Generalized Minimal Residual Research Articles

Multistep matrix splitting iteration preconditioning for singular linear systems

Multistep matrix splitting iterations serve as preconditioning for Krylov subspace methods for solving singular linear systems. The preconditioner is applied to the generalized minimal residual (GMRES) method and the flexible GMRES (FGMRES) method. We present theoretical and practical justifications for using this approach. Numerical experiments show that the multistep generalized shifted splitting (GSS) and Hermitian and skew-Hermitian splitting (HSS) iteration preconditioning are more robust and efficient compared to standard preconditioners for some test problems of large sparse singular linear systems.

Finite Volume Method for Pricing European and American Options under Jump-Diffusion Models

AbstractA class of finite volume methods is developed for pricing either European or American options under jump-diffusion models based on a linear finite element space. An easy to implement linear interpolation technique is derived to evaluate the integral term involved, and numerical analyses show that the full discrete system matrices are M-matrices. For European option pricing, the resulting dense linear systems are solved by the generalised minimal residual (GMRES) method; while for American options the resulting linear complementarity problems (LCP) are solved using the modulus-based successive overrelaxation (MSOR) method, where the H+-matrix property of the system matrix guarantees convergence. Numerical results are presented to demonstrate the accuracy, efficiency and robustness of these methods.

A fast and automatic full-potential finite volume solver on Cartesian grids for unconventional configurations

To meet the requirements of fast and automatic computation of subsonic and transonic aerodynamics in aircraft conceptual design, a novel finite volume solver for full potential flows on adaptive Cartesian grids is developed in this paper. Cartesian grids with geometric adaptation are firstly generated automatically with boundary cells processed by cell-cutting and cell-merging algorithms. The nonlinear full potential equation is discretized by a finite volume scheme on these Cartesian grids and iteratively solved in an implicit fashion with a generalized minimum residual (GMRES) algorithm. During computation, solution-based mesh adaptation is also applied so as to capture flow features more accurately. An improved ghost-cell method is proposed to implement the non-penetration wall boundary condition where the velocity-potential of a ghost cell is modified by an analytic method instead. According to the characteristics of the Cartesian grids, the Kutta condition is applied by specially computing the gradients on Kutta-faces without directly assigning the potential jump to cells adjacent wake faces, which can significantly improve the solution converging speed. The feasibility and accuracy of the proposed method are validated by several typical cases of sub/transonic flows around an ONERA M6 wing, a DLR-F4 wing-body, and an unconventional figuration of a blended wing body (BWB). The validation cases demonstrate a fast convergence with fully automatic grid treatment and computation, and the results suggest its capacity in application for aircraft conceptual design.

Frequency-domain optical tomographic image reconstruction algorithm with the simplified spherical harmonics (SP3) light propagation model

We introduce here the finite volume formulation of the frequency-domain simplified spherical harmonics model with n-th order absorption coefficients (FD-SPN) that approximates the frequency-domain equation of radiative transfer (FD-ERT). We then present the FD-SPN based reconstruction algorithm that recovers absorption and scattering coefficients in biological tissue. The FD-SPN model with 3rd order absorption coefficient (i.e., FD-SP3) is used as a forward model to solve the inverse problem. The FD-SP3 is discretized with a node-centered finite volume scheme and solved with a restarted generalized minimum residual (GMRES) algorithm. The absorption and scattering coefficients are retrieved using a limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm. Finally, the forward and inverse algorithms are evaluated using numerical phantoms with optical properties and size that mimic small-volume tissue such as finger joints and small animals.The forward results show that the FD-SP3 model approximates the FD-ERT (S12) solution within high accuracy; the average errors in the phase (<3.7%) and the amplitude (<7.1%) of the partial current at the boundary are reported. From the inverse results we find that the absorption and scattering coefficient maps are more accurately reconstructed with the SP3 model than those with the SP1 model. Therefore, this work shows that the FD-SP3 is an efficient model for optical tomographic imaging of small-volume media with non-diffuse properties both in terms of computational time and accuracy as it requires significantly lower CPU time than the FD-ERT (S12) and also it is more accurate than the FD-SP1.

On the m-step two-parameter generalized Hermitian and skew-Hermitian splitting preconditioning method

In this paper, the non-Hermitian positive definite linear systems are solved via preconditioned Krylov subspace methods such as the generalized minimal residual (GMRES) method. To do so, the two-parameter generalized Hermitian and skew-Hermitian splitting (TGHSS) iteration method is applied to establish an m-step polynomial preconditioner. Some theoretical results are also given to investigate the convergence properties of the preconditioned method. Three numerical examples are presented to demonstrate the performance of the new method and to compare it with a recently proposed method.

