Iterative Procedure Research Articles

On applying deflation and flexible preconditioning to the adaptive Simpler GMRES method for Sylvester tensor equations

Due to their multidimensional structure, the use of tensors when solving linear matrix equations has been one of the most explored topics over the last decade. In particular, the development of Krylov subspace methods for solving tensor equations is currently attracting a great deal of interest. In this work, we develop the adaptive simpler GMRES method based on tensor format (Ad-SGMRES−BTF) for solving Sylvester tensor equations, in order to address the numerical instability of the simpler version of GMRES based on tensor format (SGMRES−BTF). In addition, an efficient Krylov subspace method, formed by the integration of adaptive deflation and preconditioning and called FAd-SGMRES-D−BTF(m,k) -in which m is the restart parameter and k is the number of harmonic Ritz pairs-, is proposed. Note that the idea behind the FAd-SGMRES-D−BTF(m,k) method is to reuse certain key information, which can be obtained from the current tensor Krylov subspace in the next cycle of the FAd-SGMRES-D−BTF(m) method. It is also important to note that FAd-SGMRES-D−BTF(m,k) represents the first attempt to synchronously combine flexible preconditioning and deflation techniques, with the aim of improving the convergence rate and reducing the computational cost of Ad-SGMRES−BTF(m). Finally, the computational cost of each cycle of the proposed methods when applied to a three-dimensional Sylvester tensor equation is estimated. Moreover, numerical experiments on synthetic and real-world applications demonstrate the potential of Ad-SGMRES−BTF and FAd-SGMRES-D−BTF(m) to solve Sylvester tensor equations.

Krylov subspace methods based quaternion tensor form for generalized Sylvester quaternion tensor equation with application to color video restoration

In this paper, we consider Krylov subspace methods based quaternion tensor form for a class of generalized Sylvester quaternion tensor equation. A novel quaternion tensor product and a global quaternion Arnoldi process based on the tensor form are proposed. Then, the quaternion tensor Krylov subspaces and their orthogonal bases are constructed via the proposed global quaternion Arnoldi process. The quaternion full orthogonalization method based on tensor format (denoted by QFOM−BTF) and the quaternion generalized minimal residual method based on tensor format (denoted by QGMRES−BTF) are established to solve the quaternion tensor equation. The theoretically analyze results of the proposals are discussed on the quaternion tensor form. Finally, we demonstrate the feasibility of QFOM−BTF and QGMRES−BTF methods on some numerical examples including the color video restoration.

A New High-Order Fractional Parallel Iterative Scheme for Solving Nonlinear Equations

Solving fractional-order nonlinear equations is crucial in engineering, where precision and accuracy are essential. This study introduces a novel fractional parallel technique for solving nonlinear equations. To enhance convergence, we incorporate a simple root-finding method of order 3γ + 1 as a correction term in the parallel scheme. Theoretical analysis shows that the parallel scheme achieves a convergence order of 6γ + 3. Using a dynamical system approach, we identify optimal parameter values, and the symmetry in the dynamical planes for different fractional parameters demonstrates the method’s stability and consistency in handling nonlinear problems. These parameter values are applied to the parallel scheme, yielding highly consistent results. Several engineering problems are examined to assess the method’s efficiency, stability, and consistency compared to existing methods.

Transport-of-intensity differential phase contrast imaging: defying weak object approximation and matched-illumination condition

Quantitative phase imaging (QPI) has emerged as a promising label-free imaging technique with growing importance in biomedical research, optical metrology, materials science, and other fields. Partially coherent illumination provides resolution twice that of the coherent diffraction limit, along with improved robustness and signal-to-noise ratio, making it an increasingly significant area of study in QPI. Partially coherent QPI, represented by differential phase contrast (DPC), linearizes the phase-to-intensity transfer process under the weak object approximation (WOA). However, the nonlinear errors caused by WOA in DPC can lead to phase underestimation. Additionally, DPC requires strict matching of the illumination numerical aperture (NA) to ensure the complete transmission of low-frequency information. This necessitates precise alignment of the optical system and limits the flexible use of objective and illumination. In this study, the applicability of the WOA under different coherence parameters is explored, and a method to defy WOA by reducing the illumination NA is proposed. The proposed method uses the transport-of-intensity equation through an additional defocused intensity image to recover the lost low-frequency information due to illumination mismatch, without requiring any iterative procedure. This method overcomes the limitations of DPC being unable to recover large phase objects and does not require the strict illumination matching conditions. The accurate quantitative morphological characterization of customized artifact and microlens arrays that do not satisfy WOA under non-matched-illumination conditions demonstrated the precise quantitative capability of the proposed method and its excellent performance in the field of measurement. Meanwhile, the phase retrieval of tongue slices and oral epithelial cells demonstrated its application potential in the biomedical field. The ability to accurately recover phase under a concise and implementable optical setup makes it a promising solution for widespread application in various label-free imaging domains.

