Accelerate Convergence Research Articles

Review of numerical methods for NumILPT with computational accuracy assessment for fractional calculus

Abstract In the paper we present results of accuracy evaluation of numerous numerical algorithms for the numerical approximation of the Inverse Laplace Transform. The selected algorithms represent diverse lines of approach to this problem and include methods by Stehfest, Abate and Whitt, Vlach and Singhai, De Hoog, Talbot, Zakian and a one in which the FFT is applied for the Fourier series convergence acceleration. We use C++ and Python languages with arbitrary precision mathematical libraries to address some crucial issues of numerical implementation. The test set includes Laplace transforms considered as difficult to compute as well as some others commonly applied in fractional calculus. Evaluation results enable to conclude that the Talbot method which involves deformed Bromwich contour integration, the De Hoog and the Abate and Whitt methods using Fourier series expansion with accelerated convergence can be assumed as general purpose high-accuracy algorithms. They can be applied to a wide variety of inversion problems.

A Further Property of Functions in Class B(m): An Application of Bell Polynomials

We say that a function $\alpha(x)$ belongs to the set ${\bf A}^{(\gamma)}$ if it has an asymptotic expansion of the form $\alpha(x)\sim \sum^\infty_{i=0}\alpha_ix^{\gamma-i}$ as $x\to\infty$, which can be differentiated term by term infinitely many times. A function  $f(x)$ is in the class ${\bf B}^{(m)}$ if it satisfies a linear homogeneous differential equation of the form $f(x)=\sum^m_{k=1}p_k(x)f^{(k)}(x)$, with $p_k\in {\bf A}^{(i_k)}$, $i_k$ being integers satisfying $i_k\leq k$. These functions appear in many problems of applied mathematics and other scientific disciplines. They have been shown to have many interesting properties,  and their integrals $\int^\infty_0 f(x)\,dx$, whether convergent or divergent,  can be evaluated very efficiently via the Levin--Sidi $D^{(m)}$-transformation,  a most effective convergence acceleration method. (In case of divergence, these integrals  are defined in some summability sense, such as Abel summability or Hadamard finite part or a mixture of these two.) In this note, we show that if $f(x)$ is in ${\bf B}^{(m)}$, then so is $(f\circ g)(x)=f(g(x))$, where $g(x)>0$ for all large $x$ and $g\in {\bf A}^{(s)}$,  $s$ being a positive integer. This enlarges the scope of the $D^{(m)}$-transformation considerably to include functions of complicated arguments. We demonstrate  the validity of our result with an application of the $D^{(3)}$ transformation to two integrals $I[f]$ and $I[f\circ g]$, for some $f\in{\bf B}^{(3)}$ and $g\in{\bf A}^{(2)}$. The Fa\`{a} di Bruno formula and Bell polynomials play a central role in our study.

Acceleration of Later Convergence in a Density-Based Solver Using Adaptive Time Stepping

The convergence to the steady state of an implicit unstructured density-based compressible Navier-Stokes flow solver shows many cases exhibiting stalling or slower convergence in the later part mainly due to fluctuation in the local error. This causes a delay in convergence and often prevents machine zero convergence from being achieved. This paper presents the use of local error control for acceleration of later convergence. This method is tested against standard benchmark problems, and then on a more complex case of flow over an airfoil taken from recent published work. The results show that the method is capable of very rapid acceleration to machine zero convergence. The acceleration has been found to be of an order of 500–1000% faster in comparison to the same implicit scheme with Courant–Friedrichs–Lewy-based time stepping and no local error control. The present method acts on top of the implicit solver, and thus its implementation requires very little modification to an existing code.

A partitioned solution approach for the simulation of the dynamic behaviour of flexible marine propellers

ABSTRACTModern designs of ship-propellers make use of novel materials such as composites, which may show a much lower stiffness than classical material. In order to accurately predict the hydrodynamic characteristics of these propellers in real operating conditions, the structural deformation must be accounted for, which leads to a fluid-structure interaction problem. In the present paper, a partitioned solution approach for such a strongly coupled two-field problem is presented. Owing to the separate modelling of the fluid and the structural problem in such approaches, specialised numerical methods can be applied on either side. Thanks to their superior convergence properties, we use high-order finite elements to discretise the structural problem. On the fluid side, a boundary element method, which is specifically designed to simulate the hydrodynamic behaviour of ship propellers is employed. In order to stabilise the solution procedure, different coupling algorithms and convergence acceleration methods are investigated.

