Gauss-Seidel Smoother Research Articles

An Efficient Multigrid Solver With Two-Color Plane Gauss-Seidel Smoother for 3D Low-Frequency Electromagnetic Modeling

An Efficient Multigrid Solver With Two-Color Plane Gauss-Seidel Smoother for 3D Low-Frequency Electromagnetic Modeling

A multigrid Waveform Relaxation Method for solving the Pennes bioheat equation

This work proposes a multigrid Waveform Relaxation Method (WRMG) that uses the finite difference method for the discretization to solve Pennes’ bioheat equation. There is no evidence in the literature of using the WRMG to solve this equation. The proposed algorithm is based on the red-black Gauss-Seidel smoother in space and the line Gauss-Seidel smoother in time. Verification of code is presented. The performance analysis of the multigrid method confirms the convergence and efficiency of the algorithm. The method, which favors parallel architecture, is tested in two-dimensional numerical experiments on an academic problem and for the thermal analysis of human skin.

A fast unsmoothed aggregation algebraic multigrid framework for the large-scale simulation of incompressible flow

Multigrid methods are quite efficient for solving the pressure Poisson equation in simulations of incompressible flow. However, for viscous liquids, geometric multigrid turned out to be less efficient for solving the variational viscosity equation. In this contribution, we present an Unsmoothed Aggregation Algebraic MultiGrid (UAAMG) method with a multi-color Gauss-Seidel smoother, which consistently solves the variational viscosity equation in a few iterations for various material parameters. Moreover, we augment the OpenVDB data structure with Intel SIMD intrinsic functions to perform sparse matrix-vector multiplications efficiently on all multigrid levels. Our framework is 2.0 to 14.6 times faster compared to the state-of-the-art adaptive octree solver in commercial software for the large-scale simulation of both non-viscous and viscous flow. The code is available at http://computationalsciences.org/publications/shao-2022-multigrid.html.

An efficient multigrid solver based on a four-color cell-block Gauss-Seidel smoother for 3D magnetotelluric forward modeling

Practical application of 3D magnetotelluric inversion requires efficient forward modeling of electromagnetic (EM) fields in the earth. To resolve realistic 3D structures, large computational domains and extremely large linear systems of equations are required. The iterative solvers, which are almost exclusively used to solve these systems, can be inefficient due to the abundant null space of the curl-curl operator. Multigrid (MG) solvers are considered a potentially efficient technique for solving such problems. However, due to the abundant null solution space and existence of the air layer, MG solvers can still converge slowly or even diverge. We have developed an efficient MG solver for finite-difference frequency-domain EM solution. In this algorithm, the excellent smoothing property of an efficient four-color cell-block Gauss-Seidel (GS) is exploited to remove the short-range errors effectively, and the interpolation and prolongation operators are used to handle the long-range errors. They work as a whole to speed the convergence of our algorithm remarkably. Because all of the nodes for the four-color cell-block GS are grouped into four colors and the edge components attached to different nodes in each color are completely decoupled, this can be used to develop a highly vectorized or parallelized algorithm. Another important property is that our algorithm is locally current divergence free, effectively eliminating spurious solutions in the null space of the curl-curl operator. The accuracy and efficiency of the algorithm are verified by comparing the numerical solutions obtained with our MG solver to those from the biconjugate gradient stabilized solver with different preconditioners based on synthetic models and a model from 3D inversion. Comparisons, in terms of iteration number and computational time, indicate that our algorithm is extremely stable and efficient relative to the other solvers. Our MG algorithm will be suitable for massively parallel computing as well.

Robust Preconditioning and Error Estimates for Optimal Control of the Convection--Diffusion--Reaction Equation with Limited Observation in Isogeometric Analysis

In this paper we analyze an optimization problem with limited observation governed by a convection--diffusion--reaction equation. Motivated by a Schur complement approach, we arrive at continuous norms that enable analysis of well-posedness and subsequent derivation of error analysis and a preconditioner that is robust with respect to the parameters of the problem. We provide conditions for inf-sup stable discretizations and present one such discretization for box domains with constant convection. We also provide a priori error estimates for this discretization. The preconditioner requires a fourth order problem to be solved. For this reason, we use Isogeometric Analysis as a method of discretization. To efficiently realize the preconditioner, we consider geometric multigrid with a standard Gauss-Seidel smoother as well as a new macro Gauss-Seidel smoother. The latter smoother provides good results with respect to both the geometry mapping and the polynomial degree.

