Search results

In this paper, various regularization methods are numerically implemented using the boundary element method (BEM) in order to solve the Cauchy steady‐state heat conduction problem in an anisotropic medium. The convergence and the stability of the numerical methods are investigated and compared. The numerical results obtained confirm that stable numerical results can be obtained by various regularization methods, but if high accuracy is required for the temperature, or if the heat flux is also required, then care must be taken when choosing the regularization method since the numerical results are substantially improved by choosing the appropriate method.

In this paper, the Cauchy-type problem for the Laplace equation was solved in the rectangular domain with the use of the Chebyshev polynomials. The purpose of this paper is to present an optimal choice of the regularization parameter for the inverse problem, which allows determining the stable distribution of temperature on one of the boundaries of the rectangle domain with the required accuracy.

The Cauchy-type problem is ill-posed numerically, therefore, it has been regularized with the use of the modified Tikhonov and Tikhonov–Philips regularization. The influence of the regularization parameter choice on the solution was investigated. To choose the regularization parameter, the Morozov principle, the minimum of energy integral criterion and the L-curve method were applied.

Numerical examples for the function with singularities outside the domain were solved in this paper. The values of results change significantly within the calculation domain. Next, results of the sought temperature distributions, obtained with the use of different methods of choosing the regularization parameter, were compared. Methods of choosing the regularization parameter were evaluated by the norm N_max.

Calculation model described in this paper can be applied to determine temperature distribution on the boundary of the heated wall of, for instance, a boiler or a body of the turbine, that is, everywhere the temperature measurement is impossible to be performed on a part of the boundary.

The paper presents a new method for solving the inverse Cauchy problem with the use of the Chebyshev polynomials. The choice of the regularization parameter was analyzed to obtain a solution with the lowest possible sensitivity to input data disturbances.

Electrical capacitance tomography (ECT) and electrical resistance tomography (ERT) are promising techniques for multiphase flow measurement due to their high speed, low cost, non-invasive and visualization features. There are two major difficulties in image reconstruction for ECT and ERT: the “soft-field”effect, and the ill-posedness of the inverse problem, which includes two problems: under-determined problem and the solution is not stable, i.e. is very sensitive to measurement errors and noise. This paper aims to summarize and evaluate various reconstruction algorithms which have been studied and developed in the word for many years and to provide reference for further research and application.

In the past 10 years, various image reconstruction algorithms have been developed to deal with these problems, including in the field of industrial multi-phase flow measurement and biological medical diagnosis.

This paper reviews existing image reconstruction algorithms and the new algorithms proposed by the authors for electrical capacitance tomography and electrical resistance tomography in multi-phase flow measurement and biological medical diagnosis.

The authors systematically summarize and evaluate various reconstruction algorithms which have been studied and developed in the word for many years and to provide valuable reference for practical applications.

The purpose of this paper is to demonstrate the applicability of swarm and evolutionary techniques for regularized machine learning. Generally, by defining a proper penalty function, regularization laws are embedded into the structure of common least square solutions to increase the numerical stability, sparsity, accuracy and robustness of regression weights. Several regularization techniques have been proposed so far which have their own advantages and disadvantages. Several efforts have been made to find fast and accurate deterministic solvers to handle those regularization techniques. However, the proposed numerical and deterministic approaches need certain knowledge of mathematical programming, and also do not guarantee the global optimality of the obtained solution. In this research, the authors propose the use of constraint swarm and evolutionary techniques to cope with demanding requirements of regularized extreme learning machine (ELM).

To implement the required tools for comparative numerical study, three steps are taken. The considered algorithms contain both classical and swarm and evolutionary approaches. For the classical regularization techniques, Lasso regularization, Tikhonov regularization, cascade Lasso-Tikhonov regularization, and elastic net are considered. For swarm and evolutionary-based regularization, an efficient constraint handling technique known as self-adaptive penalty function constraint handling is considered, and its algorithmic structure is modified so that it can efficiently perform the regularized learning. Several well-known metaheuristics are considered to check the generalization capability of the proposed scheme. To test the efficacy of the proposed constraint evolutionary-based regularization technique, a wide range of regression problems are used. Besides, the proposed framework is applied to a real-life identification problem, i.e. identifying the dominant factors affecting the hydrocarbon emissions of an automotive engine, for further assurance on the performance of the proposed scheme.

Through extensive numerical study, it is observed that the proposed scheme can be easily used for regularized machine learning. It is indicated that by defining a proper objective function and considering an appropriate penalty function, near global optimum values of regressors can be easily obtained. The results attest the high potentials of swarm and evolutionary techniques for fast, accurate and robust regularized machine learning.

