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So far the literature on inverse shape design in aerodynamics is still confined to the single‐point (nominal design point) design and to steady flow. This situation cannot cope with the modern development of internal and external aerodynamics and aerothermoelasticity, especially turbomachinery and aircraft flows. Accordingly, in recent years a new generation of inverse shape design problem has been suggested and investigated theoretically and computationally, consisting mainly of: unsteady inverse and hybrid problems; multipoint inverse and hybrid problems; and inverse problem in aerothermoelasticity. It opens a new area of research in fluid mechanics and aerothermoelasticity. An overview of its status and perspective is given herein, emphasizing the new concepts, theory and methods of solution involved.

We compare a recently proposed generalized eigensystem approach and a new modified generalized eigensystem approach to more widely used truncated singular value decomposition and zero‐order Tikhonov regularization for solving multidimensional elliptic inverse problems. As a test case, we use a finite element representation of a homogeneous eccentric spheres model of the inverse problem of electrocardiography. Special attention is paid to numerical issues of accuracy, convergence, and robustness. While the new generalized eigensystem methods are substantially more demanding computationally, they exhibit improved accuracy and convergence compared with widely used methods and offer substantially better robustness.

The main purpose of this study, reflecting mainly the content of the authors’ plenary lecture, is to make a brief overview of several approaches developed by the author and his colleagues to the solution to ill-posed inverse heat transfer problems (IHTPs) with their possible extension to a wider class of inverse problems of mathematical physics and, most importantly, to show the wide possibilities of this methodology by examples of aerospace applications. In this regard, this study can be seen as a continuation of those applications that were discussed in the lecture.

The application of the inverse method was pre-tested with experimental investigations on a special test equipment in laboratory conditions. In these studies, the author used the solution to the nonlinear inverse problem in the conjugate (conductive and convective) statement. The corresponding iterative algorithm has been developed and tested by a numerical and experimental way.

It can be stated that the theory and methodology of solving IHTPs combined with experimental simulation of thermal conditions is an effective tool for various fundamental and applied research and development in the field of heat and mass transfer.

With the help of the developed methods of inverse problems, the investigation was conducted for a porous cooling with a gaseous coolant for heat protection of the re-entry vehicle in the natural environment of hypersonic flight. Moreover, the analysis showed that the inverse methods can make a useful contribution to the study of heat transfer at the surface of a solid body under the influence of the hypersonic heterogeneous (dusty) gas stream and in many other aerospace applications.

Magnetic material properties of an electromagnetic device (EMD) can be recovered by solving a coupled experimental numerical inverse problem. In order to ensure the highest possible accuracy of the inverse problem solution, all physics of the EMD need to be perfectly modeled using a complex numerical model. However, these fine models demand a high computational time. Alternatively, less accurate coarse models can be used with a demerit of the high expected recovery errors. The purpose of this paper is to present an efficient methodology to reduce the effect of stochastic modeling errors in the inverse problem solution.

The recovery error in the electromagnetic inverse problem solution is reduced using the Bayesian approximation error approach coupled with an adaptive Kriging-based model. The accuracy of the forward model is assessed and adapted a priori using the cross-validation technique.

The adaptive Kriging-based model seems to be an efficient technique for modeling EMDs used in inverse problems. Moreover, using the proposed methodology, the recovery error in the electromagnetic inverse problem solution is largely reduced in a relatively small computational time and memory storage.

The proposed methodology is capable of not only improving the accuracy of the inverse problem solution, but also reducing the computational time as well as the memory storage. Furthermore, to the best of the authors knowledge, it is the first time to combine the adaptive Kriging-based model with the Bayesian approximation error approach for the stochastic modeling error reduction.

Inverse problems deal with determining the causes on the basis of knowing their effects. The object of the inverse parameter estimation problem is to fix the thermal material parameters (the cause) on the strength of a given observation of the temperature history at one or more interior points (the effect). This paper demonstrates two novel approaches to the inverse problems. These approaches use two artificial intelligence mechanisms: neural network and genetic algorithm. Examples shown in this paper give a comparison of results obtained by both of these methods. The numerical technique of neural networks evolved from the effort to model the function of the human brain and the genetic algorithms model the evolutional process of nature. Both of the presented approaches can lead to a solution without having problems with the stability of the inverse task. Both methods are suitable for parallel processing and are advantageous for a multiprocessor computer architecture.