Time-implicit gas-kinetic scheme

Abstract The construction of time-implicit gas-kinetic scheme (GKS) is investigated to enhance its efficiency in engineering applications, especially with large time scales. GKS is based on BGK equation to simulate compressible flows, which describes particles evolution with mesoscopic distribution function. To develop an implicit GKS, classical time-implicit techniques can be directly adopted, with the employment of upwind approximate factorization for Euler equations. However, the characteristic relation of BGK equation is quite different to that of macroscopic compressible Euler/Navier-Stokes equations, which may decrease the gain from implicit techniques, especially in high-speed flows. To further improve the performance, the preconditioned generalized minimum residual (GMRES) based on numerical Jacobian matrix is considered, and the kinetic flux vector splitting (KFVS) is suggested to generate the precondition matrix. The proposed new time-implicit GKS, GMRES-GKS shows great improvements in temporal accuracy, resolution and convergence in several typical high-speed benchmark tests, ranging from transonic to hypersonic, steady to unsteady and inviscid to turbulent flows.

AOR–Uzawa iterative method for a class of complex symmetric linear system of equations

In this paper, based on accelerated overrelaxation (AOR) method and Uzawa method, we present AOR–Uzawa iterative method for solving a broad class of complex symmetric linear systems. We investigate its convergence properties. In addition, the resulting AOR–Uzawa preconditioner is used to precondition Krylov subspace methods such as the restarted generalized minimal residual (GMRES) method for solving the real equivalent formulation of the complex symmetric system. Finally, two numerical examples are given to show the effectiveness of the proposed method.

Numerical solution of 2 × 2 block linear systems by block Gram–Schmidt methods

ABSTRACTIn this paper we propose and analyse some algorithms for solving block linear systems which are based upon the block Gram–Schmidt method. In particular, we prove that the algorithm BCGS2 (Reorthogonalized Block Classical Gram–Schmidt) using Householder Q–R decomposition implemented in floating-point arithmetic is backward stable, under the mild assumptions. Numerical tests were done in MATLAB to illustrate our theoretical results. A particular emphasis is on symmetric saddle-point problems, which arise in many important practical applications. We compare the results with the generalized minimal residual (GMRES) algorithm.

The inexact-Newton via GMRES subspace method without line search technique for solving symmetric nonlinear equations

In this paper, we propose an inexact-Newton via GMRES (generalized minimal residual) subspace method without line search technique for solving symmetric nonlinear equations. The iterative direction is obtained by solving the Newton equation of the system of nonlinear equations with the GMRES algorithm. The global convergence and local superlinear convergence rate of the proposed algorithm are established under some reasonable conditions. Finally, the numerical results are reported to show the effectiveness of the proposed algorithm.

Convergence rate of GMRES on tridiagonal block Toeplitz linear systems

Iterative methods such as generalized minimal residual (GMRES) method are used to solve large sparse linear systems. This paper is considered the GMRES method for solving tridiagonal block Toeplitz linear systems with diagonal blocks, and establishes upper bounds for GMRES residuals. The coefficient matrix becomes an m-tridiagonal Toeplitz matrix, and tridiagonal toeplitz systems are subcases of these systems. Also, we show that the GMRES method on linear system computes the exact solution in at most N steps.

Anderson acceleration of the Jacobi iterative method: An efficient alternative to Krylov methods for large, sparse linear systems

We employ Anderson extrapolation to accelerate the classical Jacobi iterative method for large, sparse linear systems. Specifically, we utilize extrapolation at periodic intervals within the Jacobi iteration to develop the Alternating Anderson–Jacobi (AAJ) method. We verify the accuracy and efficacy of AAJ in a range of test cases, including nonsymmetric systems of equations. We demonstrate that AAJ possesses a favorable scaling with system size that is accompanied by a small prefactor, even in the absence of a preconditioner. In particular, we show that AAJ is able to accelerate the classical Jacobi iteration by over four orders of magnitude, with speed-ups that increase as the system gets larger. Moreover, we find that AAJ significantly outperforms the Generalized Minimal Residual (GMRES) method in the range of problems considered here, with the relative performance again improving with size of the system. Overall, the proposed method represents a simple yet efficient technique that is particularly attractive for large-scale parallel solutions of linear systems of equations.

