Broyden's Method Research Articles

Derivation and Performance Analysis of Composite Trapezoidal Iterative Methods

The pursuit of more efficient and reliable numerical methods to solve nonlinear systems of equations has long intrigued many researchers. Among these, the Broyden method has stood out since its introduction, serving as a foundational technique from which various derivative methods have evolved. These derivative methods, commonly referred to as Broyden-like iterative methods, often surpass the traditional Broyden method in terms of both the number of iterations required and the computational time needed. This study aimed to develop new Broyden-like methods by incorporating weighted combinations of different quadrature rules. Specifically, the research focused on leveraging the Composite Trapezoidal rule with n=3n=3, and comparing it against the Midpoint, Trapezoidal, and Simpson quadrature rules. By integrating these approaches, three novel methods were formulated. The findings revealed that several of these new methods demonstrated enhanced efficiency and robustness compared to their established counterparts. In a detailed comparative analysis with the classical Broyden method and other improved versions, the Midpoint–Composite Trapezoidal (M<i>T_3</i>) method emerged as the top performer. This method consistently provided superior numerical outcomes across all benchmark problems examined in the study. The results highlight the potential of these new methods to significantly advance the field of numerical analysis, offering more powerful tools for researchers and practitioners dealing with complex nonlinear systems of equations. Through this innovative approach, the study not only broadens the understanding of Broyden-like methods but also sets the stage for further advancements in the development of efficient numerical solutions.

Quadratic functions for efficient load balancing in the terminals of a substation of a three-phase asymmetric network with power loss reduction capabilities

This research addresses the problem of optimal load balancing in terminals of the three-phase substation by proposing three quadratic objective functions. These objective functions are formulated considering active, reactive, and apparent power consumptions aggregated at the terminals of the substation. The proposed formulation belongs to the mixed-integer quadratic models’ family, which can be solved globally with specialized mixed-integer convex tools. To evaluate the effect of load redistribution in the substation terminals, the 15- and 35-bus grids are tested using each of the proposed quadratic functions. In addition, Broyden's unbalanced power flow method is used to determine the extent of power loss reduction and enhancement of voltage profile. Numerical results confirm the effectiveness of the proposed mixed-integer quadratic model in enhancing electrical performance in three-phase asymmetric networks through load balancing at the substation terminals. After solving each quadratic function for the 15-bus grid, power losses were reduced between 12.9624% and 17.2550%, and these reductions were between 5.0771% and 7.7389% in the 35-bus grid.

Prediction of boundary heat flux in mold by inverse problem method

This study utilizes inverse heat transfer techniques to analyze heat flux distribution within a mold. By collecting temperature data from thermocouples, a two-dimensional heat transfer model of the mold’s cross section is developed to investigate the transverse heat flux behavior, temperature distribution, relationship between heat flux and height, and three-dimensional heat flux within the mold. The direct problem model is solved using the finite volume method, while the inverse problem is tackled with the Levenberg-Marquardt Method (LMM) in conjunction with the Broyden method (BM). The model’s validity is confirmed through experimentation, which includes assessing the impact of initial parameter estimations, thermocouple placements and quantities, and measurement inaccuracies.

A New Iterative Broyden Legendre Wavelet Galerkin FEM Applied to Unsteady State Model of Two-Dimensional Elliptic Fin

Abstract The novelty of this paper is the investigation of numerical study of a mathematical model which deals with time dependent heat flow in elliptic fin ( dry, wet and partially wet). In this paper, we developed a nonlinear model of second-order heat equations in unsteady state condition. A new iterative Broyden Legendre Wavelet Galerkin Finite Element Method is used for the solution. The central difference approximation used for discretization of second order derivatives and then utilization of Hadamard, Khatri Rao and Face splitting matrices product with Legendre Wavelet Galerkin Method transfers our main problem in to system of non-linear algebraic equations. The iterative Broyden Method provides the solution for this system. In a particular case, present solution is compared with the exact solution and is approximately the same. Effect of different parameters such as Biot number, Latent heat, Kirpichev number, Fin thickness, μ, η and ξ on the temperature distribution are discussed in detail.

