一种变系数扩散问题有限体积格式的高效预条件子

本文针对一类含变跳系数的扩散问题,在矩形网格下构造了一种节点型MACH类有限体积格式.将相应常跳系数辅助扩散问题离散格式的系数矩阵的逆作为其预条件子.利用该系数矩阵的特殊代数结构,通过降维处理技术和快速傅里叶变换等,为预条件子(该矩阵逆)的数学行为设计了一种低运算复杂度(O(Nln(N)))的直接法.数值实验验证了基于该预条件子的PCG算法的高效性和稳健性.     

一种求解H(curl)型椭圆问题的高效并行预条件子及并行实现

本文为一类H(curl)型椭圆问题的线性棱有限元方程,构造了一种基于节点辅助空间预条件子(HX预条件子)和基于简单粗空间的非重叠区域分解相结合的预条件子,并为该预条件子设计了并行算法,编制了基于MPI+OpenMP二级并行架构的并行程序.数值实验结果表明基于该预条件子的并行PCG法具有良好的算法可扩展能力和并行可扩展能力.     

求解Maxwell线性棱元鞍点系统的并行Uzawa算法

本文针对一类Maxwell方程组鞍点问题的第一类N啨d啨lec线性棱元离散系统,设计了一种基于节点辅助空间预条件子的并行Uzawa算法(HX-Uzawa-p)。数值实验结果表明,不论是对光滑系数还是对有无浮动子区域及有无内交叉点的跳系数情形,我们所设计的并行算法HX-Uzawa-p的迭代次数都基本不依赖于网格规模及系数跳幅,且具有很好的并行可扩展性。     

一种针对大波数Helmholtz方程的高性能并行预条件迭代求解算法

针对传统串行迭代法求解大波数Helmholtz方程存在效率低下且受限于单机内存的问题,提出了一种基于消息传递接口(Message Passing Interface,MPI) 的并行预条件迭代法。该算法利用复移位拉普拉斯算子对Helmholtz方程进行预条件处理,联合稳定双共轭梯度法和基于矩阵的多重网格法来求解预条件方程离散后的大规模线性系统,在Linux集群系统上基于 MPI环境实现了求解算法的并行计算,重点解决了多重网格的并行划分、信息传递和多重网格组件的构建问题。数值实验表明,对于大波数问题,提出的算法具有良好的并行加速比,相较于串行算法极大地提高了计算效率。     

自顶向下聚集型代数多重网格预条件的边权选择

针对基于图划分的自顶向下聚集型代数多重网格预条件,考察了利用METIS软件包进行多重网格构建的方法,并就该软件包只能处理整型权重,不能处理实型权重的问题,提出了一种将实型边权转化为整型边权的有效方法。之后将这种转化方法应用到METIS图划分软件中的边权选择,并用其给出了对自顶向下聚集型代数多重网格预条件的一种改进算法。通过对二维与三维模型偏微分方程离散所得稀疏线性方程组的数值实验表明,带边权的改进型算法大大提高了多重网格预条件共轭斜量法的迭代效率,特别是对各向异性问题,改进效果更加显著。     

求解二维椭圆型界面问题的外推完全多重网格法

针对二维椭圆型界面问题的离散化方程,应用外推插值技巧和样条插值方法在细网格层上构造合适的迭代初始值,加快V型多重网格法求解离散化系统的速度,设计了外推完全多重网格(EXFMG)法.数值实验表明新算法有效降低了迭代次数,计算量更少.     

针对对称对角占优线性系统的组合预条件算法

本文针对对角占优的对称矩阵(SDD)构成的稀疏线性系统,采用组合预处理技术从谱逼近角度分析并实现一种新型的预条件子.其与ILU类预条件子和AMG类预条件子相比,具有更高的并行可扩展性,满足通量守恒或者等效电阻原理.SDD矩阵通过数学上的规约手段,可以约化为标准的Laplace矩阵,其对应于图论中的无向图.基于此我们首先利用Ofer等提出的算法建立具有low stretch度量的一类生成树.然后采用树分解算法将生成树分解为子树,通过对子树选择合适的连接边进行加边修正得到相应的增广子图.最后将增广子图对应的Laplace矩阵转化为SDD矩阵,该矩阵即为原系数矩阵的预条件子.数值实验表明,与不完全Cholesky分解预条件子相比,该类预条件子更高效,其收敛速度对问题边界类型以及矩阵排序算法不敏感,并且其效率对矩阵规模增长不太敏感.     