Pipelined, Flexible Krylov Subspace Methods

We present variants of the Conjugate Gradient (CG), Conjugate Residual (CR), and Generalized Minimal Residual (GMRES) methods which are both pipelined and flexible. These allow computation of inner products and norms to be overlapped with operator and nonlinear or nondeterministic preconditioner application.The methods are hence aimed at hiding network latencies and synchronizations which can become computational bottlenecks in Krylov methods on extreme-scale systems or in the strong-scaling limit. The new variants are not arithmetically equivalent to their base flexible Krylov methods, but are chosen to be similarly performant in a realistic use case, the application of strong nonlinear preconditioners to large problems which require many Krylov iterations. We provide scalable implementations of our methods as contributions to the PETSc package and demonstrate their effectiveness with practical examples derived from models of mantle convection and lithospheric dynamics with heterogeneous viscosity structure. These represent challenging problems where multiscale nonlinear preconditioners are required for the current state-of-the-art algorithms, and are hence amenable to acceleration with our new techniques. Large-scale tests are performed in the strong-scaling regime on a contemporary leadership supercomputer, where speedups approaching, and even exceeding $2\times$ can be observed. We conclude by analyzing our new methods with a performance model targeted at future exascale machines.

A low frequency elastodynamic fast multipole boundary element method in three dimensions

This paper presents a fast multipole boundary element method (FMBEM) for the 3-D elastodynamic boundary integral equation in the `low frequency' regime. New compact recursion relations for the second-order Cartesian partial derivatives of the spherical basis functions are derived for the expansion of the elastodynamic fundamental solutions. Numerical solution is achieved via a novel combination of a nested outer---inner generalized minimum residual (GMRES) solver and a sparse approximate inverse preconditioner. Additionally translation stencils are newly applied to the elastodynamic FMBEM and an implementation of the 8, 4 and 2-box stencils is presented, which is shown to reduce the number of translations per octree level by up to $$60\,\%$$60%. This combination of strategies converges 2---2.5 times faster than the standard GMRES solution of the FMBEM. Numerical examples demonstrate the algorithmic and memory complexities of the model, which are shown to be in good agreement with the theoretical predictions.

High order finite difference method for time-space fractional differential equations with Caputo and Riemann-Liouville derivatives

We consider high order finite difference methods for two-dimensional fractional differential equations with temporal Caputo and spatial Riemann-Liouville derivatives in this paper. We propose a scheme and show that it converges with second order in time and fourth order in space. The accuracy of our proposed method can be improved by Richardson extrapolation. Approximate solution is obtained by the generalized minimal residual (GMRES) method. A preconditioner is proposed to improve the efficiency for the implementation of the GMRES method.

Parallel GMRES solver for fast analysis of large linear dynamic systems on GPU platforms

In this paper, we propose an efficient parallel dynamic linear solver, called GPU-GMRES, for transient analysis of large linear dynamic systems such as large power grid networks. The new method is based on the preconditioned generalized minimum residual (GMRES) iterative method implemented on heterogeneous CPU–GPU platforms. The new solver is very robust and can be applied to power grids with different structures as well as for general analysis problems for large linear dynamic systems with asymmetric matrices. The proposed GPU-GMRES solver adopts the very general and robust incomplete LU based preconditioner. We show that by properly selecting the right amount of fill-ins in the incomplete LU factors, a good trade-off between GPU efficiency and convergence rate can be achieved for the best overall performance. Such tunable feature can make this algorithm very adaptive to different problems. GPU-GMRES solver properly partitions the major computing tasks in GMRES solver to minimize the data traffic between CPU and GPUs to enhance performance of the proposed method. Furthermore, we propose a new fast parallel sparse matrix–vector (SpMV) multiplication algorithm to further accelerate the GPU-GMRES solver. The new algorithm, called segSpMV, can enjoy full coalesced memory access compared to existing approaches. To further improve the scalability and efficiency, segSpMV method is further extended to multi-GPU platforms, which leads to more scalable and faster multi-GPU GMRES solver. Experimental results on the set of the published IBM benchmark circuits and mesh-structured power grid networks show that the GPU-GMRES solver can deliver order of magnitudes speedup over the direct LU solver, UMFPACK. The resulting multi-GPU-GMRES can also deliver 3–12×speedup over the CPU implementation of the same GMRES method on transient analysis.

Optimized dwell time algorithm in magnetorheological finishing

An optimized dwell time algorithm for magnetorheological finishing (MRF) is discussed. Based on the D-shape of the removal function of MRF, an optimized non-negative least-squares method is introduced to get dwell time from a linear matrix equation transferred from the de-convolution operation. Moreover, one kind of general surface error map extension is developed for any shape of optics to obtain a more precise optical surface in MRF. The simulation results show that the non-negative least-squares method of the constrained generalized minimal residual (GMRES) method with adaptive Tikhonov regulation is much faster to get highly stable dwell time distribution. In combination with the general surface error map extension, the peak to valley (PV) and root mean square (RMS) of the surface error of the diameter 400 mm converge from 184.41 and 21.26 nm to 7.56 and 0.632 nm with the consistency of the edge and the aperture inside. Finally, a fabricating experiment proves the validity of the optimized algorithm. Therefore, the algorithm developed and presented in this paper can facilitate the MRF process effectively.

