Convergence Of Iterative Methods Research Articles

Increasing in three units the order of convergence of iterative methods for solving nonlinear systems

In this paper, we present a technique aimed at the implementation of high accuracy iterative procedures in nonlinear systems. The main result of this research is a general proof that guarantees an increase in the order of convergence to order p+3 units for any method of order p. This methodology has as its main objective not only this increase, but also the optimization in the use of resources by requiring only one additional functional evaluation per iteration, thus keeping a tight control over the computational cost. Although there are previous proposals in the literature that approach this goal, in this work we consider such approaches as particular cases of our general proposal. This approach encompasses many methods described by other authors and can be applied to any p-order scheme to increase it by three units. Numerical evaluations, supported by a rigorous theoretical framework, validate the robustness of the proposal, highlighting its advantage over conventional techniques and demonstrating its effectiveness in the face of numerical challenges of considerable magnitude. Summing up, with this research, we offer a substantial contribution to the field of numerical methods, proposing a paradigm that consistently enhances the convergence order of iterative methods.

A Globally Convergent Iterative Method for Matrix Sign Function and Its Application for Determining the Eigenvalues of a Matrix Pencil

In this research article, we propose a new matrix iterative method with a convergence order of five for computing the sign of a complex matrix by examining the different patterns and symmetry of existing methods. Analysis of the convergence of the method was explored on a global scale, and attraction basins were demonstrated. In addition to this, the asymptotic stability of the scheme was explored.Then, an algorithm for determing thegeneralized eigenvalues for the case of regular matrix pencils was investigated using the matrix sign computation. We performed a series of numerical experiments using numerous matrices to confirm the usefulness and superiority of the proposed method.

Operator Smith Algorithm for Coupled Stein Equations from Jump Control Systems

Consider a class of coupled Stein equations arising from jump control systems. An operator Smith algorithm is proposed for calculating the solution of the system. Convergence of the algorithm is established under certain conditions. For large-scale systems, the operator Smith algorithm is extended to a low-rank structured format, and the error of the algorithm is analyzed. Numerical experiments demonstrate that the operator Smith iteration outperforms existing linearly convergent iterative methods in terms of computation time and accuracy. The low-rank structured iterative format is highly effective in approximating the solutions of large-scale structured problems.

Optimal quantum control of charging quantum batteries

Quantum control allows us to address the problem of engineering quantum dynamics for special purposes. While recently the field of quantum batteries has attracted much attention, optimization of their charging has not benefited from the quantum control methods. Here we fill this gap by using an optimization method. We apply for the first time the convergent iterative method for the control of the population of a bipartite quantum system in two cases, starting with a qubit-qubit case. The quantum charger-battery system is considered here, where the energy is pumped into the charger by an external classical electromagnetic field. Secondly, we systematically extend our investigation to a second case involving two harmonic oscillators in the Gaussian regime, presenting an original formulation of the method. In both cases, the charger is considered to be an open dissipative system, as its interaction with the drive may require a more pronounced exposure to general interaction with environment. A key consideration in our optimization strategy is the practical concern of turning the charging external field on and off. We find that optimizing the pulse shape yields a substantial enhancement in both the power and efficiency of the charging process compared to a sinusoidal drive. The harmonic oscillator configuration of quantum batteries is particularly intriguing, as the optimal driving pulse remains effective regardless of the environmental temperature. This study introduces a novel approach to quantum battery charging optimization, opening avenues for enhanced performance in real-world applications.

Non-monotonic and self-adaptive strongly convergent iterative methods for efficiently solving variational inequalities with pseudomonotone operators

Non-monotonic and self-adaptive strongly convergent iterative methods for efficiently solving variational inequalities with pseudomonotone operators

Full operator preconditioning and the accuracy of solving linear systems

Abstract Unless special conditions apply, the attempt to solve ill-conditioned systems of linear equations with standard numerical methods leads to uncontrollably high numerical error and often slow convergence of an iterative solver. In many cases, such systems arise from the discretization of operator equations with a large number of discrete variables and the ill-conditioning is tackled by means of preconditioning. A key observation in this paper is the sometimes overlooked fact that while traditional preconditioning effectively accelerates convergence of iterative methods, it generally does not improve the accuracy of the solution. Nonetheless, it is sometimes possible to overcome this barrier: accuracy can be improved significantly if the equation is transformed before discretization, a process we refer to as full operator preconditioning (FOP). We highlight that this principle is already used in various areas, including second kind integral equations and Olver–Townsend’s spectral method. We formulate a sufficient condition under which high accuracy can be obtained by FOP. We illustrate this for a fourth order differential equation which is discretized using finite elements.

