Splitting Preconditioner Research Articles

A modified improved alternating positive semi-definite splitting preconditioner for double saddle point problems

A modified improved alternating positive semi-definite splitting preconditioner for double saddle point problems

A generalized relaxed block positive-semidefinite splitting preconditioner for generalized saddle point linear system

A generalized relaxed block positive-semidefinite splitting preconditioner for generalized saddle point linear system

Rotated block diagonal preconditioners for Navier-Stokes control problems

Fluid flow control problems play a crucial role in various industrial applications. The optimal control of the Navier-Stokes equations poses a significant challenge. By employing Oseen's approximation to linearize the problem, we encounter a series of large sparse structured linear systems. These coefficient matrices exhibit a two-by-two block structure with square blocks. Leveraging this block structure, we exploit matrix splitting preconditioners. Theoretical analysis indicates that the real parts of all eigenvalues of preconditioned matrix are equal to 1/2. A large number of eigenvalues are clustered in (1±i)/2. The imaginary part of the eigenvalues is bounded explicitly. The eigenvalue distribution predicts the fast convergence of Krylov subspace methods. To avoid solving the saddle point subsystems in the preconditioning procedure, we propose a practical variant of the preconditioner. The theoretical analysis demonstrates that if the approximation is sufficiently close to Schur complement, the eigenvalues of the modified preconditioned matrix will lie within a circle centered at the original eigenvalues with a radius less than 1. This implies that the modified preconditioner exhibits excellent performance in terms of preconditioning. The numerical experiments demonstrate that the generalized minimal residual method, when combined with the proposed preconditioners, proves to be an efficient and effective solution for Oseen's control optimization with a variable viscosity coefficient. The number of iterations required by the preconditioned methods remains unaffected by the mesh size used in finite element discretization and only marginally depends on the regularization parameter.

Efficient Learning of Transform-Domain LMS Filter Using Graph Laplacian.

Transform-domain least mean squares (TDLMS) adaptive filters encompass the class of learning algorithms where the input data are subjected to a data-independent unitary transform followed by a power normalization stage as preprocessing steps. Because conventional transformations are not data-dependent, this preconditioning procedure was shown theoretically to improve the convergence of the least mean squares (LMS) filter only for certain classes of input data. So, one can tailor the transformation to the class of data. However, in reality, if the class of input data is not known beforehand, it is difficult to decide which transformation to use. Thus, there is a need to devise a learning framework to obtain such a preconditioning transformation using input data prior to applying on the input data. It is hypothesized that the underlying topology of the data affects the selection of the transformation. With the input modeled as a weighted finite graph, our method, called preconditioning using graph (PrecoG), adaptively learns the desired transform by recursive estimation of the graph Laplacian matrix. We show the efficacy of the transform as a generalized split preconditioner on a linear system of equations and in Hebbian-LMS learning models. In terms of the improvement of the condition number after applying the transformation, PrecoG performs significantly better than the existing state-of-the-art techniques that involve unitary and nonunitary transforms.

A fast ADI based matrix splitting preconditioning method for the high dimensional space fractional diffusion equations in conservative form

In this paper, we consider the high-dimensional two-sided space fractional diffusion equations with monotonic variable diffusion coefficients, which are derived from the fractional Fick's law. We apply the implicit Euler scheme to discretize the temporal derivative and certain difference operator to discretize the space fractional derivatives. For the discretized linear systems, we apply the alternating direction implicit (ADI) scheme to split the linear systems into two sub-systems of linear equations. For both coefficient matrices of the sub-systems of linear equations, we propose the lopsided scaled diagonal and Toeplitz splitting preconditioner. The generalized minimal residual (GMRES) method combined with the proposed preconditioner is applied to solve both sub-systems of linear equations. The spectral distributions of the preconditioned matrices are analyzed, and the theoretical results are given. Numerical results demonstrate that the proposed preconditioner is efficient in accelerating the convergence rate of the GMRES method for solving the discretized system of linear equations.

Multi-parameter dimensional split preconditioner for three-by-three block system of linear equations

Multi-parameter dimensional split preconditioner for three-by-three block system of linear equations

Recycling basic columns of the splitting preconditioner in interior point methods

Recycling basic columns of the splitting preconditioner in interior point methods

A GENERALIZED SIMPLIFIED HERMITIAN AND SKEW-HERMITIAN SPLITTING PRECONDITIONER FOR DOUBLE SADDLE POINT PROBLEMS

In this work, we mainly propose a generalized simplified Hermitian and skew-Hermitian splitting (GSHSS) preconditioner for solving double saddle point problems and the eigenvalue distribution of the GSHSS preconditioner is analyzed in detail. In addition, we also study the eigenvector distribution and the degree of the minimal polynomial of the preconditioned matrix. Finally, numerical experiments show the effectiveness of the proposed preconditioner.

A modified new matrix splitting preconditioner for double saddle point problems

A modified new matrix splitting preconditioner for double saddle point problems

Preconditioning Sparse Matrices with Alternating and Multiplicative Operator Splittings

We present an algebraic framework for operator splitting preconditioners for general sparse matrices. The framework leads to four different approaches: two with alternating splittings and two with a multiplicative ansatz. The ansatz generalizes ADI and ILU methods to multiple factors and to a more general factor form. The factors may be computed directly from the matrix coefficients or adaptively by incomplete sparse inversions. The special case of tridiagonal splittings is examined in more detail. We decompose the adjacency graph of the sparse matrix into multiple (almost) disjoint linear forests, and each linear forest (a union of disjoint paths) leads to a tridiagonal splitting. We obtain specialized variants of the four general approaches. Parallel implementations for all steps are provided on a GPU. We demonstrate the effectiveness and efficiency of these preconditioners combined with GMRES on various matrices.

