================
if scale is None:
if numLevelSets == 1:
scale = 1.
else:
scale = np.ones((numLevelSets, ))
else:
scale = np.asarray(scale)
if numLevelSets == 1:
if scale.shape == ():
if not scale > 0:
raise ValueError("Value for scale must be positive.")
else:
raise ValueError("Unexpected shape %s for scale." %
scale.shape)
else:
if scale.shape is (numLevelSets, ):
if not min(scale) > 0:
raise ValueError(
"All values for scale must be positive.")
else:
raise ValueError("Unexpected shape %s for scale." %
scale.shape)
if scale_c is None or numLevelSets < 2:
scale_c = np.ones((numLevelSets, numLevelSets))
else:
scale_c = np.asarray(scale_c)
if scale_c.shape == (numLevelSets, numLevelSets):
if not all([[scale_c[l, k] > 0. for l in range(k)]
for k in range(1, numLevelSets)]):
raise ValueError(
"All values in the lower triangle of scale_c must be positive."
)
else:
raise ValueError("Unexpected shape %s for scale." %
scale_c.shape)
# ===== check the shape of the weights: =============================
if w0 is not None:
w0 = escript.interpolate(
w0, self.__pde.getFunctionSpaceForCoefficient('D'))
s0 = w0.getShape()
if numLevelSets == 1:
if not s0 == ():
raise ValueError("Unexpected shape %s for weight w0." %
(s0, ))
else:
if not s0 == (numLevelSets, ):
raise ValueError("Unexpected shape %s for weight w0." %
(s0, ))
if not self.__trafo.isCartesian():
w0 *= self.__trafo.getVolumeFactor()
if not w1 is None:
w1 = escript.interpolate(
w1, self.__pde.getFunctionSpaceForCoefficient('A'))
s1 = w1.getShape()
if numLevelSets == 1:
if not s1 == (DIM, ):
raise ValueError("Unexpected shape %s for weight w1." %
(s1, ))
else:
if not s1 == (numLevelSets, DIM):
raise ValueError("Unexpected shape %s for weight w1." %
(s1, ))
if not self.__trafo.isCartesian():
f = self.__trafo.getScalingFactors(
)**2 * self.__trafo.getVolumeFactor()
if numLevelSets == 1:
w1 *= f
else:
for i in range(numLevelSets):
w1[i, :] *= f
if numLevelSets == 1:
wc = None
else:
wc = escript.interpolate(
wc, self.__pde.getFunctionSpaceForCoefficient('A'))
sc = wc.getShape()
if not sc == (numLevelSets, numLevelSets):
raise ValueError("Unexpected shape %s for weight wc." % (sc, ))
if not self.__trafo.isCartesian():
raise ValueError(
"Non-cartesian coordinates for cross-gradient term is not supported yet."
)
# ============= now we rescale weights: =============================
L2s = np.asarray(escript.boundingBoxEdgeLengths(domain))**2
L4 = 1 / np.sum(1 / L2s)**2
if numLevelSets == 1:
A = 0
if w0 is not None:
A = escript.integrate(w0)
if w1 is not None:
A += escript.integrate(inner(w1, 1 / L2s))
if A > 0:
f = scale / A
if w0 is not None:
w0 *= f
if w1 is not None:
w1 *= f
else:
raise ValueError("Non-positive weighting factor detected.")
else: # numLevelSets > 1
for k in range(numLevelSets):
A = 0
if w0 is not None:
A = escript.integrate(w0[k])
if w1 is not None:
A += escript.integrate(inner(w1[k, :], 1 / L2s))
if A > 0:
f = scale[k] / A
if w0 is not None:
w0[k] *= f
if w1 is not None:
w1[k, :] *= f
else:
raise ValueError(
"Non-positive weighting factor for level set component %d detected."