ALTERNATING DIRECTION IMPLICIT METHOD FOR POISSON EQUATION WITH INTEGRAL CONDITIONS

In this paper, we investigate the convergence of the Peaceman-Rachford Alternating Direction Implicit method for the system of difference equations, approximating the two-dimensional elliptic equations in rectangular domain with nonlocal integral conditions. The main goal of the paper is the analysis of spectrum structure of difference eigenvalue problem with nonlocal conditions. The convergence of iterative method is proved in the case when the system of eigenvectors is complete. The main results are generalized for the system of difference equations, approximating the differential problem with truncation error O(h4).

A new block preconditioner for weighted Toeplitz regularized least-squares problems

We introduce a new block preconditioner for the solution of weighted Toeplitz regularized least-squares problems written in augmented system form. The proposed preconditioner is obtained based on the new splitting of coefficient matrix which results in an unconditionally convergent stationary iterative method. Spectral analysis of the preconditioned matrix is investigated. In particular, we show that the preconditioned matrix has a very nice eigenvalue distribution which can lead to fast convergence of the preconditioned Krylov subspace methods such as GMRES. Numerical experiments are reported to demonstrate the performance of preconditioner used with GMRES method in the solution of augmented system form of weighted Toeplitz regularized least-squares problems.

Generalized Iterative Method of Order Four with Divided Differences

Numerous applications from diverse disciplines are formulated as an equation or system of equations in abstract spaces such as Euclidean multidimensional, Hilbert, or Banach, to mention a few. Researchers worldwide are developing methodologies to handle the solutions of such equations. A plethora of these equations are not differentiable. These methodologies can also be applied to solve differentiable equations. A particular method is utilized as a sample via which the methodology is described. The same methodology can be used on other methods utilizing inverses of linear operators. The problem with existing approaches on the local convergence of iterative methods is the usage of Taylor expansion series. This way, the convergence is shown but by assuming the existence of high-order derivatives which do not appear on the iterative methods. Moreover, bounds on the error distances that can be computed are not available in advance. Furthermore, the isolation of a solution of the equation is not discussed either. These concerns reduce the applicability of iterative methods and constitute the motivation for developing this article. The novelty of this article is that it positively addresses all these concerns under weaker convergence conditions. Finally, the more important and harder to study semi-local analysis of convergence is presented using majorizing scalar sequences. Experiments are further performed to demonstrate the theory.

A Convergent Iteration Method for 3-D AOA Localization

Using angle-of-arrival (AOA) measurements from several sensors to locate a target in 3-D space is one of the commonly used wireless localization techniques. A challenge of this technique is that the weighted least-squares (WLS) needed for it tend to incur the threshold effect when the noise level is high or when the localization geometry is poor, because its implementation requires the inversion of a matrix whose condition number will be large under such conditions. This paper presents an iterative method that does not require any matrix operation and is proved to be convergent with any initial value. Additionally, it outperforms the WLS-based methods as validated by comprehensive simulation results.

Regularization Solver Guided FISTA for Electrical Impedance Tomography.

Electrical impedance tomography (EIT) is non-destructive monitoring technology that can visualize the conductivity distribution in the observed area. The inverse problem for imaging is characterized by a serious nonlinear and ill-posed nature, which leads to the low spatial resolution of the reconstructions. The iterative algorithm is an effective method to deal with the imaging inverse problem. However, the existing iterative imaging methods have some drawbacks, such as random and subjective initial parameter setting, very time consuming in vast iterations and shape blurring with less high-order information, etc. To solve these problems, this paper proposes a novel fast convergent iteration method for solving the inverse problem and designs an initial guess method based on an adaptive regularization parameter adjustment. This method is named the Regularization Solver Guided Fast Iterative Shrinkage Threshold Algorithm (RS-FISTA). The iterative solution process under the L1-norm regular constraint is derived in the LASSO problem. Meanwhile, the Nesterov accelerator is introduced to accelerate the gradient optimization race in the ISTA method. In order to make the initial guess contain more prior information and be independent of subjective factors such as human experience, a new adaptive regularization weight coefficient selection method is introduced into the initial conjecture of the FISTA iteration as it contains more accurate prior information of the conductivity distribution. The RS-FISTA method is compared with the methods of Landweber, CG, NOSER, Newton-Raphson, ISTA and FISTA, six different distributions with their optimal parameters. The SSIM, RMSE and PSNR of RS-FISTA methods are 0.7253, 3.44 and 37.55, respectively. In the performance test of convergence, the evaluation metrics of this method are relatively stable at 30 iterations. This shows that the proposed method not only has better visualization, but also has fast convergence. It is verified that the RS-FISTA algorithm is the better algorithm for EIT reconstruction from both simulation and physical experiments.

