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documentation for FDEM.py
This commit is contained in:
Lindsey Heagy
+217
-83
@@ -36,7 +36,11 @@ class BaseFDEMProblem(BaseEMProblem):
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def fields(self, m=None):
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"""
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Solve the forward problem for the fields.
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Solve the forward problem for the fields.
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:param numpy.array m: inversion model (nP,)
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:rtype numpy.array:
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:return F: forward solution
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"""
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self.curModel = m
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@@ -55,7 +59,13 @@ class BaseFDEMProblem(BaseEMProblem):
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def Jvec(self, m, v, u=None):
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"""
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Sensitivity times a vector
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Sensitivity times a vector.
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:param numpy.array m: inversion model (nP,)
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:param numpy.array v: vector which we take sensitivity product with (nP,)
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:param SimPEG.EM.FDEM.Fields u: fields object
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:rtype numpy.array:
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:return: Jv (ndata,)
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"""
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if u is None:
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@@ -83,6 +93,7 @@ class BaseFDEMProblem(BaseEMProblem):
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df_dmFun = getattr(u, '_%sDeriv_m'%rx.projField, None)
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df_dm = df_dmFun(src, v, adjoint=False)
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Df_Dm = np.array(df_dudu_dm + df_dm,dtype=complex)
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P = lambda v: rx.projectFieldsDeriv(src, self.mesh, u, v) # wrt u, also have wrt m
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@@ -94,7 +105,13 @@ class BaseFDEMProblem(BaseEMProblem):
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def Jtvec(self, m, v, u=None):
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"""
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Sensitivity transpose times a vector
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Sensitivity transpose times a vector
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:param numpy.array m: inversion model (nP,)
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:param numpy.array v: vector which we take adjoint product with (nP,)
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:param SimPEG.EM.FDEM.Fields u: fields object
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:rtype numpy.array:
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:return: Jv (ndata,)
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"""
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if u is None:
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@@ -133,6 +150,7 @@ class BaseFDEMProblem(BaseEMProblem):
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du_dmT += dfT_dm
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# TODO: this should be taken care of by the reciever
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real_or_imag = rx.projComp
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if real_or_imag is 'real':
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Jtv += np.array(du_dmT,dtype=complex).real
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@@ -147,11 +165,11 @@ class BaseFDEMProblem(BaseEMProblem):
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def getSourceTerm(self, freq):
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"""
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Evaluates the sources for a given frequency and puts them in matrix form
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Evaluates the sources for a given frequency and puts them in matrix form
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:param float freq: Frequency
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:rtype: numpy.ndarray (nE or nF, nSrc)
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:return: S_m, S_e
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:param float freq: Frequency
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:rtype: (numpy.ndarray, numpy.ndarray)
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:return: S_m, S_e (nE or nF, nSrc)
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"""
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Srcs = self.survey.getSrcByFreq(freq)
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if self._eqLocs is 'FE':
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@@ -175,20 +193,22 @@ class BaseFDEMProblem(BaseEMProblem):
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class Problem_e(BaseFDEMProblem):
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"""
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By eliminating the magnetic flux density using
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.. math ::
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\mathbf{b} = \\frac{1}{i \omega}\\left(-\mathbf{C} \mathbf{e} + \mathbf{s_m}\\right)
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we can write Maxwell's equations as a second order system in \\\(\\\mathbf{e}\\\) only:
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By eliminating the magnetic flux density using
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.. math ::
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\\left(\mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} \mathbf{C}+ i \omega \mathbf{M^e_{\sigma}} \\right)\mathbf{e} = \mathbf{C}^T \mathbf{M_{\mu^{-1}}^f}\mathbf{s_m} -i\omega\mathbf{M^e}\mathbf{s_e}
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\mathbf{b} = \\frac{1}{i \omega}\\left(-\mathbf{C} \mathbf{e} + \mathbf{s_m}\\right)
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which we solve for \\\(\\\mathbf{e}\\\).