% k)
# and now the cross-gradient:
if wc is not None:
for l in range(k):
A = escript.integrate(wc[l, k]) / L4
if A > 0:
f = scale_c[l, k] / A
wc[l, k] *= f
# else:
# raise ValueError("Non-positive weighting factor for cross-gradient level set components %d and %d detected."%(l,k))
self.__w0 = w0
self.__w1 = w1
self.__wc = wc
self.__pde_is_set = False
if self.__numLevelSets > 1:
self.__useDiagonalHessianApproximation = useDiagonalHessianApproximation
else:
self.__useDiagonalHessianApproximation = True
self._update_Hessian = True
self.__num_tradeoff_factors = numLevelSets + (
(numLevelSets - 1) * numLevelSets) // 2
self.setTradeOffFactors()
self.__vol_d = escript.vol(self.__domain)
def __init__(self,
domain,
omega,
x,
Z,
eta=None,
w0=1.,
mu=4 * math.pi * 1e-7,
sigma0=0.01,
airLayerLevel=None,
fixAirLayer=False,
coordinates=None,
tol=1e-8,
saveMemory=False,
directSolver=True):
"""
initializes a new forward model.
:param domain: domain of the model
:type domain: `Domain`
:param omega: frequency
:type omega: positive ``float``
:param x: coordinates of measurements
:type x: ``list`` of ``tuple`` with ``float``
:param Z: measured impedance (possibly scaled)
:type Z: ``list`` of ``complex``
:param eta: spatial confidence radius
:type eta: positive ``float`` or ``list`` of positive ``float``
:param w0: confidence factors for meassurements.
:type w0: ``None`` or a list of positive ``float``
:param mu: permeability
:type mu: ``float``
:param sigma0: background conductivity
:type sigma0: ``float``
:param airLayerLevel: position of the air layer from to bottom of the domain. If
not set the air layer starts at the top of the domain
:type airLayerLevel: ``float`` or ``None``
:param fixAirLayer: fix air layer (TM mode)
:type fixAirLayer: ``bool``
:param coordinates: defines coordinate system to be used (not supported yet)
:type coordinates: `ReferenceSystem` or `SpatialCoordinateTransformation`
:param tol: tolerance of underlying PDE
:type tol: positive ``float``
:param saveMemory: if true stiffness matrix is deleted after solution
of the PDE to minimize memory use. This will
require more compute time as the matrix needs to be
reallocated at each iteration.
:type saveMemory: ``bool``
:param directSolver: if true a direct solver (rather than an iterative
solver) will be used to solve the PDE
:type directSolver: ``bool``
"""
super(MT2DBase, self).__init__()
self.__trafo = coords.makeTransformation(domain, coordinates)
if not self.getCoordinateTransformation().isCartesian():
raise ValueError(
"Non-Cartesian coordinates are not supported yet.")
if len(x) != len(Z):
raise ValueError(
"Number of data points and number of impedance values don't match."
)
if eta is None:
eta = escript.sup(domain.getSize()) * 0.45
if isinstance(eta, float) or isinstance(eta, int):
eta = [float(eta)] * len(Z)
elif not len(eta) == len(Z):
raise ValueError(
"Number of confidence radii and number of impedance values don't match."
)
if isinstance(w0, float) or isinstance(w0, int):
w0 = [float(w0)] * len(Z)
elif not len(w0) == len(Z):
raise ValueError(
"Number of confidence factors and number of impedance values don't match."
)
self.__domain = domain
self._omega_mu = omega * mu
self._ks = escript.sqrt(self._omega_mu * sigma0 / 2.)
xx = escript.Function(domain).getX()
totalS = 0
self._Z = [
escript.Scalar(0., escript.Function(domain)),
escript.Scalar(0., escript.Function(domain))
]
self._weight = escript.Scalar(0., escript.Function(domain))
for s in range(len(Z)):
chi = self.getWeightingFactor(xx, 1., x[s], eta[s])
f = escript.integrate(chi)
if f < eta[s]**2 * 0.01:
raise ValueError(
"Zero weight (almost) for data point %s. Change eta or refine mesh."
% (s, ))
w02 = w0[s] / f
totalS += w02
self._Z[0] += chi * Z[s].real
self._Z[1] += chi * Z[s].imag
self._weight += chi * w02 / (abs(Z[s])**2)
if not totalS > 0:
raise ValueError(
"Scaling of weight factors failed as sum is zero.")