On mathematical modelling of measles disease via collocation approach.

Measles, a highly contagious viral disease, spreads primarily through respiratory droplets and can result in severe complications, often proving fatal, especially in children. In this article, we propose an algorithm to solve a system of fractional nonlinear equations that model the measles disease. We employ a fractional approach by using the Caputo operator and validate the model's by applying the Schauder and Banach fixed-point theory. The fractional derivatives, which constitute an essential part of the model can be treated precisely by using the Broyden and Haar wavelet collocation methods (HWCM). Furthermore, we evaluate the system's stability by implementing the Ulam-Hyers approach. The model takes into account multiple factors that influence virus transmission, and the HWCM offers an effective and precise solution for understanding insights into transmission dynamics through the use of fractional derivatives. We present the graphical results, which offer a comprehensive and invaluable perspective on how various parameters and fractional orders influence the behaviours of these compartments within the model. The study emphasizes the importance of modern techniques in understanding measles outbreaks, suggesting the methodology's applicability to various mathematical models. Simulations conducted by using MATLAB R2022a software demonstrate practical implementation, with the potential for extension to higher degrees with minor modifications. The simulation's findings clearly show the efficiency of the proposed approach and its application to further extend the field of mathematical modelling for infectious illnesses.

Investigation of combustion model via the local collocation technique based on moving Taylor polynomial (MTP) approximation/domain decomposition method with error analysis

In this paper, we develop a new meshless numerical procedure for simulating the combustion model. To that end, we employ a local meshless collocation method according to the moving Taylor polynomial (MTP) approximation. The space derivative is approximated by using the local approach and then the Crank–Nicolson algorithm is utilized to approximate the time derivative. The stability and convergence of the time-discrete formulation are discussed, analytically and numerically. The Broyden method is applied to solve this nonlinear system. Since the size of the physical domain is large, we employ the non-overlapping domain decomposition method (DDM) to obtain a faster numerical algorithm. The local meshless approaches are efficient numerical techniques to simulate models in the fluid flow. The obtained results show that the proposed numerical formulation has efficient results for solving this mathematical model.

Research on the Secondary Motion of Engine Piston Considering the Transport of Lubricating Oil

<div>At present, it is generally considered in the analysis of the secondary motion of engine piston that the piston skirt–cylinder liner friction pair is fully lubricated in an engine operating cycle. However, in practice, when the piston moves upward, the amount of lubricating oil at the inlet may not ensure that the friction pair is fully lubricated. In this article, the secondary motion of piston is studied when the transport of lubricating oil is considered to determine the lubrication condition of piston skirt–cylinder liner friction pair. The secondary motion of piston is solved based on the combined piston motion model, hydrodynamic lubrication model, asperity contact model, and lubricating oil flow model. The secondary motion equation of piston is solved by the Broyden method. The hydrodynamic lubrication equation is solved by the finite difference method. The asperity contact between piston skirt and cylinder liner is calculated by the Greenwood model. The flow of lubricating oil is analyzed based on the theory of fluid mechanics. The results indicate that, when the actual transport of lubricating oil is considered to determine the lubrication condition of piston skirt–cylinder liner friction pair, the secondary motion of piston is remarkably different from that in which the flooded lubrication is assumed in an engine operating cycle. Therefore, it is helpful to improve the accuracy and make the analysis closer to the actual engine operating situation that the transport of lubricating oil is considered in the analysis of the secondary motion of engine piston.</div>

Broyden's method assisted differential evolution (BMADE) based selective harmonic elimination in a cascaded H‐bridge multilevel inverter