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we can write Maxwell's equations as a second order system in \\\(\\\mathbf{e}\\\) only:
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.. math ::
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\\left(\mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} \mathbf{C}+ i \omega \mathbf{M^e_{\sigma}} \\right)\mathbf{e} = \mathbf{C}^T \mathbf{M_{\mu^{-1}}^f}\mathbf{s_m} -i\omega\mathbf{M^e}\mathbf{s_e}
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which we solve for :math:`\mathbf{e}`.
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:param SimPEG.Mesh mesh: mesh
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"""
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_fieldType = 'e'
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@@ -200,13 +220,16 @@ class Problem_e(BaseFDEMProblem):
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def getA(self, freq):
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"""
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.. math ::
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\mathbf{A} = \mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} \mathbf{C} + i \omega \mathbf{M^e_{\sigma}}
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System matrix
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.. math ::
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\mathbf{A} = \mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} \mathbf{C} + i \omega \mathbf{M^e_{\sigma}}
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:param float freq: Frequency
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:rtype: scipy.sparse.csr_matrix
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:return: A
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:param float freq: Frequency
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:rtype: scipy.sparse.csr_matrix
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:return: A
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"""
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MfMui = self.MfMui
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MeSigma = self.MeSigma
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C = self.mesh.edgeCurl
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@@ -215,6 +238,20 @@ class Problem_e(BaseFDEMProblem):
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def getADeriv_m(self, freq, u, v, adjoint=False):
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"""
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Product of the derivative of our system matrix with respect to the model and a vector
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.. math ::
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\\frac{\mathbf{A}(\mathbf{m}) \mathbf{v}}{d \mathbf{m}} = i \omega \\frac{d \mathbf{M^e_{\sigma}}\mathbf{v} }{d\mathbf{m}}
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:param float freq: frequency
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:param numpy.ndarray u: solution vector (nE,)
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:param numpy.ndarray v: vector to take prodct with (nP,) or (nD,) for adjoint
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:param bool adjoint: adjoint?
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:rtype: numpy.ndarray
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:return: derivative of the system matrix times a vector (nP,) or adjoint (nD,)
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"""
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dsig_dm = self.curModel.sigmaDeriv
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dMe_dsig = self.MeSigmaDeriv(u)
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@@ -225,12 +262,14 @@ class Problem_e(BaseFDEMProblem):
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def getRHS(self, freq):
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"""
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.. math ::
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\mathbf{RHS} = \mathbf{C}^T \mathbf{M_{\mu^{-1}}^f}\mathbf{s_m} -i\omega\mathbf{M_e}\mathbf{s_e}
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Right hand side for the system
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:param float freq: Frequency
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:rtype: numpy.ndarray (nE, nSrc)
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:return: RHS
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.. math ::
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\mathbf{RHS} = \mathbf{C}^T \mathbf{M_{\mu^{-1}}^f}\mathbf{s_m} -i\omega\mathbf{M_e}\mathbf{s_e}
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:param float freq: Frequency
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:rtype: numpy.ndarray
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:return: RHS (nE, nSrc)
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"""
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S_m, S_e = self.getSourceTerm(freq)
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@@ -242,6 +281,17 @@ class Problem_e(BaseFDEMProblem):
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return RHS
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def getRHSDeriv_m(self, freq, src, v, adjoint=False):
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"""
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Derivative of the right hand side with respect to the model
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:param float freq: frequency
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:param SimPEG.EM.FDEM.Src src: FDEM source
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:param numpy.ndarray v: vector to take product with
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:param bool adjoint: adjoint?
