Aitken’s Δ2 method extended

A general iterative scheme including relaxation and a corresponding problem class are presented. Some global convergence results are given. The acceleration of convergence is discussed, The scheme comprises a lot of known iterative methods such as subgradient methods and methods of successive orthogonal projections with relaxation. Applications to convex optimization, convex feasibility problems, systems of convex inequalities, variational inequalities, operator equations and systems of linear equations are given.

A new, nonclassical convergence acceleration concept, called \(\mu \)-acceleration of convergence (where \(\mu \) is a positive monotonically increasing sequence), is introduced and compared with the classical convergence acceleration concept. It is shown that this concept allows to compare the speeds of convergence for a larger set of sequences than the classical convergence acceleration concept. Also, regular matrix methods that improve and accelerate the convergence of sequences and series are studied. Some problems related to the speed of convergence of sequences and series with respect to matrix methods are discussed. Several theorems on the improvement and acceleration of the convergence are proved. As an application, the results obtained are used to increase the order of approximation of Fourier expansions and Zygmund means of Fourier expansions in certain Banach spaces.

Downward continuation of potential field is a useful tool in the processing and interpretation of magnetic and gravity data, but its direct calculation in space or spectral domain is unstable even in the presence of a small level of the noise and consequently restricts its practical application. This paper introduces a new method that combines upward continuation, vertical gradient via smoothing filters (Tikhonov regularization or Gaussian filters) and an iterative method into a single scheme to improve the stability and accuracy of the downward continuation. The optimal interval of iteration numbers for our developed approach is estimated by analysing the correlation coefficient curve between consecutive iterations. Applications to synthetic and real magnetic data show that this method can yield a more stable downward continuation of potential field data.

Stochastic iterative algorithms such as the Kaczmarz and Gauss--Seidel methods have gained recent attention because of their speed, simplicity, and the ability to approximately solve large-scale linear systems of equations without needing to access the entire matrix. In this work, we consider the setting where we wish to solve a linear system in a large matrix $X$ that is stored in a factorized form, $X = UV$; this setting either arises naturally in many applications or may be imposed when working with large low-rank datasets for reasons of space required for storage. We propose a variant of the randomized Kaczmarz method for such systems that takes advantage of the factored form and avoids computing $X$. We prove an exponential convergence rate and supplement our theoretical guarantees with experimental evidence demonstrating that the factored variant yields significant acceleration in convergence.

A numerical method is described, which is used for static and dynamic calculations of electromagnetic actuators. The mathematical model consists of three systems representing the electric, magnetic, and mechanical components, which are governed by coupled differential equations for coil current, magnetic vector potential, and armature displacement. As yet, only magnets with rotational symmetry have been considered and eddy currents have been neglected. The non-linear partial differential equation for the magnetic vector potential is solved by finite difference techniques using the line iteration method with overrelaxation of the vector potential and underrelaxation of the reluctivity. Acceleration of convergence is achieved by the multiplicative method based on Ampere's law. For the solution of the electric and mechanical differential equations, inductances and forces are calculated from the magnetic field distribution. A computer program based on these techniques is outlined. It automatically adapts the finite difference grid to the time-varying geometry and chooses the surface of integration for the evaluation of the magnetic force. Some calculations are discussed, which were performed to assist the design of print needle actuators.

<p style='text-indent:20px;'>Data completion known as Cauchy problem is one most investigated inverse problems. In this work we consider a Cauchy problem associated with Helmholtz equation. Our concerned is the convergence of the well-known alternating iterative method [<xref ref-type="bibr" rid="b25">25</xref>]. Our main result is to restore the convergence for the classical iterative algorithm (KMF) when the wave numbers are considerable. This is achieved by, some simple modification for the Neumann condition on the under-specified boundary and replacement by relaxed Neumann ones. Moreover, for the small wave number <inline-formula><tex-math id="M111111">\begin{document}$ k $\end{document}</tex-math></inline-formula>, when the convergence of KMF algorithm's [<xref ref-type="bibr" rid="b25">25</xref>] is ensured, our algorithm can be used as an acceleration of convergence.</p><p style='text-indent:20px;'>In this case, we present theoretical results of the convergence of this relaxed algorithm. Meanwhile it, we can deduce the convergence intervals related to the relaxation parameters in different situations. In contrast to the existing results, the proposed algorithm is simple to implement converges for all choice of wave number.</p><p style='text-indent:20px;'>We approach our algorithm using finite element method to obtain an accurate numerical results, to affirm theoretical results and to prove it's effectiveness.</p>