AbstractThis paper implements Broyden's Method Assisted Differential Evolution (BMADE) algorithm for eliminating third and fifth harmonics in the output voltage of 7‐level cascaded H‐Bridge multilevel inverter (CHB‐MLI) using selective harmonic elimination technique. Calculations of switching angles are done for various modulation index values (M). Initial guess for Broyden's method is calculated by Differential Evolution (DE). Exact value of switching angles is then calculated by Broyden's method using initial guess obtained from DE. Code of BMADE algorithm for obtaining the required switching angles is written in MATLAB. Results obtained with BMADE algorithm are compared with various other techniques that is, Genetic Algorithm (GA), DE, and Whale Optimization Algorithm (WOA). BMADE is able to successfully eliminate third and fifth order harmonics in the output voltage. Along with removing the desired harmonics, BMADE also provides lower THD than GA, DE, and WOA for certain range of M. The results also showed that BMADE is faster than GA, DE, and WOA as the execution time of BMADE is lesser than GA, DE, and WOA. Moreover, laboratory experiments are used to test the simulation's findings. The experimental results support the simulation results.

Solving the Power Flow Problem in Bipolar DC Asymmetric Distribution Networks Using Broyden's Method.

This research addresses the power flow analysis in bipolar asymmetric direct current (DC) networks by applying Broyden's numerical method. This general successive approximations method allows for a simple Newton-based recursive formula to reach the roots of multiple nonlinear equations. The main advantage of Broyden's approach is its simple but efficient structure which can be applied to real complex nonlinear equations.The power flow problem in bipolar DC networks is still challenging, as multiple operating options must be considered, e.g., the possibility of having a solidly grounded or floating neutral wire. The main goal of this research is to contribute with a generalization of Broyden's method for the power flow solution in bipolar DC networks, with the main advantage that, under well-defined conditions, this is a numerical method equivalent to the matricial backward/forward power flow, which is equivalent to the successive approximations power flow method. Numerical results in the 21-, 33-, and 85-bus grids while considering two connections for the neutral wire (i.e., solidly grounded at any node or floating) show the effectiveness of Broyden's method in the power flow solution for bipolar asymmetric DC networks. All numerical simulations were carried out in the MATLAB programming environment.

Inverse heat transfer prediction of the thermal parameters of tumors during cryosurgery

An inverse heat transfer approach for predicting the moving freezing front and the temperature distribution inside tumors during cryosurgery is presented. By recording the time-varying temperature at the surface of the diseased tissues, the inverse method is able to estimate simultaneously the blood perfusion rate and the metabolic heat generation of the tumor. Once these thermal properties are determined, the temperature field and the moving freezing front are predicted with a direct method. The direct problem is solved with the one-dimension Pennes bioheat equation and the enthalpy method. The inverse model rests on the Levenberg-Marquardt Method (LMM) combined to the Broyden Method (BM). The following effects on the predictions of the inverse method are investigated: (i) the initialization of the unknown thermal properties; (ii) the thermal properties of the diseased tissues; (iii) the duration of cryosurgery; (iv) the temperature of the cryoprobe; (v) the noise level on the recorded temperatures; and (vi) the total number of recorded data. It is shown that the proposed inverse method remains accurate and stable for all cases investigated. Recommendations are made for the initialization of the parameters of the inverse method and for the total number of measurements.

Error Analysis of Newly Developed Numerical Methods for Solving System of Nonlinear Equations

