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In this article, a continuous-time complex-valued projection neural network (CCPNN) in a matrix state space is first proposed for a general complex-variable basis pursuit problem. The proposed CCPNN is proved to be stable in the sense of Lyapunov and to be globally convergent to the optimal solution under the condition that the sensing matrix is not row full rank. Furthermore, an improved discrete-time complex projection neural network (IDCPNN) is proposed by discretizing the CCPNN model. The proposed IDCPNN consists of a two-step stop strategy to reduce the calculational cost. The proposed IDCPNN is theoretically guaranteed to be global convergent to the optimal solution. Finally, the proposed IDCPNN is applied to the reconstruction of sparse signals based on compressed sensing. Computed results show that the proposed IDCPNN is superior to related complex-valued neural networks and conventional basis pursuit algorithms in terms of solution quality and computation time.

PurposeThe purpose of this paper is to make multivariable gray model to be available for the application on interval gray number sequences directly, the matrix form of interval multivariable gray model (IMGM(1,m,k) model) is constructed to simulate and forecast original interval gray number sequences in this paper.Design/methodology/approachFirstly, the interval gray number is regarded as a three-dimensional column vector, and the parameters of multivariable gray model are expressed in matrix form. Based on the dynamic gray action and optimized background value, the interval multivariable gray model is constructed. Finally, two examples and comparisons are carried out to verify the effectiveness of IMGM(1,m,k) model.FindingsThe model is applied to simulate and predict expert value, foreign direct investment, automobile sales and steel output, respectively. The results show that the proposed model has better simulation and prediction performance than another two models.Practical implicationsDue to the uncertainty information and continuous changing of reality, the interval gray numbers are used to characterize full information of original data. And the IMGM(1,m,k) model not only considers the characteristics of parameters changing with time but also takes into account information on lower, middle and upper bounds of interval gray numbers simultaneously to make better suitable for practical application.Originality/valueThe main contribution of this paper is to propose a new interval multivariable gray model, which considers the interaction between the lower, middle and upper bounds of interval numbers and need not to transform interval gray number sequences into real sequences. According to combining different characteristics of each bound of interval gray numbers, the matrix form of interval multivariable gray model is established to simulate and forecast interval gray numbers. In addition, the model introduces dynamic gray action to reflect the changes of parameters over time. Instead of white equation of classic MGM(1,m), the difference equation is directly used to solve the simulated and predicted values.

As one application of the Cauchy–Binet formula, the Ball–Barthe inequality plays a key role in solving reverse isoperimetric inequalities. Recently, a new Grassmannian Ball–Barthe inequality was es...

This article deals with the generalization of the matrix form of the Oresme sequence, extending it to the field of integers. In addition, the two valid generating matrices were discussed, through the permutation of the rows and columns, respectively, totaling two valid matrices of Oresme for the left-hand side, and two more generating matrices of Oresme for the right-hand side. In addition, a new Oresme sequence was introduced, given through the hybridization process of these numbers, obtaining mathematical properties and theorems, inherent to this process.

This paper presents the dynamic model of multibody systems driven by electric motors, the so-called electromechanical systems. The mechanical systems considered in this study include an open loop and/or a closed loop, a full-actuated and an under-actuated one. The dynamic model of this electromechanical systems is established in matrix form by applying the Lagrangian equation with and without multipliers and substructure method. With this approach it is easy to obtain the differential equation of motion of the electro-mechanical systems based on the corresponding differential equations of the purely available mechanical system. These obtained equations describe the electromechanical systems in engineering better in case the systems are purely described by mechanical equations. The differential equations of serial and parallel manipulators, slider-crank mechanism, and overhead crane driven by electric motors are established as illustrated examples. In addition, a simplified dynamic model obtained by neglecting of current variation is also validated by numerical simulations.

