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

Returns:

scipy.sparse.csr_matrix

Derivative 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.:

$$\frac{\partial \mathbf{u}}{\partial \mathbf{m}}=\frac{\partial {\mathbf{f}}_{\mathbf{n}}}{\partial {\mathbf{f}}_{\mathbf{n}\mathbf{-}\mathbf{1}}}\cdots \frac{\partial {\mathbf{f}}_{\mathbf{2}}}{\partial {\mathbf{f}}_{\mathbf{1}}}\frac{\partial {\mathbf{f}}_{\mathbf{1}}}{\partial \mathbf{m}}$$