A simplified relaxed alternating positive semi-definite splitting preconditioner for saddle point problems with three-by-three block structure

A simplified relaxed alternating positive semi-definite splitting preconditioner for saddle point problems with three-by-three block structure

A generalized variant of modified relaxed positive-semidefinite and skew-Hermitian splitting preconditioner for generalized saddle point problems

A generalized variant of modified relaxed positive-semidefinite and skew-Hermitian splitting preconditioner for generalized saddle point problems

A modified variant of HSS preconditioner for generalized saddle point problems

Recently, Zhang [Numerical Linear Algebra with Applications, 2018: e2166] constructed an efficient variant of Hermitian and skew-Hermitian splitting (HSS) preconditioner for generalized saddle point problems, and gave the corresponding theoretical analysis and numerical experiment. In this paper, based on the efficient variant of the HSS (EVHSS) preconditioner, we relax the strong condition on the iterative parameter [Formula: see text], generalize the algorithms and further present the modified variant of Hermitian and skew-Hermitian splitting (HSS) (MVHSS) preconditioner for solving the generalized saddle point problem. Moreover, by similar theoretical analysis, we obtain that the MVHSS iterative method is convergent under weaker conditions. Finally, an example is given to verify the effectiveness of the proposed method. MSC: 65F10; 65F15; 65F50

A New Approach to the Splitting Factor Preconditioner Applied to Linear Programming Problems

In this paper, we present the results of a new approach to the splitting factor preconditioner, which is a preconditioner based on the Incomplete Cholesky factorization and the splitting preconditioner. In previous work for small linear programming problems, the preconditioner was applied in all iterations of the interior point method and compared with the splitting preconditioner also applied in all iterations. In this paper, we will do a hybrid approach, in which in the first iterations the preconditioner is the Incomplete Cholesky Factorization, and in the last iterations, the preconditioner used is the splitting factor preconditioner or the splitting preconditioner. The results obtained show that even in the hybrid approach, the splitting factor preconditioner achieved better performance.

A preconditioner based on sine transform for space fractional diffusion equations

The spatial fractional diffusion equations are discretized by the finite difference method and linear systems are obtained. Transforming the coefficient matrices of these linear systems to obtain the structure of diagonal-plus-Toeplitz. In order to solve this class of linear systems, we construct a two-parameter DTS iteration method on the basis of the DTS iteration method and give its asymptotic convergence conditions. On this basis, we introduce a two-parameter sine-transform-based splitting preconditioner to improve the convergence rate of the generalized minimal residual (GMRES) iteration method, and theoretically prove that the eigenvalues of the corresponding preconditioned matrix are clustered around 1. Finally, numerical experiments demonstrate the efficiency of the new method.

A modified alternating positive semidefinite splitting preconditioner for block three-by-three saddle point problems

We propose a modified alternating positive semidefinite splitting (MAPSS) preconditioner for solving block three-by-three saddle point problems that arise in linear programming and the finite element discretization of Maxwell equations. Spectral properties of the MAPSS-preconditioned matrix are discussed and analyzed in detail. As the efficiency of the MAPSS preconditioner depends on its parameters, we derive fast and effective formulas to compute the quasi-optimal values of these parameters. Numerical examples show that the MAPSS preconditioner performs better than the APSS preconditioner.

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

Correction to: A variant of relaxed alternating positive semi-definite splitting preconditioner for double saddle point problems

Correction to: A variant of relaxed alternating positive semi-definite splitting preconditioner for double saddle point problems

A variant of relaxed alternating positive semi-definite splitting preconditioner for double saddle point problems

Based on the relaxed alternating positive semi-definite splitting (RAPSS) preconditioner, in this paper, a new preconditioner, called variant of relaxed alternating positive semi-definite splitting (VRAPSS) preconditioner, is presented and discussed. Spectral properties of the VRAPSS preconditioned matrix are analyzed in detail. Numerical experiments are provided to verify the efficiency of the VRAPSS preconditioner.

Scaled diagonal‐times‐Toeplitz splitting iteration methods for solving discretized spatial fractional diffusion equations

In this paper, we firstly give a scaled diagonal‐times‐Toeplitz splitting (SDTS) iteration method for solving the discretization system from one‐dimensional spatial fractional diffusion equations of variable coefficients. The SDTS iteration methods can be used to solve the one‐dimensional spatial fractional diffusion equations of both anisotropic and isotropic, while the respectively scaled HSS (RSHSS) iteration method can be used only to solve anisotropic spatial fractional diffusion equations. Therefore, the SDTS iteration method is of wider range of applications than the RSHSS iteration method. Furthermore, the SDTS iteration method will naturally lead to a scaled diagonal‐times‐circulant splitting (SDCS) preconditioner. Theoretical analysis shows that the eigenvalues of the corresponding preconditioned matrix are clustered around 1. Numerical experiments are provided, which demonstrate the feasibility and effectiveness of the SDCS preconditioned GMRES method for solving the proposed examples.