The talk focuses on the theoretical investigation of the inverse problem for the Pennes' bioheat wave equation. Uniqueness and existence are the main questions under consideration.

In the report, we consider an inverse problem for a parabolic equation with unknown coefficient depending from only one independent variable: space or time variable.

We consider the following problem of determining unknown coefficient $C_{0} \left(x\right)$ of the linear parabolic equation: $$ \begin{array}{l} {\frac{\partial v\left(x,t\right)}{\partial t} =a_{0} \frac{\partial ^{2} v\left(x,t\right)}{\partial x^{2} } +a_{1} \frac{\partial v\left(x,t\right)}{\partial x} +a_{2} v\left(x,t\right)+} \\ {+f\left(x,t\right)+F\left(x,t\right),\, \, \, \, \, \, \, \, \, \, \left(x,t\right)\in \Omega =\left\{\left(x,t\right):0<x<l,\, \, 0<t\le T\right\},} \end{array} $$ under conditions: $$ k_{1} v\left(x,0\right)+\int _{0}^{T}e^{k\tau } v\left(x,\tau \right)d\tau =\varphi _{0} \left(x\right),\, \, \, \, v\left(x,T\right)=\varphi _{T} \left(x\right),\, \, \, x\in \left[0,\, l\right], $$ $$ v\left(0,t\right)=\psi _{0} \left(t\right),\, \, \, \, \, v\left(l,t\right)=\psi _{1} \left(t\right),\, \, \, \, t\in \left[0,\, T\right], $$ and where $F\left(x,t\right)=B_{0} \left(x,t\right)C_{0} \left(x\right)$ and $k,\, \, k_{1} \ne 0$ are constants.

Two cases are considered. In the first case, the known coefficients $a_{i} ,\, \, i=0,\, ...,2$ are functions of $x$, i.e. $a_{i} =a_{i} \left(x\right)$. The functions $a_{0} (x)>0,$ $a_{1} \left(x\right)$, $a_{2} \left(x\right)$, $\varphi _{0} \left(x\right),\, \varphi _{T} \left(x\right)$, $f\left(x,t\right),\, B_{0} \left(x,t\right)$, $\psi _{0} \left(t\right),\, \psi _{1} \left(t\right)$ are given and satisfy all the conditions of existence and uniqueness of the functions $v\left(x,t\right),\, \, C_{0} \left(x\right)$, which are the solutions to the problem.

We propose a numerical method of solution to the problem, which is based on the use of the method of lines. The initial problem is reduced to the parametric inverse problems with respect to ordinary differential equations. Then, we propose a non-iterative method based on using a special representation of the solutions to the obtained problems [1, 2]. Some of the results of the carried out numerical experiments are given. The obtained results show the efficiency of the proposed approach.

In the second case, similar approaches to numerical solution to the problem of identifying $C_{0} \left(t\right)$ in case $F\left(x,t\right)=B\left(x,t\right)C_{0} \left(t\right)$ are proposed. In this case, the known coefficients $a_{i} ,\, \, i=0,\, ...,2$ are functions of $t$, and instead of the first conditions, we use the following ones: $$\int _{0}^{l}e^{k\xi } v\left(\xi ,t\right)d\xi =\psi \left(t\right),\, \, \, \, t\in \left[0,\, T\right],$$ $$v\left(x,0\right)=\varphi _{0} \left(x\right),\, \, \, \, x\in \left[0,\, l\right].$$

[1] K.R. Aida-zade, A.B. Rahimov. An approach to numerical solution of some inverse problems for parabolic equations, Inverse Probl. Sci. Eng. 22: 96-111, 2014.

[2] K.R. Aida-zade, A.B. Rahimov. On recovering space or time-dependent source functions for a parabolic equation with nonlocal conditions, Appl. Math. Comp. 419, 2022.

The location and identification of hidden conducting threat objects using metal detection is an important yet difficult task. Applications include security screening at transport hubs and finding landmines and unexploded ordnance in areas of former conflict. Based on an asymptotic expansion of the perturbed magnetic field, we have derived an economical object description called a magnetic polarizability tensor (MPT), which is a function of the object’s size, shape, material properties and the exciting frequency [1]. The MPT provides the object characterisation in the leading order term of the asymptotic expansion of the perturbed magnetic field as the object size tends to 0 and we have extended this to a complete asymptotic expansion with improved object characterisation provided by generalised MPTs expressed in terms of tensorial and multi-indices [2].

To compute object characterisations, we employ a hp-finite element method, accelerated by an adaptive reduced order model and scaling results [3], to efficiently compute a large dictionary spectral signature characterisations of realistic threat and non-threat objects relevant for security screening [4]. To accurately capture small skin depths and realistic in-homogeneous objects with thin material layers, this involves including thin layers of prismatic boundary layers, which we have shown are in close agreement with measurements [5]. In this talk, we review our latest developments and discuss possible classification strategies [6].

References

[1] P. D. Ledger, W. R. B. Lionheart. The spectral properties of the magnetic polarizability tensor for metallic object characterisation. Mathematical Methods in the Applied Sciences, 43:78–113, 2020.

[2] P.D. Ledger, W.R.B. Lionheart. Properties of generalized magnetic polarizability tensors. Mathematical Methods in the Applied Sciences, To appear 2023. DOI:10.1002/mma.8856

[3] B. A. Wilson, P. D. Ledger. Efficient computation of the magnetic polarizability tensor spectral signature using proper orthogonal decomposition. International Journal for Numerical Methods in Engineering, 122:1940–1963, 2021.

[4] P. D. Ledger, B. A. Wilson, A. A. S. Amad, W. R. B. Lionheart. Identification of metallic objects using spectral magnetic polarizability tensor signatures: Object characterisation and invariants. International Journal for Numerical Methods in Engineering, 122:3941–3984, 2021.

[5] J. Elgy, P.D. Ledger, J.L. Davidson, T. Ozdeger, A.J. Peyton, Computations and measurements of the magnetic polarizability tensor characterisation of highly conducting and magnetic objects, submitted 2023.

[6] B. A. Wilson, P. D. Ledger, and W. R. B. Lionheart. Identification of metallic objects using spectral magnetic polarizability tensor signatures: Object classification. International Journal for Numerical Methods in Engineering 123: 2076-2111, 2022.