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This paper presents a novel approach to solve nonlinear second-order Volterra integro-differential equations (VIDEs) using the Iterative Collocation Method. We focus on specific VIDEs with smooth functions, aiming to obtain numerical solutions without the need to solve algebraic equation systems. Our method, based on Lagrange polynomials within a spline space, offers direct implementation, high convergence order, and efficient determination of approximation solution coefficients. Through theoretical analysis and numerical experiments, we illustrate the efficacy and accuracy of our approach, contributing to the improvement of numerical techniques for solving nonlinear VIDEs.

In this paper we consider five types of parallel preconditioners for solving large sparse nonsymmetric linear systems on the CRAY-T3E. They are ILU(0) in the wavefront ordering, ILU(0) in the multi-coloring ordering, SSOR in the wavefront ordering, the SPAI(SParse Approximate Inverse) preconditioner, and finally Multi-color Block SOR preconditioner. The ILU(0) is known to be robust and the wavefront ordering naturally exploits the parallelism but has a limited speedup due to the nonuniform lengths of the wavefronts. Multi-coloring is an efficient way of introducing the parallelism of order(N), where N is the order of the matrix but the convergence rate often deteriorates. The SPAI type preconditioner is inherently parallel and is gaining popularity. Finally, for the 5-point Laplacian matrix SOR method is known to have a nondeteriorating rate of convergence when the multi-coloring order is adopted. Also, Block SOR is expected to incur less communication overheads in a message-passing machine. Hence, Multi-Color Block SOR method is expected to have a good performance. Experiments were conducted for the Finite Difference discretizations of two problems with various meshsizes varying up to 1024×1024. MPI library was used for interprocess communications. The results show that ILU(0) in the multi-coloring ordering gives the best performance.

We give a practical and provably good Monte Carlo algorithm for approximating center points. Let P be a set of n points in R^d. A point c∈R^d is a β-center point of P if every closed halfspace containing c contains at least βn points of P. Every point set has a 1/(d+1)-center point; our algorithm finds an Ω(1/d²)-center point with high probability. Our algorithm has a small constant factor and is the first approximate center point algorithm whose complexity is subexponential in d. Moreover, it can be optimally parallelized to require O(log²d log log n) time. Our algorithm has been used in mesh partitioning methods and can be used in the construction of high breakdown estimators for multivariate datasets in statistics. It has the potential to improve results in practice for constructing weak ∊-nets. We derive a variant of our algorithm whose time bound is fully polynomial in d and linear in n, and show how to combine our approach with previous techniques to compute high quality center points more quickly.

We show the existence of strong solutions for a fluid model with Korteweg tensor, which is obtained as limit of two iterative linear schemes. The different unknowns are sequentially decoupled in the first scheme and in parallel form in the second one. In both cases, the whole sequences are bounded in strong norms and convergent towards the strong solution of the system, by using a generalization of Banach's fixed point theorem. Moreover, we explicit a priori and a posteriori error estimates (respect to the weak norms), which let us to compare both schemes.

The convergence behavior of the Krylov subspace iterative solvers towards the systems with the 3D acoustical BEM is investigated through numerical experiments. The fast multipole BEM, which is an efficient BEM based on the fast multipole method, is used for solving problems with up to about 100,000 DOF. It is verified that the convergence behavior of solvers is much affected by the formulation of the BEM (singular, hypersingular, and Burton-Miller formulation), the complexity of the shape of the problem, and the sound absorption property of the boundaries. In BiCG-like solvers, GPBiCG and BiCGStab2 have more stable convergence than others, and these solvers are useful when solving interior problems in basic singular formulation. When solving exterior problems with greatly complex shape in Burton-Miller formulation, all solvers hardly converge without preconditioning, whereas the convergence behavior is much improved with ILU-type preconditioning. In these cases GMRes is the fastest, whereas CGS is one of the good choices, when taken into account the difficulty of determining the timing of restart for GMRes. As for calculation for rigid thin objects in hypersingular formulation, much more rapid convergence is observed than ordinary interior/exterior problems, especially using BiCG-like solvers.