In this paper, we propose a new matrix form to generate all 3 × 3 involutory and MDS matrices over F 2 m and prove that the number of all 3 × 3 involutory and MDS matrices over F 2 m is ( 2 m − 1 ) 2 ⋅ ( 2 m − 2 ) ⋅ ( 2 m − 4 ) , where m > 2 . Moreover, we give 3 × 3 involutory and MDS matrices over F 2 3 , F 2 4 and F 2 8 defined by the irreducible polynomials x 3 + x + 1 , x 4 + x + 1 and x 8 + x 7 + x 6 + x + 1 , respectively, by considering the minimum XOR count, which is a metric used in the estimation of hardware implementation cost. Finally, we provide the maximum number of 1s in 3 × 3 involutory MDS matrices.

Two-phase composites with n equal non-overlapping inclusions randomly embedded in the matrix are investigated. It is considered the case when the inclusions are bounded by some Lyapunov’s boundary curve. The problem is reduced to a vector-matrix problem of dimension n for one inclusion. The generalized alternating method of Schwarz applied to the vector-matrix problem is decomposed onto n scalar problems for one inclusion which are solved numerically by the method of integral equations for any smooth shape of the inclusions. A symbolic computation method is developed to solve the same problem by means of conformal mapping and functional equations. As a purpose, the effective conductivity of such models is exactly expressed through all geometrical and mechanical properties of its components.

We explore a case example of hybridity between a large worker-owned cooperative and a union through three lenses: organizational forms, multiple institutional logics, and organizational identity. We delineate three types of organizational hybridity: (1) stretching an existing organizational form; (2) creating a new organizational form; and (3) and retaining multiple discrete organizational forms in a common venture. The cooperative–union hybrid shares members from the two contributing organizations, and so can be classified as a matrix sub-form of multi-organizational hybridity. This study describes how the coop-union hybrid manages the multiple logics and identities retained from both contributing organizations. It considers the hazards of combining these logics and identities, and offers some suggestions on how to avoid potential difficulties. Finally, given the complexity and inefficiencies of the matrix form, we explore whether matrix hybridity is a transitional or permanent form in this particular instance of a cooperative–union venture.

A way of detecting local anisotropies of the muon flux using the matrix-form data of the URAGAN hodoscope (MEPhI) is proposed. Confidence intervals for estimating the matrix-data mathematical expectations in observations with reference and sliding data-sampling periods are selected, indicator functions are constructed, and a procedure for space–time filtering is developed. The results are presented from a search for local anisotropy in the muon flux recorded in the matrix-form data of the URAGAN hodoscope.

An algorithm for the numerical solution of the Fredholm equation of the second kind with a continuous kernel by the method of integral replacement and the matrix solution of SLAE with a quadrature formula of the twelfth order of error with a number of integration intervals divisible by ten is proposed. The new formula, compared to Simpson's formula, gives 15 significant digits for the nodal values of the solution function, even with a small number of intervals of 10.20 on a segment in a finite number of elementary operations. The resulting algorithm has double precision and minimal computation time. While Simpson's formula,together with the matrix method for solving SLAE, gives only 6 significant digits with twenty integration intervals. Moreover, double precision is not available for Simpson's formula (15 zeros in the infinite norm of the solution residual), since the FORTRAN language allows maximum matrix arrays of 200×200. Estimates are obtained for the upper bound of the admissible parameter |λ| for the matrix of the Fredholm equation with strict diagonal dominance or with a small norm of the integral kernel.