The originality of the research paper lies behind the use of a novel constraint metaheuristic computing scheme which can be used for effective regularized optimally pruned extreme learning machine (OP-ELM). The self-adaption of the proposed method alleviates the user from the knowledge of the underlying system, and also increases the degree of the automation of OP-ELM. Besides, by using different types of metaheuristics, it is demonstrated that the proposed methodology is a general flexible scheme, and can be combined with different types of swarm and evolutionary-based optimization techniques to form a regularized machine learning approach.

The purpose of this paper is to present the method for solving the inverse Cauchy-type problem for the Laplace’s equation. Calculations were made for the rectangular domain with the target temperature on three boundaries and, additionally, on one of the boundaries, the heat flux distribution was selected. The purpose of consideration was to determine the distribution of temperature on a section of the boundary of the investigated domain (boundary Γ₁) and find proper method for the problem regularization.

The solution of the direct and the inverse (Cauchy-type) problems for the Laplace’s equation is presented in the paper. The form of the solution is noted as the linear combination of the Chebyshev polynomials. The collocation method in which collocation points had been determined based on the Chebyshev nodes was applied. To solve the Cauchy problem, the minimum of functional describing differences between the target and the calculated values of temperature and the heat flux on a section of the domain’s boundary was sought. Various types of the inverse problem regularization, based on Tikhonov and Tikhonov–Philips regularizations, were analysed. Regularization parameter α was chosen with the use of the Morozov discrepancy principle.

Calculations were performed for random disturbances to the heat flux density of up to 0.01, 0.02 and 0.05 of the target value. The quality of obtained results was next estimated by means of the norm. Effect of the disturbance to the heat flux density and the type of regularization on the sought temperature distribution on the boundary Γ₁ was investigated. Presented methods of regularization are considerably less sensitive to disturbances to measurement data than to Tikhonov regularization.

Discussed in this paper is an example of solution of the Cauchy problem for the Laplace’s equation in the rectangular domain that can be applied for determination of the temperature distribution on the boundary of the heated element where it is impossible to measure temperature or the measurement is subject to a great error, for instance on the inner wall of the boiler. Authors investigated numerical examples for functions with singularities outside the domain, for which values of gradients change significantly within the calculation domain what corresponds to significant changes in temperature on the wall of the boiler during the fuel combustion.

In this paper, a new method for solving the Cauchy problem for the Laplace’s equation is described. To solve this problem, the Chebyshev polynomials and nodes were used. Various types of regularization of this problem were considered.

In this paper, the exact solutions of the Schlömilch’s integral equation and its linear and non-linear generalized formulas with application are solved by using two efficient iterative methods. The Schlömilch’s integral equations have many applications in atmospheric, terrestrial physics and ionospheric problems. They describe the density profile of electrons from the ionospheric for awry occurrence of the quasi-transverse approximations. The paper aims to discuss these issues.

First, the authors apply a regularization method combined with the standard homotopy analysis method to find the exact solutions for all forms of the Schlömilch’s integral equation. Second, the authors implement the regularization method with the variational iteration method for the same purpose. The effectiveness of the regularization-Homotopy method and the regularization-variational method is shown by using them for several illustrative examples, which have been solved by other authors using the so-called regularization-Adomian method.

The implementation of the two methods demonstrates the usefulness in finding exact solutions.

The authors have applied the developed methodology to the solution of the Rayleigh equation, which is an important equation in fluid dynamics and has a variety of applications in different fields of science and engineering. These include the analysis of batch distillation in chemistry, scattering of electromagnetic waves in physics, isotopic data in contaminant hydrogeology and others.

In this paper, two reliable methods have been implemented to solve several examples, where those examples represent the main types of the Schlömilch’s integral models. Each method has been accompanied with the use of the regularization method. This process constructs an efficient dealing to get the exact solutions of the linear and non-linear Schlömilch’s integral equation which is easy to implement. In addition to that, the accompanied regularization method with each of the two used methods proved its efficiency in handling many problems especially ill-posed problems, such as the Fredholm integral equation of the first kind.

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.

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.

Solving the inverse heat conduction using Tikhonov regularization requires the selection of an optimal smoothing parameter. One popular method for choosing the smoothing parameter is the generalized cross‐validation method. This method works very well but is computationally expensive. In this paper we investigate the L‐curve method for selecting an optimal smoothing parameter. This L‐curve is easily computed and may prove very useful for large systems which preclude other methods.