Convergence Acceleration Methods Research Articles

A Coupled Ordinates Method for Convergence Acceleration of the Phonon Boltzmann Transport Equation

Sequential numerical solution methods are commonly used for solving the phonon Boltzmann transport equation (BTE) because of simplicity of implementation and low storage requirements. However, they exhibit poor convergence for low Knudsen numbers. This is because sequential solution procedures couple the phonon BTEs in physical space efficiently but the coupling is inefficient in wave vector (K) space. As the Knudsen number decreases, coupling in K space becomes dominant and convergence rates fall. Since materials like silicon have K-resolved Knudsen numbers that span two to five orders of magnitude at room temperature, diffuse-limit solutions are not feasible for all K vectors. Consequently, nongray solutions of the BTE experience extremely slow convergence. In this paper, we develop a coupled-ordinates method for numerically solving the phonon BTE in the relaxation time approximation. Here, interequation coupling is treated implicitly through a point-coupled direct solution of the K-resolved BTEs at each control volume. This implicit solution is used as a relaxation sweep in a geometric multigrid method which promotes coupling in physical space. The solution procedure is benchmarked against a traditional sequential solution procedure for thermal transport in silicon. Significant acceleration in computational time, between 10 and 300 times, over the sequential procedure is found for heat conduction problems.

Generalized pseudopolar format algorithm for radar imaging with highly suboptimal aperture length

Different from conventional spaceborne or airborne synthetic aperture radar (SAR) with optimal aperture length, an imaging radar with highly suboptimal aperture length acquires the data in short bursts by a geometry spreading over a large range. A polarlike or pseudopolar format grid is introduced to sample data close to the resolution, which presents the design of a separable kernel for efficient FFT implementation. The proposed imaging algorithm formulates the reflectivity image of the target scene as an interpolation-free double image series expansion with two usual approximation-induced phase error terms being taken into account, whereby more generalized application scenarios with high frequency, large bandwidth or larger aperture length for imaging a target scene located within either the far-field or the near-field of the radar aperture are processable with high accuracy. In addition, convergence acceleration methods in computational mathematics are introduced to accelerate the convergence of the image series expansion, so as to make the algorithm more efficient. The proposed algorithm has been validated both qualitatively and quantitatively with an extensive collection of numerical simulations and field measurements of ground-based SAR (GB-SAR) data set.

Iterated Bernstein operators for distribution function and density estimation: Balancing between the number of iterations and the polynomial degree

Despite its slow convergence, the use of the Bernstein polynomial approximation is becoming more frequent in Statistics, especially for density estimation of compactly supported probability distributions. This is due to its numerous attractive properties, from both an approximation (uniform shape-preserving approximation, etc.) and a statistical (bona fide estimation, low boundary bias, etc.) point of view. An original method for estimating distribution functions and densities with Bernstein polynomials is proposed, which takes advantage of results about the eigenstructure of the Bernstein operator to refine a convergence acceleration method. Furthermore, an original data-driven method for choosing the degree of the polynomial is worked out. The method is successfully applied to two data-sets which are important benchmarks in the field of Density Estimation.

The Fourier space restricted Hartree-Fock method for the electronic structure calculation of linear poly(tetrafluoroethylene)

Building on the pioneering work of Jean-Marie Andre and working in the laboratory he founded, the authors have developed a code called FT-1D to make Hartree-Fock electronic structure computations for stereoregular polymers using Ewald-type convergence acceleration methods. That code also takes full advantage of all line-group symmetries to calculate only the minimal set of two-electron integrals and to optimize the computation of the Fock matrix. The present communication reports a benchmark study of the FT-1D code using polytetrafluoroethylene (PTFE) as a test case. Our results not only confirm the algorithmic correctness of the code through agreement with other studies where they are applicable, but also show that the use of convergence acceleration enables accurate results to be obtained in situations where other widely-used codes (e.g., PLH and Crystal) fail. It is also found that full attention to the line-group symmetry of the PTFE polymer leads to an increase of between one and two orders of magnitude in the speed of computation. The new code can therefore be viewed as extending the range of electronic-structure computations for stereoregular polymers beyond the present scope of the successful and valuable code Crystal.