We study matrix forms of quaternionic versions of the Fourier Transform and Convolution operations. Quaternions offer a powerful representation unit, however they are related to difficulties in their use that stem foremost from non-commutativity of quaternion multiplication, and due to that $\mu^2 = -1$ posseses infinite solutions in the quaternion domain. Handling of quaternionic matrices is consequently complicated in several aspects (definition of eigenstructure, determinant, etc.). Our research findings clarify the relation of the Quaternion Fourier Transform matrix to the standard (complex) Discrete Fourier Transform matrix, and the extend on which well-known complex-domain theorems extend to quaternions. We focus especially on the relation of Quaternion Fourier Transform matrices to Quaternion Circulant matrices (representing quaternionic convolution), and the eigenstructure of the latter. A proof-of-concept application that makes direct use of our theoretical results is presented, where we produce a method to bound the spectral norm of a Quaternionic Convolution., Comment: 22 pages, 2 figures

Quantization of the noncommutative geometric spectral action has so far been performed on the final component form of the action where all traces over the Dirac matrices and symmetry algebra are carried out. In this work, in order to preserve the noncommutative geometric structure of the formalism, we derive the quantization rules for propagators and vertices in matrix form. We show that the results in the case of a product of a four-dimensional Euclidean manifold by a finite space, could be cast in the form of that of a Yang-Mills theory. We illustrate the procedure for the toy electroweak model., 16 pages, 2 figures, changed equation numbering

The present Corrigendum contains a list of corrections applicable to the authors’ paper [1].

Inspired by an old result by Georg Frobenius, we show that the unbiased version of Hall’s marriage theorem is more transparent when reformulated in the language of matrices. At the same time, we ob...

This research presents a methodology for creating mivar knowledge bases in tabular-matrix form for ground intelligent vehicle control systems. At its core, this methodology is kind of instruction for analysts facing the task of formalizing knowledge in a given subject area. The result of this formalization is a “knowledge map” created according to a special proposed template. In the future, this template allows forming a knowledge base for a given subject area in the formalism of bipartite oriented mivar networks. As an example, the subject area of ground-based intelligent vehicle control systems is used as a template. The proposed methodology of knowledge formalization makes it possible to simplify the process of creating models in Wi!Mi “Razumator-Consultant” 2.1 and also to level the probability of logical collisions when designing a knowledge model in the formalism of bipartite oriented mivar networks.

Chalcogenide-based quantum dots are useful for the application of memory-switching devices because of the control in the trap states in the materials. The control in the trap states can be achieved using a hot-injection colloidal synthesis method that produces temperature-dependent size-variable quantum dots. In addition to this, formation of a nanoscale heterostructure with an insulating material adds to the charge-trapped switching mechanism. Here, we have shown that the colloidal monodispersed CdSe quantum dots and poly(4-vinylpyridine) (PVP) formed a nanoscale heterostructure between themselves when taken in a suitable ratio to fabricate a device. This heterostructure helps realize memory-switching in the device with a maximum on-off current ratio of 105. The switching in the device is mainly due to the trap states in the CdSe quantum dots. The conduction in the off state is due to thermal charge injection and space charge injection conduction and in the on state, due to the Ohmic conduction mechanism.

A stable and accurate finite-difference discretization of first-order elastic wave equations is derived in this work. To simplify the origin and proof of the formulas, a symmetric matrix form (SMF) for elastic wave equations is presented. The curve domain is discretized using summation-by-parts (SBP) operators, and the boundary conditions are weakly enforced using the simultaneous-approximation-term (SAT) technique, which gave rise to a provably stable high-order SBP-SAT method via the energy method. In addition, SMF can be extended to wave equations of different types (SH wave and P-SV wave) and dimensions, which can simplify the boundary derivation process and improve its applicability. Application of this approximation can divide the domain into a multiblock context for calculation, and the interface boundary conditions of blocks can also be used to simulate cracks and other structures. Several numerical simulation examples, including actual elevation within the area of Lushan, China, are presented, which verifies the viability of the framework present in this paper. The applicability of simulating elastic wave propagation and the application potential in the seismic numerical simulation of this method are also revealed.

