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Inverse problems in electromagnetism, namely, the recovery of sources (currents or charges) or system data from measured effects, are usually ill-posed or, in the numerical formulation, ill-conditioned and require suitable regularization to provide meaningful results. To test new regularization methods, there is the need of benchmark problems, which numerical properties and solutions should be well known. Hence, this study aims to define a benchmark problem, suitable to test new regularization approaches and solves with different methods.

To assess reliability and performance of different solving strategies for inverse source problems, a benchmark problem of current synthesis is defined and solved by means of several regularization methods in a comparative way; subsequently, an approach in terms of an artificial neural network (ANN) is considered as a viable alternative to classical regularization schemes. The solution of the underlying forward problem is based on a finite element analysis.

The paper provides a very detailed analysis of the proposed inverse problem in terms of numerical properties of the lead field matrix. The solutions found by different regularization approaches and an ANN method are provided, showing the performance of the applied methods and the numerical issues of the benchmark problem.

The value of the paper is to provide the numerical characteristics and issues of the proposed benchmark problem in a comprehensive way, by means of a wide variety of regularization methods and an ANN approach.

The purpose of this paper is to apply the meshless method of fundamental solutions (MFS) to the two‐dimensional time‐dependent heat equation in order to locate an unknown internal inclusion.

The problem is formulated as an inverse geometric problem, using non‐invasive Dirichlet and Neumann exterior boundary data to find the internal boundary using a non‐linear least‐squares minimisation approach. The solver will be tested when locating a variety of internal formations.

The method implemented was proven to be both stable and reasonably accurate when data were contaminated with random noise.

Owing to limited computational time, spatial resolution of internal boundaries may be lower than some similar case investigations.

This research will have practical implications to the modelling and monitoring of crystalline deposit formations within the nuclear industry, allowing development of future designs.

Similar work has been completed in regards to the steady state heat equation, however to the best of the authors' knowledge no previous work has been completed on a time‐dependent inverse inclusion problem relating to the heat equation, using the MFS. Preliminary results presented here will have value for possible future design and monitoring within the nuclear industry

A promising approach for the solution of the electrocardiographic inverse problem is the calculation of the cardiac activation sequence from body surface potential (BSP) mapping data. Here, a two‐fold regularization scheme is applied in order to stabilize the inverse solution of this intrinsically ill‐posed problem. The solution of the inverse problem is defined by the minimum of a non‐linear cost function. The L‐curve method can be applied for regularization parameter determination. Solving the optimization problem by a Newton‐like method, the L‐curve may be of pronged shape. Then a numerically unique determination of the optimal regularization parameter will become difficult. This problem can be avoided applying an iterative linearized algorithm. It is shown that activation time imaging due to temporal and spatial regularization is stable with respect to large model errors. Even neglecting cardiac anisotropy in activation time imaging results in an acceptable inverse solution.

The purpose of this paper is to develop an inverse approach for 3D thermal sources determination.

The developed approach is based on the Green's function for Poison's equation. Forward and inverse couple electromagnetic‐thermal field problems are formulated. Finite elements models are built and applied. Thermal field data are acquired by thermo vision camera. The thermal field sources are determined inside of the investigated inaccessible volume object using modeled and measured data with the developed approach.

The presented method and implemented examples demonstrate the possibilities of the developed approach for inverse source problem solution and determination of thermal field distributions of electrical devices.

The proposed inverse method uses the Green's function for Poison's equation for solution of thermal field problem taking into account the couple electromagnetic‐thermal problems. Proposed inverse method is very fast, accurate and can be used in many practical activities for electrical current determination and visualization in inaccessible regions only by measured external thermal field. Thermal field data needed for the method are easily acquired by thermo vision camera.

This paper attempts to develop an efficient algorithm to solve the inverse problem of identifying constitutive parameters in VFG (viscoelastic functionally graded) materials/structures.

An adaptive recursive algorithm with high fidelity is developed to acquire the derivatives of displacements with respect to constitutive parameters, which are required for the accurate and stable gradient based inverse analysis. A two-step strategy is presented in the process of identification, by which the unknown parameters can be separately identified and the scale and complexity of the inverse VFG problem are reduced. At each step, the process of identification is treated as an optimization problem that is solved by the Levenberg–Marquardt method.

The solution accuracy of forward problems and derivatives of displacements can be stably achieved with different step sizes, and constitutive parameters of homogenous/regional-inhomogeneous VFG materials/structures can be effectively and accurately identified. By examining the reliability, resolution, impacts of reference information and noisy data, the effectiveness of the proposed approach is numerically verified via three numerical examples.

An adaptive recursive algorithm is developed for derivatives computing with high fidelity, providing a solid platform for the sensitivity analysis and thereby a two-step strategy in conjunction with Levenberg–Marquardt method is presented in the process of identification. Consequently, an effective algorithm is developed to identify constitutive parameters of homogenous/regional-inhomogeneous VFG materials/structures.

The purpose of this paper is to demonstrate an inverse problem approach for the determination of stress zones in steel plates of electrical machines. Steel plates of electrical machines suffer large mechanical stress by processes like cutting or punching during the fabrication. The mechanical stress has effects on the electrical properties of the steel, and thus on the losses of the machine.

