Chebyshev Acceleration Research Articles

Descent methods for elastic body simulation on the GPU

We show that many existing elastic body simulation approaches can be interpreted as descent methods, under a nonlinear optimization framework derived from implicit time integration. The key question is how to find an effective descent direction with a low computational cost. Based on this concept, we propose a new gradient descent method using Jacobi preconditioning and Chebyshev acceleration. The convergence rate of this method is comparable to that of L-BFGS or nonlinear conjugate gradient. But unlike other methods, it requires no dot product operation, making it suitable for GPU implementation. To further improve its convergence and performance, we develop a series of step length adjustment, initialization, and invertible model conversion techniques, all of which are compatible with GPU acceleration. Our experiment shows that the resulting simulator is simple, fast, scalable, memory-efficient, and robust against very large time steps and deformations. It can correctly simulate the deformation behaviors of many elastic materials, as long as their energy functions are second-order differentiable and their Hessian matrices can be quickly evaluated. For additional speedups, the method can also serve as a complement to other techniques, such as multi-grid.

Mathematical modeling of two-phase compressible fluid filtration based on modified adaptive method of minimum amendments

The work objective is to build and investigate the modified adaptive method of minimum amendments (MAMMA) which is destined for the numerical simulation of the two-phase compressible fluid filtration in porous media. This approach allows overcoming the known use limitations of other methods of the finite-difference equations solution, such as: crucial differential pressures acting on the oil-and-water bearing formation; and the compressibility of the medium at the considerable gas content in the oil phase. An approximation method - an explicit one for defining the function of water saturation, and an implicit one for the pressure function computation - is selected as the research basis. When setting the initial boundary value problem and its sampling, the process of the two-phase compressible fluid filtration in the space-dimensional domain with the lateral area bounded below by the subface of stratum, and above - by the bed top, is considered. A two-layer iterative method of the variational type - a modified method of minimal amendments adapted for solving finite-difference equations of the two-phase compressible fluid with a non-selfadjoint operator under the most general assumptions on the properties of the grid-problem operator is built. It is shown that a MAMMA has the asymptotic convergence rate characteristic of the “classical” alternate triangular method that does not use the Chebyshev acceleration technique and can be applied to the problems with a self-adjoint operator. Numerical experiments have confirmed the high efficiency of MAMMA. It is established that to achieve the specified accuracy, the number of iterations at the MAMMA reduces to 3-20 times as compared to the method of Seidel and the overrelaxation method.

Benchmarks of 3D Laplace Equation Solvers in a Cubic Configuration for Streamer Simulation

The aim of this paper is to test a developed SOR R&B method using the Chebyshev accelerator algorithm to solve the Laplace equation in a cubic 3D configuration. Comparisons are made in terms of precision and computing time with other elliptic equation solvers proposed in the open source LIS library. The first results, obtained by using a single core on a HPC, show that the developed SOR R&B method is efficient when the spectral radius needed for the Chebyshev acceleration is carefully pre-estimated. Preliminary results obtained with a parallelized code using the MPI library are also discussed when the calculation is distributed over one hundred cores.

Spectral recycling strategies for the solution of nonlinear eigenproblems in thermoacoustics

SummaryIn this work, we consider the numerical solution of large nonlinear eigenvalue problems that arise in thermoacoustic simulations involved in the stability analysis of large combustion devices. We briefly introduce the physical modeling that leads to a nonlinear eigenvalue problem that is solved using a nonlinear fixed point iteration scheme. Each step of this nonlinear method requires the solution of a complex non‐Hermitian linear eigenvalue problem. We review a set of state‐of‐the‐art eigensolvers and discuss strategies to recycle spectral information from one nonlinear step to the next. More precisely, we consider the Jacobi–Davidson algorithm, the Implicitly Restarted Arnoldi method, the Krylov–Schur solver and its block‐variant, and the subspace iteration method with Chebyshev acceleration. On a small test example, we study the relevance of the different approaches and illustrate on a large industrial test case the performance of the parallel solvers best suited to recycle spectral information for large‐scale thermoacoustic stability analysis. Copyright © 2015 John Wiley & Sons, Ltd.

Chebyshev acceleration of iterative refinement

It is well known that the FGMRES algorithm can be used as an alternative to iterative refinement and, in some instances, is successful in computing a backward stable solution when iterative refinement fails to converge. In this study, we analyse how variants of the Chebyshev algorithm can also be used to accelerate iterative refinement without loss of numerical stability and at a computational cost at each iteration that is less than that of FGMRES and only marginally greater than that of iterative refinement. A component-wise error analysis of the procedure is presented and numerical tests on selected sparse test problems are used to corroborate the theory.