For many problems in Physics and Computational Fluid Dynamics (CFD), providing an accurate approximation of derivatives is a challenging task. This paper presents a class of high order numerical schemes for approximating the first derivative. These approximations are derived based on solving a special system of equations with some unknown coefficients. The construction method provides numerous types of schemes with different orders of accuracy. The accuracy of each scheme is analyzed by using Fourier analysis, which illustrates the dispersion and dissipation of the scheme. The polynomial technique is used to verify the order of accuracy of the proposed schemes by obtaining the error terms. Dispersion and dissipation errors are calculated and compared to show the features of high order schemes. Furthermore, there is a plan to study the stability and accuracy properties of the present schemes and apply them to standard systems of time dependent partial differential equations in CFD.

In this paper, an integral representation is given for special bounded solutions of pseudoparabolic equations of the form $$ Lw:=\frac{\partial}{\partial t}\left(w_{\overline \phi}+aw+b\overline{w} \right) +cw+d\overline{w} $$ by means of a generating pair of the corresponding class of the generalized $Q$-holomorphic functions in $L_{p,2}(\mathbb{C})$, for $p>2$, where $a,\ b, \ c, \ d$ are functions of $z$ alone.

The Invariant Eigen-operator (IEO) method is widely used in solving the eigenfrequencies of the coulped quantum mesoscopic circuits. The previous IEO method is complicated but stylized, we always wasted much time in this boring processes. Here we extended the IEO method to the matrix form based on Lagrangian of the complex mesoscopic circuits, and express the ideas and processes of the previous calculations of the IEO method in a very simple matrix form. The mathematical methods we used is the indicator representation of the matrix, and we got a very simple and convenient matrix form of the IEO method. This form has important significance for the calculation of large-scale complex multi-loop mesoscopic circuits. Moreover, the matrix form of the IEO method is very friendly to the programming implementation of the complex quantum mesoscopic L−C circuits, it is probably a most optimal algorithm for calculating the eigenfrequencies of the quantum mesoscopic L−C circuits. In addition, with some help of computer programs, we used this method to calculate the eigenfrequencies of three L−C mesoscopic circuits, including two cases with and without mutual inductance. We revealed some relevant properties of these circuits by calculating results, indicating that the eigenfrequency is only related to the element properties of the mesoscopic circuit itself. Finally, we found that this method can also be used in other areas like atom-light coupling systems and solid state physics.

When a model output is a linear function of the model parameters, the estimation process is significantly simplified, since the optimal estimates can be determined without the usage of a numerical optimization method. Moreover, some types of nonlinear models w.r.t. their parameters can be interpreted as linear (obviously introducing a discrepancy). This is the main premise behind the linear approach for parameter estimation, where the Least Squares (LS) method is used for parameters estimation. As this assumption contradicts with the non-linear parameterized model structure, the estimation process becomes iterative. In spite of this, the linear approach is frequently preferable due to the reduced number of computations, compared with the non-linear approach, where the model is correctly considered as non-linear. Moreover, some issues with the starting point selection, stuck at a local minima, etc., natural for the non-linear approach, are avoided. In this paper estimators are presented, based on the linear approach, for both MIMO linear and non-linear parameterized models in a parameter matrix form. The representatives of the first group are LS and Weighted LS (WLS). For non-linear models, this approach is presented in terms of Extended LS (ELS). The topic regarding the efficient realizations of the estimators is also discussed