Convergence acceleration of number-machine sequences

The symmetric alternating direction method of multipliers is an efficient algorithm, which updates the Lagrange multiplier twice at each iteration and the variables are treated in a symmetric manner. Considering that the convergence range of the parameters plays an important role in the implementation of the algorithm. In this paper, we analyze the convergence rate of the symmetric ADMM with a more relaxed parameter range for solving the two block nonconvex separable optimization problem under the assumption that the generated sequence is bounded. Two cases are considered. In the first case, both components of the objective function are nonconvex, we prove the convergence of the augmented Lagrangian function sequence, and establish the $ O(1/\sqrt{k}) $ worst-case complexity measured by the difference of two consecutive iterations. In the second case, one component of the objective function is convex and the error bound condition is assumed, then we can prove that the iterative sequence converges locally to a KKT point in a R-linear rate; and an auxiliary sequence converges in a Q-linear rate. Furthermore, a practical inexact symmetric ADMM with relative error criteria is proposed, and the associated convergence analysis is established under the same conditions.

Iterative solution of the incremental problem for elastic-plastic structures with associated flow laws

In this paper, we study the triple hierarchical variational inequality problem (THVIP, for short) with the hierarchical variational inequality constraint for finitely many nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. Based on Korpelevich's extragradient method, hybrid steepest-descent method, Mann's implicit iteration method and Halpern's iteration method, a general implicit hybrid iterative algorithm for solving this THVIP is proposed. Under mild conditions, the strong convergence of the iterative sequences generated by the algorithm is established. Our results improve and extend the corresponding results in the earlier and recent literature.

Vector sequence transformations for the acceleration of logarithmic convergence

Link-based similarity measures play a significant role in many graph based applications. Consequently, measuring node similarity in a graph is a fundamental problem of graph datamining. Personalized pagerank (PPR) and simrank (SR) have emerged as the most popular and influential link-based similarity measures. Recently, a novel link-based similarity measure, penetrating rank (P-Rank), which enriches SR, was proposed. In practice, PPR, SR and P-Rank scores are calculated by iterative methods. As the number of iterations increases so does the overhead of the calculation. The ideal solution is that computing similarity within the minimum number of iterations is sufficient to guarantee a desired accuracy. However, the existing upper bounds are too coarse to be useful in general. Therefore, we focus on designing an accurate and tight upper bounds for PPR, SR, and P-Rank in the paper. Our upper bounds are designed based on the following intuition: the smaller the difference between the two consecutive iteration steps is, the smaller the difference between the theoretical and iterative similarity scores becomes. Furthermore, we demonstrate the effectiveness of our upper bounds in the scenario of top-k similar nodes queries, where our upper bounds helps accelerate the speed of the query. We also run a comprehensive set of experiments on real world data sets to verify the effectiveness and efficiency of our upper bounds.

Eleven different methods for accelerating convergence of sequences and series have been tested and compared on a wide range of test problems, including both linearly and logarithmically convergent series, monotone and alternating series. All but one of these methods are already in the literature, and they include both linear and nonlinear methods. The only methods found to accelerate convergence across the board were the u and v transforms of Levin and the $\theta $ algorithm of Brezinski. The paper gives detailed comparisons of all the tested methods on the basis of number of correct digits in the answer as a function of number of terms of the series used. A theorem of Germain-Bonne states that methods of a certain form which are exact on geometric series will accelerate linear convergence. The theorem applies to $\theta _2 $, and we have extended it to apply to Levin's transforms. No corresponding theorem is known for logarithmic convergence, but u, v, and $\theta $ are exact on certain large classes of logarithmic series, and all tested methods lacking this property failed to accelerate some logarithmically convergent series.

A necessary and sufficient condition for the strong convergence of Lipschitzian pseudocontractive mapping in Banach spaces

On an application of symbolic computation and computer graphics to root-finders: The case of multiple roots of unknown multiplicity