Generalized Minimal Residual Research Articles

On the preconditioning of three-by-three block saddle point problems

We establish a new iterative method for solving a class of large and sparse linear systems of equations with three-by-three block coefficient matrices having saddle point structure. Convergence properties of the proposed method are studied in details and its induced preconditioner is examined for accelerating the convergence speed of generalized minimal residual (GMRES) method. More precisely, we analyze the eigenvalue distribution of the preconditioned matrix. Numerical experiments are reported to demonstrate the effectiveness of the proposed preconditioner.

Solving Coupled Cluster Equations by the Newton Krylov Method.

We describe using the Newton Krylov method to solve the coupled cluster equation. The method uses a Krylov iterative method to compute the Newton correction to the approximate coupled cluster amplitude. The multiplication of the Jacobian with a vector, which is required in each step of a Krylov iterative method such as the Generalized Minimum Residual (GMRES) method, is carried out through a finite difference approximation, and requires an additional residual evaluation. The overall cost of the method is determined by the sum of the inner Krylov and outer Newton iterations. We discuss the termination criterion used for the inner iteration and show how to apply pre-conditioners to accelerate convergence. We will also examine the use of regularization technique to improve the stability of convergence and compare the method with the widely used direct inversion of iterative subspace (DIIS) methods through numerical examples.

Convergence of a generalized PMHSS method for a class of singular block two-by-two linear systems

In this work, we introduce a generalization of the preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method, named as generalized PMHSS (GPMHSS) iteration method, to solve a class of singular block two-by-two system of linear equations. Theoretical analyses show that the GPMHSS iteration method converges unconditionally to the minimum norm least squares solution for any initial guess no matter the system is consistent or inconsistent. Besides, with the preconditioner derived from the GPMHSS iteration method, the preconditioned generalized minimal residual (GMRES) method also determines the minimum norm least squares solution of the consistent singular block two-by-two linear systems at breakdown. Numerical experiments are presented to show the effectiveness and the robustness of the GPMHSS iteration method and the corresponding preconditioner.

Scattering of seismic waves by three-dimensional large-scale hill topography simulated by a fast parallel IBEM

To solve seismic wave scattering by a large-scale three-dimensional (3-D) hill topography, a fast parallel indirect boundary element method (IBEM) is developed by proposing a new construction method for the wave field, modifying the generalized minimum residual (GMRES) algorithm and constructing an OpenMP plus MPI parallel model. The validations of accuracy and efficiency show that this method can solve 3-D seismic response of a large-scale hill topography for broadband waves, and overcome the weakness of large storage and low efficiency of the traditional IBEM. Based on this new algorithm architecture, taking the broadband scattering of plane SV waves by a large-scale Gaussian-shaped hill of thousands-meters height as an example, the influence of several important parameters is investigated, including the incident frequency, the incident angle and the height-width and length-width ratio of the hill. The numerical results illustrate that the amplification effect on the ground motion by a near-hemispherical hill is more significant than the narrow hill. For low-frequency waves, the scattering effect of the higher hill is more pronounced, and there is only a single peak near the top of the hill. However, for high-frequency waves, rapid spatial variation of displacement amplitude appears on the hill surface.

Implicit discrete ordinates discontinuous Galerkin method for radiation problems on shared-memory multicore CPU/many-core GPU computation architecture

The implicit discrete ordinates (SN ) discontinuous Galerkin (DG)/generalized minimal residual (GMRES) method has been implemented on shared-memory, multicore CPU/many-core GPU heterogeneous computation architecture via standard application programming interfaces (APIs) of open multiprocessing (OpenMP) and computer unified device architecture (CUDA). The compiler derivative based OpenMP parallelization has been performed to port the sequential SN DG/GMRES implementation to run in parallel on multiple CPU cores that share global memory. Benchmarked on a workstation comprising 12 Intel(R) Xeon(R) CPU cores, a parallel efficiency of ∼90% has been achieved for all reference problems. The ILUT preconditioning has been performed to improve algebraic conditions of associated linear systems, and it has been demonstrated that preconditioning has accelerated simulations of all reference problems by a factor of 10. Finally, it has been showcased that a K20m GPU has boosted the computation capacity of an Intel(R) Xeon(R) CPU core by a modest factor of 2–4 for all the reference problems.