Contribution of ultra-processed food and animal-plant protein intake ratio to the environmental impact of Belgian diets

There is growing concern about the various impacts of food consumption, both on human and planetary health. Given the context-specific nature of consumption patterns, evaluating their national-level impacts is crucial for proactive policy development. This research aims to evaluate the environmental impact of current Belgian diets, with particular attention to the contribution of food groups, ultra-processed foods (UPF), and the animal-to-plant protein ratio. The methodology consists of three key stages. Firstly, the Belgian diet was summarised, based on data from the Belgian National Food Consumption Survey 2014/2015. Secondly, the origin of the most frequently consumed foods was traced using trade databases. Finally, a cradle-to-grave life cycle assessment was conducted to determine the impact of Belgian diets on climate change, water use, land use, and fossil resource scarcity. In this third step, an iterative procedure for selecting the food items to be included in the study was performed. The iterative approach resulted in the inclusion of 227 food items in the analysis. The results indicate greenhouse gas (GHG) emissions of 4.4 [4.27–4.54] kg CO2-equivalent per person per day. Red meat (35 %), beverages (16 %), dairy products (16 %) and snacks (10 %) are identified as primary contributors to climate change. Similar results were observed for land use impacts. Water use and fossil resource scarcity exhibited different trends, with beverages being the most impactful food group. Moreover, UPF account for 50 % of the total climate change and land use impacts, with a linear relationship observed between increased UPF consumption and GHG emissions and land use. A similar linear trend is observed between the ratio of animal-to-plant protein intake and both climate change and land use impact categories. A shift from the current protein ratio to a ratio of 40/60, as suggested in the Flemish Green Deal Protein Shift has been shown to result in a reduction in GHG emissions of the diet by 31 %. This study emphasises the need to target the consumption of high-impact foods such as UPF and animal-based products. Future research will investigate the relationship between environmental and health impacts.

Refining uniform approximation algorithm for low-rank Chebyshev embeddings

Abstract Nowadays, low-rank approximations are a critical component of many numerical procedures. Traditionally the problem of low-rank approximation of matrices is solved in unitary invariant norms such as Frobenius or spectral norm due to the existence of efficient methods for constructing approximations. However, recent results discover the potential of low-rank approximations in the Chebyshev norm, which naturally arises in many applications. In this paper, we investigate the problem of uniform approximation of vectors, which is the main component in the low-rank approximations of matrices in the Chebyshev norm. The principal novelty of this paper is the accelerated algorithm for solving uniform approximation problems. We also analyze the iterative procedure of the proposed algorithm and demonstrate that it has a geometric convergence rate. Finally, we provide an extensive numerical evaluation, which demonstrates the effectiveness of the proposed procedures.

Phase synthesis method in multi-aperture optical systems based on iterative image processing algorithms

Background. Modern technologies in the field of optics and photonics place high demands on the quality and output power of the radiation source. The fulfillment of these requirements can currently be achieved by creating multi-aperture laser systems with coherent beam addition. The main problem of creating such systems is the development of phase synthesis methods in a multichannel optical system. Aim. In this paper, we consider a phase synchronization method without a reference beam for a coherent addition system with active feedback, which is based on the Gerchberg–Saxton algorithm. This algorithm makes it possible to reconstruct complex field amplitudes in the aperture and focal planes from intensity distributions in these fields. The application of this algorithm for multichannel systems is analyzed and its features, such as the occurrence of stagnation and ambiguity of the solution, are revealed. Methods. Solutions are proposed to eliminate the problem of convergence of the algorithm to side solutions and getting the iterative procedure into the local extremum using global optimization algorithms, methods of reducing the dimension of the problem and the introduction of antisymmetric amplitude modulation. Results. The paper demonstrates the results of phase field reconstruction for a large number of optical sources. For a seven-aperture system, a physical simulation was performed to restore phase information, confirming the results of numerical modeling. Conclusion. The proposed approach is a reasonable alternative to currently existing methods and can be used in the problem of coherent addition in multichannel optical systems.

