Symmetric Linear Systems Research Articles

Task-parallel tiled direct solver for dense symmetric indefinite systems

This paper proposes a direct solver for symmetric indefinite linear systems. The program is parallelized via the OpenMP task construct and outperforms existing programs. The proposed solver avoids pivoting, which requires a lot of data movement, during factorization with preconditioning using the symmetric random butterfly transformation. The matrix data layout is tiled after the preconditioning to more efficiently use cache memory during factorization. Given the low-rank property of the input matrices, an adaptive crossing approximation is used to make a low-rank approximation before the update step to reduce the computation load. Iterative refinement is then used to improve the accuracy of the final result. Finally, the performance of the proposed solver is compared to that of various symmetric indefinite linear system solvers to show its superiority.

A hybrid-mixed finite element method for single-phase Darcy flow in fractured porous media

We present a hybrid-mixed finite element method for a novel hybrid-dimensional model of single-phase Darcy flow in a fractured porous media. In this model, the fracture is treated as a (d−1)-dimensional interface within the d-dimensional fractured porous domain, for d=2,3. Two classes of fracture are distinguished based on the permeability magnitude ratio between the fracture and its surrounding medium: when the permeability in the fracture is (significantly) larger than in its surrounding medium, it is considered as a conductive fracture; when the permeability in the fracture is (significantly) smaller than in its surrounding medium, it is considered as a blocking fracture. The conductive fractures are treated using the classical hybrid-dimensional approach of the interface model where pressure is assumed to be continuous across the fracture interfaces, while the blocking fractures are treated using the recent Dirac-δ function approach where normal component of Darcy velocity is assumed to be continuous across the interface. Due to the use of Dirac-δ function approach for the blocking fractures, our numerical scheme allows for nonconforming meshes with respect to the blocking fractures. This is the major novelty of our model and numerical discretization. Moreover, our numerical scheme produces locally conservative velocity approximations and leads to a symmetric positive definite linear system involving pressure degrees of freedom on the mesh skeleton only. As an application, we extend the idea to a simple transport model. The performance of the proposed method is demonstrated by various benchmark test cases in both two- and three-dimensions. Numerical results indicate that the proposed scheme is highly competitive with existing methods in the literature

Rothe–Legendre pseudospectral method for a semilinear pseudoparabolic equation with nonclassical boundary condition

A semilinear pseudoparabolic equation with nonlocal integral boundary conditions is studied in the present paper. Using Rothe method, which is based on backward Euler finitedifference schema, we designed a suitable semidiscretization in time to approximate the original problem by a sequence of standard elliptic problems. The questions of convergence of the approximation scheme as well as the existence and uniqueness of the solution are investigated. Moreover, the Legendre pseudospectral method is employed to discretize the time-discrete approximation scheme in the space direction. The main advantage of the proposed approach lies in the fact that the full-discretization schema leads to a symmetric linear algebraic system, which may be useful for theoretical and practical reasons. Finally, numerical experiments are included to illustrate the effectiveness and robustness of the presented algorithm.

An efficient relaxed shift-splitting preconditioner for a class of complex symmetric indefinite linear systems

<abstract><p>In this work, by introducing a scalar matrix $ \alpha I $, we transform the complex symmetric indefinite linear systems $ (W+i T)x = b $ into a block two-by-two complex equations equivalently, and propose an efficient relaxed shift-splitting (ERSS) preconditioner. By adopting the relaxation technique, the ERSS preconditioner is not only a computational advantage but also closer to the original two-by-two of complex coefficient matrix. The eigenvalue distributions of the preconditioned matrix are analysed. An efficient and practical formula for computing the parameter value $ \alpha $ is also derived by computing the Frobenius norm of symmetric indefinite matrix $ T $. Numerical examples on a few model problems are illustrated to verify the performances of the ERSS preconditioner.</p></abstract>

The asymptotic solution of a singularly perturbed Cauchy problem for Fokker-Planck equation

The asymptotic method is a very attractive area of applied mathematics. There are many modern research directions which use a small parameter such as statistical mechanics, chemical reaction theory and so on. The application of the Fokker-Planck equation (FPE) with a small parameter is the most popular because this equation is the parabolic partial differential equations and the solutions of FPE give the probability density function. In this paper we investigate the singularly perturbed Cauchy problem for symmetric linear system of parabolic partial differential equations with a small parameter. We assume that this system is the Tikhonov non-homogeneous system with constant coefficients. The paper aims to consider this Cauchy problem, apply the asymptotic method and construct expansions of the solutions in the form of two-type decomposition. This decomposition has regular and border-layer parts. The main result of this paper is a justification of an asymptotic expansion for the solutions of this Cauchy problem. Our method can be applied in a wide variety of cases for singularly perturbed Cauchy problems of Fokker-Planck equations.

