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Gives introductory remarks about chapter 1 of this group of 31 papers, from ISEF 1999 Proceedings, in the methodologies for field analysis, in the electromagnetic community. Observes that computer package implementation theory contributes to clarification. Discusses the areas covered by some of the papers ‐ such as artificial intelligence using fuzzy logic. Includes applications such as permanent magnets and looks at eddy current problems. States the finite element method is currently the most popular method used for field computation. Closes by pointing out the amalgam of topics.

The dynamics of coupling between spectrum and resolvent under ε‐perturbations of operator and matrix spectra are studied both theoretically and numerically. The phenomenon of non‐trivial pseudospectra encountered in these dynamics is treated by relating information in the complex plane to the behaviour of operators and matrices. On a number of numerical results we show how an intrinsic blend of theory with symbolic and numerical computations can be used effectively for the analysis of spectral problems arising from engineering applications.

To identify the properties of novel discrete differential operators of the first- and the second-order for periodic and two-periodic time functions.

The development of relations between the values of first and second derivatives of periodic and two-periodic functions, as well as the values of the functions themselves for a set of time instants. Numerical tests of discrete operators for selected periodic and two-periodic functions.

Novel discrete differential operators for periodic and two-periodic time functions determining their first and the second derivatives at very high accuracy basing on relatively low number of points per highest harmonic.

Reduce the complexity of creation difference equations for ordinary non-linear differential equations used to find periodic or two-periodic solutions, when they exist.

Application to steady-state analysis of non-linear dynamic systems for solutions predicted as periodic or two-periodic in time.

Identify novel discrete differential operators for periodic and two-periodic time functions engaging a large set of time instants that determine the first and second derivatives with very high accuracy.

X‐ray irradiation of photoresists, such as polymethylmethacrylate (PMMA), on a silicon substrate is an important technique in micro fabrication used to obtain structures and devices with a high aspect ratio. The process is composed of a mask and a photoresist deposited on a substrate (with a gap between mask and resist). Predictions of the temperature distribution in three dimensions in the different layers (mask, gap, photoresist and substrate) and of the potential temperature rise are essential for determining the effect of high flux X‐ray exposure on distortions in the photoresist due to thermal expansion. In this study, we develop a numerical method for obtaining the steady state temperature profile in an X‐ray irradiation process by using a preconditioned Richardson method for the Poisson equation in the micro‐scale. A domain decomposition algorithm is then obtained based on the parallel “divide and conquer” procedure for tridiagonal linear systems. Numerical results show that such a method is efficient.

The purpose of this paper is to describe a numerical inversion technology developed to reconstruct endocardial electric potential maps on the internal surface of heart chambers utilizing intracavitary multi‐electrode catheter measurements. The objective is to perform the reconstruction real time with high accuracy, thereby allowing the incorporation of the technology into medical imaging systems.

Electrode potential points from several beats are merged in order to maximize the information extracted from the catheter measurements. To solve the ill‐posed inverse problem fast, numerically stable solution algorithms based on generalized Tikhonov regularization and bidiagonalization are developed. The latter algorithm also provides an efficient framework for choosing the regularization parameter optimally.

Results of three examples are presented to thoroughly illustrate the performance of the algorithm: one with synthetic data generated in a computational electromagnetics (virtual lab) environment, thereby allowing exact error analysis; another with measured data from a phantom‐bench human heart model where the effect of measurement errors can be investigated in a controlled environment; and a third example that illustrates how the algorithm performs when the catheter data are collected in vivo in a swine heart.

The speed and accuracy in the three examples successfully prove that the inversion technology can be a key component of medical imaging systems.

While some elements of these computational models and techniques presented have been used for decades, the authors achieve speed and accuracy that have not been reported before by combining multi‐beat catheter measurements, the generalized Tikhonov regularization technique, a bidiagonalization algorithm and other top‐notch linear algebra techniques.

There are shortcomings of modern methods of ensuring the stability of Kalman filtration in unmanned vehicles’ (UVs) navigation systems under the condition of a priori uncertainty of the dispersion matrix of measurement interference. First, it is the absence of strict criteria for the selection of adaptation coefficients in the calculation of the a posteriori covariance matrix. Secondly, it is the impossibility of adaptive estimation in real time from the condition of minimum covariance of the updating sequence due to the necessity of its preliminary calculation.