Adaptive complex frequency with V-cycle GMRES for preconditioning 3D Helmholtz equation

Solving the Helmholtz equation has important applications in various areas, such as acoustics and electromagnetics. Using an iterative solver together with a proper preconditioner is key for solving large 3D Helmholtz equation. The performance of existing Helmholtz preconditioners usually deteriorates when the minimum spatial sampling density is small (approximately four points per wavelength [PPW]). To improve the efficiency of the Helmholtz preconditioner at a small minimum spatial sampling density, we have adopted a new preconditioner. In our scheme, the preconditioning matrix is constructed based on an adaptive complex frequency that varies with the minimum spatial sampling density in terms of the number of PPWs. Furthermore, the multigrid V-cycle with a GMRES smoother is adopted to effectively solve the corresponding preconditioning linear system. The adaptive complex frequency together with a GMRES smoother can work stably and efficiently at different minimum spatial sampling densities. Numerical results of four typical 3D models show that our scheme is more efficient than the multilevel GMRES method and shifted Laplacian with multigrid full V-cycle and a symmetric Gauss-Seidel smoother for preconditioning the 3D Helmholtz linear system, especially when the minimum spatial sampling density is large (approximately 120 PPW) or small (approximately 4 PPW).

Multilevel augmented Lagrangian solvers for overconstrained contact formulations

Multigrid methods for two-body contact problems are mostly based on special mortar discretizations, nonlinear Gauss-Seidel solvers, and solution-adapted coarse grid spaces. Their high computational efficiency comes at the cost of a complex implementation and a nonsymmetric master-slave discretization of the nonpenetration condition. Here we investigate an alternative symmetric and overconstrained segment-to-segment contact formulation that allows for a simple implementation based on standard multigrid and a symmetric treatment of contact boundaries, but leads to nonunique multipliers. For the solution of the arising quadratic programs, we propose augmented Lagrangian multigrid with overlapping block Gauss-Seidel smoothers. Approximation and convergence properties are studied numerically at standard test problems.

Fast multigrid solution of high-order accurate multiphase Stokes problems

A fast multigrid solver is presented for high-order accurate Stokes problems discretised by local discontinuous Galerkin (LDG) methods. The multigrid algorithm consists of a simple V-cycle, using an element-wise block Gauss-Seidel smoother. The efficacy of this approach depends on the LDG pressure penalty stabilisation parameter -- provided the parameter is suitably chosen, numerical experiment shows that: (i) for steady-state Stokes problems, the convergence rate of the multigrid solver can match that of classical geometric multigrid methods for Poisson problems; (ii) for unsteady Stokes problems, the convergence rate further accelerates as the effective Reynolds number is increased. An extensive range of two- and three-dimensional test problems demonstrates the solver performance as well as high-order accuracy -- these include cases with periodic, Dirichlet, and stress boundary conditions; variable-viscosity and multi-phase embedded interface problems containing density and viscosity discontinuities several orders in magnitude; and test cases with curved geometries using semi-unstructured meshes.

An efficient multiscale‐like multigrid computation for 2D convection‐diffusion equations on nonuniform grids

An efficient multiscale‐like multigrid (MSLMG) method combined with a high‐order compact (HOC) difference scheme on nonuniform grids is presented to solve the two‐dimensional (2D) convection‐diffusion equations. The discrete systems with given appropriate initial solutions on two finest grids are solved to obtain the MSLMG solutions with discretization‐level accuracy by performing fewer multigrid cycles; it is implemented with alternating line Gauss–Seidel smoother, interpolation, and restriction operators on the nonuniform grids. Numerical experiments of boundary layer or local singularity problems are conducted to show that the proposed algorithm with the HOC scheme on nonuniform grids is efficient and effective to decrease the computational cost and time, and the computed approximation on the nonuniform grids has fourth order accuracy.

An efficient extrapolation full multigrid method for elliptic problems in two and three dimensions

An extrapolation full multigrid (EXFMG) method is proposed for solving the large linear systems arising from linear finite element discretization of two- (2D) and three-dimensional (3D) elliptic boundary value problems. A good initial guess on the next finer grid is constructed through combining Richardson extrapolation and quadratic FE interpolation for the numerical solutions on two-level of grids (current and previous grids). And the FE linear system on the next finer grid is solved to obtain the EXFMG solution with discretization-level accuracy by performing one or two multigrid cycles with Gauss–Seidel smoother. Moreover, Richardson extrapolation is further used to achieve higher-order accuracy. The numerical results of five typical elliptic problems including one 2D problem and four 3D problems are conducted to show the effectiveness and efficiency of the EXFMG algorithm.