An Efficient Hartree-Fock Implementation Based on the Contraction of Integrals in the Primitive Basis.

A new approach to implement the restricted closed-shell Hartree-Fock equation is proposed. In the ansatz presented, the explicit transformation of integrals from the primitive to the atomic-orbital basis is omitted. Instead, the density matrix is transformed to the primitive basis, in which it is contracted with the untransformed integrals. Obtained is the two-electron part of the Fock matrix, which is transformed back to the atomic orbital basis. The remaining steps of the self-consistent field algorithm are then performed as usual. The program presented here incorporates the most important standard techniques, such as integral prescreening, convergence acceleration (via the direct inversion of the iterative subspace ansatz), and the differential density scheme. Test calculations on standard Hartree-Fock problems were compared to the commercially available MOLPRO and TURBOMOLE program packages. Except in a few special cases, the performance of the program presented here is superior, in comparison to those two programs. Accelerations by up to a factor of 5 were found, with respect to MOLPRO calculations, and up to 3 for TURBOMOLE (in the latter case, up to 55 for generalized contracted basis sets). The program structure is independent of the type of radial contraction; however, the best results are obtained for generalized radial contraction basis sets of low contraction. The program is written in C++ and utilizes code generation engines to automatically generate the routines for the integration and density contraction. Streaming SIMD extensions are used explicitly.

A multiscale method for semi-linear elliptic equations with localized uncertainties and non-linearities

A multiscale numerical method is proposed for the solution of semi-linear elliptic stochastic partial differential equations with localized uncertainties and non-linearities, the uncertainties being modeled by a set of random parameters. It relies on a domain decomposition method which introduces several subdomains of interest (called patches) containing the different sources of uncertainties and non-linearities. An iterative algorithm is then introduced, which requires the solution of a sequence of linear global problems (with deterministic operators and uncertain right-hand sides), and non-linear local problems (with uncertain operators and/or right-hand sides) over the patches. Non-linear local problems are solved using an adaptive sampling-based least-squares method for the construction of sparse polynomial approximations of local solutions as functions of the random parameters. Consistency, convergence and robustness of the algorithm are proved under general assumptions on the semi-linear elliptic operator. A convergence acceleration technique (Aitken’s dynamic relaxation) is also introduced to speed up the convergence of the algorithm. The performances of the proposed method are illustrated through numerical experiments carried out on a stationary non-linear diffusion-reaction problem.

Middle Pleistocene seismically induced clay diapirism in an intraplate zone, western Brittany, France

AbstractThe Brittany region of France is located in a low seismicity intraplate zone. Most of the instrumented earthquakes are limited to a shallow crustal depth without surface rupture. A paleoseismological analysis was performed on deposits on the Crozon Peninsula and in the Elorn estuary. We highlight hydroplastic deformations induced by liquefaction leading to clay diapirism, which were likely triggered by past earthquakes. This diapirism seems to be frequent in continental nonconsolidated sediments and to develop on the inherited tectonic structures, when a shallow water table and confining layers exist. Timing of deformation is dated using paleoenvironmental data, and electron spin resonance and infrared-stimulated luminescence dating methods. Two seismic periods were identified in western Europe during early Marine Oxygen Isotope Stage (MIS) 10 (~380 ka) and early MIS 8 (~280–265 ka). The lack of similar deformations affecting the Holocene tidal deposits in the Bay of Brest suggests that the magnitude of the triggering paleoearthquakes is probably higher (Mw~6) than the recent events (Mw5.4). These unusual intraplate major paleoearthquakes need specific factors affecting the far-field crustal stress loading to be triggered, such as a brief acceleration of the Africa-Eurasia lithospheric plate convergence, glacio-isostatic stress perturbations associated with the onset of major glaciations in northern Europe, or other processes induced by orbital forcing.

Regularized nonlinear acceleration

We describe a convergence acceleration technique for unconstrained optimization problems. Our scheme computes estimates of the optimum from a nonlinear average of the iterates produced by any optimization method. The weights in this average are computed via a simple linear system, whose solution can be updated online. This acceleration scheme runs in parallel to the base algorithm, providing improved estimates of the solution on the fly, while the original optimization method is running. Numerical experiments are detailed on classical classification problems.

