The present monograph is devoted to selected numerical aspects of tomographic imaging, formulated as partial differential equations and solved primarily by the Boundary Element Method. The book focuses on practical and computational issues that arise when BEM is applied to forward and inverse problems in electrical impedance tomography, diffuse optical tomography, acoustic tomography, ultrasound-related modelling, and wave scattering problems.
The motivation for preparing this work comes from the observation that many tomographic modalities require accurate modelling of physical fields inside domains that are only partially accessible from the boundary. In such problems, the measured information is usually obtained from sensors located on the external boundary of the object, whereas the internal distribution of material parameters must be reconstructed indirectly. This naturally leads to the formulation of inverse problems, in which the quality of image reconstruction strongly depends on the reliability and numerical stability of the forward model.
Although the Finite Element Method is often more convenient for modelling heterogeneous domains, the Boundary Element Method offers several advantages that are particularly important in selected classes of tomographic and wave propagation problems. These include the reduction of discretisation to the boundary, the natural treatment of unbounded or semi-unbounded domains, and the simultaneous approximation of the state variable and its normal derivative. At the same time, BEM requires careful treatment of boundary conditions, interfaces between subregions, singular integrals, and fundamental solutions of the governing equations. These issues constitute one of the main themes of the monograph.
The book does not aim to provide an elementary introduction to BEM or FEM. It is intended for readers with a basic knowledge of numerical methods, mathematical modelling, and partial differential equations. The main objective is to discuss those aspects of BEM implementation that become essential in tomographic applications, especially when the analysed domain contains material heterogeneities, interfaces, complex boundary conditions, or wave phenomena.
A substantial part of the monograph is devoted to the comparison of zero-order and second-order boundary elements. The authors analyse how the choice of approximation order influences the accuracy of the forward solution, the behaviour of the numerical model near interfaces and boundaries, and the effectiveness of inverse reconstruction. Particular attention is paid to practical consequences for tomographic imaging, where computational cost, discretisation density, measurement protocol, noise level, and stability of optimisation procedures are of fundamental importance.
The monograph also presents the role of benchmark solutions and reference calculations. In several examples, FEM solutions are used as reference results for validating BEM calculations. This comparative approach allows the reader to assess the numerical reliability of the proposed formulations and to understand the circumstances under which BEM may become an effective alternative or complement to FEM in tomographic modelling.
The scope of the book extends from electrical impedance tomography to diffuse optical tomography and acoustic tomography. The selected examples show that a common mathematical and numerical framework can be adapted to different physical fields, provided that the governing equations, material parameters, boundary conditions, and measurement protocols are properly defined. This multidisciplinary perspective is important because modern tomographic systems increasingly combine multiple physical modalities and require flexible numerical tools for forward modelling and inverse analysis.
The authors also address selected practical difficulties, including the implementation of Dirichlet, Neumann, Robin, periodicity, and interface conditions, the treatment of singularities in integral formulations, the use of fundamental solutions, the definition of objective functions for inverse problems, and the influence of measurement noise on reconstruction accuracy. These topics are presented not only from a theoretical perspective, but also through numerical examples intended to clarify their practical consequences.
The book is therefore addressed to researchers, doctoral students, engineers, and specialists working in computational tomography, numerical modelling, inverse problems, non-destructive testing, biomedical engineering, process tomography, acoustic field analysis, and related areas [29]. It may also be useful for developers of tomographic systems who need to understand how numerical modelling choices affect image reconstruction quality and computational efficiency.
The authors hope that this monograph will contribute to a better understanding of the practical applicability of the Boundary Element Method in tomographic imaging. The intention is not to demonstrate that BEM should replace FEM in all tomographic problems, but rather to show where BEM provides distinctive computational advantages, what limitations must be considered, and how the method can be effectively used in selected forward and inverse modelling tasks.
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