DIM = domain.getDim()
z = domain.getX()[DIM - 1]
self._ztop = escript.sup(z)
self._zbottom = escript.inf(z)
if airLayerLevel is None:
airLayerLevel = self._ztop
self._airLayerLevel = airLayerLevel
# botton:
mask0 = escript.whereZero(z - self._zbottom)
r = mask0 * [
escript.exp(self._ks * (self._zbottom - airLayerLevel)) *
escript.cos(self._ks * (self._zbottom - airLayerLevel)),
escript.exp(self._ks * (self._zbottom - airLayerLevel)) *
escript.sin(self._ks * (self._zbottom - airLayerLevel))
]
#top:
if fixAirLayer:
mask1 = escript.whereNonNegative(z - airLayerLevel)
r += mask1 * [1, 0]
else:
mask1 = escript.whereZero(z - self._ztop)
r += mask1 * [
self._ks * (self._ztop - airLayerLevel) + 1, self._ks *
(self._ztop - airLayerLevel)
]
self._q = (mask0 + mask1) * [1, 1]
self._r = r
#====================================
self.__tol = tol
self._directSolver = directSolver
self._saveMemory = saveMemory
self.__pde = None
if not saveMemory:
self.__pde = self.setUpPDE()
def __init__(
self,
domain,
numLevelSets=1,
w0=None,
w1=None,
wc=None,
location_of_set_m=Data(),
useDiagonalHessianApproximation=False,
tol=1e-8,
coordinates=None,
scale=None,
scale_c=None,
):
"""
initialization.
:param domain: domain
:type domain: `Domain`
:param numLevelSets: number of level sets
:type numLevelSets: ``int``
:param w0: weighting factor for the m**2 term. If not set zero is assumed.
:type w0: ``Scalar`` if ``numLevelSets`` == 1 or `Data` object of shape
(``numLevelSets`` ,) if ``numLevelSets`` > 1
:param w1: weighting factor for the grad(m_i) terms. If not set zero is assumed
:type w1: ``Vector`` if ``numLevelSets`` == 1 or `Data` object of shape
(``numLevelSets`` , DIM) if ``numLevelSets`` > 1
:param wc: weighting factor for the cross gradient terms. If not set
zero is assumed. Used for the case if ``numLevelSets`` > 1
only. Only values ``wc[l,k]`` in the lower triangle (l<k)
are used.
:type wc: `Data` object of shape (``numLevelSets`` , ``numLevelSets``)
:param location_of_set_m: marks location of zero values of the level
set function ``m`` by a positive entry.
:type location_of_set_m: ``Scalar`` if ``numLevelSets`` == 1 or `Data`
object of shape (``numLevelSets`` ,) if ``numLevelSets`` > 1
:param useDiagonalHessianApproximation: if True cross gradient terms
between level set components are ignored when calculating
approximations of the inverse of the Hessian Operator.
This can speed-up the calculation of the inverse but may
lead to an increase of the number of iteration steps in the
inversion.
:type useDiagonalHessianApproximation: ``bool``
:param tol: tolerance when solving the PDE for the inverse of the
Hessian Operator
:type tol: positive ``float``
:param coordinates: defines coordinate system to be used
:type coordinates: ReferenceSystem` or `SpatialCoordinateTransformation`
:param scale: weighting factor for level set function variation terms.
If not set one is used.
:type scale: ``Scalar`` if ``numLevelSets`` == 1 or `Data` object of
shape (``numLevelSets`` ,) if ``numLevelSets`` > 1
:param scale_c: scale for the cross gradient terms. If not set
one is assumed. Used for the case if ``numLevelSets`` > 1
only. Only values ``scale_c[l,k]`` in the lower triangle
(l<k) are used.
:type scale_c: `Data` object of shape (``numLevelSets``,``numLevelSets``)
"""
if w0 is None and w1 is None:
raise ValueError("Values for w0 or for w1 must be given.")
if wc is None and numLevelSets > 1:
raise ValueError("Values for wc must be given.")
self.__pre_input = None
self.__pre_args = None
self.logger = logging.getLogger("inv.%s" % self.__class__.__name__)
self.__domain = domain
DIM = self.__domain.getDim()
self.__numLevelSets = numLevelSets
self.__trafo = makeTransformation(domain, coordinates)
self.__pde = LinearPDE(self.__domain, numEquations=self.__numLevelSets, numSolutions=self.__numLevelSets)
self.__pde.getSolverOptions().setTolerance(tol)
self.__pde.setSymmetryOn()
self.__pde.setValue(A=self.__pde.createCoefficient("A"), D=self.__pde.createCoefficient("D"))
try:
self.__pde.setValue(q=location_of_set_m)
except IllegalCoefficientValue:
raise ValueError("Unable to set location of fixed level set function.")