Structure Preprocessing Method for the System of Unclosed Linear Algebraic Equations

The complexity of open linear algebraic equations makes it difficult to obtain analytical solutions, and preprocessing techniques can be applied to coefficient matrices, which has become an effective method to accelerate the convergence of iterative methods. Therefore, it is important to preprocess the structure of open linear algebraic equations to reduce their complexity. Open linear algebraic equations can be divided into symmetric linear equations and asymmetric linear equations. The former is based on 2 × 2. The latter is preprocessed by the improved QMRGCGS method, and the applications of the two methods are analyzed, respectively. The experimental results show that when the step is 500, the pretreatment time of quasi-minimal residual generalized conjugate gradient square 2 method is 34.23 s, that of conjugate gradient square 2 method is 35.14 s, and that of conjugate gradient square method is 45.20 s, providing a new reference method and idea for solving and preprocessing non-closed linear algebraic equations.

Low-rank tensor methods for Markov chains with applications to tumor progression models

Cancer progression can be described by continuous-time Markov chains whose state space grows exponentially in the number of somatic mutations. The age of a tumor at diagnosis is typically unknown. Therefore, the quantity of interest is the time-marginal distribution over all possible genotypes of tumors, defined as the transient distribution integrated over an exponentially distributed observation time. It can be obtained as the solution of a large linear system. However, the sheer size of this system renders classical solvers infeasible. We consider Markov chains whose transition rates are separable functions, allowing for an efficient low-rank tensor representation of the linear system’s operator. Thus we can reduce the computational complexity from exponential to linear. We derive a convergent iterative method using low-rank formats whose result satisfies the normalization constraint of a distribution. We also perform numerical experiments illustrating that the marginal distribution is well approximated with low rank.

Parallel Dynamic Sparse Approximate Inverse Preconditioning Algorithm on GPU

The dynamic sparse approximate inverse (SPAI) preconditioner has proven to be effective in accelerating the convergence of iterative methods for large linear systems. Recently, accelerating it on graphics processing unit (GPU) has attracted considerable attention due to the fact that the cost of constructing the preconditioner is high. However, the existing parallel dynamic SPAI preconditioning algorithms on GPU are usually ineffective because of the out-of-memory error for large matrices. This motivates us to investigate how to accelerate the construction of dynamic SPAI preconditioners on GPU. In this article, we propose an efficient dynamic SPAI preconditioning algorithm on GPU, called GDSPAI. For our proposed GDSPAI, there are the following novelties: (1) a well-known dynamic SPAI preconditioning algorithm is substantially modified to address the main challenges of parallelization on GPU, (2) a parallel framework of constructing the dynamic SPAI preconditioner on GPU is presented on the basis of the modified dynamic SPAI preconditioning algorithm; and (3) each component of the preconditioner is computed in parallel inside a group of threads. Experimental results show that the proposed GDSPAI is effective for large matrices, and outperforms the popular preconditioning algorithms in three public libraries, as well as a recent parallel static SPAI preconditioning algorithm.

PENELITIAN DESAIN MEBEL BERBASIS PANGKALAN DATA DENGAN METODE DIVERGEN KONVERGEN ITERATIF SEBAGAI STRATEGI R&D & MANUFAKTUR PERUSAHAAN (STUDI KASUS: CORPORATE SPECIALISTS, MALAYSIA & HOMELEGANCE, USA)