Iterative approach for photonic crystal devices design

Background. An approach to the design of photonic crystal (PhC) devices is proposed, which differs from the known general-purpose optimization methods (for example, a genetic algorithm or gradient procedures) by using information about diffraction patterns at different frequencies when optimizing an element designed to operate at the selected wavelength. The following is an approach to the design of photonic crystal elements. The proposed approach differs from known general-purpose optimization methods (e.g., genetic algorithm or gradient procedures) by using diffraction pattern information at different frequencies while optimizing an element designed to operate at the selected wavelength. The design of functional PhC structures with expected characteristics (for example, waveguides) for a certain wavelength (albeit given by a monochromatic radiation source) is described. Aim. Development based on the FDTD method and confirmation of the functionality of the iterative procedure for calculating the characteristics of metal-dielectric PhC lattices. Methods. The study is based on an iterative approach to the design of PhC elements based on the use of the FDTD method. Results. Model examples were used for demonstration of practical convergence and applicability of the developed iterative procedure. The efficiency of the PhC waveguide, understood as the ratio of the output energy to the input energy, was increasing at each iteration up to 97.2%. Conclusion. The method of synthesis of metal-dielectric PhC structures with preset properties based on application of developed iterative procedure is proposed and argued. The results of the analysis of the 2D PhC waveguide based on a set of round copper rods show the applicability of the proposed method.

Tridiagonal Weighting‐Based Interference Cancellation Method in High‐Mobility OFDM Systems

ABSTRACTIn orthogonal frequency division multiplexing (OFDM) systems, the time‐selective fading channel causes the signal Doppler expansion in high‐mobility scenarios, leading to intercarrier interference (ICI) in the received signals. People had employed a full‐dimensional weighting matrix method to reduce the Doppler‐assisted ICI, though it required a high implementation complexity. In order to reduce the ICI and produce lower complexities, this paper proposes to exploit the tridiagonal weighting matrix to suppress ICIs, in which an optimization problem is constructed to design the tridiagonal weighting matrix, and the nonzero coefficients within the tridiagonal matrix denote optimization variables. Moreover, the optimization objective exploits the system bit error rate (BER) instead of the conventional signal to interference plus noise ratio (SINR), which results in a highly nonlinear optimization problem. Therefore, an iterative solving approach is proposed according to the successive convex approximation (SCA). Simulations show that the proposed method can achieve a good balance between the complexity and BER degradation, especially in the scenario with moderate Doppler spreads.

Application of CR Iteration Scheme in the Generation of Mandelbrot Sets of zp+logct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z^p + \\log c^t$$\\end{document} Function

Mandelbrot set is one of the most fascinating objects in mathematics, so in the literature, one can find many studies related to this set. Moreover, there are many different extensions and generalizations of the classical Mandelbrot set. The two most popular extensions are the use of various kinds of functions and iterations. In this paper, firstly, we replace the constant c in the classical zp+c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z^p + c$$\\end{document} function used in the definition of Mandelbrot set with logct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\log c^t$$\\end{document}, where t∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t \\in \\mathbb {R}$$\\end{document} and t≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t \\ge 1$$\\end{document}. Secondly, we replace the Picard iteration with the CR iteration scheme. For this combination of function and iteration, we prove the escape criterion for the escape-time algorithm used to generate some images of the proposed sets. Moreover, we study the dependency between the iteration’s parameters and two numerical measures (average number of iterations and generation time). We show that this dependency is complex and non-linear.