Accelerated convergence method for fast Fourier transform simulation of coupled cavities

Fast Fourier transform (FFT) simulation was used to calculate the power and spatial distribution of resonant fields in optical cavities. This is an important tool when characterizing the effect of imperfect geometry and mirror aberrations. This method is, however, intrinsically slow when the cavities are of relatively high finesse. When this is the case, an accelerated convergence scheme may be used to calculate the steady-state cavity field with a speed that is orders of magnitude faster. The rate of convergence of this method, however, is unpredictable, as many different factors may detrimentally affect its performance. In addition, its use in multiple cavity configurations is not well understood. An in-depth study of the limitations and optimization of this method is presented, together with a formulation of its use in multiple cavity configurations. This work has not only resulted in consistent improvement in performance and stability of the accelerated convergence method but also allows the simulation of optical configurations, which would not previously have been possible.

Coarse mesh finite difference formulation for accelerated Monte Carlo eigenvalue calculation

An efficient Monte Carlo (MC) eigenvalue calculation method for source convergence acceleration and stabilization is developed by employing the Coarse Mesh Finite Difference (CMFD) formulation. The detailed methods for constructing the CMFD system using proper MC tallies are devised such that the coarse mesh homogenization parameters are dynamically produced. These involve the schemes for tally accumulation and periodic reset of the CMFD system. The method for feedback which is to adjust the MC fission source distribution (FSD) using the CMFD global solution is then introduced through a weight adjustment scheme. The CMFD accelerated MC (CMFD-MC) calculation is examined first for a simple one-dimensional multigroup problem to investigate the effectiveness of the accelerated fission source convergence process and also to analyze the sensitivity of the CMFD-MC solutions on the size of coarse meshes and on the number of CMFD energy groups. The performance of CMFD acceleration is then assessed for a set of two-dimensional and three-dimensional multigroup (3D) pressurized water reactor core problems. It is demonstrated that very rapid convergence of the MC FSD is possible with the CMFD formulation in that a sufficiently converged MC FSD can be obtained within 20 cycles even for large three-dimensional problems which would require more than 600 inactive cycles with the standard MC fission source iteration scheme. It is also shown that the optional application of the CMFD formulation in the active cycles can stabilize FSDs such that the real-to-apparent variance ratio of the local tallies can be reduced. However, due to the reduced importance of the variance bias in fine local tallies of 3D MC eigenvalue problems, the effectiveness of CMFD in tally stabilization turns out to be not so great.

Multilevel correction for collocation solutions of Volterra integral equations with proportional delays

In this paper, we propose a convergence acceleration method for collocation solutions of the linear second-kind Volterra integral equations with proportional delay qt $(0<q<1)$ . This convergence acceleration method called multilevel correction method is based on a kind of hybrid mesh, which can be viewed as a combination between the geometric meshes and the uniform meshes. It will be shown that, when the collocation solutions are continuous piecewise polynomials whose degrees are less than or equal to ${m} (m \leqslant 2)$ , the global accuracy of k level corrected approximation is $O(N^{-(2m(k+1)-\varepsilon)})$ , where N is the number of the nodes, and $\varepsilon$ is an arbitrary small positive number.