Higher-order adaptive finite-element methods for Kohn–Sham density functional theory

We present an efficient computational approach to perform real-space electronic structure calculations using an adaptive higher-order finite-element discretization of Kohn–Sham density-functional theory (DFT). To this end, we develop an a priori mesh-adaption technique to construct a close to optimal finite-element discretization of the problem. We further propose an efficient solution strategy for solving the discrete eigenvalue problem by using spectral finite-elements in conjunction with Gauss–Lobatto quadrature, and a Chebyshev acceleration technique for computing the occupied eigenspace. The proposed approach has been observed to provide a staggering 100–200-fold computational advantage over the solution of a generalized eigenvalue problem. Using the proposed solution procedure, we investigate the computational efficiency afforded by higher-order finite-element discretizations of the Kohn–Sham DFT problem. Our studies suggest that staggering computational savings—of the order of 1000-fold—relative to linear finite-elements can be realized, for both all-electron and local pseudopotential calculations, by using higher-order finite-element discretizations. On all the benchmark systems studied, we observe diminishing returns in computational savings beyond the sixth-order for accuracies commensurate with chemical accuracy, suggesting that the hexic spectral-element may be an optimal choice for the finite-element discretization of the Kohn–Sham DFT problem. A comparative study of the computational efficiency of the proposed higher-order finite-element discretizations suggests that the performance of finite-element basis is competing with the plane-wave discretization for non-periodic local pseudopotential calculations, and compares to the Gaussian basis for all-electron calculations to within an order of magnitude. Further, we demonstrate the capability of the proposed approach to compute the electronic structure of a metallic system containing 1688 atoms using modest computational resources, and good scalability of the present implementation up to 192 processors.

Fast Convergence of Iterative Computation for Incompressible-Fluid Reynolds Equation

When a discretized Reynolds equation is to be solved iteratively at least three subjects have to be determined first. These are the iterative solution method, the size of gridwork for the numerical model, and the stopping criterion for the iterative computing. The truncation error analysis of the Reynolds equation is used to provide the stopping criterion, as well as to estimate an adequate grid size based on a required relative precision or grid convergence index. In the simulated lubrication analyses, the convergent rate of the solution is further improved by combining a simple multilevel computing, the modified Chebyshev acceleration, and multithreaded computing. The best case is obtained by using the parallel three-level red-black successive-over-relaxation (SOR) with Chebyshev acceleration. The speedups of the best case relative to the case using sequential SOR with optimal relaxation factor are around 210 and 135, respectively, for the slider and journal bearing simulations.

Chebyshev acceleration of criticality calculations in boundary element applications to neutron diffusion

Abstract In this study, Chebyshev polynomial method has been applied in the acceleration of the outer iterations for the numerical solution of the two dimensional neutron diffusion equation by the boundary element method. Boundary element discretization of the diffusion equation results in a full and nonsymmetric coefficient matrix which is quite different than the sparse and symmetrical matrices of the finite element and finite difference methods to which Chebyshev acceleration has been classically applied. To assess the merit of the Chebyshev method in the case of boundary element discretization of the multigroup diffusion equation constitutes the main objective of the study. Numerical experimentation has established that the Chebyshev acceleration is effective also in case of boundary element discretization despite the dissimilarity of the coefficient matrices compared to the more conventional methods.

A nonhydrostatic numerical model for density stratified flow and its applications

A modular numerical model was developed for simulating density-stratified flow in domains with irregular bottom topography. The model was designed for examining interactions between stratified flow and topography, e.g., tidally driven flow over two-dimensional sills or internal solitary waves propagating over a shoaling bed. The model was based on the non-hydrostatic vorticity-stream function equations for a continuously stratified fluid in a rotating frame. A self-adaptive grid was adopted in the vertical coordinate, the Alternative Direction Implicit (ADI) scheme was used for the time marching equations while the Poisson equation for stream-function was solved based on the Successive Over Relaxation (SOR) iteration with the Chebyshev acceleration. The numerical techniques were described and three applications of the model were presented.