Research and Analysis of the Small Disturbance Equation in Subsonic, Transonic, and Supersonic Regimes

This paper explores the application of the Small Disturbance Equation (SDE) across subsonic, transonic, and supersonic flow regimes. Derived from the Euler and Navier-Stokes equations, the SDE offers an efficient framework for analyzing aerodynamic behaviors, particularly through the utilization of discretization techniques and iterative solving methods executed in Python. The study assesses the accuracy and limitations of the SDE in detailing essential flow characteristics, revealing that while the equation performs effectively in subsonic and transonic flows, it encounters challenges in supersonic regimes. Nonlinear effects such as shock waves significantly hinder its performance at high speeds. Compared with conventional computational fluid dynamics (CFD) methods, the SDE stands out in scenarios where computational efficiency is paramount. However, its limitations in handling high-speed flows must be carefully considered, highlighting the need for further refinement in its application to supersonic dynamics. This analysis suggests that while the SDE is beneficial for certain aerodynamic studies, its scope and utility are constrained by the inherent complexities of high-speed fluid dynamics.

3D Controlled‐source electromagnetic modelling in anisotropic media using secondary potentials and a cascadic multigrid solver

AbstractQuantitative interpretation of the data from controlled‐source electromagnetic methods, whether via forward modelling or inversion, requires solving a considerable number of forward problems, and multigrid methods are often employed to accelerate the solving process. In this study, a new extrapolation cascadic multigrid method is employed to solve the large sparse complex linear system arising from the finite element approximation of Maxwell's equations using secondary potentials. The equations using secondary potentials are discretized by the classic nodal finite element method on nonuniform rectilinear grids. The resulting linear systems are solved by the extrapolation cascadic multigrid method with a new prolongation operator and preconditioned Stabilized bi‐conjugate gradient method smoother. High‐order interpolation and global extrapolation formulas are utilized to construct the multigrid prolongation operator. The extrapolation cascadic multigrid method with the new prolongation operator is easier to implement and more flexible in application than the original one. Finally, several synthetic examples including layered models, models with anisotropic anomalous bodies or layers, are used to validate the accuracy and efficiency of the proposed method. Numerical results show that the extrapolation cascadic multigrid method improves the efficiency of 3D controlled‐source electromagnetic forward modelling a lot, compared with traditional iterative solvers and some state‐of‐the‐art methods or software (e.g., preconditioned flexible generalized minimal residual method, emg3d) in the considered models and grid settings. The efficiency benefit is more evident as the number of unknowns increases, and the proposed method is more efficient at low frequencies. The extrapolation cascadic multigrid method can also be used to solve systems of equations arising from related applications, such as induction logging, airborne electromagnetic, etc.

Local Convergence Study for an Iterative Scheme with a High Order of Convergence

In this paper, we address a key issue in Numerical Functional Analysis: to perform the local convergence analysis of a fourth order of convergence iterative method in Banach spaces, examining conditions on the operator and its derivatives near the solution to ensure convergence. Moreover, this approach provides a local convergence ball, within which initial estimates lead to guaranteed convergence with details about the radii of domain of convergence and estimates on error bounds. Next, we perform a comparative study of the Computational Efficiency Index (CEI) between the analyzed scheme and some known iterative methods of fourth order of convergence. Our ultimate goal is to use these theoretical findings to address practical problems in engineering and technology.

Transcendence and Normality of Complex Numbers via Hurwitz Continued Fractions

Abstract We study the topological, dynamical, and descriptive set-theoretic properties of Hurwitz continued fractions. Hurwitz continued fractions associate an infinite sequence of Gaussian integers to every complex number that is not a Gaussian rational. The resulting space of sequences of Gaussian integers $\Omega $ is not closed. Using an iterative procedure, we show that $\Omega $ contains a natural subset whose closure $\overline{\textsf{R}}$ encodes continued fraction expansions of complex numbers that are not Gaussian rationals. We prove that $(\overline{\textsf{R}}, \sigma )$ is a subshift with a feeble specification property. As an application, we determine the rank in the Borel hierarchy of the set of Hurwitz normal numbers with respect to the complex Gauss measure. We also construct a family of complex transcendental numbers with bounded partial quotients.

Low‐rank approximations of frequency‐limited discrete‐time Gramians

AbstractWe consider the approximate computation of Gramians of large‐scale linear discrete‐time state‐space systems. Of particular interest are the Gramians which arise in frequency‐limited balanced truncation model order reduction. While there are many results for the continuous‐time situation, the discrete‐time scenario is usually considered to a much lesser extent. The Gramians are solutions of discrete‐time Lyapunov matrix equations (also called Stein matrix equations) which, if the reduction is restricted to smaller frequency intervals, also incorporate matrix functions in the form of matrix logarithms. We discuss the efficient numerical handling of these matrix function as well as the computation of low‐rank Gramian approximation by rational Krylov subspace and related iterative methods.

