In the following, we derive the expression for the gradient in the discrete case. The sampled forward model can be expressed as a vector Display Formula
(12)where is a sparse matrix indexed by , where the support of the basis functions , , overlap. Taking the derivative of Eq. (6) with respect to , we have Display Formula
(13)Using the expression for the absorbed energy density [Eq. (12)], Display Formula
(14)where is a vector of zeros with a single 1 in position . Substituting into Eq. (13) gives Display Formula
(15)The first term in Eq. (15) is Display Formula
(16)where is given by a reordering of Display Formula
(17)Note that while is symmetric, in general, is not.