Convergence Acceleration Methods Research Articles

Enhanced and generalized one–step Neville algorithm: Fractional powers and access to the convergence rate

The recursive Neville algorithm allows one to calculate interpolating functions recursively. Upon a judicious choice of the abscissas used for the interpolation (and extrapolation), this algorithm leads to a method for convergence acceleration. For example, one can use the Neville algorithm in order to successively eliminate inverse powers of the upper limit of the summation from the partial sums of a given, slowly convergent input series. Here, we show that, for a particular choice of the abscissas used for the extrapolation, one can replace the recursive Neville scheme by a simple one-step transformation, while also obtaining access to subleading terms for the transformed series after convergence acceleration. The matrix-based, unified formulas allow one to estimate the rate of convergence of the partial sums of the input series to their limit. In particular, Bethe logarithms for hydrogen are calculated to 100 decimal digits. Generalizations of the method to series whose remainder terms can be expanded in terms of inverse factorial series, or series with half-integer powers, are also discussed.

A novel accelerated convergence method for solving adjoint equations based on modal reduction

The efficiency of adjoint-based aerodynamic shape optimization depends critically on the solution efficiency of adjoint equations. In this letter, we employ the Proper Orthogonal Decomposition (POD) method to analyze the adjoint field samples and project them from the physical space into a low-order modal space. Subsequently, the full-order adjoint equations are reduced to low-order equations using the POD modes. Thus, we can efficiently predict the initial values for pseudo-time marching, thereby accelerating the solution of adjoint equations. Results indicate that the high-order POD modes are crucial for constructing the low-dimensional system. Moreover, this method can be seamlessly integrated with our previously established Dynamic Mode Decomposition (DMD) acceleration method to form a POD+DMD acceleration approach. Application of this approach to the flow past a National Advisory Committee for Aeronautics 0012 airfoil demonstrates a noteworthy 80.9% reduction in iteration numbers when solving the adjoint equations. Even for the airfoil located on the upper boundary of sampling space, the number of iterations is still reduced by 72.6%. Therefore, we believe that the proposed method holds significant promise for improving the efficiency of adjoint-based aerodynamic shape optimization in future research.

The Guggenheim equation for a system of hard spheres and its generalization

In the present work, for the first time, the generalized Guggenheim equation for a system of hard spheres is obtained based on the use of the Euler accelerated convergence method. This equation allows taking into account an arbitrary number of known virial coefficients. For a metastable region, this equation is generalized to the case of taking into account the asymptotic behavior of the free energy at high densities. The resulting expression for the compressibility of the homogeneous phase of a system of hard spheres describes the data of a numerical experiment within their accuracy.

Lubrication approximation of pressure-driven viscoelastic flow in a hyperbolic channel

We investigate theoretically the steady incompressible viscoelastic flow in a hyperbolic contracting channel. The fluid viscoelasticity is modeled using the upper convected Maxwell (UCM), Oldroyd-B, Phan-Thien and Tanner (PTT), Giesekus, and the finite elasticity non-linear elastic dumbbell with the Peterlin approximation (FENE-P) models. We first develop the general governing equations for flow within a non-deformable channel whose cross section varies with the distance from the inlet. We then exploit the classic lubrication approximation, assuming a small aspect ratio of the channel to simplify the original governing equations. The final equations, which we formulate in terms of the stream unction, are then solved analytically using a high-order asymptotic scheme in terms of the Deborah number, De, and the formulas for the average pressure drop are derived up to eight orders in De. The accuracy of the original perturbation solution is enhanced and extended over a wide range of parameters by implementing a convergence acceleration method for truncated series. Furthermore, convergence of the transformed solutions for the average pressure drop is demonstrated. The validity and accuracy of the theoretical results is independently confirmed through comparison with numerical results from simulations performed using high-order finite differences and pseudospectral methods. The results reveal the decrease in the average pressure drop with increasing the Deborah number, the polymer viscosity ratio, and the ratio of the inlet to the outlet height. We also show that the fundamental UCM and Oldroyd-B models can predict the major viscoelastic phenomena for this type of internal and confined lubrication flows, while the effect of the rheological parameters of the PTT, Giesekus, and FENE-P models on the results is minor.

