simpeg.potential_fields.magnetics.Simulation3DDifferential.Jvec

Simulation3DDifferential.Jvec(m, v, u=None)

Computing Jacobian multiplied by vector

By setting our problem as

\[\mathbf{C}(\mathbf{m}, \mathbf{u}) = \mathbf{A}\mathbf{u} - \mathbf{rhs} = 0\]

And taking derivative w.r.t m

\[ \begin{align}\begin{aligned}\nabla \mathbf{C}(\mathbf{m}, \mathbf{u}) = \nabla_m \mathbf{C}(\mathbf{m}) \delta \mathbf{m} + \nabla_u \mathbf{C}(\mathbf{u}) \delta \mathbf{u} = 0\\\frac{\delta \mathbf{u}}{\delta \mathbf{m}} = - [\nabla_u \mathbf{C}(\mathbf{u})]^{-1}\nabla_m \mathbf{C}(\mathbf{m})\end{aligned}\end{align} \]

With some linear algebra we can have

\[ \begin{align}\begin{aligned}\nabla_u \mathbf{C}(\mathbf{u}) = \mathbf{A}\\\nabla_m \mathbf{C}(\mathbf{m}) = \frac{\partial \mathbf{A}} {\partial \mathbf{m}} (\mathbf{m}) \mathbf{u} - \frac{\partial \mathbf{rhs}(\mathbf{m})}{\partial \mathbf{m}}\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\frac{\partial \mathbf{A}}{\partial \mathbf{m}}(\mathbf{m})\mathbf{u} = \frac{\partial \mathbf{\mu}}{\partial \mathbf{m}} \left[\Div \diag (\Div^T \mathbf{u}) \dMfMuI \right]\\\dMfMuI = \diag(\MfMui)^{-1}_{vec} \mathbf{Av}_{F2CC}^T\diag(\mathbf{v})\diag(\frac{1}{\mu^2})\\\frac{\partial \mathbf{rhs}(\mathbf{m})}{\partial \mathbf{m}} = \frac{\partial \mathbf{\mu}}{\partial \mathbf{m}} \left[ \Div \diag(\M^f_{\mu_{0}^{-1}}\mathbf{B}_0) \dMfMuI \right] - \diag(\mathbf{v}) \mathbf{D} \mathbf{P}_{out}^T \frac{\partial B_{sBC}}{\partial \mathbf{m}}\end{aligned}\end{align} \]

In the end,

\[\frac{\delta \mathbf{u}}{\delta \mathbf{m}} = - [ \mathbf{A} ]^{-1} \left[ \frac{\partial \mathbf{A}}{\partial \mathbf{m}}(\mathbf{m})\mathbf{u} - \frac{\partial \mathbf{rhs}(\mathbf{m})}{\partial \mathbf{m}} \right]\]

A little tricky point here is we are not interested in potential (u), but interested in magnetic flux (B). Thus, we need sensitivity for B. Now we take derivative of B w.r.t m and have

\[ \begin{align}\begin{aligned}\frac{\delta \mathbf{B}} {\delta \mathbf{m}} = \frac{\partial \mathbf{\mu} } {\partial \mathbf{m} } \left[ \diag(\M^f_{\mu_{0}^{-1} } \mathbf{B}_0) \dMfMuI \ - \diag (\Div^T\mathbf{u})\dMfMuI \right ]\\ - (\MfMui)^{-1}\Div^T\frac{\delta\mathbf{u}}{\delta \mathbf{m}}\end{aligned}\end{align} \]

Finally we evaluate the above, but we should remember that

Note

We only want to evaluate

\[\mathbf{J}\mathbf{v} = \frac{\delta \mathbf{P}\mathbf{B}} {\delta \mathbf{m}}\mathbf{v}\]

Since forming sensitivity matrix is very expensive in that this monster is “big” and “dense” matrix!!