A method of convergence acceleration of some continued fractions II

Most of the methods for convergence acceleration of continued fractions K(a m /b m ) are based on the use of modified approximants S m (ω m ) in place of the classical ones S m (0), where ω m are close to the tails f (m) of the continued fraction. Recently (Nowak, Numer Algorithms 41(3):297–317, 2006), the author proposed an iterative method producing tail approximations whose asymptotic expansion accuracies are being improved in each step. This method can be successfully applied to a convergent continued fraction K(a m /b m ), where \(a_m = \alpha_{-2} m^2 + \alpha_{-1} m + \ldots\), b m = β − 1 m + β 0 + ... (α − 2 ≠ 0, \(|\beta_{-1}|^2+|\beta_{0}|^2\neq 0\), i.e. \(\deg a_m=2\), \(\deg b_m\in\{0,1\}\)). The purpose of this paper is to extend this idea to the class of two-variant continued fractions K (a n /b n + a n ′/b n ′) with a n , a n ′, b n , b n ′ being rational in n and \(\deg a_n=\deg a_n'\), \(\deg b_n=\deg b_n'\). We give examples involving continued fraction expansions of some elementary and special mathematical functions.

Faster convergence for MoM analysis of resonant circular arrays of cylindrical dipoles

Method-of-moments analyses of resonant circular arrays of cylindrical dipoles present unique and particular difficulties. A previous publication overcame the major difficulties by using a set of improved kernels in the usual Hallen-type integral equations and by applying an appropriate convergence acceleration method. Based on previous works dealing with the isolated cylindrical dipole, proposed is a better method of calculating the constant C (m) appearing in the aforementioned equations; this results both in improved accuracy and in faster overall convergence of the authors' moment method.

The early history of convergence acceleration methods

The history of convergence acceleration methods in the 17th century is surveyed. Acceleration methods are classified into three categories.

Local stress singularities in mixed axisymmetric problems of the bending of circular cylinders

A method of analyzing the near-edge stress state in mixed problems of the deformation of an isotropic cylindrical body is proposed. The method is based on the expansion of the solution of three-dimensional problems of elasticity into a series of Lurie–Vorovich homogeneous basis functions. An asymptotic analysis is performed to find the principal part of the solution of the infinite systems of linear algebraic systems to which the problems are reduced. The type of the stress singularity at the edge of the cylinder is the same as in the mixed problems for a quarter plane. Kummer’s convergence acceleration method is used. The obtained results are validated by testing the boundary conditions and by comparing with results obtained by other authors

A p-multigrid spectral difference method for two-dimensional unsteady incompressible Navier–Stokes equations

This paper presents the development of a 2D high-order solver with spectral difference method for unsteady incompressible Navier–Stokes equations accelerated by a p-multigrid method. This solver is designed for unstructured quadrilateral elements. Time-marching methods cannot be applied directly to incompressible flows because the governing equations are not hyperbolic. An artificial compressibility method (ACM) is employed in order to treat the inviscid fluxes using the traditional characteristics-based schemes. The viscous fluxes are computed using the averaging approach (Sun et al., 2007; Kopriva, 1998) [29,12]. A dual time stepping scheme is implemented to deal with physical time marching. A p-multigrid method is implemented (Liang et al., 2009) [16] in conjunction with the dual time stepping method for convergence acceleration. The incompressible SD (ISD) method added with the ACM (SD-ACM) is able to accurately simulate 2D steady and unsteady viscous flows.

Acceleration of convergence of general linear sequences by the Shanks transformation