A splitting iterative method and preconditioner for complex symmetric linear system via real equivalent form

A splitting iterative method and preconditioner for complex symmetric linear system via real equivalent form

Gradient based iterative methods for solving symmetric tensor equations

AbstractThe steepest descent and conjugate gradient methods are classical gradient based iterative methods for solving symmetric positive definite linear system . In this article, we are concerned with the numerical solution of the tensor equation by those well‐known iterations in which is an mth order n‐dimensional symmetric tensor. Then we prove that the developed iterative methods converge locally linearly under some appropriate conditions. Finally, some numerical experiments are provided to illustrate the feasibility and effectiveness of the proposed methods.

Convergence analysis of some tent-based schemes for linear hyperbolic systems

Finite element methods for symmetric linear hyperbolic systems using unstructured advancing fronts (satisfying a causality condition) are considered in this work. Convergence results and error bounds are obtained for mapped tent pitching schemes made with standard discontinuous Galerkin discretizations for spatial approximation on mapped tents. Techniques to study semidiscretization on mapped tents, design fully discrete schemes, prove local error bounds, prove stability on spacetime fronts, and bound error propagated through unstructured layers are developed.

Direct optimization of BPX preconditioners

We consider an automatic construction of locally optimal preconditioners for positive definite linear systems. To achieve this goal, we introduce a differentiable loss function that does not explicitly include the estimation of minimal eigenvalue. Nevertheless, the resulting optimization problem is equivalent to a direct minimization of the condition number. To demonstrate our approach, we construct a parametric family of modified BPX preconditioners. Namely, we define a set of empirical basis functions for coarse finite element spaces and tune them to achieve better condition number. For considered model equations (that includes Poisson, Helmholtz, Convection–diffusion, Biharmonic, and others), we achieve from two to twenty times smaller condition numbers for symmetric positive definite linear systems.

Double parameter splitting (DPS) iteration method for solving complex symmetric linear systems

Complex linear equation systems appear in many branches of mathematical sciences and have many applications in science and engineering. Therefore, there have been developed many methods to solve them. Here a new method based on decomposition of coefficient matrix for the solution of a positive symmetric linear system is studied. Our method uses two parameters. We obtain a domain on which this new method converges without any condition. We will also discuss how to calculate optimal parameters. Moreover we will prove that there exists a region in R2 such that DPS method is faster than CRI method (Wang et al. (2017) [48]). Finally some test problems will be given and simulation results will be reported to support the theoretical results.

A fully structured preconditioner for a class of complex symmetric indefinite linear systems

A fully structured preconditioner for a class of complex symmetric indefinite linear systems

EM-WaveHoltz: How to find Time-Harmonic Solutions by Time-Domain Solvers and Positive Definite Systems

We present EM-WaveHoltz, a new method for computing frequency domain solutions to Maxwell's equations in linear complex media. EM-WaveHoltz is based on a fixed point iteration which is applied to the initial data of a time domain solution. The fixed point iteration converges to a single time-harmonic component and can be formulated as a symmetric positive definite linear system of equations. The method is agnostic to the material model and to the time-domain solver used.

Efficient second-order ADI difference schemes for three-dimensional Riesz space-fractional diffusion equations

In this paper, a three-dimensional time-dependent Riesz space-fractional diffusion equation is considered, and an alternating direction implicit (ADI) difference scheme is proposed, in which a weighted and shifted Grünwald difference scheme (see Tian et al. (2015) [33]) is utilized for the discretizations of space-fractional derivatives, and a fractional-Douglas-Gunn type ADI method is utilized for the discretization of time derivative. The method is proved to be unconditionally stable and convergent with second-order accuracy both in time and space with respect to a weighted discrete energy norm. Efficient implementation of the method is carefully discussed, and then based on fast matrix-vector multiplications, a fast conjugate gradient (FCG) solver for the resulting symmetric positive definite linear algebraic system is developed. Numerical experiments support the theoretical analysis and show strong effectiveness and efficiency of the method for large-scale modeling and simulations. Finally, a linearized ADI scheme based on second-order extrapolation method is developed and tested for the nonlinear Riesz space-fractional diffusion equation.

HILUCSI: Simple, robust, and fast multilevel ILU for large‐scale saddle‐point problems from PDEs

AbstractIncomplete factorization is a widely used preconditioning technique for Krylov subspace methods for solving large‐scale sparse linear systems. Its multilevel variants, such as ILUPACK, are more robust for many symmetric or unsymmetric linear systems than the traditional, single‐level incomplete LU (or ILU) techniques. However, the previous multilevel ILU techniques still lacked robustness and efficiency for some large‐scale saddle‐point problems, which often arise from systems of PDEs. We introduce HILUCSI, or Hierarchical Incomplete LU‐Crout with Scalability‐oriented and Inverse‐based dropping. As a multilevel preconditioner, HILUCSI statically and dynamically permutes individual rows and columns to the next level for deferred factorization. Unlike ILUPACK, HILUCSI applies symmetric preprocessing techniques at the top levels but always uses unsymmetric preprocessing and unsymmetric factorization at the coarser levels. The deferring combined with mixed preprocessing enabled a unified treatment for nearly or partially symmetric systems and simplified the implementation by avoiding mixed and pivots for symmetric indefinite systems. We show that this combination improves robustness for indefinite systems without compromising efficiency. Furthermore, to enable superior efficiency for large‐scale systems with millions or more unknowns, HILUCSI introduces a scalability‐oriented dropping in conjunction with a variant of inverse‐based dropping. We demonstrate the effectiveness of HILUCSI for dozens of benchmark problems, including those from the mixed formulation of the Poisson equation, Stokes equations, and Navier–Stokes equations. We also compare its performance with ILUPACK and the supernodal ILUTP in SuperLU.