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.

In this paper the authors review the recent numerical techniques proposed to solve the forward and inverse problems concerning the electromagnetic casting and electromagnetic levitation techniques of the metallurgical industry. In addition, the authors present a new topology optimization method to solve the inverse axisymmetric electromagnetic levitation problem.

The proposed method is based on an exact second-order topological expansion of a Kohn–Vogelius-like functional specially devised for this problem.

Through some examples the authors show that it can find suitable solutions efficiently.

The new method completes the set of efficient methods available to solve the inverse electromagnetic casting and the inverse axisymmetric electromagnetic levitation problems.

The study of heat transfer mechanisms is an area of great interest because of various applications that can be developed. Mathematically, these phenomena are usually represented by partial differential equations associated with initial and boundary conditions. In general, the resolution of these problems requires using numerical techniques through discretization of boundary and internal points of the domain considered, implying a high computational cost. As an alternative to reducing computational costs, various approaches based on meshless (or meshfree) methods have been evaluated in the literature. In this contribution, the purpose of this paper is to formulate and solve direct and inverse problems applied to Laplace’s equation (steady state and bi-dimensional) considering different geometries and regularization techniques. For this purpose, the method of fundamental solutions is associated to Tikhonov regularization or the singular value decomposition method for solving the direct problem and the differential Evolution algorithm is considered as an optimization tool for solving the inverse problem. From the obtained results, it was observed that using a regularization technique is very important for obtaining a reliable solution. Concerning the inverse problem, it was concluded that the results obtained by the proposed methodology were considered satisfactory, as even with different levels of noise, good estimates for design variables in proposed inverse problems were obtained.

In this contribution, the method of fundamental solution is used to solve inverse problems considering the Laplace equation.

In general, the proposed methodology was able to solve inverse problems considering different geometries.

The association between the differential evolution algorithm and the method of fundamental solutions is the major contribution.

The purpose of this paper is to present topological derivatives-based reconstruction algorithms to solve an inverse scattering problem for penetrable obstacles.

The method consists in rewriting the inverse reconstruction problem as a topology optimization problem and then to use the concept of topological derivatives to seek a higher-order asymptotic expansion for the topologically perturbed cost functional. Such expansion is truncated and then minimized with respect to the parameters under consideration, which leads to noniterative second-order reconstruction algorithms.

In this paper, the authors develop two different classes of noniterative second-order reconstruction algorithms that are able to accurately recover the unknown penetrable obstacles from partial measurements of a field generated by incident waves.

The current paper is a pioneer work in developing a reconstruction method entirely based on topological derivatives for solving an inverse scattering problem with penetrable obstacles. Both algorithms proposed here are able to return the number, location and size of multiple hidden and unknown obstacles in just one step. In summary, the main features of these algorithms lie in the fact that they are noniterative and thus, very robust with respect to noisy data as well as independent of initial guesses.

This paper aims to discuss the inverse problems that arise in various practical heat transfer processes. The purpose of this paper is to provide an identification method for predicting the internal boundary conditions for thermal analysis of mechanical structure. A few examples of heat transfer systems are given to illustrate the applicability of the method and the challenges that must be addressed in solving the inverse problem.

In this paper, the thermal network method and the finite difference method are used to model the two-dimensional heat conduction inverse problem of the tube structure, and the heat balance equation is arranged into an explicit form for heat load prediction. To solve the matrix ill-conditioned problem in the process of solving the inverse problem, a Tikhonov regularization parameter selection method based on the inverse computation-contrast-adjustment-approach was proposed.

The applicability of the proposed method is illustrated by numerical examples for different dynamically varying heat source functions. It is proved that the method can predict dynamic heat source with different complexity.

The modeling calculation method described in this paper can be used to predict the boundary conditions for the inner wall of the heat transfer tube, where the temperature sensor cannot be placed.

This paper presents a general method for the direct prediction of heat sources or boundary conditions in mechanical structure. It can directly obtain the time-varying heat flux load and thtemperature field of the machine structure.