ЭКГ, матричный вид, информационные технологии

Economics is a science based on the choice of business alternatives in the circumstances that characterize the analyzed business environment. Various circumstances are not under control of business decision-makers, but they must be considered and respected by them. It is known that the environment is an indispensable part of the business system and that their permanent interaction is present. The environment can be convinient and inconvinient for certain business systems, where top business ingenuity means adapting the business system to the environment and finding opportunities for functional progress, regardless of the environment. Business circumstances are experiencing revolutionary changes caused by the emergency situation related to the spread of COVID - 19 virus, with almost all business systems being affected by new and unexpected circumstances. The degree and form of business conditionality ranges from disastrous to favorable, but it is also determined by the ingenuity of business management, and not only by the appearance of a global pandemic. Modeling the business strategies according to the degree of impact conditioned by the nature of the activity requires the formulation of a payment matrix and thus the selection of the optimal business strategy in line with the capabilities of the business system. Marketing commitments are an instrument of rapprochement with users of products and / or services of the business system and logically arise from strategic commitments. The research task is to formalize the modelling of business systems strategic behaviour in the field of composing marketing activities cohering with the impact degree of economy conditions on business and the need for additional efforts to establish a new form of business to amortize the consequences of new circumstances. The matrix formulation of strategic solutions, intersected with the newly created circumstances expressed in the form of the conditionality degree of business activities to epidemiological circumstances, enables the selection of optimal business in accordance with business opportunities. More developed business systems that already use internet marketing and social media marketing have risen and positioned themselves in the market and as such their business will have the best chance of surviving a pandemic period. Marketing activities are to be adapted to the target market, where it is clear that the online market operates perfectly in circumstances other than analysed, globally more significant than in the analysed territory, where marketing would enable market expansion and preparation for standardization of created online or contactless offer, and thus prepare the market and users for the future business. The world will operate on the principle of raising awareness in various ways to overcome the negative impact formed in the global pandemic and this includes openness to: new technologies, online business, new techniques for changing beliefs, classical education, basic education in business economics, psychological topics to strengthen the mental state and immune system, as well as many others, which will have a positive impact on each individual as well as on companies as systems.

The article proposes a mathematical matrix entry of stoichiometric relations for an arbitrary chemical formula. This allows you to simply formulate the problem of solving the equations of spectral analysis as a vector problem of estimating parameters in the presence of such restrictions

Data processing algorithms are important parts of modern measurement systems. These algorithms are often delivered to the user as complex program and their numerical structure is not known. Therefore also influence of algorithm on processed data accuracy is not known. One of the methods to evaluate uncertainty propagation through algorithm is based on matrix form of algorithm. Coefficient matrix of algorithm represents its numerical operations and it is a basis to algorithm accuracy evaluation. The paper presents a method how to identify this coefficient matrix when algebraic form of the algorithm is known or is difficult to use. Identified matrix form of algorithm is then used to estimate uncertainty propagation through exemplary algorithms. Results are compared with experiments.

Datasets for the classification task are usually encoded by a matrix of numbers, the order of rows and columns does not matter. Swapping any two objects or features in it does not change the hidden target function and performance of the machine learning algorithms train of the dataset. However, in the dataset generation problem solution such symmetry is an obstacle. In this paper, we study several methods of the inverse transformation of classification dataset aiming to break the symmetry. We experimented with it in the meta-learning problems of datasets generation and algorithm selection which were solved by conditional generative adversarial nets with convolutional networks.

We show that the r-by-r minors of a d-by-n matrix of variables form a tropical basis of the ideal they generate if and only if one of the following conditions holds: (i) r ⩽ 3 , (ii) r = min { d , n } , (iii) r = 4 and min { d , n } ⩽ 6 . This answers a question raised by M. Chan, A. Jensen, and E. Rubei.

In this paper a formalism, called perturbational crystal orbital theory, is proposed and examined. This formalism is based on a recent formulation of the perturbational molecular orbital theory in matrix form, and could be considered as an extension of it to periodic lattices. The perturbational crystal orbital theory is applied to the study of the energy band changes due to geometrical and electronegative perturbations of a 1D lattice. For these cases, the exact results are compared with those obtained by applying the proposed formalism up to second order, and a discussion of those results is carried out.