A GPU-Accelerated Parallel Shooting Algorithm for Analysis of Radio Frequency and Microwave Integrated Circuits

This paper presents a new parallel shooting-Newton method based on a graphic processing unit (GPU)-accelerated periodic Arnoldi shooting solver (GAPAS) for fast periodic steady-state analysis of radio frequency/millimeter-wave integrated circuits. The new algorithm first explores a periodic structure of the state matrix by using a periodic Arnoldi algorithm for computing the resulting structured Krylov subspace in the generalized minimal residual (GMRES) solver. The resulting periodic Arnoldi shooting method is very amenable for massive parallel computing, such as GPUs. Second, the periodic Arnoldi-based GMRES solver in the shooting-Newton method is parallelized on the recent NVIDIA Tesla GPU platforms. We further explore CUDA GPUs features, such as coalesced memory access and overlapping transfers with computation to boost the efficiency of the resulting parallel GAPAS method. Experimental results from several industrial examples show that when compared with the state-of-the-art implicit GMRES method under the same accuracy, the new parallel shooting-Newton method can lead up to $8\times$ speedup.

The upper and lower bounds for generalized minimal residual method on a tridiagonal Toeplitz linear system

The generalized minimal residual (GMRES) method is widely used to solve a linear system . This paper establishes upper and lower bounds for GMRES residuals for solving an tridiagonal Toeplitz linear system. For normal matrix A, this problem has been studied previously by Li [Convergence of CG and GMRES on a tridiagonal Toeplitz linear system, BIT 47(3) (2007), 577–599.]. Also, Li and Zhang [The rate of convergence of GMRES on a tridiagonal Toeplitz linear system, Numer. Math. 112 (2009), pp. 267–293.] for non-symmetric matrix A, presented upper bound for GMRES residuals. In fact, our main goal in this paper is to find the upper and lower bounds for GMRES residuals on normal tridiagonal Toeplitz linear systems, and lower bounds for residuals of GMRES on solving non-normal tridiagonal Toeplitz linear systems.

General constraint preconditioning iteration method for singular saddle-point problems

For the singular saddle-point problems with nonsymmetric positive definite (1,1) block, we present a general constraint preconditioning (GCP) iteration method based on a singular constraint preconditioner. Using the properties of the Moore–Penrose inverse, the convergence properties of the GCP iteration method are studied. In particular, for each of the two different choices of the (1,1) block of the singular constraint preconditioner, a detailed convergence condition is derived by analyzing the spectrum of the iteration matrix. Numerical experiments are used to illustrate the theoretical results and examine the effectiveness of the GCP iteration method. Moreover, the preconditioning effects of the singular constraint preconditioner for restarted generalized minimum residual (GMRES) and quasi-minimal residual (QMR) methods are also tested.

Overlapping Schwarz Methods with a Standard Coarse Space for Almost Incompressible Linear Elasticity

Low-order finite element discretizations of the linear elasticity system suffer increasingly from locking effects and ill-conditioning, when the material approaches the incompressible limit, if only the displacement variables are used. Mixed finite elements using both displacement and pressure variables provide a well-known remedy, but they yield larger and indefinite discrete systems for which the design of scalable and efficient iterative solvers is challenging. Two-level overlapping Schwarz preconditioners for the almost incompressible system of linear elasticity, discretized by mixed finite elements with discontinuous pressures, are constructed and analyzed. The preconditioned systems are accelerated either by a GMRES (generalized minimum residual) method applied to the resulting discrete saddle point problem or by a PCG (preconditioned conjugate gradient) method applied to a positive definite, although extremely ill-conditioned, reformulation of the problem obtained by eliminating all pressure variables on the element level. A novel theoretical analysis of the algorithm for the positive definite reformulation is given by extending some earlier results by Dohrmann and Widlund. The main result of the paper is a bound on the condition number of the algorithm which is cubic in the relative overlap and grows logarithmically with the number of elements across individual subdomains but is otherwise independent of the number of subdomains, their diameters and mesh sizes, the incompressibility of the material, and possible discontinuities of the material parameters across the subdomain interfaces. Numerical results in the plane confirm the theory and also indicate that an analogous result should hold for the saddle point formulation, as well as for spectral element discretizations.