SimPEG.regularization.SmoothnessFirstOrder.deriv2

SmoothnessFirstOrder.deriv2(m, v=None) → csr_matrix

Second derivative

Parameters

• m (numpy.ndarray) – geophysical model

• v (numpy.ndarray) – vector to multiply

Return type

scipy.sparse.csr_matrix

Returns

WtW, or if v is supplied WtW*v (numpy.ndarray)

The regularization is:

$$R(m)=\frac{1}{2}\mathbf{(}\mathbf{m}\mathbf{-}{\mathbf{m}}_{\mathbf{\text{ref}}}{\mathbf{)}}^{\mathrm{\top }}{\mathbf{W}}^{\mathrm{\top }}\mathbf{W}\mathbf{(}\mathbf{m}\mathbf{-}{\mathbf{m}}_{\mathbf{\text{ref}}}\mathbf{)}$$

So the second derivative is straight forward:

$$R(m)={\mathbf{W}}^{\mathrm{\top }}\mathbf{W}$$