Solution methods are the major tools in research especially in the area of applied mathematics. This is because, most real-life problems result into system of nonlinear equations, and the right solution method with less computational error is required to obtain an approximated solution to these system of nonlinear equations. The introduction of the Broyden method set the groundwork for the development of several other methods, many of which are referred to as Broyden-like approaches by various researchers. In most cases, these methods have proven to be superior to the original classical Broyden method in terms of the number of iterations and CPU time needed to acquire a solution. Using the solutions of the traditional Broyden method as a point of comparison, this study aimed to examine the error associated with two newly developed numerical methods, the Trapezoidal-Simpson-3/8 (TS-3/8) and Midpoint-Simpson-3/8 (MS-3/8) methods. Results gathered after applying the classical Broyden, MS-3/8 and TS-3/8 methods to solve some bench-mark problems involving system of nonlinear equations and estimating the errors associated with each of the methods considered in the study, using the formula of the approximate error, showed that the error associated with the MS-3/8 method was minimal compared to that of the Broyden and the TS-3/8 methods. At the end of the study, the results gathered suggested the MS-3/8 technique as the most highly advised numerical approach among the other methods. This means that, MS-3/8 method is a more accurate solution method for solving system of nonlinear equations considering the results in this paper.

Analysis of GMRES for Low‐Rank and Small‐Norm Perturbations of the Identity Matrix

AbstractIn many applications, linear systems arise where the coefficient matrix takes the special form I + K + E, where I is the identity matrix of dimension n, rank(K) = p ≪ n, and ∥E∥ ≤ ϵ < 1. GMRES convergence rates for linear systems with coefficient matrices of the forms I+K and I+E are guaranteed by well‐known theory, but only relatively weak convergence bounds specific to matrices of the form I + K + E currently exist. In this paper, we explore the convergence properties of linear systems with such coefficient matrices by considering the pseudospectrum of I + K. We derive a bound for the GMRES residual in terms of ϵ when approximately solving the linear system (I + K + E)x = b and identify the eigenvalues of I + K that are sensitive to perturbation. In particular, while a clustered spectrum away from the origin is often a good indicator of fast GMRES convergence, that convergence may be slow when some of those eigenvalues are ill‐conditioned. We show there can be at most 2p eigenvalues of I + K that are sensitive to small perturbations. We present numerical results when using GMRES to solve a sequence of linear systems of the form (I + Kj + Ej)xj = bj that arise from the application of Broyden's method to solve a nonlinear partial differential equation.

AI-Enhanced Performance Evaluation of Python, MATLAB, and Scilab for Solving Nonlinear Systems of Equations: A Comparative Study Using the Broyden Method

AI-Enhanced Performance Evaluation of Python, MATLAB, and Scilab for Solving Nonlinear Systems of Equations: A Comparative Study Using the Broyden Method

Solving nonlinear equations with a direct Broyden method and its acceleration

Solving nonlinear equations with a direct Broyden method and its acceleration

On the effect of nonlinearity and Jacobian initialization on the convergence of the generalized Broyden quasi‐Newton method

AbstractThis article investigates two aspects of the generalized Broyden quasi‐Newton method that have a major impact on its convergence: the initial approximation of the Jacobian and the presence of nonlinearities in the secant conditions. After reformulating the common representation of generalized Broyden, a straightforward interpretation is given. This leads to a natural extension of the method in which an application‐dependent physics‐based surrogate model is used as initial approximation of the (inverse) Jacobian. A carefully chosen surrogate has the potential to greatly reduce the required number of iterations. The behavior of generalized Broyden depends strongly on the parameter that determines how many secant conditions are satisfied by the Jacobian approximation. Respecting all secant conditions reduces it to Anderson acceleration; a single one to Broyden's original method. An analysis demonstrates that these two variants behave very differently when nonlinearities are present in the secant conditions: they are ignored by Broyden, but can destabilize Anderson. On the other hand, the analysis shows that Broyden tends to neglect small linear information, possibly reducing convergence speed. To mitigate stability problems with Anderson acceleration, a practical method to detect and remove nonlinear secant information is introduced next. Finally, we solve a steady free‐surface‐flow problem using several generalized Broyden variants, testing the influence of the surrogate, the nonlinearities and the combination thereof. The results agree with the theoretical predictions, showing large differences in convergence behavior. Furthermore, the proposed method effectively negates the problems related to nonlinearities in this case.