In this paper, the authors present a sensor arrangement and an appropriate algorithm for determining the spatial permeability distribution in steel plates. The forward problem for stress zone imaging is explained and an appropriate numerical solution technique is proposed. Then an inverse problem formulation is introduced and the nature of the problem is analyzed.

Based on sensitivity analysis, different measurement procedures are compared and a measurement setup is suggested. Further the ill‐posed nature of the inverse problem is analyzed by the Picard condition.

Because of the increased losses due to stress zones, the quantification of stress effects is of interest to adjust the production process. Stress zone imaging is a first approach for the application of an imaging system to quantify these material defects.

This paper presents a simulation study about the applicability of an inverse problem for stress zone imaging and presents first reconstruction results. Further, the paper discusses several issues about stress zone imaging for the ongoing research.

Most of the redundant dual-arm robots are singular free, dexterous and collision free compared to other robotic arms. This paper aims to analyse the workspace of redundant arms to study the manipulability. Furthermore, multi-layer perceptron (MLP) algorithm is used to determine the various joint parameters of both the upper body redundant arms. Trajectory planning of robotic arms is carried out with the help of inverse solutions obtained from the MLP algorithm.

In this paper, the kinematic equations are derived from screw theory approach and inverse kinematic solutions are determined using MLP algorithm. Levenberg–Marquardt (LM) and Bayesian regulation (BR) techniques are used as the backpropagation algorithms. The results from two backpropagation techniques are compared for determining the prediction accuracy. The inverse solutions obtained from the MLP algorithm are then used to optimize the cubic spline trajectories planned for avoiding collision between arms with the help of convex optimization technique. The dexterity of the redundant arms is analysed with the help of Cartesian workspace of arms.

Dexterity of redundant arms is analysed by studying the voids and singular spaces present inside the workspace of arms. MLP algorithms determine unique solutions with less computational effort using BR backpropagation. The inverse solutions obtained from MLP algorithm effectively optimize the cubic spline trajectory for the redundant dual arms using convex optimization technique.

Most of the MLP algorithms used for determining the inverse solutions are used with LM backpropagation technique. In this paper, BR technique is used as the backpropagation technique. BR technique converges fast with less computational time than LM method. The inverse solutions of arm joints for traversing optimized cubic spline trajectory using convex optimization technique are computed from the MLP algorithm.

The inverse problem of identifying the time-dependent potential coefficient along with the temperature in the fourth-order Boussinesq–Love equation (BLE) with initial and boundary conditions supplemented by mass measurement is, for the first time, numerically investigated. From the literature, the authors already know that this inverse problem has a unique solution. However, the problem is still ill-posed by being unstable to noise in the input data.

For the numerical discretization, the authors apply the Crank–Nicolson finite difference method along with the Tikhonov regularization for finding a stable and accurate approximate solution. The resulting nonlinear minimization problem is solved using the MATLAB routine lsqnonlin. Both exact and numerically simulated noisy input data are inverted.

The present computational results demonstrate that obtained solutions are stable and accurate.

The inverse problem presented in this paper was already showed to be locally uniquely solvable, but no numerical identification has been studied yet. Therefore, the main aim of the present work is to undertake the numerical realization. The von Neumann stability analysis is also discussed.

The purpose of this paper is to revisit the recursive computation of closed-form expressions for the topological derivative of shape functionals in the context of time-harmonic acoustic waves scattering by sound-soft (Dirichlet condition), sound-hard (Neumann condition) and isotropic inclusions (transmission conditions).

The elliptic boundary value problems in the singularly perturbed domains are equivalently reduced to couples of boundary integral equations with unknown densities given by boundary traces. In the case of circular or spherical holes, the spectral Fourier and Mie series expansions of the potential operators are used to derive the first-order term in the asymptotic expansion of the boundary traces for the solution to the two- and three-dimensional perturbed problems.

As the shape gradients of shape functionals are expressed in terms of boundary integrals involving the boundary traces of the state and the associated adjoint field, then the topological gradient formulae follow readily.

The authors exhibit singular perturbation asymptotics that can be reused in the derivation of the topological gradient function in the iterated numerical solution of any shape optimization or imaging problem relying on time-harmonic acoustic waves propagation. When coupled with converging Gauss−Newton iterations for the search of optimal boundary parametrizations, it generates fully automatic algorithms.

This study aims to investigate the possible use of a deep neural network (DNN) as an inverse solver.

Different models based on DNNs are designed and proposed for the resolution of inverse electromagnetic problems either as fast solvers for the direct problem or as straightforward inverse problem solvers, with reference to the TEAM 25 benchmark problem for the sake of exemplification.

Using DNNs as straightforward inverse problem solvers has relevant advantages in terms of promptness but requires a careful treatment of the underlying problem ill-posedness.

This work is one of the first attempts to exploit DNNs for inverse problem resolution in low-frequency electromagnetism. Results on the TEAM 25 test problem show the potential effectiveness of the approach but also highlight the need for a careful choice of the training data set.