Estimation of the geometric measure of entanglement with Wehrl moments through artificial neural networks

In recent years, artificial neural networks (ANNs) have become an increasingly popular tool for studying problems in quantum theory, and in particular entanglement theory. In this work, we analyse to what extent ANNs can accurately predict the geometric measure of entanglement of symmetric multiqubit states using only a limited number of Wehrl moments (moments of the Husimi function of the state) as input, which represents partial information about the state. We consider both pure and mixed quantum states. We compare the results we obtain by training ANNs with the informed use of convergence acceleration methods. We find that even some of the most powerful convergence acceleration algorithms do not compete with ANNs when given the same input data, provided that enough data is available to train these ANNs. We also provide an experimental protocol for measuring Wehrl moments, which is state-independent. More generally, this work opens up perspectives for the estimation of entanglement measures and other SU(2)-invariant quantities, such as the Wehrl entropy, in a way that is more accessible in experiments than by means of full state tomography.

Self-sustained Oscillations and Limit Cycles in Rayleigh System with Cubic Return Force

An oscillatory system with an excitation mechanism as in a Rayleigh oscillator, but with a nonlinear (cubic) returning force, is investigated. Using the accelerated convergence method and the continuation procedure for the parameter, limit cycles are constructed and the amplitudes and periods of self-oscillations are calculated. This is done for a wide range of feedback coefficient values, in which this coefficient is not asymptotically small or large. The proposed iterative procedure allows to achieve the specified accuracy of calculations. The analysis of the features of the limit cycle caused by an increase in the self-excitation coefficient is carried out. The results obtained are compared with the self-oscillations of a classical Rayleigh oscillator with a linear returning force.

Using the Accelerated Convergence Method for Solving the Singular Sturm–Liouville Problem

This article is dedicated to the memory of L.D. Akulenko, with whom the author of the article worked for more than 40 years. Within the framework of the accelerated convergence method developed jointly, a number of classes of problems related to the Sturm–Liouville problems were solved. Based on the research results, several dozen articles and a generalizing monograph [1] were published. In this paper, we describe the adaptation of the method to solving singular Sturm–Liouville problems.

Approximation of the Tikhonov regularization parameter through Aitken's extrapolation

In the present work, we study the determination of the regularization parameter and the computation of the regularized solution in Tikhonov regularization, by the Aitken's extrapolation method. In particular, this convergence acceleration method is adjusted for the approximation of quadratic forms that appear in regularization methods, such as the generalized cross-validation method, the quasi-optimality criterion, the Gfrerer/Raus method and the Morozov's discrepancy principle. We present several numerical examples to illustrate the effectiveness of the derived estimates for approximating the regularization parameter for several linear discrete ill-posed problems and we compare the described method with further existing methods, for the determination of the regularized solution.

A modified dual time integration technique for aerodynamic quasi-static and dynamic stall hysteresis

Simulation of the aerodynamic stall phenomenon in both quasi-static and dynamic conditions requires expensive computational resources. The computations become even more costly for static stall hysteresis using an unsteady solver with very slow variation of angle of attack at low reduced frequencies. In an explicit time-marching solver that satisfies the low Courant number condition, that is, [Formula: see text], the computational cost for such simulations becomes prohibitive, especially at higher Reynolds numbers due to the presence of thin-stretched cells with large aspect ratio in the boundary layer. In this paper, a segregated solver method such as the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) is modified as a dual pseudo-time marching method so that the unsteady problem at each time step is reformulated as a steady state problem. The resulting system of equations in the discretized finite volume formulation is then reduced to zero or near-zero residuals using available convergence acceleration methods such as local time stepping, multi-grid acceleration and residual smoothing. The performance and accuracy of the implemented algorithm was tested for three different airfoils at low to moderate Reynolds numbers in both incompressible and compressible flow conditions covering both attached and separated flow regimes. The results obtained are in close agreement with the published experimental and computational results for both quasi-static and dynamic stall and have demonstrated significant savings in computational time.

A hybrid neutronics method with novel fission diffusion synthetic acceleration for criticality calculations

A novel Fission Diffusion Synthetic Acceleration (FDSA) method is developed and implemented as a part of a hybrid neutronics method for source convergence acceleration and variance reduction in Monte Carlo (MC) criticality calculations. The acceleration of the MC calculation stems from constructing a synthetic operator and solving a low-order problem using information obtained from previous MC calculations. By applying the P1 approximation, two correction terms, one for the scalar flux and the other for the current, can be solved in the low-order problem and applied to the transport solution. A variety of one-dimensional (1-D) and two-dimensional (2-D) numerical tests are constructed to demonstrate the performance of FDSA in comparison with the standalone MC method and the coupled MC and Coarse Mesh Finite Difference (MC-CMFD) method on both intended purposes. The comparison results show that the acceleration by a factor of 3–10 can be expected for source convergence and the reduction in MC variance is comparable to CMFD in both slab and full core geometries, although the effectiveness of such hybrid methods is limited to systems with small dominance ratios.