Furniture design for mass production & export orientation is complex innature, tend to have a lot of revisions, thus require a lot investment in fund,time, and effort. There are 2 research issue: 1) Mistakes in design proposalsoften occurred (size, construction, style, finishing, price), can not match withfactory production capacity, or buyer’s market target. In this case, factoriesare suppliers of research partner 1 (CS), and buyer is research partner 2(Homelegance). Second issue: 2) Designer’s idea often surrounds in designaesthetic alone, disregard the A-Z aspect in supply & demand chain of amass-produced furniture (production, marketing, packing, shipping, etc).This research uses qualitative research model with many case studies, andapplies 3 methods: 1) divergent convergent iterative design method; 2) 2D &3D database from CS; 3) considering US market response fromHomelegance. This research aims to: 1) providing ways and recommendationfor stakeholder to reduce revisions (time & cost saving), 2) producingdesigns with export oriented quality, 3) providing design insights foracademics about furniture design from early to final phase. Research output(furniture samples) are stored in multiple supplier’s warehouse.

Hybrid absorbing scheme based on hyperelliptical layers with non-reflecting boundary conditions in scalar wave equations

Modeling unbounded or semi-infinite media with numerical methods, such as finite elements, requires avoiding that waves pass through the boundaries of the truncated computational domain without reflection. For this reason, in areas such as Geophysics or Acoustics, absorbing layers are used to circumvent this issue. Nevertheless, sizing them is an open problem due to the necessary calibration of their parameters for an adequate performance and to prevent numerical instabilities. Absorbing layers are dependent on both wave frequency and material properties set up inside them and, affect the convergence of iterative methods in inverse problems by the inherent variation of domain properties within the process, especially in the transient regime. On the other hand, hybrid absorbing boundary conditions are the combination of absorbing layers and the application of non-reflecting boundary conditions on the layer boundary with the purpose of increasing their effectiveness. In this work, an analytical approach is presented to size the portion added to the original domain and to determine its damping parameters simultaneously. Hyperelliptical absorbing layers are introduced and, for generating the hybrid absorbing boundary conditions scheme, Sommerfeld or 1st-Higdon boundary conditions are imposed on the outer layer boundary. Therefore, this methodology is adaptive according to the intermediate solutions of the transient optimization process for reducing possible numerical instabilities and spurious information for the inverse problem. The transient equilibrium equations are implemented by using the finite element method and an implicit time integration method. Results are presented to show the potential of the proposal.

Convergence of a Constrained Vector Extrapolation Scheme

Among extrapolation methods, Anderson acceleration (AA) is a popular technique for speeding up convergence of iterative processes toward their limit points. AA proceeds by extrapolating a better approximation of the limit using a weighted combination of previous iterates. Whereas AA was originally developed in the context of nonlinear integral equations, or to accelerate the convergence of iterative methods for solving linear systems, it is also used to extrapolate the solution of nonlinear systems. Simple additional stabilization strategies can be used in this context to control conditioning issues. In this work, we study a constrained vector extrapolation scheme based on an offline version of AA with fixed window size, for solving nonlinear systems arising in optimization problems, where the stabilization strategy consists in bounding the magnitude of the extrapolation weights. We provide explicit convergence bounds for this method and, as a by-product, upper bounds on a constrained version of the Chebyshev problem on polynomials.

A closed-form bound on the asymptotic linear convergence of iterative methods via fixed point analysis

In many iterative optimization methods, fixed-point theory enables the analysis of the convergence rate via the contraction factor associated with the linear approximation of the fixed-point operator. While this factor characterizes the asymptotic linear rate of convergence, it does not explain the non-linear behavior of these algorithms in the non-asymptotic regime. In this letter, we take into account the effect of the first-order approximation error and present a closed-form bound on the convergence in terms of the number of iterations required for the distance between the iterate and the limit point to reach an arbitrarily small fraction of the initial distance. Our bound includes two terms: one corresponds to the number of iterations required for the linearized version of the fixed-point operator and the other corresponds to the overhead associated with the approximation error. With a focus on the convergence in the scalar case, the tightness of the proposed bound is proven for positively quadratic first-order difference equations.

Connection between the existence of a priori estimate for a flux and the convergence of iterative methods for diffusion equation with highly varying coefficients

Abstract An iterative method with the number of iterations independent of the coefficient jumps is proposed for the boundary value problem for a diffusion equation with highly varying coefficient. The method applies one solution of the Poisson equation at each step of iteration. In the present paper we extend the class of domains the iterative method is justified for.

Development of cubically convergent iterative derivative free methods for computing multiple roots

Development of cubically convergent iterative derivative free methods for computing multiple roots