Dynamics of the Transition from Spiral Rainbands to a Secondary Eyewall in Hurricane Earl (2010)

Abstract Airborne Doppler radar captured the inner core of Hurricane Earl during the early stages of secondary eyewall formation (SEF), providing needed insight into the SEF dynamics. An organized rainband complex outside of the primary eyewall transitioned into an axisymmetric secondary eyewall containing a low-level tangential wind maximum. During this transition, the downshear-left quadrant of the storm exhibited several notable features. A mesoscale descending inflow (MDI) jet persistently occurred across broad stretches of stratiform precipitation in a pattern similar to previous studies. This negatively buoyant jet traveled radially inward and descended into the boundary layer. Farther inward, enhanced low-level inflow and intense updrafts appeared. The updraft adjacent to the MDI was likely triggered by a region of convergence and upward acceleration (induced by the negatively buoyant MDI) entering the high-θe boundary layer. This updraft and the MDI in the downshear-left quadrant accelerated the tangential winds in a radial range where the axisymmetric wind maximum of the secondary eyewall soon developed. This same quadrant eventually exhibited the strongest overturning circulation and wind maximum of the forming secondary eyewall. Given these features occurring in succession in the downshear-left quadrant, we hypothesize that the MDI plays a significant dynamical role in SEF. The MDI within a mature rainband complex persistently perturbs the boundary layer, which locally forces enhanced convection and tangential winds. These perturbations provide steady low-level forcing that projects strongly onto the axisymmetric field, and forges the way for secondary eyewall development via one of several SEF theories that invoke axisymmetric dynamical processes.

Hybrid Approach with Improved Genetic Algorithm and Simulated Annealing for Thesis Sampling

Sampling inspection uses the sample characteristics to estimate that of the population, and it is an important method to describe the population, which has the features of low cost, strong applicability and high scientificity. This paper aims at the sampling inspection of the master’s degree thesis to ensure their quality, which is commonly estimated by random sampling. Since there are disadvantages in random sampling, a hybrid algorithm combined with an improved genetic algorithm and a simulated annealing algorithm is proposed in this paper. Furthermore, a novel mutation strategy is introduced according to the specialty of Shanghai’s thesis sampling to improve the efficiency of sampling inspection; the acceleration of convergence of the algorithm can also take advantage of this. The new algorithm features the traditional genetic algorithm, and it can obtain the global optimum in the optimization process and provide the fairest sampling plan under the constraint of multiple sampling indexes. The experimental results on the master’s thesis dataset of Shanghai show that the proposed algorithm well meets the requirements of the sampling inspection in Shanghai with a lower time-complexity.

Convergence acceleration of alternating series

We propose a new simple convergence acceleration method for a wide range class of convergent alternating series. It has some common features with Smith’s and Ford’s modification of Levin’s and Weniger’s sequence transformations, but its computational and memory cost is lower. We compare all three methods and give some common theoretical results. Numerical examples confirm a similar performance of all of them.

Performance Evaluation of Lower Complexity Hybrid-Fix-and-Round-LLL Algorithm for MIMO System

Lenstra-Lenstra-Lovasz (LLL) is an effective receiving algorithm for Multiple-Input-Multiple-Output (MIMO) systems, which is believed can achieve full diversity in MIMO detection of fading channels. However, the LLL algorithm features polynomial complexity and shows poor performance in terms of convergence. The reduction of algorithmic complexity and the acceleration of convergence are key problems in optimizing the LLL algorithm. In this paper, a variant of the LLL algorithm, the Hybrid-Fix-and-Round LLL algorithm, which combines both fix and round measurements in the size reduction procedure, is proposed. By utilizing fix operation, the algorithmic procedure is altered and the size reduction procedure is skipped by the hybrid algorithm with significantly higher probability. As a consequence, the simulation results reveal that the Hybrid-Fix-and-Round-LLL algorithm carries a faster rate of convergence compared to the original LLL algorithm, and its algorithmic complexity is at most one order lower than original LLL algorithm in real field. Comparing to other families of LLL algorithm, Hybrid-Fix-and-Round-LLL algorithm can make a better compromise in performance and algorithmic complexity.