求解鞍点问题的一类广义SSOR预条件子

本文研究了鞍点问题的预条件子.在SSOR型预处理方法的基础上,通过引入新的松弛参数,提出了一种广义的SSOR型预条件子,该预条件子需要选择一个预处理矩阵和2个待定参数.文中分析了预处理后系数矩阵特征值的性质及收敛性,最后用数值例子验证了新预条件子的有效性.     

自顶向下聚集型代数多重网格预条件的健壮性与参数敏感性研究

本文针对自顶向下聚集型代数多重网格预条件,进行了健壮性与参数敏感性研究。对从各向同性与各向异性偏微分方程边值问题离散所得的多种稀疏线性方程组,首先对问题规模敏感性进行了研究,并与基于强连接的经典聚集型算法进行了系统比较,发现虽然对沿不同坐标轴具有强各向异性的问题,基于坐标分割的自顶向下聚集型算法不如基于强连接的经典聚集算法,但对其它所有情形,自顶向下聚集型算法都具有明显优势,特别是在采用Jacobi光滑时,优势更加显著。之后,对最粗网格层的分割数与每次每个子图进行分割时的分割数这两个参数进行了敏感性分析,发现在采用Jacobi光滑求解五点差分离散所得的稀疏线性方程组时,自顶向下聚集型算法对这两个参数存在敏感性,虽然大部分情形下,迭代次数比较稳定,但在少量几种情形下,迭代次数明显增加。而对从九点差分离散得到的稀疏线性方程组,以及在采用Gauss-Seidel光滑的情况下,算法对这两个参数的选取不再具有敏感性,迭代次数都比较稳定。综合分析表明,自顶向下聚集型代数多重网格预条件具有较好的健壮性,特别是在采用Gauss-Seidel光滑,或采用九点差分离散时,健壮性表现更加充分。     

Cahn-Hilliard方程多重网格求解器收敛性分析

Cahn-Hilliard(CH)方程是相场模型中的一个基本的非线性方程，通常使用数值方法进行分析。在对CH方程进行数值离散后会得到一个非线性的方程组，全逼近格式(Full Approximation Storage, FAS)是求解这类非线性方程组的一个高效多重网格迭代格式。目前众多的求解CH方程主要关注数值格式的收敛性，而没有论证求解器的可靠性。文中给出了求解CH方程离散得到的非线性方程组的多重网格算法的收敛性证明，从理论上保证了计算过程的可靠性。针对CH方程的时间二阶全离散差分数值格式，利用快速子空间下降(Fast Subspace Descent, FASD)框架给出其FAS格式多重网格求解器的收敛常数估计。为了完成这一目标，首先将原本的差分问题转化为完全等价的有限元问题，再论证有限元问题来自一个凸泛函能量形式的极小化，然后验证能量形式及空间分解满足FASD框架假设，最终得到原多重网格算法的收敛系数估计。结果显示，在非线性情形下，CH方程中的参数ε对网格尺度添加了限制，太小的参数会导致数值计算过程不收敛。最后通过数值实验验证了收敛系数与方程参数及网格尺度的依赖关系。     

Towards a rigorously justified algebraic preconditioner for high-contrast diffusion problems