# =========== check the shape of the scales: ========================
if scale is None:
if numLevelSets == 1:
scale = 1.0
else:
scale = np.ones((numLevelSets,))
else:
scale = np.asarray(scale)
if numLevelSets == 1:
if scale.shape == ():
if not scale > 0:
raise ValueError("Value for scale must be positive.")
else:
raise ValueError("Unexpected shape %s for scale." % scale.shape)
else:
if scale.shape is (numLevelSets,):
if not min(scale) > 0:
raise ValueError("All values for scale must be positive.")
else:
raise ValueError("Unexpected shape %s for scale." % scale.shape)
if scale_c is None or numLevelSets < 2:
scale_c = np.ones((numLevelSets, numLevelSets))
else:
scale_c = np.asarray(scale_c)
if scale_c.shape == (numLevelSets, numLevelSets):
if not all([[scale_c[l, k] > 0.0 for l in range(k)] for k in range(1, numLevelSets)]):
raise ValueError("All values in the lower triangle of scale_c must be positive.")
else:
raise ValueError("Unexpected shape %s for scale." % scale_c.shape)
# ===== check the shape of the weights: =============================
if w0 is not None:
w0 = interpolate(w0, self.__pde.getFunctionSpaceForCoefficient("D"))
s0 = w0.getShape()
if numLevelSets == 1:
if not s0 == ():
raise ValueError("Unexpected shape %s for weight w0." % (s0,))
else:
if not s0 == (numLevelSets,):
raise ValueError("Unexpected shape %s for weight w0." % (s0,))
if not self.__trafo.isCartesian():
w0 *= self.__trafo.getVolumeFactor()
if not w1 is None:
w1 = interpolate(w1, self.__pde.getFunctionSpaceForCoefficient("A"))
s1 = w1.getShape()
if numLevelSets == 1:
if not s1 == (DIM,):
raise ValueError("Unexpected shape %s for weight w1." % (s1,))
else:
if not s1 == (numLevelSets, DIM):
raise ValueError("Unexpected shape %s for weight w1." % (s1,))
if not self.__trafo.isCartesian():
f = self.__trafo.getScalingFactors() ** 2 * self.__trafo.getVolumeFactor()
if numLevelSets == 1:
w1 *= f
else:
for i in range(numLevelSets):
w1[i, :] *= f
if numLevelSets == 1:
wc = None
else:
wc = interpolate(wc, self.__pde.getFunctionSpaceForCoefficient("A"))
sc = wc.getShape()
if not sc == (numLevelSets, numLevelSets):
raise ValueError("Unexpected shape %s for weight wc." % (sc,))
if not self.__trafo.isCartesian():
raise ValueError("Non-cartesian coordinates for cross-gradient term is not supported yet.")
# ============= now we rescale weights: =============================
L2s = np.asarray(boundingBoxEdgeLengths(domain)) ** 2
L4 = 1 / np.sum(1 / L2s) ** 2
if numLevelSets == 1:
A = 0
if w0 is not None:
A = integrate(w0)
if w1 is not None:
A += integrate(inner(w1, 1 / L2s))
if A > 0:
f = scale / A
if w0 is not None:
w0 *= f
if w1 is not None:
w1 *= f
else:
raise ValueError("Non-positive weighting factor detected.")