The computer-aided diagnosis for histopathological images has attracted considerable attention. Principal component analysis network (PCANet) is a novel deep learning algorithm for feature learning with the simple network architecture and parameters. In this study, a color pattern random binary hashing-based PCANet (C-RBH-PCANet) algorithm is proposed to learn an effective feature representation from color histopathological images. The color norm pattern and angular pattern are extracted from the principal component images of R, G, and B color channels after cascaded PCA networks. The random binary encoding is then performed on both color norm pattern images and angular pattern images to generate multiple binary images. Moreover, we rearrange the pooled local histogram features by spatial pyramid pooling to a matrix-form for reducing the dimension of feature and preserving spatial information. Therefore, a C-RBH-PCANet and matrix-form classifier-based feature learning and classification framework is proposed for diagnosis of color histopathological images. The experimental results on three color histopathological image datasets show that the proposed C-RBH-PCANet algorithm is superior to the original PCANet and other conventional unsupervised deep learning algorithms, while the best performance is achieved by the proposed feature learning and classification framework that combines C-RBH-PCANet and matrix-form classifier.

In this paper, we aim to evaluate the distribution of the aggregate claims in the collective risk model. The claim count distribution is firstly assumed to belong to a generalised (a, b, 0) family. A matrix form recursive formula is then derived to evaluate the related compound distribution when individual claim amounts follow a discrete distribution on non-negative integers. The corresponding formula is also given for continuous individual claim amounts. Secondly, we pay particular attention to the recursive formula for compound phase-type distributions, since only certain types of discrete phase-type distributions belong to the generalised (a, b, 0) family. Similar recursive formulae are obtained for discrete and continuous individual claim amount distributions. Finally, numerical examples are presented for three counting distributions.

Traditional three dimension (3D) bicubic Bezier surface requires to fix Bezier surface parameters to accurately generate the 3D surface from the two-dimension (2D) curve. In this paper, a modified mathematical technique presents to measure and control of the time of 3D bicubic Bezier surface using De-Casteljau and Blossoming algorithms in matrix form. The approach modifies the Blossoming algorithm to construct a cubic Bezier curve in matrix form. The De-Casteljau method harnessed to upgrade the matrix form of 2D to bicubic 3D-Bezier surface depending on t and v values in matrix form that provides numerous solutions for designers to control of 3D design time. It is an efficient technique that can be used in two-dimensional curves and is well suited to many computer graphics and geometric fields. The approach tested based on changing the coefficient of parametric surfaces of Bezier surface. Results of the suggested method indicate that the proposed approach accurately measures and controls the design time to find the optimal time of 3D surfaces. The proposed method produces a better and optimal time of surface generation design leading to more dominant over a design which provides flexibility to the designers to smooth a design from the time points of surface generation.

In this paper, we establish an extension of the matrix form of the Brunn-Minkowski inequality. As applications, we give generalizations on the metric addition inequality of Alexander.2000 Mathematics subject classification: primary 52A40.

In this work, a matrix form of numerical algorithm for spectral shift is presented based on the novel nonequilibrium solvation model that is established by introducing the constrained equilibrium manipulation. This form is convenient for the development of codes for numerical solution. By means of the integral equation formulation polarizable continuum model (IEF-PCM), a subroutine has been implemented to compute spectral shift numerically. Here, the spectral shifts of absorption spectra for several popular chromophores, N,N-diethyl-p-nitroaniline (DEPNA), methylenecyclopropene (MCP), acrolein (ACL) and p-nitroaniline (PNA) were investigated in different solvents with various polarities. The computed spectral shifts can explain the available experimental findings reasonably. Discussions were made on the contributions of solute geometry distortion, electrostatic polarization and other non-electrostatic interactions to spectral shift.

There are important relationships between the discrete-time linear periodic descriptor systems and the discrete-time periodic matrix equations. In the present paper, we introduce the matrix form of the biconjugate residual (BCR) algorithm for solving the discrete-time periodic Sylvester matrix equations AiXiBi+CiXi+1Di=Ei,i=1,2,.... It is shown that the introduced algorithm converges to the solutions within a finite number of iterations in the absence of round-off errors. Finally, three numerical examples are given to demonstrate the efficiency and the performance of the algorithm.