In this paper we present a new preconditioner suitable for solving linear systems arising from finite element approximations of elliptic PDEs with high-contrast coefficients. The construction of the preconditioner consists of two phases. The first phase is an algebraic one which partitions the degrees of freedom into “high” and “low” permeability regions which may be of arbitrary geometry. This partition yields a corresponding blocking of the stiffness matrix and hence a formula for the action of its inverse involving the inverses of both the high permeability block and its Schur complement in the original matrix. The structure of the required sub-block inverses in the high contrast case is revealed by a singular perturbation analysis (with the contrast playing the role of a large parameter). This shows that for high enough contrast each of the sub-block inverses can be approximated well by solving only systems with constant coefficients. The second phase of the algorithm involves the approximation of these constant coefficient systems using multigrid methods. The result is a general method of algebraic character which (under suitable hypotheses) can be proved to be robust with respect to both the contrast and the mesh size. While a similar performance is also achieved in practice by algebraic multigrid (AMG) methods, this performance is still without theoretical justification. Since the first phase of our method is comparable to the process of identifying weak and strong connections in conventional AMG algorithms, our theory provides to some extent a theoretical justification for these successful algebraic procedures. We demonstrate the advantageous properties of our preconditioner using experiments on model problems. Our numerical experiments show that for sufficiently high contrast the performance of our new preconditioner is almost identical to that of the Ruge and Stüben AMG preconditioner, both in terms of iteration count and CPU-time.     

Additive Schwarz with aggregation-based coarsening for elliptic problems with highly variable coefficients

Summary We develop a new coefficient-explicit theory for two-level overlapping domain decomposition preconditioners with non-standard coarse spaces in iterative solvers for finite element discretisations of second-order elliptic problems. We apply the theory to the case of smoothed aggregation coarse spaces introduced by Vanek, Mandel and Brezina in the context of algebraic multigrid (AMG) and are particularly interested in the situation where the diffusion coefficient (or the permeability) α is highly variable throughout the domain. Our motivating example is Monte Carlo simulation for flow in rock with permeability modelled by log–normal random fields. By using the concept of strong connections (suitably adapted from the AMG context) we design a two-level additive Schwarz preconditioner that is robust to strong variations in α as well as to mesh refinement. We give upper bounds on the condition number of the preconditioned system which do not depend on the size of the subdomains and make explicit the interplay between the coefficient function and the coarse space basis functions. In particular, we are able to show that the condition number can be bounded independent of the ratio of the two values of α in a binary medium even when the discontinuities in the coefficient function are not resolved by the coarse mesh. Our numerical results show that the bounds with respect to the mesh parameters are sharp and that the method is indeed robust to strong variations in α. We compare the method to other preconditioners and to a sparse direct solver.      

Robust multigrid preconditioners for cell-centered finite volume discretization of the high-contrast diffusion equation

We study a conservative 5-point cell-centered finite volume discretization of the high-contrast diffusion equation. We aim to construct preconditioners that are robust with respect to the magnitude of the coefficient contrast and the mesh size simultaneously. For that, we prove and numerically demonstrate the robustness of the preconditioner proposed by Aksoylu et al. (Comput Vis Sci 11:319–331, 2008) by extending the devised singular perturbation analysis from linear finite element discretization to the above discretization. The singular perturbation analysis is more involved than that of finite element case because all the subblocks in the discretization matrix depend on the diffusion coefficient. However, as the diffusion coefficient approaches infinity, that dependence is eliminated. This allows the same preconditioner to be utilized due to similar limiting behaviours of the submatrices; leading to a narrowing family of preconditioners that can be used for different discretizations. Therefore, we have accomplished a desirable preconditioner design goal. We compare our numerical results to standard cell-centered multigrid implementations and observe that performance of our preconditioner is independent of the utilized smoothers and prolongation operators. As a side result, we also prove a fundamental qualitative property of solution of the high-contrast diffusion equation. Namely, the solution over the highly-diffusive island becomes constant asymptotically. Integration of this qualitative understanding of the underlying PDE to our preconditioner is the main reason behind its superior performance. Diagonal scaling is probably the most basic preconditioner for high-contrast coefficients. Extending the matrix entry based spectral analysis introduced by Graham and Hagger, we rigorously show that the number of small eigenvalues of the diagonally scaled matrix depends on the number of isolated islands comprising the highly-diffusive region. This indicates that diagonal scaling creates a significant clustering of the spectrum, a favorable property for faster convergence of Krylov subspace solvers.     