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:rtype: numpy.ndarray
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:return: product of rhs deriv with a vector
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"""
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C = self.mesh.edgeCurl
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MfMui = self.MfMui
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S_mDeriv, S_eDeriv = src.evalDeriv(self, adjoint)
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@@ -256,20 +306,22 @@ class Problem_e(BaseFDEMProblem):
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class Problem_b(BaseFDEMProblem):
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"""
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We eliminate \\\(\\\mathbf{e}\\\) using
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We eliminate :math:`\mathbf{e}` using
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.. math ::
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.. math ::
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\mathbf{e} = \mathbf{M^e_{\sigma}}^{-1} \\left(\mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} \mathbf{b} - \mathbf{s_e}\\right)
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\mathbf{e} = \mathbf{M^e_{\sigma}}^{-1} \\left(\mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} \mathbf{b} - \mathbf{s_e}\\right)
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and solve for \\\(\\\mathbf{b}\\\) using:
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and solve for :math:`\mathbf{b}` using:
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.. math ::
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.. math ::
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\\left(\mathbf{C} \mathbf{M^e_{\sigma}}^{-1} \mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} + i \omega \\right)\mathbf{b} = \mathbf{s_m} + \mathbf{M^e_{\sigma}}^{-1}\mathbf{M^e}\mathbf{s_e}
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\\left(\mathbf{C} \mathbf{M^e_{\sigma}}^{-1} \mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} + i \omega \\right)\mathbf{b} = \mathbf{s_m} + \mathbf{M^e_{\sigma}}^{-1}\mathbf{M^e}\mathbf{s_e}
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.. note ::
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The inverse problem will not work with full anisotropy
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.. note ::
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The inverse problem will not work with full anisotropy
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:param SimPEG.Mesh mesh: mesh
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"""
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_fieldType = 'b'
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@@ -281,12 +333,14 @@ class Problem_b(BaseFDEMProblem):
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def getA(self, freq):
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"""
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.. math ::
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\mathbf{A} = \mathbf{C} \mathbf{M^e_{\sigma}}^{-1} \mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} + i \omega
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System matrix
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:param float freq: Frequency
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:rtype: scipy.sparse.csr_matrix
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:return: A
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.. math ::
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\mathbf{A} = \mathbf{C} \mathbf{M^e_{\sigma}}^{-1} \mathbf{C}^T \mathbf{M_{\mu^{-1}}^f} + i \omega
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:param float freq: Frequency
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:rtype: scipy.sparse.csr_matrix
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:return: A
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"""
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MfMui = self.MfMui
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@@ -302,6 +356,20 @@ class Problem_b(BaseFDEMProblem):
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def getADeriv_m(self, freq, u, v, adjoint=False):
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"""
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Product of the derivative of our system matrix with respect to the model and a vector
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.. math ::
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\\frac{\mathbf{A}(\mathbf{m}) \mathbf{v}}{d \mathbf{m}} = \mathbf{C} \\frac{\mathbf{M^e_{\sigma}} \mathbf{v}}{d\mathbf{m}}
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:param float freq: frequency
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:param numpy.ndarray u: solution vector (nF,)
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:param numpy.ndarray v: vector to take prodct with (nP,) or (nD,) for adjoint
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:param bool adjoint: adjoint?
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:rtype: numpy.ndarray
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:return: derivative of the system matrix times a vector (nP,) or adjoint (nD,)
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"""
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MfMui = self.MfMui
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C = self.mesh.edgeCurl
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MeSigmaIDeriv = self.MeSigmaIDeriv
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@@ -321,12 +389,14 @@ class Problem_b(BaseFDEMProblem):
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def getRHS(self, freq):
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"""
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.. math ::
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\mathbf{RHS} = \mathbf{s_m} + \mathbf{M^e_{\sigma}}^{-1}\mathbf{s_e}
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Right hand side for the system
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:param float freq: Frequency
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:rtype: numpy.ndarray (nE, nSrc)
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:return: RHS
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.. math ::
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\mathbf{RHS} = \mathbf{s_m} + \mathbf{M^e_{\sigma}}^{-1}\mathbf{s_e}
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:param float freq: Frequency
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:rtype: numpy.ndarray
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:return: RHS (nE, nSrc)
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"""
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S_m, S_e = self.getSourceTerm(freq)
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@@ -342,6 +412,17 @@ class Problem_b(BaseFDEMProblem):
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return RHS
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def getRHSDeriv_m(self, freq, src, v, adjoint=False):
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"""
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Derivative of the right hand side with respect to the model
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:param float freq: frequency
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:param SimPEG.EM.FDEM.Src src: FDEM source
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:param numpy.ndarray v: vector to take product with
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:param bool adjoint: adjoint?