Matrix-free multigrid solvers for phase-field fracture problems

In this work, we present a framework for the matrix-free solution to a monolithic quasi-static phase-field fracture model with geometric multigrid methods. Using a standard matrix-based approach within the Finite Element Method requires lots of memory, which eventually becomes a serious bottleneck. A matrix-free approach overcomes this problem and greatly reduces the amount of required memory, allowing to solve larger problems on available hardware. One key challenge is concerned with the crack irreversibility for which a primal–dual active set method is employed. Here, the active set values of fine meshes must be available on coarser levels of the multigrid algorithm. The developed multigrid method provides a preconditioner for a generalized minimal residual (GMRES) solver. This method is used for solving the linear equations inside Newton’s method for treating the overall nonlinear-monolithic discrete displacement/phase-field formulation. Several numerical examples demonstrate the performance and robustness of our solution technology. Mesh refinement studies, variations in the phase-field regularization parameter, iterations numbers of the linear and nonlinear solvers, and some parallel performances are conducted to substantiate the efficiency of the proposed solver for single fractures, multiple pressurized fractures, and a L-shaped panel test in three dimensions.

Scalable Block Preconditioners for Linearized Navier-Stokes Equations at High Reynolds Number

We review a number of preconditioners for the advection-diffusion operator and for the Schur complement matrix, which, in turn, constitute the building blocks for Constraint and Triangular Preconditioners to accelerate the iterative solution of the discretized and linearized Navier-Stokes equations. An intensive numerical testing is performed onto the driven cavity problem with low values of the viscosity coefficient. We devise an efficient multigrid preconditioner for the advection-diffusion matrix, which, combined with the commuted BFBt Schur complement approximation, and inserted in a 2×2 block preconditioner, provides convergence of the Generalized Minimal Residual (GMRES) method in a number of iteration independent of the meshsize for the lowest values of the viscosity parameter. The low-rank acceleration of such preconditioner is also investigated, showing its great potential.

Convergence acceleration for high-order shock-fitting methods in hypersonic flow applications with efficient implicit time-stepping schemes

The research on hypersonic boundary layer transition is a recent focal point in fluid dynamics. One of the effective numerical methods for transition study is the high-order shock-fitting method, which is especially suitable for dealing with shock-related interactions and removing post-shock oscillations as compared with shock-capturing methods. Currently, explicit time-stepping schemes are mainly used for shock-fitting methods. In comparison, implicit schemes have received less attention due to the extra complexity arising from dynamic shock and moving mesh, even though they can accelerate the convergence speeds for steady flows by up to an order of magnitude. In this paper, two implicit time-stepping methods, GMRES (generalized minimum residual) and LR (line-relaxation), are realized for the efficient calculation of steady flow fields using the 5th-order shock-fitting method. The gas models include both calorically perfect gas and 5-species thermal-chemical non-equilibrium gas required for high-speed and high-enthalpy flows. The results of test cases showed that the convergence rates of GMRES+LR and LR are 4 to 20 times faster than the explicit/semi-implicit Runge-Kutta schemes. Therefore, the usage of implicit GMRES and LR as time-stepping schemes raises the computational efficiency for obtaining steady boundary layer flows, which helps the important high-order shock-fitting methods play a greater role in the study of hypersonic boundary layer stability and transition.

An augmented immersed finite element method for variable coefficient elliptic interface problems in two and three dimensions

An augmented immersed finite element method is proposed for solving elliptic interface problems with discontinuous and variable coefficients. By introducing the jump of normal derivative of the solution along the interface as an augmented variable, the interface problem is solved iteratively on Cartesian meshes using the generalized minimal residual (GMRES) iterative method. Within each iteration, we need to solve a system of linear equations with the same symmetric and positive definite matrix but with different right-hand sides. The sequence of linear equations is solved efficiently by the multi-grid solvers since the set up of coarse grid matrices and decompositions is only computed once. Extensive numerical examples show that the number of GMRES iterations is nearly independent of the mesh size when a diagonal-like preconditioning is applied. Thus the computational work of the proposed method is linearly proportional to the number of grid nodes. Implementation details are given and extensive numerical examples in two and three dimensions are provided to show the accuracy and efficiency of the proposed method.

