## 2 Basic Elements
### FPGAs and Inverse Source Problems  
Inverse source problems appear in many applications. One such application is the inverse electroencephalographic problem, which involves determining bioelectrical sources of the brain from scalp measurements of the electrical potential produced by these sources. Identifying dipolar sources associated with epileptic foci is particularly important given that epilepsy affects around 50 million people worldwide according to the World Health Organization. Another application is the inverse electrocardiography problem, which involves determining the epicardial potential from measurements of the potential on the chest. These examples are multichannel problems for which the architectures developed in this work can be applied. These are generally ill-posed problems in the sense of Hadamard due to the non-uniqueness of the solution and the numerical instability, which can produce large changes to the location of the source when errors are presented. Regularization methods are employed to handle this instability. The Tikhonov regularization method is the best known among them; it depends on a parameter that must be chosen in terms of the measurement error. The identification problem studied in this work only presents numerical instability since the problem has a unique solution [7; 9]. The regularization in this work considers two parameters. The former is the Tikhonov regularization parameter. The latter is the term in which the series is truncated. From numerical tests, we found these parameters. This is a contribution of this work. FPGA devices offer several advantages for the implementation of algorithms: the primary one being their capability to accelerate algorithms. Inverse source identification problems, which appear in many applications, are one such area where these advantagescan be leveraged. In this work, we have developed two architectures for an algorithm designed to solve inverse source identification problems. A natural question arises: How are these problems and the architectures related? The answer lies in the stable source identification algorithms (one for each of them). This work presents two roll-up architectures in which a stable algorithm was implemented to recover sources located on the interface that separates two homogeneous media that make up a non-homogeneous medium. The algorithm is developed in the bidimensional case, but the tridimensional case corresponds to the identification of sources located in the cerebral cortex. In the tridimensional case, we must use other elements for the base (spherical harmonics instead of circular harmonics), but the methodology presented here can be extended without problem. One of the key advantages of the architectures developed in this work is our ability to determine the number of coefficients required, as this number significantly influences the numerical stability of the algorithm. Additionally, we can select the number of bits to achieve accurate approximations while minimizing hardware consumption. For instance, the tridimensional version of the algorithm is presented here. These developments can assist us with constructing portable devices for source and potential identification. Furthermore, they can facilitate the design and implementation of reconfigurable devices for real-time imaging of sources. It is crucial to highlight that these architectures are designed to serve as a foundation for other algorithms. We can deduce that the architectures developed in this work can be repurposed for other similar algorithms by leveraging their reconfigurable features. In the following, we provide some insights into FPGAs and the loop unrolling architectures. Broadly speaking, the design of systems based on FPGAs is geared towards enhancing performance and optimizing loops. The loop unrolling technique, which expands loops into an iterative version, has shown promising results in this area [10].