On sparse forward solutions in non-stationary domains for the EIT imaging problem

In the forward EIT-problem numerical solutions of an elliptic partial differential equation are required. Given the arbitrary geometries encountered, the Finite Element Method (FEM) is, naturally, the method of choice. Nowadays, in EIT applications, there is an increasing demand for finer Finite Element mesh models. This in turn results to a soaring number of degrees of freedom and an excessive number of unknowns. As such, only piece-wise linear basis functions can practically be employed to maintain inexpensive computations. In addition, domain reduction and/or compression schemes are often sought to further counteract for the growing number of unknowns. In this paper, we replace the piece-wise linear with wavelet basis functions (coupled with the domain embedding method) to enable sparse approximations of the forward computations. Given that the forward solutions are repeatedly, if not extensively, utilised during the image reconstruction process, considerable computational savings can be recorded whilst maintaining O(N) forward problem complexity. We verify with numerical results that, in practice, less than 5% of the involved coefficients are actually required for computations and, hence, needs to be stored. We finalise this work by addressing the impact to the inverse problem. It is worth underlining that the proposed scheme is independent of the actual family of wavelet basis functions of compact support.