Krylov subspace iterative solvers are applied to large-scale finite element sound-field analyses of architectural rooms. First, convergence behaviors are compared among four iterative solvers. Results show that the Conjugate Orthogonal Conjugate Gradient (COCG) method offers the best characteristics for finite-element (FE) analysis from the viewpoint of robustness of convergence and computation time. Two investigations to reduce the computation time of the COCG method were carried out. Results show the following. (1) The mean residual of sound pressure levels between COCG method and direct method is less than 0.1 dB if the convergence criterion is set to 10^-4 and the maximum residual of those between COCG method and direct method is less than 0.2 dB if the convergence criterion is set to 10^-6. (2) The computation time of the COCG method with diagonal preconditioning is about 30% shorter than that of COCG method without preconditioning. Finally, sound pressure level distributions obtained using the authors' FEM are compared to those obtained using fast multipole BEM (FMBEM) and measurements.

In this paper, we propose a computing algorithm to obtain optimal checking times which minimize the expected cost rate of sequential inspection models. This algorithm is applied to two modified inspection models with finite intervals such that the unit is checked at times $T_k$ ($k=1,2,\dots ,N$) for given $N$ and given $S\equiv T_N$. The modified algorithms are proposed and optimal checking times to minimize the expected cost rates are obtained. Numerical examples of each model are given to show how to use the algorithm. This method would be applied to other sequential inspection models by its suitable modifications.

Network-on-chip multicore architectures with a large number of processing elements are becoming a reality with the recent developments in technology. In these modern systems the processing elements are interconnected with regular network-on-chip (NoC) topologies such as meshes and trees. In this paper we propose a parallel Gauss-Seidel (GS) iterative algorithm for solving large systems of linear equations on a torus NoC architecture. The proposed parallel algorithm is O(Nn²/k²) time complexity for solving a system with matrix of order n on a k × k torus NoC architecture with N iterations assuming n and N are large compared to k (i.e. for large linear systems that require a large number of iterations). We show that under these conditions the proposed parallel GS algorithm has near optimal speedup.

The three-parameter Weibull distribution is popular in reliability analysis. However, in the context of a small sample size, the issue of parameter estimation of the three-parameter Weibull distribution is difficult to address. If only a small sample size is available, a reasonable method is to empirically determine the shape parameter, and the emphasis is placed on location parameter estimation. This paper presents a theoretical model to establish the relationship between the location parameter, the minimal sample value, and the sample size. Moreover, an iterative method is proposed to estimate the Weibull location parameter and scale parameter with an empirical value for the shape parameter. Compared with the maximum likelihood method (MLE), the Weibull plot with the sample correlation coefficient and TL-moment, the proposed method can more effectively estimate the parameters of the three-parameter Weibull distribution.

This paper deals with the problem of image reconstruction from incomplete projections. A novel fuzzy vector objective optimization model is developed by integrating the fuzzy set theory and vector objective optimization (multi-objective decision-making). The objective function is expressed as a membership function, and the minimum operator is taken as a fuzzy operator. Furthermore, a novel iterative method is proposed to resolve the fuzzy optimization problem. The images reconstructed from simulated noise projections and real projections obtained from an industrial scanner show that the new algorithm can provide higher resolution and better smoothness than the images reconstructed by the transformation method and the conventional iterative method, so it is more feasible for image reconstruction from incomplete projections.

We study the existence and the regularity of the solutions of some nonlinear partial differential system arising in the study of free convection in a two-dimensional bounded domain, modeling a porous medium saturated with a fluid. By introducing an iterative method, the closeness of such solutions by solution of linear elliptic problems is given with an exponential rate of convergence.

From some modifications of Chebyshev's method, we consider a uniparametric family of iterative methods that are more efficient than Newton's method, and we then construct two iterative methods in a similar way to the Secant method from Newton's method. These iterative methods do not use derivatives in their algorithms and one of them is more efficient than the Secant method, which is the classical method with this feature.

A time-dependent formulation of quantum control is employed to investigate optimally controlled vibrational population transfer in a diatomic quantum system. The problem of finding the optimal laser field needed to achieve a specific quantum transition from an initial state to the desired target goal is formulated using an iterative method and the conjugate gradient method (CGM). The time-dependent Schrödinger equation is solved with interaction of laser radiation with matter included within a dipole approximation in the Hamiltonian. Appropriate boundary conditions are chosen for the evolution problem. The control objective is chosen as the value of transition probability from an initial state to a target state. A comparison is made between the results obtained using the iterative method and the CGM for optimization. Finally, quantum bits are encoded using the vibrational states of the diatomic in the regime of low-vibrational excitation.

