For solving a class of block two-by-two real linear system, a new single-step iteration method based on triangular splitting scheme is proposed in this paper. Then the convergence properties of this method are carefully investigated. In addition, we determine its optimal iteration parameters and give the corresponding optimal convergence factor. It is worth mentioning that the SSTS iteration method is robust and superior to SBTS and PSBTS iteration methods under suitable conditions. Finally, some numerical experiments are carried out to validate the theoretical results and evaluate this new method.
In this paper, we propose a simple sparse approximate inverse for triangular matrices (SAIT). Using the Jacobi iteration method, we obtain an expression of the exact inverse of triangular matrix, which is a finite series. The SAIT is constructed based on this series. We apply the SAIT matrices to iterative methods with ILU preconditioners. The two triangular solvers in the ILU preconditioning procedure are replaced by two matrix-vector multiplications, which can be fine-grained parallelized. We test this method by solving some linear systems and eigenvalue problems with preconditioned iterativ…
For solving the continuous Sylvester equation, a class of the multiplicative splitting iteration method is presented. We consider two symmetric positive definite splittings for each coefficient matrix of the continuous Sylvester equations and it can be equivalently written as two multiplicative splitting matrix equations. When both coefficient matrices in the continuous Sylvester equation are (non-symmetric) positive semi-definite, and at least one of them is positive definite; we can choose Hermitian and skew-Hermitian (HS) splittings of matrices $A$ and $B$, in the first equation, and the sp…
By applying the minimal residual technique to the Hermitian and skew-Hermitian (HSS) iteration scheme, we introduce a non-stationary iteration method named minimal residual Hermitian and skew-Hermitian (MRHSS) iteration method to solve the continuous Sylvester equation. Numerical results verify the effectiveness and robustness of the MRHSS iteration method versus the HSS method for the continuous Sylvester equation. Moreover, by numerical computation, we show that the MRHSS splitting can be used as a splitting preconditioner and induce accurate, robust and effective preconditioned Krylov subsp…
This paper presents enhancement strategies for the Hermitian and skew-Hermitian splitting method based on gradient iterations. The spectral properties are exploited for the parameter estimation, often resulting in a better convergence. In particular, steepest descent with early stopping can generate a rough estimate of the optimal parameter. This is better than an arbitrary choice since the latter often causes stability problems or slow convergence. Additionally, lagged gradient methods are considered as inner solvers for the splitting method. Experiments show that they are competitive with co…
The dynamic iteration method with a restricted additive Schwarz splitting is investigated to co-simulate linear differential algebraic equations system coming from RLC electrical circuit with linear components. We show the pure linear convergence or divergence of the method with respect to the linear operator belonging to the restricted additive Schwarz interface. It allows us to accelerate it toward the true solution with the Aitken's technique for accelerating convergence. This provides a dynamic iteration method less sensitive to the splitting. Numerical examples with convergent and diverge…
For a linear complementarity problem, we present a relaxaiton accelerated two-sweep matrix splitting iteration method. The convergence analysis illustrates that the proposed method converges to the exact solution of the linear complementarity problem when the system matrix is an $H_+$-matrix and the convergence conditions are given. Numerical experiments show that the proposed method is more efficient than the existing ones.
The fixed-stress splitting scheme is a popular method for iteratively solving the Biot equations. The method successively solves the flow and mechanic subproblems while adding a stabilizing term to the flow equation, which includes a parameter that can be chosen freely. However, the convergence properties of the scheme depend significantly on this parameter and choosing it carelessly might lead to a very slow, or even diverging, method. In this paper, we present a way to exploit the matrix structure arizing from discretizing the equations in the regime of impermeable porous media in order to o…
Diagonal and normal with Toeplitz-block splitting iteration method for space fractional coupled nonlinear Schrödinger equations with repulsive nonlinearities
We propose an improved version of the Hermitian/skew-Hermitian splitting (HSS) iterative method, which we call HSS(0), to solve non-Hermitian linear systems with a positive definite Hermitian part. The improvement is based on solving the Hermitian half iteration without a shift, and applying a shift only for the skew-Hermitian solve. An optimal parameter is derived analytically, and a corresponding upper bound on the convergence speed is obtained. Using a combination of analytical proofs and numerical validations, we show that HSS(0) yields a dramatically faster convergence speed than standard…
Inner-iteration preconditioning with a symmetric splitting matrix for rank-deficient least squares problems
Institute of Computer Science, The Czech Academy of Sciences（计算机科学研究所，捷克科学院）；Faculty of Engineering, Information and Systems, University of Tsukuba（筑波大学工程信息系统学部）
A dynamic iteration scheme for linear infinite-dimensional port-Hamiltonian systems is proposed. The dynamic iteration is monotone in the sense that the error is decreasing, it does not require any stability condition and is in particular applicable to port-Hamiltonian formulations arising from domain decompositions.