A dynamically load-balanced parallel p-adaptive implicit high-order flux reconstruction method for under-resolved turbulence simulation

We present a dynamically load-balanced parallel p-adaptive implicit high-order flux reconstruction method for under-resolved turbulence simulation. The high-order explicit first stage, singly diagonal implicit Runge–Kutta (ESDIRK) method is employed to circumvent the restriction on the time step size. The pseudo transient continuation is coupled with the matrix-free restarted generalized minimal residual (GMRES) method to solve the nonlinear equations at each stage, except the first one, of ESDIRK. We use the spectral decay smoothness indicator as the refinement/coarsening indicator for p-adaptation. A dynamic load balancing technique is developed with the aid of the open-source library ParMETIS. The trivial cost, compared to implicit time stepping, of mesh repartitioning and data redistribution enables us to conduct p-adaptation and load balancing every time step. An isentropic vortex propagation case is employed to study the impact of element weights used in mesh repartitioning on parallel efficiency. We apply the p-adaptive solver for implicit large eddy simulation (ILES) of the transitional flows over a cylinder when Reynolds number (Re) is 3900 and the SD7003 wing when Re is 60000. Numerical experiments demonstrate that a significant reduction in the run time (up to 70%) and total number of solution points (up to 76%) can be achieved with p-adaptation.

Matrix MOC: The method of characteristics in matrix form

Conventionally, tens to hundreds times of repeated characteristic sweeping is required for the Method of Characteristics (MOC) to converge within the inner-outer iteration scheme, even with the acceleration techniques like CMFD and CMR employed to improve the convergence rate. Therefore, the Matrix MOC, which converts the characteristic sweeping into a linear system, was proposed attempting to accelerate the characteristic sweeping itself. However, it is computationally intensive if not prohibitive to construct and store the coefficient matrices in the explicit form. This study proposes four principle numerical properties of the coefficient matrices in the Matrix MOC. Exploiting these properties reduces significantly the memory consumption and the computational time, making it possible to explicitly store the sparse coefficient matrices in CSR (Compressed Sparse Row) scheme for large problems. The multi-group coupling GMRES (Generalized Minimal RESidual) solver is then employed to solve the Matrix MOC equations of all groups simultaneously. Since the matrices are stored explicitly in CSR scheme, the operations involving sparse matrices and vectors are performed by taking advantage of the fully optimized Intel® MKL library. The Matrix MOC based 2D code named TIGER is developed, and verified by the BWR lattice benchmark and several variations of the C5G7 problem. Numerical results demonstrate that the Matrix MOC with the proposed numerical properties has adequate accuracy, and is feasible to solve a 2D small core problem on an ordinary PC with explicitly stored matrices. In addition, the optimal parameters of the multi-group coupling GMRES solver are investigated by sensitivity analysis to get the highest computational efficiency.

An implicit difference scheme with the KPS preconditioner for two-dimensional time–space fractional convection–diffusion equations

This paper deals with the numerical computation and analysis for a class of two-dimensional time–space fractional convection–diffusion equations. An implicit difference scheme is derived for solving this class of equations. It is proved under some suitable conditions that the derived difference scheme is stable and convergent. Moreover, the convergence orders of the scheme in time and space are also given. In order to accelerate the convergence rate, by combining the Kronecker product splitting (KPS) preconditioner with the generalized minimal residual (GMRES) method, a preconditioning strategy for implementing the difference scheme is introduced. Finally, several numerical examples are presented to illustrate the computational accuracy and efficiency of the methods.

Efficient Convergence for a Higher-Order Unstructured Finite Volume Solver for Compressible Flows

This Paper presents a three-dimensional higher-order-accurate finite volume algorithm for the solution of steady-state compressible flow problems. Higher-order accuracy is achieved by constructing a piecewise continuous representation of the average solution values using the -exact reconstruction scheme. The pseudo-transient continuation method is employed to reduce the solution of the discretized system of nonlinear equations into the solution of a series of linear systems, which are subsequently solved using the generalized minimal residual (GMRES) method. This Paper considers several preconditioning methods in conjunction with different matrix reordering algorithms and shows that the proposed preconditioner based on inner GMRES iterations can enhance the convergence speed and reduce the memory cost of the solver. Moreover, when starting from a lower-order solution as the initial condition, this Paper shows that ramping up the Courant–Friedrichs–Lewy (CFL) number accelerates the convergence rate. Finally, this Paper verifies the developed finite volume algorithm by solving a set of test problems, in which optimal solution convergence with mesh refinement is attained.