A front-end method based on random forest proximity distance (PD) is used to screen the test set to improve protein–protein interaction site (PPIS) prediction. The assessment of a distance metric is done under the assumption that a distance definition of higher quality leads to higher classification. On an independent test set, the numerical analysis based on statistical inference shows that the PD has the advantage over Mahalanobis and Cosine distance. Based on the fact that the proximity distance depends on the tree composition of the random forest model, an iterative method is designed to optimize the proximity distance, which adjusts the tree composition of the random forest model by adjusting the size of the training set. Two PD metrics, 75PD and 50PD, are obtained by the iterative method. On two independent test sets, compared with the PD produced by the original training set, the values of 75PD in Matthews correlation coefficient and F₁ score were higher, and the differences between them were statistically significant. All numerical experiments show that the closer the distance between the test data and the training data, the better the prediction results of the predictor. These indicate that the iterative method can optimize proximity distance definition and the distance information provided by PD can be used to indicate the reliability of prediction results.

In this paper, a new iterative method, for solving sparse nonsymmetrical systems of linear equations is proposed based on the Simultaneous Elimination and Back-Substitution Method (SEBSM), and the method is applied to solve systems resulted in engineering problems solved using Finite Element Method (FEM). First, SEBSM is introduced for solving general linear systems using the direct method. And, then an iterative method based on SEBSM is presented. In the method, the coefficient matrix $A$ is split into lower, diagonally banded and upper matrices. The iterative convergence can be controlled by selecting a suitable bandwidth of the diagonally banded matrix. And the size of the working array needing to be stored in iteration is as small as the bandwidth of the diagonally banded matrix. Finally, an accelerating strategy for this iterative method is proposed by introducing a relaxation factor, which can speed up the convergence effectively if an optimal relaxation factor is chosen. Two numerical examples are given to demonstrate the behavior of the proposed method.

In this paper, based on the basic principle of the SPH method’s kernel approximation, a new kernel approximation was constructed to compute first-order derivative through Taylor series expansion. Derivative in Newton’s method was replaced to propose a new SPH iterative method for solving nonlinear equations. The advantage of this method is that it does not require any evaluation of derivatives, which overcame the shortcoming of Newton’s method. Quadratic convergence of new method was proved and a variety of numerical examples were given to illustrate that the method has the same computational efficiency as Newton’s method.

A simple and efficient iterative method involving Green’s function is presented for solving the general $n$th-order nonlinear boundary value problems. Existence and uniqueness results for such problems are established using the fixed-point theorems. This guarantees the uniform convergence of the proposed iterative algorithm. The resulting iterative solutions does not contain any unknown constants and automatically satisfy the given boundary conditions. This gives the iterative method much wider applicability in obtaining the analytical or numerical solution of the problem. The performance of the proposed approach is assessed and tested on four examples.

This paper presents a new numerical iteration method for solving the absolute value equations. The proposed method uses the generalized Newton method as a predictor step, and the five-point open Newton–Cotes formula is considered the corrector step. The convergence of the proposed method is studied in detail. The proposed method solves large systems effectively due to its simplicity and effectiveness. In this paper, we have solved the beam equation, using the finite difference method to transform it into a system of absolute value equations, and then solved it using the proposed method. Several numerical examples were provided to demonstrate the accuracy and effectiveness of the new approach. In addition, the novel approach solves absolute value equations with greater accuracy and precision than other existing methods.

In this paper we introduce a family of one-point methods for multiple roots using weight functions. We analyze the necessary conditions for the weight function to achieve third-order accuracy, and then present a sequence of rational weight functions for a complex quadratic function. The resulting members of the family exhibit Julia sets that align with the perpendicular bisector of the roots. We compare selected members with existing methods based on basin of attraction analysis, demonstrating their competitive performance. Additionally, we investigate the influence of root multiplicity on the fractal structures of basin of attraction for equations with both simple and multiple roots.

The implementation of iterative methods using inverses to solve equations is a computationally expensive or impossible task in general. This is the case, since the analytical form of the inverse is difficult to find in practice. That is why, we replace the inverse by a sum of linear operators which is well defined. The convergence of the resulting hybrid methods is studied based on majorizing sequences under generalized continuity assumptions on the operators involved and in the setting of a Banach space. It is demonstrated by numerical experimentations that the convergence order as well as the number of iterations required to obtain a predetermined error tolerance when comparing the original to the hybrid method is essentially the same.