Numerical Algorithms for the Reconstruction of Space-Dependent Sources in Thermoelasticity

TL;DR

The paper addresses the recovery of space-dependent sources in a type-III thermoelastic system from final-time or time-averaged measurements. It develops a unified computational framework incorporating Landweber iterations and gradient-based minimisation, including Sobolev-gradient corrections, with explicit adjoint/sensitivity analysis to obtain gradients. Uniqueness and convergence results are established for the inverse problems under various data setups and noise levels, and the numerical study on a 1D copper-alloy model compares Landweber, steepest descent, and conjugate-gradient methods, both in ${\bf L}^2$ and Sobolev gradients. The Sobolev-gradient approach, while sometimes trading off some accuracy, provides boundary-robust reconstructions and avoids the boundary values being fixed during iterations, enhancing applicability to boundary-inaccessible scenarios. Overall, the work delivers a rigorous, data-driven toolkit for stable space-dependent source reconstruction in thermoelastic media with memory effects, with practical implications for materials science and biomedical sensing.

Abstract

This paper investigates the inverse problems of determining a space-dependent source for thermoelastic systems of type III under adequate time-averaged or final-in-time measurements and conditions on the time-dependent part of the sought source. Several numerical methods are proposed and examined, including a Landweber scheme and minimisation methods for the corresponding cost functionals, which are based on the gradient and conjugate gradient method. A shortcoming of these methods is that the values of the sought source are fixed ab initio and remain fixed during the iterations. The Sobolev gradient method is applied to overcome the possible inaccessibility of the source values at the boundary. Numerical examples are presented to discuss the different approaches and support our findings based on the implementation on the FEniCSx platform.

Figures (14)

• Figure 1: Exact $\chi_T(x)$ and noisy measurements $\chi_T^e(x)$ on the finer grid (left) and their projection on the working grid (right) for noise levels in $\{1\%,3\%,5\%\}.$
• Figure 2: Reconstruction using the Landweber scheme for $f^0$ (left) with corresponding relative errors $e_r$ in function of the relaxation parameter $\alpha$ (right) for ISP1.2.
• Figure 3: Reconstruction using the Landweber scheme for $f^0$ (left) with corresponding iterations $K$ in function of the relaxation parameter $\alpha$ (right) for ISP1.2.
• Figure 4: Reconstruction using the Landweber scheme for $f^1$ (left) with corresponding relative errors $e_r$ in function of the relaxation parameter $\alpha$ (right) for ISP1.2.
• Figure 5: Reconstruction using the Landweber scheme for $f^1$ (left) with corresponding iterations $K$ in function of the relaxation parameter $\alpha$ (right) for ISP1.2.

Theorems & Definitions (35)

• Lemma 2.1: Well-posedness forward problem VanBockstal2017b
• Theorem 2.1: Uniqueness ISP1.1
• Theorem 2.2: Uniqueness ISP1.2
• Theorem 2.3: Uniqueness ISP2
• Lemma 3.1
• proof
• Remark 3.1
• Theorem 3.2
• proof
• Theorem 3.3