Iterative Method Research Articles

Adaptive Factored Incomplete Inverse Matrices

ABSTRACTThe solution of large sparse linear systems is an essential part in many scientific fields. Numerical solution of such systems is usually performed using iterative methods in conjunction with effective preconditioning schemes. A new class of factored approximate inverses is proposed, namely adaptive factored incomplete inverse matrices, which are computed using a recursive Schur complement‐based approach. This class of approximate inverses does not require the knowledge of a sparsity pattern, which is formed adaptively during computation. For this to be possible, a flexible sparse storage scheme was designed. Several numerical dropping strategies are showcased and discussed, along with the effects of reordering to the preconditioner density and the corresponding convergence behavior. Dropping based on monitoring the growth of elements in the incomplete LU factorization is also presented and discussed along with improvements during computation of the successive Schur complements. A static sparsity pattern variant is also provided and discussed. Implementation details and analysis, for computing the proposed scheme, are given along with the computational complexity and memory requirements. Numerical results depicting the effectiveness and applicability of the proposed scheme are also presented.

Stochastic Gradient Descent with Pre-Conditioned Polyak Step-Size

Stochastic Gradient Descent (SGD) is one of the many iterative optimization methods that are widely used in solving machine learning problems. These methods display valuable properties and attract researchers and industrial machine learning engineers with their simplicity. However, one of the weaknesses of this type of methods is the necessity to tune learning rate (step-size) for every loss function and dataset combination to solve an optimization problem and get an efficient performance in a given time budget. Stochastic Gradient Descent with Polyak Step-size (SPS) is a method that offers an update rule that alleviates the need of fine-tuning the learning rate of an optimizer. In this paper, we propose an extension of SPS that employs preconditioning techniques, such as Hutchinson’s method, Adam, and AdaGrad, to improve its performance on badly scaled and/or ill-conditioned datasets.

Protein Design by Integrating Machine Learning and Quantum-Encoded Optimization

The protein design problem involves finding polypeptide sequences folding into a given three-dimensional structure. Its rigorous algorithmic solution is computationally demanding, involving a nested search in sequence and structure spaces. Structure searches can now be bypassed thanks to recent machine-learning breakthroughs, which have enabled accurate and rapid structure predictions. Similarly, sequence searches might be entirely transformed by the advent of quantum annealing machines and by the required new encodings of the search problem, which could be performative even on classical machines. In this work, we introduce a general protein design scheme where algorithmic and technological advancements in machine learning and quantum-inspired algorithms can be integrated, and an optimal physics-based scoring function is iteratively learned. In this first proof-of-concept application, we apply the iterative method to a lattice protein model amenable to exhaustive benchmarks, finding that it can rapidly learn a physics-based scoring function and achieve promising design performances. Strikingly, our quantum-inspired reformulation outperforms conventional sequence optimization even when adopted on classical machines. The scheme is general and can be extended, e.g., to encompass off-lattice models, and it can integrate progress on various computational platforms, thus representing a new paradigm approach for protein design. Published by the American Physical Society 2024

Analytical solutions of the Caudrey-Dodd-Gibbon equation using Khater II and variational iteration methods.

This study focuses on solving the Caudrey-Dodd-Gibbon ( ) equation using the Khater II ( ) method and the Variational Iteration ( ) method. The equation is a pivotal mathematical model in nonlinear wave dynamics, essential for understanding the evolution, interaction, and preservation of wave forms in dispersive media. Its applications span various fields, including fluid dynamics, nonlinear optics, and plasma physics, where it plays a crucial role in analyzing solitons and complex wave interactions. In this research, we meticulously implement the and methods to derive solutions for this nonlinear partial differential equation. Our findings reveal new aspects of the equation's behavior, offering deeper insights into nonlinear wave phenomena. The significance of this study lies in its contribution to advancing the understanding of these phenomena and their practical applications in the academic realm. The results underscore the effectiveness of the employed methods, their innovative contributions, and their relevance to applied mathematics.

Gradient estimates for Δpu + Δqu + a|u|s−1u + b|u|l−1u = 0 on a complete Riemannian manifold and applications

In this paper, we use the Nash–Moser iteration method to study the local and global behaviors of non-negative solutions to the nonlinear elliptic equation [Formula: see text] defined on a complete Riemannian manifold [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] are constants and [Formula: see text], with [Formula: see text], is the usual [Formula: see text]-Laplace operator. Under some assumptions on [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], we derive gradient estimates and Liouville-type theorems for non-negative solutions to the above equation. In particular, we show that, if [Formula: see text] is a non-negative entire solution to [Formula: see text] ([Formula: see text]) on a complete non-compact Riemannian manifold [Formula: see text] with non-negative Ricci curvature and [Formula: see text], and [Formula: see text] where [Formula: see text] then [Formula: see text] is a trivial constant solution.