The Shanks transformation is a powerful nonlinear extrapolation method that is used to accelerate the convergence of slowly converging, and even diverging, sequences {A n }. It generates a two-dimensional array of approximations $${A^{(j)}_n}$$ to the limit or anti-limit of {A n } defined as solutions of the linear systems$$A_l=A^{(j)}_n +\sum^{n}_{k=1}\bar{\beta}_k(\Delta A_{l+k-1}),\ \ j\leq l\leq j+n,$$ where $${\bar{\beta}_{k}}$$ are additional unknowns. In this work, we study the convergence and stability properties of $${A^{(j)}_n}$$, as j → ∞ with n fixed, derived from general linear sequences {A n }, where $${{A_n \sim A+\sum^{m}_{k=1}\zeta_k^n\sum^\infty_{i=0} \beta_{ki}n^{\gamma_k-i}}}$$ as n → ∞, where ζ k ≠ 1 are distinct and |ζ 1| = ... = |ζ m | = θ, and γ k ≠ 0, 1, 2, . . .. Here A is the limit or the anti-limit of {A n }. Such sequences arise, for example, as partial sums of Fourier series of functions that have finite jump discontinuities and/or algebraic branch singularities. We show that definitive results are obtained with those values of n for which the integer programming problems$$\begin{array}{ll}{\quad\quad\quad\quad\max\limits_{s_1,\ldots,s_m}\sum\limits_{k=1}^{m}\left[(\Re\gamma_k)s_k-s_k(s_k-1)\right],}\\ {{\rm subject\,to}\,\, s_1\geq0,\ldots,s_m\geq0\quad{\rm and}\quad \sum\limits_{k=1}^{m} s_k = n,}\end{array}$$ have unique (integer) solutions for s 1, . . . , s m . A special case of our convergence result concerns the situation in which $${{\Re\gamma_1=\cdots=\Re\gamma_m=\alpha}}$$ and n = mν with ν = 1, 2, . . . , for which the integer programming problems above have unique solutions, and it reads $${A^{(j)}_n-A=O(\theta^j\,j^{\alpha-2\nu})}$$ as j → ∞. When compared with A j − A = O(θ j j α ) as j → ∞, this result shows that the Shanks transformation is a true convergence acceleration method for the sequences considered. In addition, we show that it is stable for the case being studied, and we also quantify its stability properties. The results of this work are the first ones pertaining to the Shanks transformation on general linear sequences with m > 1.

Asymptotic behavior of the Eckhoff method for convergence acceleration of trigonometric interpolation

Convergence acceleration of the classical trigonometric interpolation by the Eckhoff method is considered, where the exact values of the “jumps” are approximated by solution of a system of linear equations. The accuracy of the “jump” approximation is explored and the corresponding asymptotic error of interpolation is derived. Numerical results validate theoretical estimates.

Cross rules of some extrapolation algorithms

In this paper, we present the cross rules of some extrapolation algorithms. These new cross rules are similar in their spirit to the cross rule for the scalar ε-algorithm and the ϱ-algorithm. The first interest of such rules is that, in the algorithms for implementing sequence transformations for the acceleration of the convergence, they allow us to compute only the quantities which have a signification for extrapolation and to jump over the intermediate ones. The second interest comes out from the link between convergence acceleration methods, soliton theory and discrete integrable systems. In some cases, cross rules directly lead to physically significant integrable systems. For example, the cross rule of the confluent qd-algorithm leads to the celebrated Toda equation, which is known to have physical applications.

REVIEWS OF MOVING GRID GENERATION TECHNIQUES AND NUMERICAL METHODS FOR UNSTEADY FLOW

CFD methods forunsteady flow simulations are reviewed in this paper, includingmoving grid generation techniques and unsteady numerical methods.Based on the topology of computational grids, two types ofunsteady numerical methods are discussed, e.g. the methods basedon structured grids and unstructured/hybrid grids, respectively.Their advantages and drawbacks are analyzed with regard toefficiency, robustness for complex geometries and other features.In the framework of structured grids, several moving gridtechniques are reviewed, including the rigid moving structuredgrid approach, the deforming structured grid technique withtrans-finite interpolation, overlapping moving structured gridtechnique, and patched moving structured grid technique. Then someunsteady numerical methods on structured grids are discussed inthe systems of inertial and non-inertial coordinates. In addition,the discretization schemes in temporal direction and the GeometricConservation Law (GCL) are emphasized for unsteady simulations onmoving grids. In the context of unstructured/hybrid grids,overlapping unstructured grid technique, re-meshing unstructuredgrid technique, node-relaxation approach based on `Spring analogy'and Delaunay mapping are reviewed and compared. And, a combineddynamic unstructured/hybrid grid method developed by the authorsis suggested, which is a kind of coupled method of `Spring'relaxation, Delaunay mapping and local re-meshing. The GCLapproach, the convergence acceleration methods, and numericalschemes for unsteady incompressible flows are discussed forunsteady flow solvers on unstructured/hybrid grids. Some examplesand applications performed by the authors are given at the end ofthe paper.

