Many materials in nature possess properties that are unknown and difficult to measure directly. In such a situation, inverse modelling offers a viable alternative where one is trying to infer those unknown properties from appropriate measurements of the main dependent variable(s) governing the physical process under investigation. Our investigation is driven by the fact that knowledge of the properties of biological tissues is essential in monitoring any abnormalities that may be forming and have a major impact on organs malfunctioning. Therefore, these disorders must be detected and treated early to save lives and improve the general health. Within the framework of thermal therapies, e.g. hyperthermia or cryoablation, the knowledge of the tissue temperature and its properties, as well as the blood perfusion rate are of utmost importance. Motivated by such a significant biomedical application, this study investigates the reconstruction of biological properties in the thermal-wave hyperbolic model of bio-heat transfer.

The support of the EPSRC grant EP/W000873/1 on “Transient Tomography for Defect Detection'' is acknowledged.

The isotropic thermoelasticity system of type-III, describing the mechanical and thermal behaviours of a body occupying a bounded domain with a Lipschitz boundary, is considered. The displacement vector and either the normal heat flux or the temperature are prescribed on the boundary.

This talk deals with the theoretical and numerical reconstruction of a time-dependent heat source from the knowledge of an additional weighted integral measurement of the temperature in the framework mentioned above. Firstly, it is proved that the measurement type depends on the available thermal boundary condition, expressed by different conditions on the associated weight function. Secondly, for both thermal boundary conditions, the existence of a unique weak solution for exact data is proved, which is achieved by employing Rothe's method. This approach has the advantage of including a time-discrete numerical scheme for computations. Hence, for each of the two inverse source problems considered in this talk, a numerical algorithm that builds upon a decoupling technique is proposed, and the convergence of these numerical schemes is proved for exact data. Furthermore, the uniqueness of a solution is obtained by using an energy estimate. Finally, using the finite element method, the numerical results obtained for various numerical examples with noisy measurements are presented to validate the convergence and stability of the proposed algorithms. The noisy data are regularised using the nonlinear least-squares method; hence, they can be used as input for the proposed numerical scheme.

The results presented in this talk are published in [1].

[1] K. Van Bockstal, L. Marin. Finite element method for the reconstruction of a time-dependent heat source in isotropic thermoelasticity systems of type-III, Zeitschrift für angewandte Mathematik und Physik 73, 2022.

A thermoelastic system describes the interaction between the changes in the shape of an object $\mathbf{u}(\mathbf{x},t)$ and the fluctuation in the temperature $\theta(\mathbf{x},t)$. We consider an isotropic thermoelastic system of type-III which links the elastic and thermal behaviors of an isotropic material occupying a bounded domain $\Omega \subset \mathbb{R}^d$ with Lipschitz continuous boundary. In this contribution, we will study and discuss uniqueness results for solutions to several inverse source problems (ISPs). Our main assumption is that either the heat source $h$ or load source $\mathbf{p}$ can be decomposed as a product of a given time-dependent and an unknown space-dependent function. The main goal is to find the spatial component given some measurement of the function(s) $\mathbf{u}(\mathbf{x},t)$ and/or $\theta(\mathbf{x},t).$

More specifically, the first ISP under consideration deals with the determination of the spatial component $\mathbf{f}(\mathbf{x})$ of the load source $\mathbf{p}(\mathbf{x},t) = g(t)\mathbf{f}(\mathbf{x})$ from the final in time measurement $\mathbf{u}(\mathbf{x},T),$ or from the time-average measurement $\int_0^T \mathbf{u}(\mathbf{x},t)\,\mathrm{d}t,$ where $T$ denotes the final time. The second ISP concerns finding $f(\mathbf{x})$ in the heat source $h(\mathbf{x},t) = g(t) f(\mathbf{x}) $ from the time-average measurement $\int_0^T \theta(\mathbf{x},t)\,\mathrm{d}t.$ The uniqueness results are formulated under suitable assumptions on the temporal component $g(t)$ and its derivative. Some examples will be provided showing the necessity of these (sign) conditions on $g.$ The results holds for (homogeneous) Dirichlet boundary conditions on $\mathbf{u}$ and $\theta$ as well as in the case a (homogeneous) Neumann boundary condition for $\theta$ is used. Finally, the in last ISP, we discuss the problem of finding both $\mathbf{f}$ and $f$ simultaneously when a combination of different measurements is available. The presented work is based on joint work with Dr. Karel Van Bockstal [1].

[1] F. Maes, K. Van Bockstal. Uniqueness for inverse source problems of determining a space-dependent source in thermoeleastic systems, J. Inverse Ill-Posed Probl. 30(6): 845-856, 2022.