Algebraic multigrid methods based on element preconditioning

This paper presents a new algebraic multigrid (AMG) solution strategy for large linear systems with a sparse matrix arising from a finite element discretization of some self-adjoint, second order, scalar, elliptic partial differential equation. The AMG solver is based on Ruge/Stübens method. Ruge/Stübens algorithm is robust for M-matrices, but unfortunately the “region of robustness“ between symmetric positive definite M-matrices and general symmetric positive definite matrices is very fuzzy.

For this reason the so-called element preconditioning technique is introduced in this paper. This technique aims at the construction of an M-matrix that is spectrally equivalent to the original stiffness matrix. This is done by solving small restricted optimization problems. AMG applied to the spectrally equivalent M-matrix instead of the original stiffness matrix is then used as a preconditioner in the conjugate gradient method for solving the original problem.

The numerical experiments show the efficiency and the robustness of the new preconditioning method for a wide class of problems including problems with anisotropic elements.     

A Uniform Additive Schwarz Preconditioner for High-Order Discontinuous Galerkin Approximations of Elliptic Problems

In this paper we design and analyze a uniform preconditioner for a class of high-order Discontinuous Galerkin schemes. The preconditioner is based on a space splitting involving the high-order conforming subspace and results from the interpretation of the problem as a nearly-singular problem. We show that the proposed preconditioner exhibits spectral bounds that are uniform with respect to the discretization parameters, i.e., the mesh size, the polynomial degree and the penalization coefficient. The theoretical estimates obtained are supported by numerical tests.     

Single scale wavelet approximations in layout optimization

The standard structural layout optimization problem in 2D elasticity is solved using a wavelet based discretization of the displacement field and of the spatial distribution of material. A fictitious domain approach is used to embed the original design domain within a simpler domain of regular geometry. A Galerkin method is used to derive discretized equations, which are solved iteratively using a preconditioned conjugate gradient algorithm. A special preconditioner is derived for this purpose. The method is shown to converge at rates that are essentially independent of discretization size, an advantage over standard finite element methods, whose convergence rate decays as the mesh is refined. This new approach may replace finite element methods in very large scale problems, where a very fine resolution of the shape is needed. The derivation and examples focus on 2D-problems but extensions to 3D should involve only few changes in the essential features of the procedure.     

A numerical investigation on the interplay amongst geometry, meshes, and linear algebra in the finite element solution of elliptic PDEs

In this paper, we study the effect of the choice of mesh quality metric, preconditioner, and sparse linear solver on the numerical solution of elliptic partial differential equations (PDEs). We smooth meshes on several geometric domains using various quality metrics and solve the associated elliptic PDEs using the finite element method. The resulting linear systems are solved using various combinations of preconditioners and sparse linear solvers. We use the inverse mean ratio and radius ratio metrics in addition to conditioning-based scale-invariant and interpolation-based size-and-shape metrics. We employ the Jacobi, SSOR, incomplete LU, and algebraic multigrid preconditioners and the conjugate gradient, minimum residual, generalized minimum residual, and bi-conjugate gradient stabilized solvers. We focus on determining the most efficient quality metric, preconditioner, and linear solver combination for the numerical solution of various elliptic PDEs with isotropic coefficients. We also investigate the effect of vertex perturbation and the effect of increasing the problem size on the number of iterations required to converge and on the solver time. In this paper, we consider Poisson’s equation, general second-order elliptic PDEs, and linear elasticity problems.     

A balanced accumulation scheme for parallel PDE solvers

We present a tailored load balancing technique that addresses specific performance issues in the boundary data accumulation algorithm for non-overlapping domain decompositions. The technique is used to speed up a parallel conjugate gradient algorithm with an algebraic multigrid preconditioner to solve a potential problem on an unstructured tetrahedral finite element mesh. The optimized accumulation algorithm significantly improves the performance of the parallel solver and we show up to 50 % runtime improvements over the standard approach in benchmark runs with up to 48 MPI processes. The load balancing problem itself is a global optimization problem that is solved approximately by local optimization algorithms in parallel that require no communication during the optimization process.