Two-Parameter Block Triangular Splitting Preconditioner for Block Two-by-Two Linear Systems

This paper proposes a two-parameter block triangular splitting(TPTS)preconditioner for the general block two-by-two linear systems.The eigenvalues of the corresponding preconditioned matrix are proved to cluster around 0 or 1 under mild conditions.The limited numerical results show that the TPTS preconditioner is more efficient than the classic block-diagonal and block-triangular preconditioners when applied to the flexible generalized minimal residual(FGMRES)method.

Construction and Analysis of Structured Preconditioners for Block Two-by-Two Matrices

For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to high-quality preconditioning matrices for some typical matrices from the real-world applications.

A multilevel preconditioner and its shared memory implementation for a new generation reservoir simulator

As a result of the interplay between advances in computer hardware, software, and algorithm, we are now in a new era of large-scale reservoir simulation, which focuses on accurate flow description, fine reservoir characterization, efficient nonlinear/linear solvers, and parallel implementation. In this paper, we discuss a multilevel preconditioner in a new-generation simulator and its implementation on multicore computers. This preconditioner relies on the method of subspace corrections to solve large-scale linear systems arising from fully implicit methods in reservoir simulations. We investigate the parallel efficiency and robustness of the proposed method by applying it to million-cell benchmark problems.

Preconditioners for Incompressible Navier-Stokes Solvers

In this paper we give an overview of the present state of fast solvers for the solution of the incompressible Navier-Stokes equations discretized by the finite element method and linearized by Newton or Picard's method.It is shown that block preconditioners form an excellent approach for the solution,however if the grids are not to fine preconditioning with a Saddle point ILU matrix(SILU) may be an attractive alternative. The applicability of all methods to stabilized elements is investigated.In case of the stand-alone Stokes equations special preconditioners increase the efficiency considerably.

Jacobian-free Newton-Krylov subspace method with wavelet-based preconditioner for analysis of transient elastohydrodynamic lubrication problems with surface asperities

This paper presents an investigation into the effect of surface asperities on the over-rolling of bearing surfaces in transient elastohydrodynamic lubrication(EHL) line contact. The governing equations are discretized by the finite difference method. The resulting nonlinear system of algebraic equations is solved by the Jacobian-free Newtongeneralized minimal residual(GMRES) from the Krylov subspace method(KSM). The acceleration of the GMRES iteration is accomplished by a wavelet-based preconditioner.The profiles of the lubricant pressure and film thickness are obtained at each time step when the indented surface moves through the contact region. The prediction of pressure as a function of time provides an insight into the understanding of fatigue life of bearings.The analysis confirms the need for the time-dependent approach of EHL problems with surface asperities. This method requires less storage and yields an accurate solution with much coarser grids. It is stable, efficient, allows a larger time step, and covers a wide range of parameters of interest.

Biorthogonal Wavelet Based Algebraic Multigrid Preconditioners for Large Sparse Linear Systems

In this article algebraic multigrid as preconditioners are designed, with biorthogonal wavelets, as intergrid operators for the Krylov subspace iterative methods. Construction of hierarchy of matrices in algebraic multigrid context is based on lowpass filter version of Wavelet Transform. The robustness and efficiency of this new approach is tested by applying it to large sparse, unsymmetric and ill-conditioned matrices from Tim Davis collection of sparse matrices. Proposed preconditioners have potential in reducing cputime, operator complexity and storage space of algebraic multigrid V-cycle and meet the desired accuracy of solution compared with that of orthogonal wavelets.

A NEW ILU PRECONDITIONER FOR BORDERED LINEAR SYSTEMS

Bordered linear systems arise from many industrial applications, such as reservoir simulation and structural engineering. Traditional ILU preconditioners which throw away the additional equations are often too crude for these systems. We describe a practical implementation of ILU preconditioners which are more accurate and more robust. The emphasis of this paper is on implementation rather than on theory.

Some Properties of the Optimal Preconditioner and the Generalized Superoptimal Preconditioner

The optimal preconditioner and the superoptimal preconditioner were proposed in 1988 and 1992 respectively. They have been studied widely since then. Recently, Chen and Jin [6] extend the superoptimal preconditioner to a more general case by using the Moore-Penrose inverse. In this paper, we further study some useful properties of the optimal and the generalized superoptimal preconditioners. Several existing results are extended and new properties are developed.