Chebyshev acceleration for SOR-like method

There are several proposals for the generalization of Young’s successive over-relaxation (SOR) method to solve the saddle-point problem or augmented system. The most practical version is the SOR-like method (G.H. Golub et al., BIT, 41, 71–85, 2001), which was further studied by Li et al. (Int. J. Comput. Math., 81, 749–765, 2004) who found that the iteration matrix of the SOR-like method has no complex eigenvalues only under certain conditions. Motivated by the results of Li and co-authors, we consider the Chebyshev acceleration of the SOR-like method (GSOR-SI). First, the convergence of the GSOR-SI method is given. Secondly, it is shown that the asymptotic rate of the convergence of the GSOR-SI method is much larger than that of the SOR-like method, which indicates that the GSOR-SI method has a faster rate of convergence than the SOR-like method. Finally, numerical comparisons are given which show the GSOR-SI method is indeed faster than the SOR-like method.

Recent Advances in Spectral Nodal Methods for Numerically Solving Neutron‐Diffusion Eigenvalue Problems

In this paper we present the recent advances in spectral nodal methods for numerically solving neutron-diffusion eigenvalue problems in Cartesian geometry. For one‐dimensional two‐energy group diffusion criticality problems, we describe the use of nonconventional albedo boundary conditions that substitute approximately the reflective properties of the nonmultiplying media around the core. We also present the Chebyshev acceleration scheme for the power method used in the outer iterations. For three‐dimensional one‐speed diffusion eigenvalue problems, we present a spectral nodal method that is based on transverse‐integrating the diffusion equation separately in X, Y, and Z directions, and then considering flat approximations for the transverse leakage terms. Numerical results for typical model problems are given to illustrate the accuracy.

Parallel computation of polynomials with minimal uniform norm and its application to large eigenproblems

We investigate the parallelization of an algorithm that computes the polynomial with minimal uniform norm on polygonal domains. The obtained polynomial is used to accelerate large non-Hermitian eigenvalue problems. We report the performance results obtained on the machine Paragon and compare this method with the one based on Chebyshev acceleration techniques.

A Dynamical Pseudo-Spectral Domain Decomposition Technique: Application to Viscous Compressible Flows

A dynamical spectral domain decomposition method is presented. In each subdomain a transformation of coordinate is used. Both the locations of the interfaces and the parameters of the mappings are dynamically adapted by minimizing theHω2-norm of the calculated solution. We show on some functions that the total norm of the Chebyshev series depends on the location of the interfaces. Moreover, there exists a minimum that defines the best location of the interface. This defines a dynamical generation of Chebyshev collocation points. This numerical method is applied on partial differential equations and it is shown that both the overall accuracy and the matching at the interfaces are improved with respect to a fixed interface calculation. This algorithm is then used for the numerical solution of the time-dependent full Navier–Stokes equations. The solution technique consists in a Fourier–Chebyshev collocation method combined with a matching method. The computational domain is decomposed into subdomains in the vertical direction. In each subdomain a coordinate transform is used and the locations of the interfaces are dynamically determined. The elliptic problems coming from the viscous terms are solved by means of the Chebyshev acceleration method. Density is matched with an upwind procedure whereas the velocities, the temperature, and the concentration are handled with the influence matrix method. Numerical examples are carried out on the compressible Kelvin–Helmholtz and Rayleigh–Taylor flows.

Resonant frequencies and field distributions for the shielded uniaxially anisotropic dielectric resonator by the FD-SIC method

New formulations for resonant modes of a shielded uniaxially anisotropic dielectric resonator (DR), such as sapphire, are proposed. They are solved by the finite-difference and simultaneous iteration with the Chebyshev (FD-SIC) acceleration method. Like an isotropic DR cavity, one azimuthal field is used for azimuthally invariant TM or TE modes and two TM fields are used for azimuthally variant hybrid modes. It is shown that the governing equation for TE modes is the same as that for the isotropic DR case. For TM and hybrid modes, more general /spl psi/(=rH/sub /spl phi//) and H/sub r/-H/sub z/ formulations than those for the isotropic DR are derived, respectively. Cylindrical cavities loaded with a rod or ring DR can be easily modeled and analyzed by the present method. Resonant frequencies and field distributions can be accurately and efficiently obtained. Numerical results of resonant frequencies of rod sapphire DR cavities are compared to those by the mode-matching method in the literature to verify the present approach. The electric- and magnetic-field distributions are also presented for hybrid modes of the uniaxially anisotropic DR cavity.