Survey of numerical stability issues in convergence acceleration

An important issue that arises in application of convergence acceleration (extrapolation) methods is that of stability in the presence of floating-point arithmetic. This issue turns out to be critical because numerical instability is inherent, even built in, when convergence acceleration methods are applied to certain types of sequences that occur commonly in practice. If methods are applied without taking this issue into account, the attainable accuracy is limited, and eventually destroyed completely, as more terms are added in the process. Therefore, it is important to understand the origin of the problem and to propose practical ways to solve it effectively. In this work, we present a general discussion of the issue of stability within the context of a generalization of the Richardson extrapolation process, and review some of the recent developments that have taken place in the theoretical study of many of the known acceleration methods. We discuss approaches that have been proposed to cope with built-in instabilities when applying various methods, and illustrate the effectiveness of these strategies with some numerical examples.

Milestones in the Development of Iterative Solution Methods

Iterative solution methods to solve linear systems of equations were originally formulated as basic iteration methods of defect‐correction type, commonly referred to as Richardson′s iteration method. These methods developed further into various versions of splitting methods, including the successive overrelaxation (SOR) method. Later, immensely important developments included convergence acceleration methods, such as the Chebyshev and conjugate gradient iteration methods and preconditioning methods of various forms. A major strive has been to find methods with a total computational complexity of optimal order, that is, proportional to the degrees of freedom involved in the equation. Methods that have turned out to have been particularly important for the further developments of linear equation solvers are surveyed. Some of them are presented in greater detail.

Application of Kummer's Transformation to the Efficient Computation of the 3-D Green's Function With 1-D Periodicity

The 3-D homogeneous Green's function with 1-D periodicity is commonly expressed as spatial and spectral infinite series that may show very slow convergence. In this work Kummer's transformation is applied to the spatial series in order to accelerate its convergence. By retaining a sufficiently large number of asymptotic terms in the application of Kummer's transformation, the spatial series is split into a set of series which can be accurately obtained with very low computational effort. The numerical results obtained show that, when the number of asymptotic terms retained in Kummer's transformation is large enough, the convergence acceleration method proposed in this work is always faster than existing acceleration methods such as the spectral Kummer-Poisson's method and Ewald's method.

Asymptotic analysis of a generalized Richardson extrapolation process on linear sequences

In this work, we give a detailed convergence and stability analysis for the author's generalized Richardson extrapolation process GREP (m) as this is being applied to linearly convergent or divergent infinite sequences {A n }, where A n ~ A + ∑ m k=1 ζ n k ∑ ∞ i=0 β ki n γk-i as n → ∞, ζ k ≠ 1 being distinct. The quantity we would like to compute is A, whether it is the limit or antilimit of {A n }. Such sequences arise, for example, as partial sums of power series and of Fourier series of functions that have algebraic and/or logarithmic branch singularities. Specifically, we define the GREP (m) approximation A n (m,j) to A, with n = (n 1, ..., n m ) and α > 0, via the linear systems A l = A n (m,j) + ∑ m k=1 ζ l k ∑ n i=0 -1 β ki (α + l) γk-1 , j ≤ l ≤ j + ∑ m k=1 n k, where β ki are additional unknowns. We study the convergence and stability properties of A n (m,j) as j → ∞. We show, in particular, that A n (m,j) -A = ∑ m k=1 O(ζ j k j γk-2nk ) as j → ∞. When compared with A j - A = ∑ m k=1 O(ζ j k j γk ) as j → ∞, this result shows that GREP (m) is a true convergence acceleration method for the sequences considered. In addition, we show that GREP (m) is stable for the case being studied, and we also quantify its stability properties. The results of this work are the first ones pertaining to GREP (m) with m > 1.