Fast multigrid solvers for conforming and non-conforming multi-patch Isogeometric Analysis

Isogeometric Analysis is a high-order discretization method for boundary value problems that uses a number of degrees of freedom which is as small as for a low-order method. Standard isogeometric discretizations require a global parameterization of the computational domain. In non-trivial cases, the domain is decomposed into patches having separate parameterizations and separate discretization spaces. If the discretization spaces agree on the interfaces between the patches, the coupling can be done in a conforming way. Otherwise, non-conforming discretizations (utilizing discontinuous Galerkin approaches) are required. The author and his coworkers have previously introduced multigrid solvers for Isogeometric Analysis for the conforming case. In the present paper, these results are extended to the non-conforming case. Moreover, it is shown that the multigrid solves get even more powerful if the proposed smoother is combined with a (standard) Gauss–Seidel smoother.

Numerical analysis of Krylov multigrid methods for stationary advection-diffusion equation

This paper outlines the problem of applying multigrid methods for the stationary scalar advection-diffusion equation for the given advection filed. We assume that all functions are smooth enough to be represented by discretization methods. Two main discretization are considered: finite difference method for Dirichlet boundary conditions and pseudo-spectral method for the periodic boundary. We test the following smoothers for multigrid methods: Jacobi smoother, Gauss-Seidel smoother, Krylov subspace smoother (GMRES method with and without preconditioners). The analysis is performed in the space formed by the cross product of discretization parameters, diffusion coefficient values, multigrid levels and smoothers.We demonstrate that the most efficient strategy depends on parameter value and given velocity field. Best variants include Gauss-Seidel smoothers which is optimal for advection-dominated problem while multigrid method is used as a preconditioner for a Krylov method. Such methods can be used for spectral or pseudo-spectral methods where explicit dense matrix storage is impossible.

Multigrid Methods for Time-Fractional Evolution Equations: A Numerical Study

In this work, we develop an efficient iterative scheme for a class of nonlocal evolution models involving a Caputo fractional derivative of order $$\alpha (0,1)$$ in time. The fully discrete scheme is obtained using the standard Galerkin method with conforming piecewise linear finite elements in space and corrected high-order BDF convolution quadrature in time. At each time step, instead of solving the linear algebraic system exactly, we employ a multigrid iteration with a Gauss–Seidel smoother to approximate the solution efficiently. Illustrative numerical results for nonsmooth problem data are presented to demonstrate the approach.

OpenFOAM에서 사용하는 Gauss-Seidel Smoother의 로버스트 코드

This paper investigates how OpenFOAM matrix solver works based on mesh information of cell and face indices for each cell. The smoother code of Gauss-Seidel in OpenFOAM uses minimum information of start- and end-indices of owner face for each cell to make it efficient in computational time, but can give rise to incorrect calculation of matrix solver in a grid construction. To complement the weakness of Gauss-Seidel smoother, OpenFOAM provides diagonal incomplete LU (DILU) decomposition smoother. The DILU smoother is robust to the face-ordering but spends a lot of computational time to solve the matrix due to an inefficient usage of owner cell and neighbour cell for all faces. This paper proposes a robust modification of Gauss-Seidel smoother in OpenFOAM maintaining its computational efficiency. The present essence is to get perfect information of adjacent cell indices regardless of face ordering.

Robust multigrid solvers for the biharmonic problem in isogeometric analysis

In this paper, we develop multigrid solvers for the biharmonic problem in the framework of isogeometric analysis (IgA). In this framework, one typically sets up B-splines on the unit square or cube and transforms them to the domain of interest by a global smooth geometry function. With this approach, it is feasible to set up H2-conforming discretizations. We propose two multigrid methods for such a discretization, one based on Gauss–Seidel smoothing and one based on mass smoothing. We prove that both are robust in the grid size, the latter is also robust in the spline degree. Numerical experiments illustrate the convergence theory and indicate the efficiency of the proposed multigrid approaches, particularly of a hybrid approach combining both smoothers.

Geometric multigrid algorithms for elliptic interface problems using structured grids

In this work, we develop geometric multigrid algorithms for the immersed finite element methods for elliptic problems with interface (Chou et al. Adv. Comput. Math. 33, 149–168 2010; Kwak and Lee, Int. J. Pure Appl. Math. 104, 471–494 2015; Li et al. Numer. Math. 96, 61–98 2003, 2004; Lin et al. SIAM J. Numer. Anal. 53, 1121–1144 2015). We need to design the transfer operators between levels carefully, since the residuals of finer grid problems do not satisfy the flux condition once projected onto coarser grids. Hence, we have to modify the projected residuals so that the flux conditions are satisfied. Similarly, the correction has to be modified after prolongation. Two algorithms are suggested: one for finite element spaces having vertex degrees of freedom and the other for edge average degrees of freedom. For the second case, we use the idea of conforming subspace correction used for P1 nonconforming case (Lee 1993). Numerical experiments show the optimal scalability in terms of number of arithmetic operations, i.e., $\mathcal {O}(N)$ for $\mathcal {V}$ -cycle and CG algorithms preconditioned with $\mathcal {V}$ -cycle. In $\mathcal {V}$ -cycle, we used only one Gauss-Seidel smoothing. The CPU times are also reported.