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:rtype: numpy.ndarray
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:return: product of rhs deriv with a vector
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"""
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C = self.mesh.edgeCurl
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S_m, S_e = src.eval(self)
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MfMui = self.MfMui
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@@ -373,21 +454,22 @@ class Problem_b(BaseFDEMProblem):
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class Problem_j(BaseFDEMProblem):
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"""
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We eliminate \\\(\\\mathbf{h}\\\) using
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We eliminate \\\(\\\mathbf{h}\\\) using
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.. math ::
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.. math ::
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\mathbf{h} = \\frac{1}{i \omega} \mathbf{M_{\mu}^e}^{-1} \\left(-\mathbf{C}^T \mathbf{M_{\\rho}^f} \mathbf{j} + \mathbf{M^e} \mathbf{s_m} \\right)
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\mathbf{h} = \\frac{1}{i \omega} \mathbf{M_{\mu}^e}^{-1} \\left(-\mathbf{C}^T \mathbf{M_{\\rho}^f} \mathbf{j} + \mathbf{M^e} \mathbf{s_m} \\right)
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and solve for \\\(\\\mathbf{j}\\\) using
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and solve for \\\(\\\mathbf{j}\\\) using
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.. math ::
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.. math ::
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\\left(\mathbf{C} \mathbf{M_{\mu}^e}^{-1} \mathbf{C}^T \mathbf{M_{\\rho}^f} + i \omega\\right)\mathbf{j} = \mathbf{C} \mathbf{M_{\mu}^e}^{-1} \mathbf{M^e} \mathbf{s_m} -i\omega\mathbf{s_e}
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\\left(\mathbf{C} \mathbf{M_{\mu}^e}^{-1} \mathbf{C}^T \mathbf{M_{\\rho}^f} + i \omega\\right)\mathbf{j} = \mathbf{C} \mathbf{M_{\mu}^e}^{-1} \mathbf{M^e} \mathbf{s_m} -i\omega\mathbf{s_e}
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.. note::
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This implementation does not yet work with full anisotropy!!
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.. note::
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This implementation does not yet work with full anisotropy!!
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:param SimPEG.Mesh mesh: mesh
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"""
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_fieldType = 'j'
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@@ -399,12 +481,14 @@ class Problem_j(BaseFDEMProblem):
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def getA(self, freq):
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"""
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.. math ::
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\\mathbf{A} = \\mathbf{C} \\mathbf{M^e_{mu^{-1}}} \\mathbf{C}^T \\mathbf{M^f_{\\sigma^{-1}}} + i\\omega
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System matrix
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:param float freq: Frequency
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:rtype: scipy.sparse.csr_matrix
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:return: A
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.. math ::
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\\mathbf{A} = \\mathbf{C} \\mathbf{M^e_{\\mu^{-1}}} \\mathbf{C}^T \\mathbf{M^f_{\\sigma^{-1}}} + i\\omega
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:param float freq: Frequency
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:rtype: scipy.sparse.csr_matrix
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:return: A
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"""
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MeMuI = self.MeMuI
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@@ -421,12 +505,20 @@ class Problem_j(BaseFDEMProblem):
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def getADeriv_m(self, freq, u, v, adjoint=False):
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"""
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In this case, we assume that electrical conductivity, \\\(\\\sigma\\\) is the physical property of interest (i.e. \\\(\\\sigma\\\) = model.transform). Then we want
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Product of the derivative of our system matrix with respect to the model and a vector
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.. math ::
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In this case, we assume that electrical conductivity, :math:`\sigma` is the physical property of interest (i.e. :math:`\sigma` = model.transform). Then we want
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\\frac{\mathbf{A(\sigma)} \mathbf{v}}{d \\mathbf{m}} &= \\mathbf{C} \\mathbf{M^e_{mu^{-1}}} \\mathbf{C^T} \\frac{d \\mathbf{M^f_{\\sigma^{-1}}}}{d \\mathbf{m}}
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&= \\mathbf{C} \\mathbf{M^e_{mu}^{-1}} \\mathbf{C^T} \\frac{d \\mathbf{M^f_{\\sigma^{-1}}}}{d \\mathbf{\\sigma^{-1}}} \\frac{d \\mathbf{\\sigma^{-1}}}{d \\mathbf{\\sigma}} \\frac{d \\mathbf{\\sigma}}{d \\mathbf{m}}
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.. math ::
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\\frac{\mathbf{A(\sigma)} \mathbf{v}}{d \mathbf{m}} = \mathbf{C} \mathbf{M^e_{mu^{-1}}} \mathbf{C^T} \\frac{d \mathbf{M^f_{\sigma^{-1}}}\mathbf{v} }{d \mathbf{m}}
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:param float freq: frequency
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||||
:param numpy.ndarray u: solution vector (nF,)
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||||
:param numpy.ndarray v: vector to take prodct with (nP,) or (nD,) for adjoint
|
||||
:param bool adjoint: adjoint?