A Two-Level Additive Schwarz Preconditioner for Local C^0 Discontinuous Galerkin Methods of Kirchhoff Plates

A two-level additive Schwarz preconditioner based on the overlapping domain decomposition approach is proposed for the local C0 discontinuous Galerkin (LCDG) method of Kirchhoff plates.Then with the help of an intergrid transfer operator and its error estimates,it is proved that the condition number is bounded by O(1 + (H4/δ4)),where H is the diameter of the subdomains and δ measures the overlap among subdomains.And for some special cases of small overlap,the estimate can be improved as O(1 + (H3/δ3)).At last,some numerical results are reported to demonstrate the high efficiency of the two-level additive Schwarz preconditioner.

Finite Difference Preconditioners for Legendre Based Spectral Element Methods on Elliptic Boundary Value Problems

Finite difference type preconditioners for spectral element discretizations based on Legendre-Gauss-Lobatto points are analyzed. The latter is employed for the approximation of uniformly elliptic partial differential problems. In this work, it is shown that the condition number of the resulting preconditioned system is bounded independently of both of the polynomial degrees used in the spectral element method and the element sizes. Several numerical tests verify the h-p independence of the proposed preconditioning.

Domain Decomposition Preconditioners for Mixed Finite-Element Discretization of High-Contrast Elliptic Problems

In this paper,we design an efficient domain decomposition(DD)preconditioner for the saddle-point problem resulting from the mixed finite-element discretization of multiscale elliptic problems.By proper equivalent algebraic operations,the original saddle-point system can be transformed to another saddle-point system which can be preconditioned by a block-diagonel matrix efficiently.Actually,the first block of this block-diagonal matrix corresponds to a multiscale H(div)problem,and thus,the direct inverse of this block is unpractical and unstable for the large-scale problem.To remedy this issue,a two-level overlapping DD preconditioner is proposed for this//(div)problem.Our coarse space consists of some velocities obtained from mixed formulation of local eigenvalue problems on the coarse edge patches multiplied by the partition of unity functions and the trivial coarse basis(e.g.,Raviart-Thomas element)on the coarse grid.The condition number of our preconditioned DD method for this multiscale H(div)system is bounded by C(1+务)(1+log4(^)),where 6 denotes the width of overlapping region,and H,h are the typical sizes of the subdomain and fine mesh.Numerical examples are presented to confirm the validity and robustness of our DD preconditioner.

A New Choice of the Preconditioner of PHSS Method for Saddle Point Problems

Bai, Golub and Pan presented a preconditioned Hermitian and skew-Hermitian splitting(PHSS) method [Numerische Mathematik, 2004, 32: 1-32] for non-Hermitian positive semidefinite linear systems. We improve the method to solve saddle point systems whose(1,1) block is a symmetric positive definite M-matrix with a new choice of the preconditioner and compare it with other preconditioners. The results show that the new preconditioner outperforms the previous ones.

An Improved Wavelet Based Preconditioner for Sparse Linear Problems

In this paper, we present the construction of purely algebraic Daubechies wavelet based preconditioners for Krylov subspace iterative methods to solve linear sparse system of equations. Effective preconditioners are designed with DWTPerMod algorithm by knowing size of the matrix and the order of Daubechies wavelet. A notable feature of this algorithm is that it enables wavelet level to be chosen automatically making it more robust than other wavelet based preconditioners and avoids user choosing a level of transform. We demonstrate the efficiency of these preconditioners by applying them to several matrices from Tim Davis collection of sparse matrices for restarted GMRES.

On the Preconditioning Properties of RHSS Preconditioner for Saddle-Point Linear Systems

In this paper,for the regularized Hermitian and skew-Hermitian splitting(RHSS)preconditioner introduced by Bai and Benzi(BIT Numer Math 57:287–311,2017)for the solution of saddle-point linear systems,we analyze the spectral properties of the preconditioned matrix when the regularization matrix is a special Hermitian positive semidefinite matrix which depends on certain parameters.We accurately describe the numbers of eigenvalues clustered at(0,0)and(2,0),if the iteration parameter is close to 0.An estimate about the condition number of the corresponding eigenvector matrix,which partly determines the convergence rate of the RHSS-preconditioned Krylov subspace method,is also studied in this work.