A novel liquid fraction iteration methodology for addressing oscillatory issues in the total enthalpy method

The total enthalpy method (TEM) has been proposed and employed for several decades to address both isothermal and gradual phase change problems. However, recent investigations into isothermal phase changes have revealed that the phase interface predicted by the TEM oscillates with variations in spatial and temporal discretizations, due to the inconsistency between liquid fraction and enthalpy. To address this issue, and drawing inspiration from the liquid fraction iteration methodology used in the widely adopted heat source method (HSM) to ensure consistency between liquid fraction and temperature, this paper introduces a novel liquid fraction iteration methodology specifically tailored for the TEM, referred to as the iterative TEM (ITEM). This approach is validated against an experimental benchmark involving the melting of a pure substance. Grid and time-step dependence studies confirm that the ITEM effectively eliminates oscillations and exhibits convergence with respect to both grid size and time step. Moreover, the ITEM achieves accuracy comparable to that of the HSM. Finally, the computational costs associated with the ITEM are examined, revealing that costs increase rapidly once the grid Fourier number (Fo) exceeds one. Maintaining the grid Fo number below approximately 0.5 and ensuring the ratio of the relaxation factor to the grid Fo number to approach one significantly improve computational efficiency. The relaxation factor is a crucial parameter within the liquid fraction iteration scheme.

Approximating Fixed Point of (α, β, γ)-Nonexpansive Mapping Using Picard-Thakur Iterative Scheme

In this paper, we explore the fixed point approximation of (α, β, γ)–nonexpansive mappings using the Picard-Thakur iterative scheme. We identify the efficiency of Picard-Thakur iterative scheme through a numerical example by comparing it with other iterative schemes. We also proved strong and weak fixed point convergence theorems for (α, β, γ)–nonexpansive mappings by using Picard-Thakur iterative scheme.

Optimization-based iterative learning control scheme for point-to-point tracking of nonlinear systems

Optimization-based iterative learning control scheme for point-to-point tracking of nonlinear systems

A new Legendre polynomial approach for computing the matrix exponential action on a vector

AbstractWe propose a new approach for computing the action of the matrix exponential over a vector. The approach is based on a Legendre polynomial expansion in the framework of the so‐called ‐product solution to ODEs. The new approach can be combined with Krylov subspace methods.

Compensation of isotope ratio mismatch in double isotope dilution inductively coupled plasma mass spectrometry (ID-ICP/MS)

Abstract Exact-matching double isotope dilution inductively coupled plasma mass spectrometry (ID-ICP/MS) is widely used to characterize reference materials (RMs) in elemental analysis. In this technique, achieving exactly matching isotope ratios for the sample and calibration blends is regarded as an important prerequisite for obtaining accurate measurement results. However, meeting this condition requires multiple time-consuming iterative measurements. In the current study, an alternative approach that can avoid lengthy iterative procedures while maintaining the accuracy of the ID-ICP/MS results was successfully investigated. We examined the effects of an extensively wide range of inexact-matching isotope ratios (approximately 75 %–130 %) in double ID-ICP/MS for elements with a wide mass range. Our experimental study, using gravimetrically prepared samples of Ca, Cd, Cu, Fe, Hg, Mg, Ni, Pb, and Zn, demonstrated that, despite deliberately introducing mismatching of isotope ratios, the accuracy of the ID-ICP/MS results remained consistent with relative measurement bias of typically less than 0.5 %. The absence of a systematic bias due to deviations in the sample blend isotope ratios from the target ratios of the calibration blends revealed that the variability due to an isotope ratio mismatch was sufficiently compensated for. Furthermore, the expanded measurement uncertainties were sufficiently small with negligible variations observed across the different matching ratios. Typically, they were less than 1 %, except for Fe, Hg, Pb, and Zn which were less than 2 %. This assertion is also supported by theoretically calculated error magnification factors. Consequently, it is feasible to directly utilize the marginally estimated mass fraction of the analyte of interest without extensive iterative measurements. The findings of this study provide robust data for ID-ICP/MS, allowing to circumvent lengthy iterative procedures while maintaining the accuracy and precision of the measurement results, particularly in the characterization of reference materials for elemental analysis.