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:rtype: numpy.ndarray
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||||
:return: derivative of the system matrix times a vector (nP,) or adjoint (nD,)
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"""
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MeMuI = self.MeMuI
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@@ -446,12 +538,14 @@ class Problem_j(BaseFDEMProblem):
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def getRHS(self, freq):
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"""
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.. math ::
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Right hand side for the system
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||||
|
||||
\mathbf{RHS} = \mathbf{C} \mathbf{M_{\mu}^e}^{-1}\mathbf{s_m} -i\omega \mathbf{s_e}
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:param float freq: Frequency
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||||
:rtype: numpy.ndarray (nE, nSrc)
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:return: RHS
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.. math ::
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\mathbf{RHS} = \mathbf{C} \mathbf{M_{\mu}^e}^{-1}\mathbf{s_m} -i\omega \mathbf{s_e}
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||||
:param float freq: Frequency
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:rtype: numpy.ndarray (nE, nSrc)
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:return: RHS
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"""
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S_m, S_e = self.getSourceTerm(freq)
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@@ -466,6 +560,17 @@ class Problem_j(BaseFDEMProblem):
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return RHS
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def getRHSDeriv_m(self, freq, src, v, adjoint=False):
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"""
|
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Derivative of the right hand side with respect to the model
|
||||
|
||||
:param float freq: frequency
|
||||
:param SimPEG.EM.FDEM.Src src: FDEM source
|
||||
:param numpy.ndarray v: vector to take product with
|
||||
:param bool adjoint: adjoint?
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||||
:rtype: numpy.ndarray
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:return: product of rhs deriv with a vector
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||||
"""
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||||
|
||||
C = self.mesh.edgeCurl
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MeMuI = self.MeMuI
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S_mDeriv, S_eDeriv = src.evalDeriv(self, adjoint)
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@@ -489,18 +594,19 @@ class Problem_j(BaseFDEMProblem):
|
||||
|
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class Problem_h(BaseFDEMProblem):
|
||||
"""
|
||||
We eliminate \\\(\\\mathbf{j}\\\) using
|
||||
We eliminate \\\(\\\mathbf{j}\\\) using
|
||||
|
||||
.. math ::
|
||||
.. math ::
|
||||
|
||||
\mathbf{j} = \mathbf{C} \mathbf{h} - \mathbf{s_e}
|
||||
\mathbf{j} = \mathbf{C} \mathbf{h} - \mathbf{s_e}
|
||||
|
||||