Monte Carlo Source Convergence Acceleration by Hybrid Multigroup and Continuous Energy Neutron Transport

A novel source convergence acceleration method for Monte Carlo eigenvalue calculations is proposed in this paper. The method consists of simulating the bulk of the inactive cycles with online-generated multigroup cross sections. Then the active cycles are simulated with continuous-energy cross sections to preserve full fidelity. The method was implemented in the Monte Carlo code SCONE and tested on several three-dimensional full-length assembly models. In some cases, the same multigroup cross sections were used for several spatially separated materials in order to limit statistical uncertainties. The method was shown to accelerate calculations by a factor of 2.5 to 5 at the cost of a slightly increased standard deviation in the flux distribution estimated across several independent simulations. The memory usage due to storing multigroup cross sections does not seem to be prohibitive for practical applications.

Forced propagation method for Monte Carlo fission source convergence acceleration in the RMC

In loosely coupled or large-scale problems with high dominance ratios, slow fission source convergence can take extremely long time, reducing Monte Carlo (MC) criticality calculation efficiency. Although various acceleration methods have been developed, some methods cannot reduce convergence times, whereas others have been limited to specific problem geometries. In this study, a new fission source convergence acceleration (FSCA) method, the forced propagation (FP) method, has been proposed, which forces the fission source to propagate and accelerate fission source convergence. Additionally, some stabilization techniques have been designed to render the method more practical. The resulting stabilized method was then successfully implemented in the MC transport code, and its feasibility and effectiveness were tested using the modified OECD/NEA, one-dimensional slab benchmark, and the Hoogenboom full-core problem. The comparison results showed that the FP method was able to achieve efficient FSCA.

On the Asymptotic Linear Convergence Speed of Anderson Acceleration, Nesterov Acceleration, and Nonlinear GMRES

We consider nonlinear convergence acceleration methods for fixed-point iteration $x_{k+1}=q(x_k)$, including Anderson acceleration (AA), nonlinear GMRES (NGMRES), and Nesterov-type acceleration (co...

ON THE EXISTENCE OF A CONDITIONALLY PERIODIC SOLUTION OF A QUASILINEAR SYSTEM DIFFERENTIAL EQUATION IN THE CRITICAL CASE

In the theory of nonlinear oscillations one often encounters conditionally periodic oscillations resulting from the superposition of several oscillations with frequencies incommensurable with each other. When finding a solution to a resonant quasilinear differential system in the form of a conditionally periodic function, the problem of a small denominator arises. Consequently, the proof of the existence and even more the construction of such a solution is not an easy task. In this article, drawing on the work of V.I. Arnold, I. Moser, and other researchers proved the existence and constructed a conditionally periodic solution of a second-order quasilinear differential system in the critical case. Accelerated convergence method by N.N. Bogolyubova, Yu.A. Mitropolsky, A.M. Samoylenko. The result can be applied to construct a conditionally periodic solution of specific differential systems.

Efficient topology optimization based on DOF reduction and convergence acceleration methods

This paper proposes a highly efficient topology optimization using two accelerated methods, which can reduce the degrees of freedom (DOFs) of the finite element equations and accelerate the iteration convergence of the topology optimization. For the DOF reduction, a method based on the empty elements and the displacement change during the topology iterations is presented to remove the DOFs from the finite element equations. For the convergence acceleration, a gray-scale suppression method is proposed to accelerate the polarization of design variables which accelerates the iteration convergence of the topology optimization. Three numerical examples including 2D and 3D cases are tested, and the results show that the proposed method can significantly improve the efficiency of the topology optimization and obtain the optimization results with the same accuracy. The computational time is only about 7% - 29% compared to the conventional topology optimization method.

Dipole Admittances Obtained via Convergence Acceleration Methods: Fast and Accurate Computations

Provided that the exact kernel is used, the application of simple and popular numerical methods to Hall?n-type integral equations for increasing numbers of basis functions (or points per wavelength) yields sequences of driving-point conductance and susceptance that can be extrapolated via properly selected convergence acceleration methods (CAMs). Such methods can lead to accurate final or converged results without the need for cumbersome and exhaustive runs or tests. Easy-to-apply CAMs are considered and highly accurate results are provided for the driving-point admittances of dipole antennas. The numerical results thus obtained are compared to well-known measured monopole admittances. These measured results are discussed in detail and some interpretations that allow a firm comparison are proposed. We thus combine good engineering practices (some of which are not novel) to obtain reliable theoretical values of drivingpoint admittance.