A preconditioner based on a splitting-type iteration method for solving complex symmetric indefinite linear systems

A preconditioner based on a splitting-type iteration method for solving complex symmetric indefinite linear systems

Modified two-step scale-splitting iteration method for solving complex symmetric linear systems

For the large sparse complex symmetric linear systems, based on the combination method of real part and imaginary part method established by Wang et al. (J Comput Appl Math 325:188–197, 2017) and the double-step scale splitting one derived by Zheng et al. (Appl Math Lett 73:91–97, 2017), we construct a modified two-step scale-splitting (MTSS) iteration method in this paper. The convergence properties of the MTSS iteration method and its quasi-optimal parameters which minimizes the upper bound for the spectral radius of the proposed method are presented. Meanwhile, we derive the inexact variant of the MTSS iteration method. On this basis, we also introduce a minimum residual MTSS iteration method and its inexact version and give their convergence analyses. Numerical results on complex symmetric linear systems support that the proposed methods are more efficient and robust than some other commonly used ones.

Newton’s method for the parameterized generalized eigenvalue problem with nonsquare matrix pencils

The l parameterized generalized eigenvalue problems for the nonsquare matrix pencils, proposed by Chu et al.in 2006, can be formulated as an optimization problem on a corresponding complex product Stiefel manifold. In this paper, an effective and efficient algorithm based on the Riemannian Newton’s method is established to solve the underlying problem. Under our proposed framework, to solve the corresponding Newton’s equation, it can be converted to solve a standard real symmetric linear system with a dimension reduction. By combining the Riemannian curvilinear search method with Barzilai–Borwein steps, a hybrid algorithm with both global and quadratic convergence is obtained. Numerical experiments are provided to illustrate the efficiency of the proposed method. Detailed comparisons with some latest methods are also provided to show the merits of the proposed approach.

Efficient block splitting iteration methods for solving a class of complex symmetric linear systems

In this paper, we first propose a new block splitting (NBS) iteration method for solving the large sparse complex symmetric linear systems. The NBS iteration method avoids complex arithmetic compared with the combination method of real part and imaginary part (CRI) one established by Wang et al. (2017). The unconditional convergence and the quasi-optimal parameter of the NBS iteration method are given. Moreover, by further accelerating the NBS one with another parameter, we construct the parameterized BS (PBS) iteration method and establish its convergence theory. Also, the spectral properties of the PBS-preconditioned matrix are analyzed and the parameter selection strategy of the PBS iteration method is given. Numerical experiments are reported to illustrate the feasibility and effectiveness of the proposed methods.

Efficiency of nonparametric finite elements for optimal-order enforcement of Dirichlet conditions on curvilinear boundaries

In recent papers (see e.g. Ruas (2020a) and Ruas (2020b)) a nonparametric technique of the Petrov–Galerkin type was analyzed, whose aim is the accuracy enhancement of higher order finite element methods to solve boundary value problems with Dirichlet conditions, posed in smooth curved domains. In contrast to parametric elements, it employs straight-edged triangular or tetrahedral meshes fitting the domain. In order to attain best-possible orders greater than one, piecewise polynomial trial-functions are employed, which interpolate the Dirichlet conditions at points of the true boundary. The test-functions in turn are defined upon the standard degrees of freedom associated with the underlying method for polytopic domains. As a consequence, when the problem at hand is self-adjoint a non symmetric linear system has to be solved. This paper is primarily aimed at showing that in this case, an efficient symmetrization of the solution procedure can be achieved by means of a fast converging iterative method. In order to illustrate the great generality of our nonparametric approach, experimentation is presented with a finite element method having degrees of freedom other than nodal values. More specifically we consider a nonconforming quadratic element in the solution of the three-dimensional Poisson equation. The performance evaluation however is conducted as well for two versions of the classical conforming quadratic method, namely, the nonparametric Petrov–Galerkin formulation considered in Ruas (2020b) and the standard isoparametric one. The study of this symmetrization is completed by an optimal error estimation in the broken H1-norm for the nonparametric version of the nonconforming method, which had not been addressed in previous work.

Parallel Newton–Chebyshev polynomial preconditioners for the conjugate gradient method

In this note, we exploit polynomial preconditioners for the conjugate gradient method to solve large symmetric positive definite linear systems in a parallel environment. We put in connection a specialized Newton method to solve the matrix equation X−1 = A and the Chebyshev polynomials for preconditioning. We propose a simple modification of one parameter which avoids clustering of extremal eigenvalues in order to speed-up convergence. We provide results on very large matrices (up to 8.6 billion unknowns) in a parallel environment showing the efficiency of the proposed class of preconditioners.