Multi-scale CLEAN for Fourier-based hard X-ray solar imaging

Abstract Multi-scale deconvolution is an ill-posed inverse problem in imaging, with applications ranging from microscopy, through medical imaging, to astronomical remote sensing. In the case of high-energy space telescopes, multi-scale deconvolution algorithms need to account for the peculiar property of native measurements, which are sparse samples of the Fourier transform of the incoming radiation. The present paper proposes a multi-scale version of CLEAN, which is the most popular iterative deconvolution method in Fourier-based astronomical imaging. Using synthetic data generated according to a simulated but realistic source configuration, we show that this multi-scale version of CLEAN performs better than the original one in terms of accuracy, photometry, and regularization. Further, the application to a data set measured by the NASA Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) shows the ability of multi-scale CLEAN to reconstruct rather complex flaring topographies.

Heteronuclear Glycinate Complexes of Fe(II), Fe(III) and Co(II), Their Model Parameters

The system Fe(II)–Fe(III)–Co(II)–Gly–H2O was studied by the Clark–Nikolsky oxidation potential method at a temperature of 298.15 K, ionic strength of the solution [Na(H)CIO4] I = 1.0 mol/l. The formation of mononuclear and heteronuclear compounds of various compositions in the system was found: [FeHL(H2O)5]3+, [Fe(HL)2(H2O)4]3+, [Fe(HL)(OH)(H2O)4]2+, [СoL(H2O)5]+, [FeIIIСoIIL(H2O)11]4+, [FeIIIСoII(L)2(H2O)10]3+, [FeL(H2O)5]+, [Fe(HL)2(H2O)4]2+, [Fe(HL)(OH)(H2O)4]+ and [Fe(HL)(OH)2(H2O)3]0. The stability and model parameters of coordination compounds were calculated by the Yusupov oxidation function iteration method, and their coordination compound distribution diagrams were constructed. It was found that the heteronuclear complex [FeIIIСoIIL(H2O)11]4+ is the most stable, with an accumulation degree of 99.50%, and exists up to pH = 9.5.

Preconditioned inexact fixed point iteration method for solving tensor absolute value equation

Preconditioned inexact fixed point iteration method for solving tensor absolute value equation

THE FIXED POINT ITERATIONS METHOD FOR THE CONTRACTION MAPPING

This paper studies the fixed point iterations method that considers the contraction mapping and one of its applications for the parameter re-estimation of the normal (Gaussian) distribution, where the presence of outliers is considered. In order to study the observed method Banach’s fixed point theorem is presented, where it is shown that the contraction property is directly related to the Jacobian of the observed mapping. Furthermore, convergence analysis is conducted in order to estimate the convergence order of the observed model.

Partial volume correction for Lu-177-PSMA SPECT.

The limited spatial resolution in SPECT images leads to partial volume effect (PVE), degrading the subsequent dosimetric accuracy. We aim to quantitatively evaluate PVE and partial volume corrections (PVC), i.e., recovery coefficient (RC)-PVC (RC-PVC), reblurred Van-Cittert (RVC) and iterative Yang (IY), in 177Lu-PSMA-617 SPECT images. We employed a geometrical cylindrical phantom containing five spheres (diameters ranging from 20 to 40mm) and 40 XCAT phantoms with various anatomical variations and activity distributions. SIMIND Monte Carlo code was used to generate realistic noisy projections. In the clinical study, sequential quantitative SPECT/CT imaging at 4 time-points post 177Lu-PSMA-617 injections were analyzed for 10 patients. Iterative statistical reconstruction methods were used for reconstruction with attenuation, scatter and geometrical collimator detector response corrections, followed by post-filters. The RC-curves were fit based on the geometrical phantom study and applied for XCAT phantom and clinical study in RC-PVC. Matched and 0.5-2.0 voxels (2.54-10.16mm) mismatched sphere masks were deployed in IY. The coefficient of variation (CoV) was measured on a uniform background on the geometrical phantom. RCs of spheres and mean absolute activity error (MAE) of kidneys and tumors were evaluated in simulation data, while the activity difference was evaluated in clinical data before and after PVC. In the simulation study, the spheres experienced significant PVE, i.e., 0.26 RC and 0.70 RC for the 20mm and 40mm spheres, respectively. RVC and IY improved the RC of the 20mm sphere to 0.37 and 0.75 and RC of the 40mm sphere to 0.96 and 1.04. Mismatch in mask increased the activity error for all spheres in IY. RVC increased noise and caused Gibbs ringing artifacts. For XCAT phantoms, both RVC and IY performed comparably and were superior to RC-PVC in reducing the MAE of the kidneys. However, IY and RC-PVC outperformed RVC for tumors. The XCAT phantom study and clinical study showed a similar trend in the kidney and tumor activity differences between non-PVC and PVC. PVE greatly impacts activity quantification, especially for small objects. All PVC methods improve the quantification accuracy in 177Lu-PSMA SPECT.