To flexibly control the main lobes number and azimuth angle of terahertz-transmitted beams, we here propose a kind of matrix-form-coding metasurface. Coding metasurfaces are artificial structures composed of periodic or nonperiodic subwavelength unit cells, which are a kind of equivalent material that is made by replacing elementary particles such as molecules and atoms of natural materials with ordered artificial "particles". By critically design of coding modes, the azimuth and elevation of the transmitted terahertz beam can be flexibly manipulated. Particularly, the 1-bit coding is realized in x-direction coding, y-direction coding, and the chessboard coding. The combination of 2-bit and 1-bit coding forms a chessboard coding in the form of matrix, so as to flexibly control the azimuth and elevation of the transmitted beam and control the number of main lobes. Our coding method can flexibly control the azimuth of the beam and effectively reduce the energy of the side lobes. This work opens up a new digital perspective for new methods of metamaterials, making it possible to combine electromagnetic wave manipulation with digital signal processing.

In this paper, we propose a new matrix form to generate all 3 x 3 involutory and MDS matrices over F-2(m) and prove that the number of all 3 x 3 involutory and MDS matrices over F-2(m) is (2(m) - 1)(2) . (2(m) - 2) . (2(m) - 4), where m > 2. Moreover, we give 3 x 3 involutory and MDS matrices over F-2(3), F-2(4) and F-2(8) defined by the irreducible polynomials x(3) +x+ 1, x(4) +x + 1 and x(8) + x(7) + x(6) + x + 1, respectively, by considering the minimum XOR count, which is a metric used in the estimation of hardware implementation cost. Finally, we provide the maximum number of 1s in 3 x 3 involutory MDS matrices. (C) 2019 Elsevier B.V. All rights reserved.

In this note, the neutron group fluxes are calculated through the implementation of the matrix form transport equation solution by the P N method and the main objective is to obtain accurate numerical results. Some results for two sample problems are graphically presented. A neutron flux to dose conversion is also proposed to simulate irradiation planning.

The Perturbational Molecular Orbital (PMO) theory for non-degenerate Hamiltonian is presented in matrix notation. As a consequence of this formulation, a practical algorithm to obtain analytical and numerical expressions up to any order is described. Degenerate perturbation theory is also discussed in the context of the present formulation of PMO theory. A simpler two state problem is examined to illustrate the convergence of the PMO expansion. Finaly, an heuristic convergence criterion is presented.

Large scale systems of algebraic equations are frequently solved by iterative solution methods, such as the conjugate gradient method for symmetric or a generalized conjugate gradient or generalized minimum residual method for nonsymmetric linear systems. In practice, to get an acceptable elapsed computing time when solving large scale problems, one shall use parallel computer platforms. However, such methods involve orthogonalization of search vectors which requires computation of many inner products and, hence, needs global communication of data, which will be costly in computer times. In this paper, we propose various inner product free methods, such as the Chebyshev acceleration method. We study the solution of linear systems arising from optimal control problems for PDEs, such as the edge element discretization of the time-periodic eddy current optimal control problem. Following a discretize-then-optimize scheme, the resulting linear system is of a three-by-three block matrix form. Various solution methods based on an approximate Schur complement and inner product free iterative solution methods for this linear system are analyzed and compared with an earlier used method for two-by-two block matrices with square blocks. The convergence properties and implementation details of the proposed methods are analyzed to show their effectiveness and practicality. Both serial and parallel numerical experiments are presented to further investigate the performance of the proposed methods compared with some other existing methods.

Recently, numerous research institution and companies conduct the investigation and the development of wireless power transfer (WPT) systems. Especially, WPT systems using intermediate frequency band, e.g. around 85kHz, magnetic field have gained attention for the charging method of electric vehicle. Therefore, there is growing concern about health effects induced by magnetic fields (MFs) of the intermediate frequency band. The 3-D impedance method (IM) is one of the techniques to evaluate the induced electric field and current density within the biological body caused by MFs exposure in the intermediate frequency band for which quasi-static approximation is applied.