A high-order cross-platform incompressible Navier–Stokes solver via artificial compressibility with application to a turbulent jet

Modern hardware architectures such as GPUs and manycore processors are characterised by an abundance of compute capability relative to memory bandwidth. This makes them well-suited to solving temporally explicit and spatially compact discretisations of hyperbolic conservation laws. However, classical pressure-projection-based incompressible Navier–Stokes formulations do not fall into this category. One attractive formulation for solving incompressible problems on modern hardware is the method of artificial compressibility. When combined with explicit dual time stepping and a high-order Flux Reconstruction discretisation, the majority of operations can be cast as compute bound matrix–matrix multiplications that are well-suited for GPU acceleration and manycore processing. In this work, we develop a high-order cross-platform incompressible Navier–Stokes solver, via artificial compressibility and dual time stepping, in the PyFR framework. The solver runs on a range of computer architectures, from laptops to the largest supercomputers, via a platform-unified templating approach that can generate/compile CUDA, OpenCL and C/OpenMP code at runtime. The extensibility of the cross-platform templating framework defined within PyFR is clearly demonstrated, as is the utility of P-multigrid for convergence acceleration. The platform independence of the solver is verified on Nvidia Tesla P100 GPUs and Intel Xeon Phi 7210 KNL manycore processors with a 3D Taylor–Green vortex test case. Additionally, the solver is applied to a 3D turbulent jet test case at Re=10,000, and strong scaling is reported up to 144 GPUs. The new software constitutes the first high-order accurate cross-platform implementation of an incompressible Navier–Stokes solver via artificial compressibility and P-multigrid accelerated dual time stepping to be published in the literature. The technology has applications in a range of sectors, including the maritime and automotive industries. Moreover, due to its cross-platform nature, the technology is well placed to remain relevant in an era of rapidly evolving hardware architectures. Program summaryProgram Title: PyFR v1.7.5Program Files doi:http://dx.doi.org/10.17632/65m665nt9c.1Licensing provisions: BSD 3-clauseProgramming language: Python, CUDA, OpenCL and CSupplementary material: Configuration and mesh files for the Taylor–Green Vortex and Turbulent jet test casesJournal reference of previous version: Comput. Phys. Commun. 185 (2014) 3028–3040Does the new version supersede the previous version?: YesReasons for the new version: Adding support for incompressible flowsSummary of revisions: Introducing a new high-order cross-platform incompressible flow solver via artificial compressibility and P-multigrid accelerated dual time stepping.Nature of problem: Incompressible Euler and Navier–Stokes equations for solving unsteady turbulent flows.Solution method: Artificial compressibility formulation discretised with a high-order Flux Reconstruction approach in space and P-multigrid accelerated dual time stepping in time.Additional comments including restrictions and unusual features: The algorithm targets modern massively parallel hardware platforms. Cross-platform capability is achieved via runtime code generation.

Efficient approximation of free-surface Green function and OpenMP parallelization in frequency-domain wave–body interactions

The evaluation of Green function without translation has always been the most time-consuming part of the hydrodynamic analysis based on potential flow theory. This research investigates the new implementations of efficient approximations of Green function and the code parallelization with OpenMP library. First, algorithms of infinite-depth case are developed with economized Chebyshev polynomials based on modified region and subdomain partitions. Second, an improved Gauss–Laguerre method for the Cauchy integral is presented to evaluate the finite-depth case with the convergence acceleration by the reduction of fraction and singularity elimination by the infinite-depth case together with the exponential integral. For the analyzed container vessel, on the premise of good numerical accuracy, we conclude that the present method has nearly obtain the same efficiency as Chen (Hydrodynamics in offshore and naval applications—Part I, Bureau Veritas, Paris, 2004) for the infinite-depth case and almost 30% higher efficiency for the finite-depth case. Moreover, we also point out that the numerical efficiency is significantly enhanced again due to the OpenMP parallelization of the serial code.

Convergence acceleration in scattering series and seismic waveform inversion using nonlinear Shanks transformation

Convergence acceleration in scattering series and seismic waveform inversion using nonlinear Shanks transformation

Three-Dimensional Elastic Analysis of a Structure with Holes Using Accelerated Coupling-Matrix-Free Iterative s-Version FEM

Some improvements of the coupling-matrix-free iterative s-version finite element method (FEM) to shorten its computational time are proposed. Then, the proposed method is applied to three-dimensional stress concentration problems. For sufficiently small computational time for practical use, two key techniques are introduced. First, the iteration is accelerated drastically by using the proposed convergence acceleration techniques. Secondly, stress transfers between global and local meshes are accelerated considerably by a bucket search algorithm. The proposed method was more than one hundred times faster than the straightforward algorithm of the coupling-matrix-free iterative s-version FEM.