Computing the field of values and pseudospectra using the Lanczos method with continuation

The field of values and pseudospectra are useful tools for understanding the behaviour of various matrix processes. To compute these subsets of the complex plane it is necessary to estimate one or two eigenvalues of a large number of parametrized Hermitian matrices; these computations are prohibitively expensive for large, possibly sparse, matrices, if done by use of the QR algorithm. We describe an approach based on the Lanczos method with selective reorthogonalization and Chebyshev acceleration that, when combined with continuation and a shift and invert technique, enables efficient and reliable computation of the field of values and pseudospectra for large matrices. The idea of using the Lanczos method with continuation to compute pseudospectra is not new, but in experiments reported here our algorithm is faster and more accurate than existing algorithms of this type.

Chebyshev acceleration techniques for large complex non hermitian eigenvalue problems

We propose an extension of the Arnoldi-Chebyshev algorithm to the large complex non Hermitian case. We demonstrate the algorithm on two applied problems.

Parallel implementation of a Monte Carlo particle simulation coupled to Maxwell's equations

AbstractPMC‐3D, a parallel three‐dimensional (3‐D) Monte Carlo device simulator for multicomputers is discussed. The parallel algorithm is and extension of the standard Monte Carlo device simulation model in three dimensions, that combines the stockastic Monte Carlo particle simulation method with a solution of Poisson equation for quasi‐static problems, and full Maxwell's equations for microwave or electro‐optic phenomena. While our implementation of the Poisson solver is based on an iterative method that uses and odd/even ordering with Chebyshev acceleration, a simple leapfrog method based upon the computational unit of Yee cell is used in a finite‐difference, time‐domain (FDTD) formulation to solve Maxwell's equations.The parallel algorithms were implemented on a 1024‐node distributed memory nCUBE multicomputer. In this work, we present the application of PMC‐3D to generate the static current‐Voltage characteristics of GaAs MESFET, and to simulate photogenerated femtosecond electrical transients produced in coplanar stripline on a GaAs substrate.

A two grid acceleration scheme for the nodal diffusion equation in hexagonal geometry

A two grid acceleration scheme for the solution of the nodal diffusion equation in hexagonal geometry has been formulated. The proposed scheme also involves the linear shifting of the keff values during each iteration in both the coarse nodes as well as the fine nodes for accelerating the initial stages of the solution. The partially converged solutions of the nodal diffusion equation in the coarse nodes and fine nodes are used to predict a parameter for acceleration of the flux and fission source convergence in the subsequent iterations in the fine node grid. The effectiveness of the iteration scheme was tested by incorporating it in the NODHEX program which solves the multigroup diffusion equation in 2-D by the Nodal Expansion Method. The VVER-1000, SNR-300 reactor core benchmarks and a proposed 500 MWe LMFBR were analysed. The results were compared with the results of the calculation using the Chebyshev two parameter acceleration scheme for flux convergence and without any acceleration scheme in the NODHEX program. The two grid acceleration scheme is found to be in most cases more efficient than the commonly used Chebyshev acceleration scheme.

Analysis of metallic waveguides with rectangular boundaries by using the finite-difference method and the simultaneous iteration with the Chebyshev acceleration

A numerical procedure based on the finite-difference method and simultaneous iteration of the power method in conjunction with the Chebyshev acceleration technique is utilized to analyze the metallic waveguides. Due to the efficiency of the present sparse matrix eigenproblem solver, lots of unknowns can be used in the domains of the waveguide cross-sections. Therefore, accurate cutoff wavenumbers or frequencies can be obtained by using the simple finite-difference method for the commonly used metallic waveguides such as the L-shaped, single-ridged, double-ridged, and rectangular coaxial waveguides. Some discrepancies with the numerical results in the recent literature are found and detailed discussions are provided to verify the correctness of the present results. >

Finite-difference analysis of dielectric-loaded cavities using the simultaneous iteration of the power method with the Chebyshev acceleration technique

A numerical procedure using the finite-difference technique, simultaneous iteration based on the power method, and the Chebyshev-polynomial preconditioning is proposed to analyze dielectric-loaded cavities. The merit of this method is that no matrix inversions are invoked and the convergence rate of the power method is greatly accelerated by the preconditioning. The TE, TM, and hybrid modes in axisymmetrical cavities loaded with a rod or ring dielectric resonator are analyzed. For the hybrid modes, an H/sub r/-H/sub z/ formulation is proposed. Accurate numerical results are obtained efficiently and no spurious solutions are found by the present method.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>