An efficient algorithm based on Broyden's and generalized-α method for large scale flexible multi-body systems

This paper designs a novel and efficient numerical strategy based on generalized-α algorithm and Broyden's method to solve dynamic equations of large scale flexible multi-body system. The dynamic models are established in view of Absolute Nodal Coordinate Formulation(ANCF) and the clearance joint model is described by nonlinear contact force model and modified Coulomb's friction model. The proposed strategy only needs a given initial Jacobian matrix and then the numerical solutions can be obtained by using Sherman-Morrison-Woodburg theorem, that can greatly increase the computational efficiency as a result of avoiding solving of the Jacobian matrix in each iteration. Newton-Raphson iteration and Trust-Region-Dogleg algorithm, as two typical numerical solutions for nonlinear dynamic equations of large scale flexible multi-body systems, are selected to be compared the effectiveness and efficiency with the proposed numerical method. Numerical results show that the proposed method is accurate and robust, and can improve the computational efficiency by several or even a dozen times than the other two common methods for large scale flexible multi-body systems including flexible double pendulum model, rigid and rigid-flexible coupling solar array systems.

A Broyden-based algorithm for multi-objective local-search optimization

In multi-objective optimization, the direction vectors in the objective space that improve the current non-dominated set are named improvement directions. The Improvement Direction Mapping (IDM) methods apply a spatial transformation to map improvement directions in the objective space to search for directions in the variable space. Jacobian based transformations can perform the mapping, however, they require the analytic expressions of the objectives and their derivatives. Hence, they serve as a reference in academic problems but are impractical in real-world applications. Furthermore, they are commonly ill-conditioned and under-determined; consequently, they deviate the search as the current non-dominated set approaches to the Pareto set. This work proposes a solution via an iterative updating based on the Broyden method. The proposal promotes parallelism between the displacement of the non-dominated solutions in the objective space and improvement directions delivered by Chebyshev scalarizing functions. Compared to other transformations reported in the specialized literature, the proposed method demonstrates a better conditioning that provokes to err at a lesser extent in providing successful search directions. This impacts the algorithm performance, mainly in locations close to the Pareto set. These advantages are demonstrated using benchmark functions and metrics.

Real Time Estimator to Perform Targeted Biopsies With a Free-Wrist Robot Despite Large Deformations of the Insertion Orifice.

In the context of keyhole surgery, and more particularly of uterine biopsy, the fine automatic movements of a surgical instrument held by a robot with 3 active DOF’s require an exact knowledge of the point of rotation of the instrument. However, this center of rotation is not fixed and moves during an examination. This paper deals with a new method of detecting and updating the interaction matrix linking the movements of the robot with the surgical instrument. This is based on the method of updating the Jacobian matrix which is named the “Broyden method”. It is able to take into account body tissue deformations in real time in order to improve the pointing task for automatic movements of a surgical instrument in an unknown environment.

Performance improvement of an oil-lubricated journal bearing using bionic-textures fused micro-pockets

Abstract This article investigates the static and dynamic performances of bionic-textures fused micro-pocketed journal bearing considering cavitation. The new bionic textures placed inside the micro-pocket towards the trailing side has been conceived inspired from the skins of freshwater fish. The mass conservation satisfying partial differential lubrication equations for the static and dynamic pressures are discretized using finite volume method (FVM) followed by simultaneous solution of the system equations using the Fischer-Burmeister-Newton-Schur (FBNS) and the Broyden's method with Sherman-Morrison formula. The minimum film thickness, friction force, temperature rise, dynamic coefficients, effective damping and stiffness, critical mass and system characteristic roots have been explored with new conceived bores and the numerical results have also been compared with conventional plain and conventional pocketed bore bearings. Significant static and dynamic performance improvements have been found in presence of bionic textures fused micro-pocketed bores in comparison to conventional bores.

An inverse free Broyden's method for solving equations

An inverse free Broyden's method for solving equations