Positive semidefinite and positive semidefinite splitting iteration methods for solving nonsingular positive semidefinite systems
Minab Higher Education Center, University of Hormozgan（霍尔木甘大学米纳布高等教育学院）；Department of Mathematics, University of Hormozgan（霍尔木甘大学数学系）；Faculty of Mathematical Sciences, University of Guilan（吉兰大学数理学院）
A shift splitting modified Newton-type (SSMN) iteration method is introduced for solving large sparse generalized absolute value equations (GAVEs). The SSMN method is established by replacing the regularized splitting of the coefficient matrix of the linear part, which is employed in the modified Newton-type (MN) iteration method, with the shift splitting of the matrix. The conditions for the convergence of the proposed method are discussed in depth for the cases when the coefficient matrix is a general matrix, a symmetric positive definite matrix, and an $H_{+}$-matrix. Through two numerical …
Structural index reduction algorithms for differential algebraic equations via fixed-point iteration
Chengdu Institute of Computer Applications, Chinese Academy of Sciences（中国科学院成都计算机应用研究所）；Automated Reasoning and Cognition Key Lab of Chongqing, CIGIT, CAS（中国科学院重庆智能计算研究院）
Some variants of the (block) Gauss--Seidel iteration for the solution of linear systems with $M$-matrices in (block) Hessenberg form are discussed. Comparison results for the asymptotic convergence rate of some regular splittings are derived: in particular, we prove that for a lower-Hessenberg M-matrix $ρ(P_{GS})\geq ρ(P_S)\geq ρ(P_{AGS})$, where $P_{GS}, P_S, P_{AGS}$ are the iteration matrices of the Gauss--Seidel, staircase, and anti-Gauss--Seidel method. This is a result that does not seem to follow from classical comparison results, as these splittings are not directly comparable. It is s…
Gauss-Seidel (GS) relaxation is often employed as a preconditioner for a Krylov solver or as a smoother for Algebraic Multigrid (AMG). However, the requisite sparse triangular solve is difficult to parallelize on many-core architectures such as graphics processing units (GPUs). In the present study, the performance of the traditional GS relaxation based on a triangular solve is compared with two-stage variants, replacing the direct triangular solve with a fixed number of inner Jacobi-Richardson (JR) iterations. When a small number of inner iterations is sufficient to maintain the Krylov conver…
Matrix splitting iteration methods play a vital role in solving large sparse linear systems. Their performance heavily depends on the splitting parameters, however, the approach of selecting optimal splitting parameters has not been well developed. In this paper, we present a multitask kernel-learning parameter prediction method to automatically obtain relatively optimal splitting parameters, which contains simultaneous multiple parameters prediction and a data-driven kernel learning. For solving time-dependent linear systems, including linear differential systems and linear matrix systems, we…
Extending the Numerical Flow Iteration to the multi-species Vlasov-Maxwell system through Hamiltonian Splitting
KU Leuven（荷语鲁汶大学）；RWTH Aachen University（亚琛工业大学）；Royal Belgian Institute for Space Aeronomy（比利时皇家空间气动力学研究所）；TU Darmstadt（达姆施塔特工业大学）；Nvidia GmbH（英伟达有限公司）
We present some accelerated variants of fixed point iterations for computing the minimal non-negative solution of the unilateral matrix equation associated with an M/G/1-type Markov chain. These variants derive from certain staircase regular splittings of the block Hessenberg M-matrix associated with the Markov chain. By exploiting the staircase profile we introduce a two-step fixed point iteration. The iteration can be further accelerated by computing a weighted average between the approximations obtained at two consecutive steps. The convergence of the basic two-step fixed point iteration an…
In this paper, we execute the shift-splitting preconditioner for asymmetric saddle point problems with its (1,2) block's transposition unequal to its (2,1) block under the removed minus of its (2,1) block. The proposed preconditioner is stemmed from the shift splitting (SS) iteration method for solving asymmetric saddle point problems, which is convergent under suitable conditions. The relaxed version of the shift-splitting preconditioner is obtained as well. The spectral distributions of the related preconditioned matrices are given. Numerical experiments from the Stokes problem are offered t…
We develop a fast and reliable method for solving large-scale optimal transport (OT) problems at an unprecedented combination of speed and accuracy. Built on the celebrated Douglas-Rachford splitting technique, our method tackles the original OT problem directly instead of solving an approximate regularized problem, as many state-of-the-art techniques do. This allows us to provide sparse transport plans and avoid numerical issues of methods that use entropic regularization. The algorithm has the same cost per iteration as the popular Sinkhorn method, and each iteration can be executed efficien…
In this paper, we present, evaluate and analyse the performance of parallel synchronous Jacobi algorithms by different partitioned procedures including band-row splitting, band-row sparsity pattern splitting and substructuring splitting, when solving sparse large linear systems. Numerical experiments performed on a set of academic 3D Laplace equation and on a real gravity matrices arising from the Chicxulub crater are exhibited, and show the impact of splitting on parallel synchronous iterations when solving sparse large linear systems. The numerical results clearly show the interest of substr…