Galilean-invariant preconditioned central-moment lattice Boltzmann method without cubic velocity errors for efficient steady flow simulations.

Lattice Boltzmann (LB) models used for the computation of fluid flows represented by the Navier-Stokes (NS) equations on standard lattices can lead to non-Galilean-invariant (GI) viscous stress involving cubic velocity errors. This arises from the dependence of their third-order diagonal moments on the first-order moments for standard lattices, and strategies have recently been introduced to restore Galilean invariance without such errors using a modified collision operator involving corrections to either the relaxation times or the moment equilibria. Convergence acceleration in the simulation of steady flows can be achieved by solving the preconditioned NS equations, which contain a preconditioning parameter that can be used to tune the effective sound speed, and thereby alleviating the numerical stiffness. In the present paper, we present a GI formulation of the preconditioned cascaded central-moment LB method used to solve the preconditioned NS equations, which is free of cubic velocity errors on a standard lattice, for steady flows. A Chapman-Enskog analysis reveals the structure of the spurious non-GI defect terms and it is demonstrated that the anisotropy of the resulting viscous stress is dependent on the preconditioning parameter, in addition to the fluid velocity. It is shown that partial correction to eliminate the cubic velocity defects is achieved by scaling the cubic velocity terms in the off-diagonal third-order moment equilibria with the square of the preconditioning parameter. Furthermore, we develop additional corrections based on the extended moment equilibria involving gradient terms with coefficients dependent locally on the fluid velocity and the preconditioning parameter. Such parameter dependent corrections eliminate the remaining truncation errors arising from the degeneracy of the diagonal third-order moments and fully restore Galilean invariance without cubic defects for the preconditioned LB scheme on a standard lattice. Several conclusions are drawn from the analysis of the structure of the non-GI errors and the associated corrections, with particular emphasis on their dependence on the preconditioning parameter. The GI preconditioned central-moment LB method is validated for a number of complex flow benchmark problems and its effectiveness to achieve convergence acceleration and improvement in accuracy is demonstrated.

A new smoothing approach for accelerating the convergence of power-law preconditioning method in steady and unsteady flows simulation

A new explicit smoothing approach is presented to improve the convergence acceleration of laminar viscous steady-state and unsteady flows simulation over airfoils (with/without considering surface air injection). This approach is developed based on Jameson's residual smoothing technique, and it is called here as a Solution Smoothing Method (SSM). This scheme is accompanied by the progressive locally power-law preconditioning method. The steady/unsteady preconditioned governing equations are solved using Jameson's cell-centered finite volume method in which the stabilization is achieved by means of the second- and fourth-order artificial dissipation terms. An implicit dual-time algorithm is used for simulation of unsteady cases and an explicit four-stage Runge–Kutta time integration with local time step is applied to achieve the steady-state condition in pseudo-time in an inner loop. Stability of SSM is analyzed on a one-dimensional linear model equation, and the effects of different Courant numbers, artificial dissipation coefficients, and smoothing parameters are investigated. Numerical results in comparison with other classic preconditioning methods show a reduction of the iteration number required to compute the steady-state and unsteady solution. Results show about 55 ∼ 77 and 40 ∼ 72% reduction in iteration number for steady and unsteady flows simulation, respectively.

Optimal neutron population growth in accelerated Monte Carlo criticality calculations

We present a source convergence acceleration method for Monte Carlo criticality calculations. The method gradually increases the neutron population size over the successive inactive as well as active criticality cycles. This helps to iterate the fission source faster at the beginning of the simulation where the source may contain large errors coming from the initial cycle; and, as the neutron population size grows over the cycles, the bias in the source gets reduced. Unlike previously suggested acceleration methods that aim at optimisation of the neutron population size, the new method does not have any significant computing overhead, and moreover it can be easily implemented into existing Monte Carlo criticality codes. The effectiveness of the method is demonstrated on a number of PWR full-core criticality calculations using a modified SERPENT 2 code.

Coupled modified KdV equations, skew orthogonal polynomials, convergence acceleration algorithms and Laurent property

In this paper, we show that the coupled modified KdV equations possess rich mathematical structures and some remarkable properties. The connections between the system and skew orthogonal polynomials, convergence acceleration algorithms and Laurent property are discussed in detail.