In the following, we derive the expression for the gradient in the discrete case. The sampled forward model can be expressed as a vector $H={Hj,j=1,\u2026,N}$Display Formula
$Hj=\u222b\Omega H(r)\Psi j(r)d\Omega =\u27e8\Psi j,H\u27e9.=\u2211ik\mu a\u2009i\phi k\u222b\Omega \Psi j(r)ui(r)uk(r)d\Omega =\phi TCj\mu a,$(12)
where $Cj$ is a sparse matrix indexed by $i$, $k$ where the support of the basis functions $\Psi j(r)$, $ui(r)$, $uk(r)$ overlap. Taking the derivative of Eq. (6) with respect to $\mu a\u2009i$, we have Display Formula$\u2202E\u2202\mu a\u2009i=\u2212\u2211j(\u2202Hj\u2202\mu a\u2009i)(djm\u2212Hj).$(13)
Using the expression for the absorbed energy density [Eq. (12)], Display Formula$\u2202Hj\u2202\mu a\u2009i=eiTCj\phi +\mu aTCj\u2202\phi \u2202\mu a\u2009i,$(14)
where $ei$ is a vector of zeros with a single 1 in position $i$. Substituting into Eq. (13) gives Display Formula$\u2202E\u2202\mu a\u2009i=\u2212\u2211j(eiTCj\phi +\mu aTCj\u2202\phi \u2202\mu a\u2009i)(djm\u2212Hj).$(15)
The first term in Eq. (15) is Display Formula$\u2211jeiTCj\phi (djm\u2212Hj)=\u2211j,i,keiCikj\phi k(djm\u2212Hj)=\u2211j,k\phi k(djm\u2212Hj)\u222b\Omega \Psi j(r)ui(r)uk(r)d\Omega =\phi TEi(dm\u2212H),$(16)
where $Ei$ is given by a reordering of $Cikj$Display Formula$Ekji=\u222b\Omega \Psi j(r)ui(r)uk(r)d\Omega .$(17)
Note that while $Cj$ is symmetric, in general, $Ei$ is not.