Automatic detection of small bowel tumors in wireless capsule endoscopy images using ensemble learning.

Wireless Capsule Endoscopy (WCE) is a minimally invasive diagnosis tool for lesion detection in the gastrointestinal tract, reaching places where conventional endoscopy is unable to. However, the significant amount of acquired data leads to difficulties in the diagnosis by the physicians; which can be eased with computer assistance. This paper addresses a method for the automatic detection of tumors in WCE by using a two-step based procedure: region of interest selection and classification. The first step aims to separate abnormal from normal tissue by using automatic segmentation based on a Gaussian Mixture Model (GMM). A modified version of the Anderson method for convergence acceleration of the expectation-maximization (EM) algorithm is proposed. The proposed features for both segmentation and classification are based on the CIELab color space, as a way of bypassing lightness variations, where the L component is discarded. Tissue variability among subjects, light inhomogeneities and even intensity differences among different devices can be overcome by using simultaneously features from both regions. In the second step, an ensemble system with partition of the training data with a new training scheme is proposed. At this stage, the gating network is trained after the experts have been trained decoupling the joint maximization of both modules. The partition module is also used at the test step, leading the incoming data to the most likely expert allowing incremental adaptation by preserving data diversity. This algorithm outperforms others based on texture features selected from Wavelets and Curvelets transforms, classified by a regular support vector machine (SVM) in more than 5%. This work shows that simpler features can outperform more elaborate ones if appropriately designed. In the current case, luminance was discarded to cope with saturated tissue, facilitating the color perception. Ensemble systems remain an open research field. In the current case, changes in both topology and training strategy have led to significant performance improvements. A system with this level of performance can be used in current clinical practice.

Transverse Oscillations of a Cantilever Rod of Rectangular Cross Section and Variable Thickness

A numerical-analytical algorithm for calculating (a type of accelerated convergence method) of eigenvalues and modes of transverse vibrations of a rod of variable cross-section is proposed. The algorithm is implemented as a software package in the Maple symbolic computations language. The use of the package was demonstrated by the example of solving problems of determining the eigen forms and natural frequencies of rods with different dependences of the cross section on the longitudinal distance.

Mode multigrid - A novel convergence acceleration method

This paper proposes a mode multigrid (MMG) method, and applies it to accelerate the convergence of the steady state flow on unstructured grids. The dynamic mode decomposition (DMD) method is used to analyze the convergence process of steady flow field according to the solution vectors from the previous time steps. Unlike the traditional multigrid method, we project the flowfield solutions from the physical space into the modal space, and truncate all the high-frequency modes but only the first-order mode are retained based on the DMD analysis. The real solutions in the physical space can be obtained simply by the inverse transformation from the modal space. The developed MMG method ingeniously avoids the complicated process of coarsening computational mesh, and does not need to make any change for the grid in physical space. Therefore, it is very convenient to be applied to any numerical schemes with just a little change for the flow solver, which is also suitable for unstructured grids and easy for parallel computing. Several typical test cases have been used to verify the effectiveness of the proposed method, which demonstrates that the MMG can dramatically reduce the number of iterative steps for the different mesh types, different accuracy of spatial discretization and different time-marching schemes. The method is 3 to 6 times faster than the baseline method while ensuring the computational accuracy.

Free Energy and the Equation of State of a System of Solid Spheres in Narrow Cylindrical Pores

In this work, the free energy and the equation of state of a system of solid spheres in narrow cylindrical pores are determined using the combined method of accelerated convergence of series. This method includes the Euler method and a method based on taking the behavior of the system into account at high densities near the dense packing and on the idea of the effective number of the nearest neighbors. The results are compared with the data of a computer experiment for three different values of pore size and in all cases good agreement between theory and experiment was obtained. These are much better than the results of virial expansion, and when the transverse pore sizes increase, better than the results found on the basis of a series in powers of pressure as well. The method makes it possible to estimate the limits of applicability of the method of convergence acceleration, based on the transition from a series in density to a series in terms of pressure. To do this, the density with close packing and the maximum allowable density found by the method of Euler are compared. If these densities are close, then only the Euler method can be used. In the case of a significant difference, a combined method of accelerated convergence is required.