mirror of
https://github.com/wassname/simpeg.git
synced 2026-06-28 14:12:43 +08:00
Rowan Cockett
116 lines
3.5 KiB
Python
116 lines
3.5 KiB
Python
import numpy as np
|
|
from SimPEG.utils import mkvc, sdiag
|
|
norm = np.linalg.norm
|
|
|
|
|
|
class Problem(object):
|
|
"""Problem is the base class for all geophysical forward problems in SimPEG"""
|
|
def __init__(self, mesh):
|
|
self.mesh = mesh
|
|
pass
|
|
|
|
def residual(self, m):
|
|
pass
|
|
|
|
def modelTransform(self, m):
|
|
"""
|
|
:param numpy.array m: model
|
|
:rtype: numpy.array
|
|
:return: transformed model
|
|
|
|
The modelTransform changes the model into the physical property.
|
|
|
|
A common example of this is to invert for electrical conductivity
|
|
in log space. In this case, your model will be log(sigma) and to
|
|
get back to sigma, you can take the exponential:
|
|
|
|
.. math::
|
|
|
|
m = \log{\sigma}
|
|
|
|
\exp{m} = \exp{\log{\sigma}} = \sigma
|
|
"""
|
|
return np.exp(mkvc(m))
|
|
|
|
def modelTransformDeriv(self, m):
|
|
"""
|
|
:param numpy.array m: model
|
|
:rtype: scipy.csr_matrix
|
|
:return: derivative of transformed model
|
|
|
|
The modelTransform changes the model into the physical property.
|
|
The modelTransformDeriv provides the derivative of the modelTransform.
|
|
|
|
If the model transform is:
|
|
|
|
.. math::
|
|
|
|
m = \log{\sigma}
|
|
|
|
\exp{m} = \exp{\log{\sigma}} = \sigma
|
|
|
|
Then the derivative is:
|
|
|
|
.. math::
|
|
|
|
\\frac{\partial \exp{m}}{\partial m} = \\text{sdiag}(\exp{m})
|
|
"""
|
|
return sdiag(np.exp(mkvc(m)))
|
|
|
|
def _test_modelTransformDeriv(self):
|
|
m = np.random.rand(5)
|
|
return checkDerivative(lambda m : [self.modelTransform(m), self.modelTransformDeriv(m)], m)
|
|
|
|
def misfit(self, field):
|
|
"""
|
|
:param numpy.array field: geophysical field of interest
|
|
:rtype: float
|
|
:return: data misfit
|
|
|
|
The data misfit using an l_2 norm is:
|
|
|
|
.. math::
|
|
|
|
\mu_\\text{data} = {1\over 2}\left| \mathbf{W} (\mathbf{Pu} - d_\\text{obs}) \\right|_2^2
|
|
|
|
Where P is a projection matrix that brings the field on the full domain to the data measurement locations;
|
|
u is the field of interest; d_obs is the observed data; and W is the weighting matrix.
|
|
"""
|
|
R = self.W*(self.P*field - self.dobs)
|
|
return 0.5*mkvc(R).inner(mkvc(R))
|
|
|
|
def misfitDeriv(self, field):
|
|
"""
|
|
TODO: Change this documentation.
|
|
|
|
:param numpy.array field: geophysical field of interest
|
|
:rtype: float
|
|
:return: data misfit derivative
|
|
|
|
The data misfit using an l_2 norm is:
|
|
|
|
.. math::
|
|
|
|
\mu_\\text{data} = {1\over 2}\left| \mathbf{W} (\mathbf{Pu} - d_\\text{obs}) \\right|_2^2
|
|
|
|
Where P is a projection matrix that brings the field on the full domain to the data measurement locations;
|
|
u is the field of interest; d_obs is the observed data; and W is the weighting matrix.
|
|
"""
|
|
|
|
R = self.W*(self.P*field - self.dobs)
|
|
# TODO: make in terms of the field and call Jt, e.g. if looping over RHSs using i: self.Jt(field[:,i],self.W[:,i]*R[:,i])
|
|
return mkvc(R)
|
|
|
|
def J(self, u):
|
|
pass
|
|
|
|
def Jt(self, v):
|
|
pass
|
|
|
|
if __name__ == '__main__':
|
|
from SimPEG.inverse import checkDerivative
|
|
|
|
p = Problem(None)
|
|
m = np.random.rand(5)
|
|
checkDerivative(lambda m : [p.modelTransform(m), p.modelTransformDeriv(m)], m)
|