A Generalised Difference Equation and Its Dynamics and Solutions

Rational difference equations have a wide range of applications in various fields of science. To illustrate, the equation xn+1=a+bxnc+dxn,n=0,1,..., known as the Riccati difference equation, has been applied in the field of optics. In this study, the global asymptotic stability of the difference equation xn+1=Axn−2k+j+1B+Cxn−(k+j)xn−2k+j+1,n=0,1,..., is proved. The solutions of this difference equation are obtained by applying the standard iteration method, and the periodicity of these solutions is determined. Furthermore, this difference equation represents a generalisation of the results obtained in previous studies.

Iterative Methods for Nonlinear Systems Applied to Grid Impedance Estimation: A Comparative Study

In this work, a comparative study is presented to evaluate the performance of iterative methods to solve nonlinear systems applied to grid impedance estimation. The iterative methods of Newton-Raphson, Potra-Pták, and Chun were embedded in the control system of a three-phase inverter supported by a photovoltaic plant connected to the grid. The adopted impedance estimation technique consists of successive variation of the power injected into the grid, more precisely, at three different levels. The voltage and current amplitudes at the point of common coupling~(PCC) are monitored and serve as input for the iterative methods, which, after processing them, provide an estimate of the grid impedance. To compare the performance between the methods, the following merit figures were listed: execution time, number of iterations required to deliver the estimates, percentage error, efficiency index, computational efficiency, computational efficiency index, and stability of the iterative method. The results presented were obtained through real-time simulations. From that, it was possible to conclude about the method with the best performance, thus contributing to greater assertiveness on the part of designers when choosing the most efficient iterative method to be embedded in a microcontroller for grid impedance estimation purposes.

Advancing nonlinear problem solutions: Newton-like iterative techniques for varied convergence orders using homotopy perturbation

Abstract Iterative methods are crucial in numerical analysis for approximating solutions to nonlinear problems encountered in various fields, such as biomathematics, thermodynamics, chemical engineering, and fluid mechanics. This paper introduces innovative iterative methods of convergence orders three and six to eight, developed using the homotopy perturbation technique (HPT). We address the limitations of traditional derivative-based methods by incorporating divided differences, resulting in derivative-free schemes with optimal convergence properties. We present four newly designed algorithms (HPM1, HPM2, HPM3, and HPM4) and provide a comprehensive convergence analysis. Numerical simulations and basins of attraction demonstrate the superior performance of the proposed methods compared to existing methods of the same order. The proposed methods exhibit faster convergence and larger basins of attraction, making them highly effective for solving nonlinear equations. Our new numerical methods rely on approximations and iterative processes, with each small step bringing us closer to the exact solution. Exploring these numerical methods encourages us to apply mathematics to solve challenging problems in physics, engineering, finance, and more.

Machine-learning-aided biochar production from aquatic biomass

AbstractModeling hydrothermal carbonization (HTC) and pyrolysis carbonization (PLC) for the conversion of biomass into high-quality biochar for various applications shows promise. Unlike the extensive modeling studies on lignocellulosic biomass, research on aquatic biomass (AB) had not been reported until now. In this study, we compiled 586 data points from existing literature and trained five tree-based models to predict the yields of hydrochar and pyrochar and their properties, including nitrogen recovery degree, energy density, energy recovery degree, and residual sulfur degree, based on 10 feedstock and process parameters. The random forest regression (RFR) model demonstrated the highest predictive accuracy among these models. It achieved R2 values ranging from 0.89 to 0.98 for hydrochar yield, nitrogen recovery degree of hydrochar, energy recovery degree of hydrochar, and residual sulfur degree of hydrochar. The extreme gradient boosting (XGB) model also showed exemplary performance, with R2 values between 0.84 and 0.94 for energy density of hydrochar, pyrochar yield, and nitrogen recovery degree of pyrochar. Results on feature importance highlighted that, beyond the well-documented impact of process parameters, the properties of biochar were significantly influenced by the elemental compositions, such as nitrogen and sulfur contents of the feedstock. The relationship between these factors was further elucidated using partial dependence plots. Finally, we used RFR model for hydrochar yield and XGB model for pyrochar yield as examples, to test generalization ability of developed models with new data, further explaining their application methods. Overall, this study provided valuable insights into predicting and understanding the HTC and PLC processes of AB to produce high-quality biochar for various applications using low resources and time costs. Besides, we presented an iterative learning application method where the developed models demonstrated exceptionally high performance with new data. This method is highly versatile and can be adopted across various directions in the field of machine learning. Graphical Abstract