and solve for \\\(\\\mathbf{h}\\\) using
|
||||
and solve for \\\(\\\mathbf{h}\\\) using
|
||||
|
||||
.. math ::
|
||||
.. math ::
|
||||
|
||||
\\left(\mathbf{C}^T \mathbf{M_{\\rho}^f} \mathbf{C} + i \omega \mathbf{M_{\mu}^e}\\right) \mathbf{h} = \mathbf{M^e} \mathbf{s_m} + \mathbf{C}^T \mathbf{M_{\\rho}^f} \mathbf{s_e}
|
||||
\\left(\mathbf{C}^T \mathbf{M_{\\rho}^f} \mathbf{C} + i \omega \mathbf{M_{\mu}^e}\\right) \mathbf{h} = \mathbf{M^e} \mathbf{s_m} + \mathbf{C}^T \mathbf{M_{\\rho}^f} \mathbf{s_e}
|
||||
|
||||
:param SimPEG.Mesh mesh: mesh
|
||||
"""
|
||||
|
||||
_fieldType = 'h'
|
||||
@@ -512,13 +618,15 @@ class Problem_h(BaseFDEMProblem):
|
||||
|
||||
def getA(self, freq):
|
||||
"""
|
||||
.. math ::
|
||||
System matrix
|
||||
|
||||
\mathbf{A} = \mathbf{C}^T \mathbf{M_{\\rho}^f} \mathbf{C} + i \omega \mathbf{M_{\mu}^e}
|
||||
.. math ::
|
||||
|
||||
:param float freq: Frequency
|
||||
:rtype: scipy.sparse.csr_matrix
|
||||
:return: A
|
||||
\mathbf{A} = \mathbf{C}^T \mathbf{M_{\\rho}^f} \mathbf{C} + i \omega \mathbf{M_{\mu}^e}
|
||||
|
||||
:param float freq: Frequency
|
||||
:rtype: scipy.sparse.csr_matrix
|
||||
:return: A
|
||||
"""
|
||||
|
||||
MeMu = self.MeMu
|
||||
@@ -528,6 +636,19 @@ class Problem_h(BaseFDEMProblem):
|
||||
return C.T * (MfRho * C) + 1j*omega(freq)*MeMu
|
||||
|
||||
def getADeriv_m(self, freq, u, v, adjoint=False):
|
||||
"""
|
||||
Product of the derivative of our system matrix with respect to the model and a vector
|
||||
|
||||
.. math::
|
||||
\\frac{\mathbf{A}(\mathbf{m}) \mathbf{v}}{d \mathbf{m}} = \mathbf{C}^{\\top} \\frac{d \mathbf{M^f_{\\rho}}\mathbf{v} }{d\mathbf{m}}
|
||||
|
||||
:param float freq: frequency
|
||||
:param numpy.ndarray u: solution vector (nE,)
|
||||
:param numpy.ndarray v: vector to take prodct with (nP,) or (nD,) for adjoint
|
||||
:param bool adjoint: adjoint?
|
||||
:rtype: numpy.ndarray
|
||||
:return: derivative of the system matrix times a vector (nP,) or adjoint (nD,)
|
||||
"""
|
||||
|
||||
MeMu = self.MeMu
|
||||
C = self.mesh.edgeCurl
|
||||
@@ -539,13 +660,15 @@ class Problem_h(BaseFDEMProblem):
|
||||
|
||||
def getRHS(self, freq):
|
||||
"""
|
||||
.. math ::
|
||||
Right hand side for the system
|
||||
|
||||
\mathbf{RHS} = \mathbf{M^e} \mathbf{s_m} + \mathbf{C}^T \mathbf{M_{\\rho}^f} \mathbf{s_e}
|
||||
.. math ::
|
||||
|
||||
:param float freq: Frequency
|
||||
:rtype: numpy.ndarray (nE, nSrc)
|
||||
:return: RHS
|
||||
\mathbf{RHS} = \mathbf{M^e} \mathbf{s_m} + \mathbf{C}^T \mathbf{M_{\\rho}^f} \mathbf{s_e}
|
||||
|
||||
:param float freq: Frequency
|
||||
:rtype: numpy.ndarray
|
||||
:return: RHS (nE, nSrc)
|
||||
"""
|
||||
|
||||
S_m, S_e = self.getSourceTerm(freq)
|
||||
@@ -557,6 +680,17 @@ class Problem_h(BaseFDEMProblem):
|
||||
return RHS
|
||||
|
||||
def getRHSDeriv_m(self, freq, src, v, adjoint=False):
|
||||
"""
|
||||
Derivative of the right hand side with respect to the model
|
||||
|
||||
:param float freq: frequency
|
||||
:param SimPEG.EM.FDEM.Src src: FDEM source
|
||||
:param numpy.ndarray v: vector to take product with
|
||||
:param bool adjoint: adjoint?
|
||||
:rtype: numpy.ndarray
|
||||
:return: product of rhs deriv with a vector
|
||||
"""
|
||||
|
||||
_, S_e = src.eval(self)
|
||||
C = self.mesh.edgeCurl
|
||||
MfRho = self.MfRho
|
||||
|
||||
Reference in New Issue
Block a user