else: # numLevelSets > 1
for k in range(numLevelSets):
A = 0
if w0 is not None:
A = integrate(w0[k])
if w1 is not None:
A += integrate(inner(w1[k, :], 1 / L2s))
if A > 0:
f = scale[k] / A
if w0 is not None:
w0[k] *= f
if w1 is not None:
w1[k, :] *= f
else:
raise ValueError("Non-positive weighting factor for level set component %d detected." % k)
# and now the cross-gradient:
if wc is not None:
for l in range(k):
A = integrate(wc[l, k]) / L4
if A > 0:
f = scale_c[l, k] / A
wc[l, k] *= f
# else:
# raise ValueError("Non-positive weighting factor for cross-gradient level set components %d and %d detected."%(l,k))
self.__w0 = w0
self.__w1 = w1
self.__wc = wc
self.__pde_is_set = False
if self.__numLevelSets > 1:
self.__useDiagonalHessianApproximation = useDiagonalHessianApproximation
else:
self.__useDiagonalHessianApproximation = True
self._update_Hessian = True
self.__num_tradeoff_factors = numLevelSets + ((numLevelSets - 1) * numLevelSets) // 2
self.setTradeOffFactors()
self.__vol_d = vol(self.__domain)
def __init__(self,
domain,
omega,
w,
data,
F,
coordinates=None,
fixAtBottom=False,
tol=1e-10,
saveMemory=True,
scaleF=True):
"""
initializes a new forward model with acoustic wave form inversion.
:param domain: domain of the model
:type domain: `Domain`
:param w: weighting factors
:type w: ``Scalar``
:param data: real and imaginary part of data
:type data: ``escript.Data`` of shape (2,)
:param F: real and imaginary part of source given at Dirac points,
on surface or at volume.
:type F: ``escript.Data`` of shape (2,)
:param coordinates: defines coordinate system to be used (not supported yet)
:type coordinates: `ReferenceSystem` or `SpatialCoordinateTransformation`
:param tol: tolerance of underlying PDE
:type tol: positive ``float``
:param saveMemory: if true stiffness matrix is deleted after solution
of PDE to minimize memory requests. This will
require more compute time as the matrix needs to be
reallocated.
:type saveMemory: ``bool``
:param scaleF: if true source F is scaled to minimize defect.
:type scaleF: ``bool``
:param fixAtBottom: if true pressure is fixed to zero at the bottom of
the domain
:type fixAtBottom: ``bool``
"""
super(AcousticWaveForm, self).__init__()
self.__trafo = edc.makeTransformation(domain, coordinates)
if not self.getCoordinateTransformation().isCartesian():
raise ValueError(
"Non-Cartesian Coordinates are not supported yet.")
if not isinstance(data, escript.Data):
raise ValueError("data must be an escript.Data object.")
if not data.getFunctionSpace() == escript.FunctionOnBoundary(domain):
raise ValueError("data must be defined on boundary")
if not data.getShape() == (2, ):
raise ValueError(
"data must have shape (2,) (real and imaginary part).")
if w is None:
w = 1.
if not isinstance(w, escript.Data):
w = escript.Data(w, escript.FunctionOnBoundary(domain))
else:
if not w.getFunctionSpace() == escript.FunctionOnBoundary(domain):
raise ValueError("Weights must be defined on boundary.")
if not w.getShape() == ():
raise ValueError("Weights must be scalar.")
self.__domain = domain
self.__omega = omega
self.__weight = w
self.__data = data
self.scaleF = scaleF
if scaleF:
A = escript.integrate(self.__weight *
escript.length(self.__data)**2)
if A > 0:
self.__data *= 1. / escript.sqrt(A)
self.__BX = escript.boundingBox(domain)
self.edge_lengths = np.asarray(escript.boundingBoxEdgeLengths(domain))
if not isinstance(F, escript.Data):
F = escript.interpolate(F, escript.DiracDeltaFunctions(domain))
if not F.getShape() == (2, ):
raise ValueError(
"Source must have shape (2,) (real and imaginary part).")
self.__F = escript.Data()
self.__f = escript.Data()
self.__f_dirac = escript.Data()
if F.getFunctionSpace() == escript.DiracDeltaFunctions(domain):
self.__f_dirac = F
elif F.getFunctionSpace() == escript.FunctionOnBoundary(domain):
self.__f = F
else:
self.__F = F
self.__tol = tol
self.__fixAtBottom = fixAtBottom
self.__pde = None
if not saveMemory:
self.__pde = self.setUpPDE()