Numerical investigation of ensemble methods with block iterative solvers for evolution problems

<p style='text-indent:20px;'>The ensemble method has been developed for accelerating a sequence of numerical simulations of evolution problems. Its main idea is, by manipulating the time stepping and grouping discrete problems, to make all members in the same group share a common coefficient matrix. Thus, at each time step, instead of solving a sequence of linear systems each of which contains only one right-hand-side vector, the ensemble method simultaneously solves a single linear system with multiple right-hand-side vectors for each group. Such a system could be solved efficiently by using direct linear solvers when the problems are of small scale, as the same LU factorization would work for the entire group members. However, for large-scale problems, iterative linear solvers often have to be used and then this appealing advantage becomes not obvious. In this paper we systematically investigate numerical performance of the ensemble method with block iterative solvers for two typical evolution problems: the heat equation and the incompressible Navier-Stokes equations. In particular, the block conjugate gradient (CG) solver is considered for the former and the block generalized minimal residual (GMRES) solver for the latter. Our numerical results demonstrate the effectiveness and efficiency of the ensemble method when working together with these block iterative solvers.

Data-driven acceleration of photonic simulations

Designing modern photonic devices often involves traversing a large parameter space via an optimization procedure, gradient based or otherwise, and typically results in the designer performing electromagnetic simulations of a large number of correlated devices. In this paper, we investigate the possibility of accelerating electromagnetic simulations using the data collected from such correlated simulations. In particular, we present an approach to accelerate the Generalized Minimal Residual (GMRES) algorithm for the solution of frequency-domain Maxwell’s equations using two machine learning models (principal component analysis and a convolutional neural network). These data-driven models are trained to predict a subspace within which the solution of the frequency-domain Maxwell’s equations approximately lies. This subspace is then used for augmenting the Krylov subspace generated during the GMRES iterations, thus effectively reducing the size of the Krylov subspace and hence the number of iterations needed for solving Maxwell’s equations. By training the proposed models on a dataset of wavelength-splitting gratings, we show an order of magnitude reduction (~10–50) in the number of GMRES iterations required for solving frequency-domain Maxwell’s equations.

Sparse multiscale representation of Galerkin method for solving linear‐mixed Volterra‐Fredholm integral equations

This paper presents an efficient method for solving the linear‐mixed Volterra‐Fredholm equations using multiscale transformation. For this purpose, by changing the variables, the Fredholm‐Volterra equation is discretized using wavelet Galerkin method. This equation reduces to a set of linear algebraic equations by using the wavelet transform matrix and the operational matrix of integration. To reach the sparse coefficients matrix for having a reduction in the computational cost, thresholding is used. This sparse system solves by generalized minimal residual (GMRES) method. If the appropriate threshold selects, the number of nonzero coefficients reduces while the error will not be less than a certain amount. The convergence analysis has been investigated. The validity and applicability of the technique are illustrated by a series of numerical tests.

Field-of-values analysis of preconditioned linearized Rayleigh–Bénard convection problems

In this paper we use the notion of field-of-values (FOV) equivalence of matrices to study a class of block-triangular preconditioners for the fixed-point linearization of the Rayleigh–Bénard convection problem discretized with inf–sup stable finite element spaces. First, sufficient conditions on the nondimensional parameters of the problem are determined in order to establish the FOV-equivalence between the system matrix and the preconditioners. Four upper triangular block preconditioners belonging to the general proposed class are then considered. Numerical experiments show that the Generalized Minimal Residual (GMRES) convergence is robust with respect to the mesh size for these preconditioned systems. We also compare the performance of the different preconditioners in terms of computational time.

Sparsity-Based Short-Circuit Analysis of Power Distribution Systems With Inverter Interfaced Distributed Generators

This paper proposes a method for calculating short-circuit current in distribution systems with inverter interfaced distributed generators (IIDG). It is capable of calculating the short circuit in a multi-phase distribution systems based on the symmetrical components without obtaining the complete impedance matrix. Furthermore, it considers the load effect and the pre-fault state of the system for the fault analysis and is able to handle unbalanced systems with mutually coupled phases. The problem is formulated as an iterative linear system calculation and is solved by generalized minimal residual (GMRES) approach. The validity of the proposed algorithm is verified through the simulation of IEEE 13-node, 123-node, 478-node, 954-node, and 1430-node distribution systems. Simulation results demonstrate the effectiveness of the proposed algorithm and the advantages of using the GMRES method to calculate the short circuit current in a distribution system with IIDG.

