simpeg.potential_fields.magnetics.Simulation3DDifferential.Jvec#

Simulation3DDifferential.Jvec(m, v, u=None)[source]#

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

$$\begin{array}{r}\begin{array}{c}\mathrm{\nabla }\mathbf{C}(\mathbf{m},\mathbf{u})={\mathrm{\nabla }}_m\mathbf{C}(\mathbf{m})\delta \mathbf{m}+{\mathrm{\nabla }}_u\mathbf{C}(\mathbf{u})\delta \mathbf{u}=0\\ \frac{\delta \mathbf{u}}{\delta \mathbf{m}}=-[{\mathrm{\nabla }}_u\mathbf{C}(\mathbf{u}){]}^{-1}{\mathrm{\nabla }}_m\mathbf{C}(\mathbf{m})\end{array}\end{array}$$

With some linear algebra we can have

$$\begin{array}{r}\begin{array}{c}{\mathrm{\nabla }}_u\mathbf{C}(\mathbf{u})=\mathbf{A}\\ {\mathrm{\nabla }}_m\mathbf{C}(\mathbf{m})=\frac{\partial \mathbf{A}}{\partial \mathbf{m}}(\mathbf{m})\mathbf{u}-\frac{\partial \mathbf{rhs}(\mathbf{m})}{\partial \mathbf{m}}\end{array}\end{array}$$

$$\begin{array}{r}\begin{array}{c}\frac{\partial \mathbf{A}}{\partial \mathbf{m}}(\mathbf{m})\mathbf{u}=\frac{\partial \mu }{\partial \mathbf{m}}[\text{Div}\text{diag}({\text{Div}}^T\mathbf{u})\text{dMfMuI}]\\ \text{dMfMuI}=\text{diag}(\text{MfMui}{)}_{vec}^{-1}{\mathbf{Av}}_{F2CC}^T\text{diag}(\mathbf{v})\text{diag}(\frac{1}{{\mu }^2})\\ \frac{\partial \mathbf{rhs}(\mathbf{m})}{\partial \mathbf{m}}=\frac{\partial \mu }{\partial \mathbf{m}}[\text{Div}\text{diag}({\text{M}}_{{\mu }_0^{-1}}^f{\mathbf{B}}_0)\text{dMfMuI}]-\text{diag}(\mathbf{v})\mathbf{D}{\mathbf{P}}_{out}^T\frac{\partial B_{sBC}}{\partial \mathbf{m}}\end{array}\end{array}$$

In the end,

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

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

$$\begin{array}{r}\begin{array}{c}\frac{\delta \mathbf{B}}{\delta \mathbf{m}}=\frac{\partial \mu }{\partial \mathbf{m}}[\text{diag}({\text{M}}_{{\mu }_0^{-1}}^f{\mathbf{B}}_0)\text{dMfMuI}-\text{diag}({\text{Div}}^T\mathbf{u})\text{dMfMuI}]\\ -(\text{MfMui}{)}^{-1}{\text{Div}}^T\frac{\delta \mathbf{u}}{\delta \mathbf{m}}\end{array}\end{array}$$

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!!