MultiGrid Preconditioners for Mixed Finite Element Methods of the Vector Laplacian

Due to the indefiniteness and poor spectral properties, the discretized linear algebraic system of the vector Laplacian by mixed finite element methods is hard to solve. A block diagonal preconditioner has been developed and shown to be an effective preconditioner by Arnold et al. (Acta Numer 15:1–155, 2006). The purpose of this paper is to propose alternative and effective block diagonal and approximate block factorization preconditioners for solving these saddle point systems. A variable V-cycle multigrid method with the standard point-wise Gauss–Seidel smoother is proved to be a good preconditioner for the discrete vector Laplacian operator. The major benefit of our approach is that the point-wise Gauss–Seidel smoother is more algebraic and can be easily implemented as a black-box smoother. This multigrid solver will be further used to build preconditioners for the saddle point systems of the vector Laplacian. Furthermore it is shown that Maxwell’s equations with the divergent free constraint can be decoupled into one vector Laplacian and one scalar Laplacian equation.

Parallel Implementation of the Coordinates-partitioning Based Aggregation-type Algebraic Multigrid Preconditioners

The coordinates-partitioning based aggregation-type algebraic multigrid preconditioners have been proven to be very efficient in the solution of sparse linear systems with conjugate gradient iterations. In this paper, a parallel algorithm for the setup is provided and the parallelization of the preconditioning process is also considered. The parallel algorithm is based on a good property of the coordinates-partitioning based aggregation, that is, the aggregation process is performed in a fashion from the coarsest level to the finest step by step. Thus, the original adjacent graph is partitioned into a number of sub-graphs, where each sub-graph is related to a node in the coarsest level and is assigned to a processor. The aggregation process can then proceed forward on each processor independently from the assigned sub-graph. When this kind of multigrid preconditioner is applied in Krylov subspace iterations, only the computation on the coarsest level and the matrix-vector multiplications related to the smoothing on each level require communication for V- and W-cycle versions, and require only some extra communication related to dot products for the K-cycle version. The size of the derived linear system on the coarsest level can be controlled by the adjustable arguments and this system can be solved again in parallel with some preconditioned Krylov subspace iterations. The structure information related to the matrix-vector multiplications is invariable to the iterations and can be derived in the setup and be stored, and is used in the latter iterations. Finally, the parallel algorithm is validated for the Gauss-Seidel smoother and some kinds of popular multigrid cycles in solving sparse linear systems from some two-dimensional model partial equations with preconditioned conjugate gradient iterations. The results show that the parallel efficiencies of both the setup and the iteration processes are satisfied.

三维薄结构热传导问题分层二次元方程的多水平方法

When the finite element method is applied to analyze the 3D thin-walled structures, some thin hexahedral elements are usually used in order to reduce the number of elements, and the corresponding higher-order elements are preferred since they have some obvious advantages in the calculation accuracy, the anti-distortion degree and so on. However, they have much higher computational complexity than the lower-order (e.g., linear) elements and the coefficient matrix of the linear algebraic equation system is severely ill-conditioned. The convergence of the commonly used solvers will deteriorate with the increasing size of the problem. An efficient and robust multi-level method was presented for the hierarchical quadratic discretizations of 3D thin-walled structures through combination of two special local block Gauss-Seidel smoothers and the DAMG algorithm based on the distance matrix. Since a hierarchical basis is used, those algebraic criteria are not needed to judge the relationships between the unknown variables and the geometric node types, and the grid transfer operators are also trivial. This makes it easy to find the coarse level (linear element) matrix derived directly from the fine level matrix, and thus the overall efficiency is greatly improved. The numerical results verify the efficiency and robustness of the proposed method.

Preconditioners for the Nonconforming Voxel Edge Element Method

The nonconforming voxel finite element method builds a structured grid of nested cube elements. It has been used both with scalar (nodal) and vector (edge) elements. For large problems, iterative methods such as the conjugate gradient method are needed to solve the global matrix equation. Preconditioning the iterative method can greatly decrease the computation time. A geometric multigrid preconditioner is investigated, for the scattering of plane waves by conducting objects, using either a Gauss-Seidel smoother or a smoother that incorporates a projection onto the gradient subspace.