Research progress of high-performance BEM and investigation on convergence of GMRES in local stress analysis of slender real thin-plate beams

Purpose Based on the error analysis, the authors proposed a new kind of high accuracy boundary element method (BEM) (HABEM), and for the large-scale problems, the fast algorithm, such as adaptive cross approximation (ACA) with generalized minimal residual (GMRES) is introduced to develop the high performance BEM (HPBEM). It is found that for slender beams, the stress analysis using iterative solver GMRES will difficult to converge. For the analysis of slender beams and thin structures, to enhance the efficiency of GMRES solver becomes a key problem in the development of the HPBEM. The purpose of this paper is study on the preconditioning method to solve this convergence problem, and it is started from the 2D BE analysis of slender beams. Design/methodology/approach The conventional sparse approximate inverse (SAI) based on adjacent nodes is modified to that based on adjacent nodes along the boundary line. In addition, the authors proposed a dual node variable merging (DNVM) preprocessing for slender thin-plate beams. As benchmark problems, the pure bending of thin-plate beam and the local stress analysis (LSA) of real thin-plate cantilever beam are applied to verify the effect of these two preconditioning method. Findings For the LSA of real thin-plate cantilever beams, as GMRES (m) without preconditioning applied, it is difficult to converge provided the length to height ratio greater than 50. Even with the preconditioner SAI or DNVM, it is also difficult to obtain the converged results. For the slender real beams, the iteration of GMRES (m) with SAI or DNVM stopped at wrong deformation state, and the computation failed. By changing zero initial solution to the analytical displacement solution of conventional beam theory, GMRES (m) with SAI or DNVM will not be stopped at wrong deformation state, but the stress error is still difficult to converge. However, by GMRES (m) combined with both SAI and DNVM preconditioning, the computation efficiency enhanced significantly. Originality/value This paper presents two preconditioners: DNVM and a modified SAI based on adjacent nodes along the boundary line of slender thin-plate beam. In the LSA, by using GMRES (m) combined with both DNVM and SAI, the computation efficiency enhanced significantly. It provides a reference for the further development of the 3D HPBEM in the LSA of real beam, plate and shell structures.

A Novel and Efficient Hybrid RSILU Preconditioner for the Parallel GMRES Solution of the Coarse Mesh Finite Difference Equations for Practical Reactor Simulations

The solution of the large, nonsymmetric, sparse linear system resulting from the discretization of the Boltzmann transport equation can be computationally intensive, even when innovative coarse mesh finite difference (CMFD) acceleration methods are applied on a high-performance computer. The research reported here describes the development and implementation of an innovative hybrid Reduced Symmetric Successive Over-Relaxation (RSOR) and Incomplete Lower Upper (ILU) (RSILU) preconditioner for the preconditioned Generalized Minimal RESidual (GMRES) solution of the three-dimensional, whole-core, pin-resolved transport calculation of a nuclear reactor core. The preconditioner was designed specifically to improve parallel computing capability and minimize the computational burden for solution of the CMFD method on a high-end computing platform. The hybrid RSILU preconditioner is applied to the preconditioned GMRES method without multicolor reordering by utilizing the ILU preconditioner for internal elements on each processor and the RSOR preconditioner for boundary elements that would require interprocessor communication. The construction of the RSILU preconditioner requires only that the diagonal elements be modified, factorized, and stored, which is identical to serial ILU. The computational cost of RSILU is minimized since the factorized diagonal block requires minimal data preparation time for interprocessor communication and exchanges information only once in parallel computation. This paper reports first the serial performance of RSOR, and numerical results show that the new proposed RSOR is an effective preconditioner even for serial computing applications. The parallel performance of RSILU is then assessed and compared to conventional multicolor ILU preconditioners. The results show that RSILU provides comparable convergence rates to the Standard Incomplete Lower Upper (SILU) preconditioners, but it is easier to implement since it does not require multicolor ordering. Although the required iterations of RSILU preconditioned GMRES increase as the number of processors increases, only slightly more iterations are required than the SILU preconditioner for a practical nuclear reactor application. The number of iterations required by the RSILU preconditioner increases only slightly and significantly less than the increased number of iterations required when using conventional Block-Jacobi Incomplete Lower Upper or Symmetric Successive Over-Relaxation preconditioners. Overall, the RSILU is shown to be an efficient and practical preconditioner for the GMRES method for improving the parallel computing performance for large-scale applications.