This paper considers a new approach to the construction of the Kalman filter adaptation algorithm. The algorithm implements the possibility of obtaining an accurate adaptive estimation of navigation parameters for integrated UVs inertial-satellite navigation systems, using the correction of non-periodic and unstable inertial estimates by high-precision satellite measurements. The problem of adaptive estimation of the noise dispersion matrix of the meter in the Kalman filter can be solved analytically using matrix methods of linear algebra. A numerical example illustrates the effectiveness of the procedure for estimating the state vector of the UVs’ navigation systems.

Adaptive estimation errors are sharply reduced in comparison with the traditional scheme to the range from 2 to 7 m in latitude and from 1.5 to 4 m in longitude.

The simplicity and accuracy of the proposed algorithm provide the possibility of its effective application to the widest class of UVs’ navigation systems.

This paper aims to propose a novel direct method for indefinite algebraic linear systems. It is well adapted for sparse linear systems, such as those of two-dimensional (2-D) finite elements problems, especially for coupled systems.

The proposed method is developed on an example of an indefinite symmetric matrix. The algorithm of the method is given next, and a comparison between the numbers of operations required by the method and the Cholesky method is also given. Finally, an application on a magnetostatic problem for classical methods (Gauss and Cholesky) shows the relative efficiency of the proposed method.

The proposed method can be used advantageously for 2-D finite elements in stepping methods without using a block decomposition of matrices.

This method is advantageous for direct linear solving for 2-D problems, but it is not recommended at this time for three-dimensional problems.

The proposed method is the first direct solver for algebraic linear systems proposed since more than a half century. It is not limited for symmetric positive systems such as many of direct and iterative methods.

A preconditioned Richardson method for solving three‐dimensional thin film elliptic problems with first order derivatives and variable coefficients has been developed based on the idea of the modified upwind difference scheme and the fact that the thickness of the thin domain is small. This method is simple because only a tridiagonal linear system is needed to solve for each iteration. The computation speed is fast since the spectral radius of the iterative operator is small. Numerical example shows the method to be efficient.

Nonlinear solution approaches for inverse problems require fast simulation techniques for the underlying sensing problem. In this work, the authors investigate finite element (FE) based sensor simulations for the inverse problem of electrical capacitance tomography. Two known computational bottlenecks are the assembly of the FE equation system as well as the computation of the Jacobian. Here, existing computation techniques like adjoint field approaches require additional simulations. This paper aims to present fast numerical techniques for the sensor simulation and computations with the Jacobian matrix.

For the FE equation system, a solution strategy based on Green’s functions is derived. Its relation to the solution of a standard FE formulation is discussed. A fast stiffness matrix assembly based on an eigenvector decomposition is shown. Based on the properties of the Green’s functions, Jacobian operations are derived, which allow the computation of matrix vector products with the Jacobian for free, i.e. no additional solves are required. This is demonstrated by a Broyden–Fletcher–Goldfarb–Shanno-based image reconstruction algorithm.

MATLAB-based time measurements of the new methods show a significant acceleration for all calculation steps compared to reference implementations with standard methods. E.g. for the Jacobian operations, improvement factors of well over 100 could be found.

The paper shows new methods for solving known computational tasks for solving inverse problems. A particular advantage is the coherent derivation and elaboration of the results. The approaches can also be applicable to other inverse problems.

The linear matrix equations have wide applications in engineering, physics, economics and statistics. The purpose of this paper is to introduce iterative methods for solving the systems of linear matrix equations.

According to the hierarchical identification principle, the authors construct alternating direction gradient-based iterative (ADGI) methods to solve systems of linear matrix equations.

The authors propose efficient ADGI methods to solve the systems of linear matrix equations. It is proven that the ADGI methods consistently converge to the solution for any initial matrix. Moreover, the constructed methods are extended for finding the reflexive solution to the systems of linear matrix equations.

This paper proposes efficient iterative methods without computing any matrix inverses, vec operator and Kronecker product for finding the solution of the systems of linear matrix equations.