Uniform Subspace Correction Preconditioners for Discontinuous Galerkin Methods with hp‑Refnement

In this paper,we develop subspace correction preconditioners for discontinuous Galerkin(DG)discretizations of elliptic problems with hp-refnement.These preconditioners are based on the decomposition of the DG fnite element space into a conforming subspace,and a set of small nonconforming edge spaces.The conforming subspace is preconditioned using a matrix-free low-order refned technique,which in this work,we extend to the hprefnement context using a variational restriction approach.The condition number of the resulting linear system is independent of the granularity of the mesh h,and the degree of the polynomial approximation p.The method is amenable to use with meshes of any degree of irregularity and arbitrary distribution of polynomial degrees.Numerical examples are shown on several test cases involving adaptively and randomly refned meshes,using both the symmetric interior penalty method and the second method of Bassi and Rebay(BR2).

Explicit Iterative Methods of Second Order and Approximate Inverse Preconditioners for Solving Complex Computational Problems

Explicit Exact and Approximate Inverse Preconditioners for solving complex linear systems are introduced. A class of general iterative methods of second order is presented and the selection of iterative parameters is discussed. The second order iterative methods behave quite similar to first order methods and the development of efficient preconditioners for solving the original linear system is a decisive factor for making the second order iterative methods superior to the first order iterative methods. Adaptive preconditioned Conjugate Gradient methods using explicit approximate preconditioners for solving efficiently large sparse systems of algebraic equations are also presented. The generalized Approximate Inverse Matrix techniques can be efficiently used in conjunction with explicit iterative schemes leading to effective composite semi-direct solution methods for solving large linear systems of algebraic equations.

A New Preconditioner with Two Variable Relaxation Parameters for Saddle Point Linear Systems with Highly Singular(1,1) Blocks

In this paper, we provide new preconditioner for saddle point linear systems with (1,1) blocks that have a high nullity. The preconditioner is block triangular diagonal with two variable relaxation paremeters and it is extension of results in [1] and [2]. Theoretical analysis shows that all eigenvalues of preconditioned matrix is strongly clustered. Finally, numerical tests confirm our analysis.

Schur Complement Based Preconditioners for Twofold and Block Tridiagonal Saddle Point Problems

In this paper,we consider using Schur complements to design preconditioners for twofold and block tridiagonal saddle point problems.One type of the preconditioners are based on the nested(or recursive)Schur complement,the other is based on an additive type Schur complement after permuting the original saddle point systems.We analyze different preconditioners incorporating the exact Schur complements.We show that some of them will lead to positively stable preconditioned systems if proper signs are selected in front of the Schur complements.These positive-stable preconditioners outperform other preconditioners if the Schur complements are further approximated inexactly.Numerical experiments for a 3-field formulation of the Biot model are provided to verify our predictions.

Projection Improved SPAI Preconditioner for FGMRES

Krylov subspace methods are widely used for solving sparse linear algebraic equations,but they rely heavily on preconditioners,and it is difficult to find an effective preconditioner that is efficient and stable for all problems.In this paper,a novel projection strategy including the orthogonal and the oblique projection is proposed to improve the preconditioner,which can enhance the efficiency and stability of iteration.The proposed strategy can be considered as a minimization process,where the orthogonal projection minimizes the energy norm of error and the oblique projection minimizes the 2-norm of the residual,meanwhile they can be regarded as approaches to correct the approximation by solving low-rank inverse of the matrices.The strategy is a wide-ranging approach and provides a way to transform the constant preconditioner into a variable one.The paper discusses in detail the projection strategy for sparse approximate inverse(SPAI)preconditioner applied to flexible GMRES and conducts the numerical test for problems from different applications.The results show that the proposed projection strategy can significantly improve the solving process,especially for some non-converging and slowly convergence systems.

A RELAXED HSS PRECONDITIONER FOR SADDLE POINT PROBLEMS FROM MESHFREE DISCRETIZATION＊

In this paper, a relaxed Hermitian and skew-Hermitian splitting （RHSS） preconditioner is proposed for saddle point problems from the element-free Galerkin （EFG） discretization method. The EFG method is one of the most widely used meshfree methods for solving partial differential equations. The RHSS preconditioner is constructed much closer to the coefficient matrix than the well-known HSS preconditioner, resulting in a RHSS fixed-point iteration. Convergence of the RHSS iteration is analyzed and an optimal parameter, which minimizes the spectral radius of the iteration matrix is described. Using the RHSS pre- conditioner to accelerate the convergence of some Krylov subspace methods （like GMRES） is also studied. Theoretical analyses show that the eigenvalues of the RHSS precondi- tioned matrix are real and located in a positive interval. Eigenvector distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are obtained. A practical parameter is suggested in implementing the RHSS preconditioner. Finally, some numerical experiments are illustrated to show the effectiveness of the new preconditioner.