Inertial Bilevel Variational Monotone Inclusion Problem in Banach Spaces

In this paper, we study accelerated Halpern-type iterative method for finding zero solution of sum of two monotone operators which solves variational inequality problem of inverse strongly monotone mapping in 2-uniformly convex and uniformly smooth real Banach spaces. The strong convergence of our proposed method is establish under some standard conditions imposed on parameters. Our theorems generalize many recently announced results in the literature.

A Parallel Monte Carlo Algorithm for the Life Cycle Asset Allocation Problem

Life cycle asset allocation is a crucial aspect of financial planning, especially for pension funds. Traditional methods often face challenges in computational efficiency and applicability to different market conditions. This study aimed to innovatively transplant an algorithm from reinforcement learning that enhances the efficiency and accuracy of life cycle asset allocation. We synergized tabular methods with Monte Carlo simulations to solve the pension problem. This algorithm was designed to correspond states in reinforcement learning to key variables in the pension model: wealth, labor income, consumption level, and proportion of risky assets. Additionally, we used cleaned and modeled survey data from Chinese consumers to validate the model’s optimal decision-making in the Chinese market. Furthermore, we optimized the algorithm using parallel computing to significantly reduce computation time. The proposed algorithm demonstrated superior efficiency compared to the traditional value iteration method. Serial execution of our algorithm took 29.88 min, while parallel execution reduced this to 1.42 min, compared to the 41.15 min required by the value iteration method. These innovations suggest significant potential for improving pension fund management strategies, particularly in the context of the Chinese market.

A Novel Approach for Time-Local Fractional Solutions of Certain Nonlinear Partial Differential Equations in Fractal Dimension

Time-local fractional approaches for nonlinear partial differential equations in fractal dimensions are essential for capturing the complex, irregular behaviors found in fractal systems. In this paper, a new modification of the local fractional Laplace variational iteration method (MLFLVIM) for obtaining analytical approximate solutions to the fractional gas dynamics equation, fractional Stefan equation, and fractional Newell-Whitehead-Segel equation within the context of fractal time space is presented. The proposed method (MLFLVIM) elegantly combines the local fractional Laplace transform (LFLT) with modified variational iteration method. Specifically, we first apply the (LFLT) to the given local fractional PDEs, yielding a transformed system of equations. We then apply modified variational iteration to this system. Finally, we use the inverse of (LFLT) to obtain the desired solution. To demonstrate the effectiveness of this approach, we implement it on three numerical physical problems. The results show that the (MLFLVIM) can successfully handle these nonlinear LFPDEs and provide accurate analytical approximation solutions.

Efficient Multiplicative Calculus-Based Iterative Scheme for Nonlinear Engineering Applications

It is essential to solve nonlinear equations in engineering, where accuracy and precision are critical. In this paper, a novel family of iterative methods for finding the simple roots of nonlinear equations based on multiplicative calculus is introduced. Based on theoretical research, a novel family of simple root-finding schemes based on multiplicative calculus has been devised, with a convergence order of seven. The symmetry in the pie graph of the convergence–divergence areas demonstrates that the method is stable and consistent when dealing with nonlinear engineering problems. An extensive examination of the numerical results of the engineering applications is presented in order to assess the effectiveness, stability, and consistency of the recently established method in comparison to current methods. The analysis includes the total number of functions and derivative evaluations per iteration, elapsed time, residual errors, local computational order of convergence, and error graphs, which demonstrate our method’s better convergence behavior when compared to other approaches.

The randomized circumcentered-reflection iteration method for solving consistent linear equations

The randomized circumcentered-reflection iteration method for solving consistent linear equations

Enhanced Ninth-Order Memory-Based Iterative Technique for Efficiently Solving Nonlinear Equations

In this article, we present a novel three-step with-memory iterative method for solving nonlinear equations. We have improved the convergence order of a well-known optimal eighth-order iterative method by converting it into a with-memory version. The Hermite interpolating polynomial is utilized to compute a self-accelerating parameter that improves the convergence order. The proposed uni-parametric with-memory iterative method improves its R-order of convergence from 8 to 8.8989. Additionally, no more function evaluations are required to achieve this improvement in convergence order. Furthermore, the efficiency index has increased from 1.6818 to 1.7272. The proposed method is shown to be more effective than some well-known existing methods, as shown by extensive numerical testing on a variety of problems.