Systems having the form \(A^\text {T}CA\) appear as governing equations in equilibrium systems, such as the nonlinear inverse problems of EIT, ECT, and ERT (electrical impedance/capacitance/resistance tomography) that motivate our study. We develop fast computation for the three tasks required for efficient iterative solution of these inverse problems: assembling the FEM (finite element method) system matrix required for numerical simulation of the forward map; operation by the Jacobian of the forward map from FEM coefficients; and also operation by the transpose of the Jacobian of the forward map from FEM coefficients. In particular, we present a vectorized assembly of the FEM matrix that gives fast computation in vectorized languages, and give sparse vectorized formulas for operation by the Jacobian and its transpose that require no additional solves beyond the forward solve. Comparison to existing adjoint state methods, that require multiple forward solves, demonstrate that these new methods are orders of magnitude faster than existing methods commonly used in EIT and ECT, such as those implemented in the EIDORS package. We derive these results in the context of ERT that is a canonical example of a \(A^\text {T}CA\) system.
