SimPEG.maps.ComboMap.deriv#
- ComboMap.deriv(m, v=None)[source]#
Derivative of the mapping with respect to the input parameters.
Any time that you create your own combination mapping, be sure to test that the derivative is correct.
- Parameters
- m(
nP)numpy.ndarray A vector representing a set of model parameters
- v(
nP)numpy.ndarray If not
None, the method returns the derivative times the vector v
- m(
- Returns
scipy.sparse.csr_matrixDerivative of the mapping with respect to the model parameters. If the input argument v is not
None, the method returns the derivative times the vector v.
Notes
Let \(\mathbf{m}\) be a set of model parameters and let [\(\mathbf{f}_n,...,\mathbf{f}_1\)] be the list of SimPEG mappings joined to create a combination mapping. Recall that the list of mappings is ordered from last applied to first applied.
Where the combination mapping acting on the model parameters can be expressed as:
\[\mathbf{u}(\mathbf{m}) = f_n(f_{n-1}(\cdots f_1(f_0(\mathbf{m}))))\]The deriv method returns the derivative of \(\mathbf{u}\) with respect to the model parameters. To do this, we use the chain rule, i.e.:
\[\frac{\partial \mathbf{u}}{\partial \mathbf{m}} = \frac{\partial \mathbf{f_n}}{\partial \mathbf{f_{n-1}}} \cdots \frac{\partial \mathbf{f_2}}{\partial \mathbf{f_{1}}} \frac{\partial \mathbf{f_1}}{\partial \mathbf{m}}\]