We present a class of reduced basis (RB) methods for the iterative solution of parametrized symmetric positive-definite (SPD) linear systems. The essential ingredients are a Galerkin projection of the underlying parametrized system onto a reduced basis space to obtain a reduced system; an adaptive greedy algorithm to efficiently determine sampling parameters and associated basis vectors; an offline-online computational procedure and a multi-fidelity approach to decouple the construction and application phases of the reduced basis method; and solution procedures to employ the reduced basis approximation as a {\em stand-alone iterative solver} or as a {\em preconditioner} in the conjugate gradient method. We present numerical examples to demonstrate the performance of the proposed methods in comparison with multigrid methods. Numerical results show that, when applied to solve linear systems resulting from discretizing the Poisson's equations, the speed of convergence of our methods matches or surpasses that of the multigrid-preconditioned conjugate gradient method, while their computational cost per iteration is significantly smaller providing a feasible alternative when the multigrid approach is out of reach due to timing or memory constraints for large systems. Moreover, numerical results verify that this new class of reduced basis methods, when applied as a stand-alone solver or as a preconditioner, is capable of achieving the accuracy at the level of the {\em truth approximation} which is far beyond the RB level.
We discuss different cases of dissipative Hamiltonian differential-algebraic equations and the linear algebraic systems that arise in their linearization or discretization. For each case we give examples from practical applications. An important feature of the linear algebraic systems is that the (non-Hermitian) system matrix has a positive definite or semidefinite Hermitian part. In the positive definite case we can solve the linear algebraic systems iteratively by Krylov subspace methods based on efficient three-term recurrences. We illustrate the performance of these iterative methods on se…
Recent advances in the field of machine learning open a new era in high performance computing. Applications of machine learning algorithms for the development of accurate and cost-efficient surrogates of complex problems have already attracted major attention from scientists. Despite their powerful approximation capabilities, however, surrogates cannot produce the `exact' solution to the problem. To address this issue, this paper exploits up-to-date ML tools and delivers customized iterative solvers of linear equation systems, capable of solving large-scale parametrized problems at any desired…
We propose a new approximate factorization for solving linear systems with symmetric positive definite sparse matrices. In a nutshell the algorithm is to apply hierarchically block Gaussian elimination and additionally compress the fill-in. The systems that have efficient compression of the fill-in mostly arise from discretization of partial differential equations. We show that the resulting factorization can be used as an efficient preconditioner and compare the proposed approach with state-of-art direct and iterative solvers.
We introduce iterative methods named TriCG and TriMR for solving symmetric quasi-definite systems based on the orthogonal tridiagonalization process proposed by Saunders, Simon and Yip in 1988. TriCG and TriMR are tantamount to preconditioned Block-CG and Block-MINRES with two right-hand sides in which the two approximate solutions are summed at each iteration, but require less storage and work per iteration. We evaluate the performance of TriCG and TriMR on linear systems generated from the SuiteSparse Matrix Collection and from discretized and stablized Stokes equations. We compare TriCG and…
In this paper, we propose and analyze iterative method based on projection techniques to solve a non-singular linear system Ax = b. In particular, for a given positive integer m, m-dimensional successive projection method (mD-SPM) for symmetric definite matrix A, is generalized for non-singular matrix A. Moreover, it is proved that mD-SPM gives better result for large values of m. Numerical experiments are carried out to demonstrate the superiority of the proposed method in comparison with other schemes in the scientific literature.
The second-order cone linear complementarity problem (SOCLCP) is a generalization of the classical linear complementarity problem. It has been known that SOCLCP, with the globally uniquely solvable property, is essentially equivalent to a zero-finding problem in which the associated function bears much similarity to the transfer function in model reduction [{\em SIAM J. Sci. Comput.}, 37 (2015), pp.~A2046--A2075]. In this paper, we propose a new rational Krylov subspace method to solve the zero-finding problem for the symmetric and positive definite SOCLCP. The algorithm consists of two stages…
Randomized methods are becoming increasingly popular in numerical linear algebra. However, few attempts have been made to use them in developing preconditioners. Our interest lies in solving large-scale sparse symmetric positive definite linear systems of equations where the system matrix is preordered to doubly bordered block diagonal form (for example, using a nested dissection ordering). We investigate the use of randomized methods to construct high quality preconditioners. In particular, we propose a new and efficient approach that employs Nyström's method for computing low rank approximat…
A numerical scheme for computing arc-length parametrized curves of low bending energy that are confined to convex domains is devised. The convergence of the discrete formulations to a continuous model and the unconditional stability of an iterative scheme are addressed. Numerical simulations confirm the theoretical results and lead to a classification of observed optimal curves within spheres.
We present a stationary iteration method, namely Alternating Symmetric positive definite and Scaled symmetric positive semidefinite Splitting (ASSS), for solving the system of linear equations obtained by using finite element discretization of a distributed optimal control problem together with time-periodic parabolic equations. An upper bound for the spectral radius of the iteration method is given which is always less than 1. So convergence of the ASSS iteration method is guaranteed. The induced ASSS preconditioner is applied to accelerate the convergence speed of the GMRES method for solvin…
In an iterative approach for solving linear systems with ill-conditioned, symmetric positive definite (SPD) kernel matrices, both fast matrix-vector products and fast preconditioning operations are required. Fast (linear-scaling) matrix-vector products are available by expressing the kernel matrix in an $\mathcal{H}^2$ representation or an equivalent fast multipole method representation. Preconditioning such matrices, however, requires a structured matrix approximation that is more regular than the $\mathcal{H}^2$ representation, such as the hierarchically semiseparable (HSS) matrix representa…
We consider the iterative solution of symmetric saddle-point matrices with a singular leading block. We develop a new ideal positive definite block diagonal preconditioner that yields a preconditioned operator with four distinct eigenvalues. We offer a few techniques for making the preconditioner practical, and illustrate the effectiveness of our approach with numerical experiments.
Combined Preconditioning with Applications in Reservoir Simulation
Kunming University of Science and Technology（昆明理工大学）；The Pennsylvania State University（宾夕法尼亚州立大学）；PetroChina Company Limited（中国石油天然气股份有限公司）；ExxonMobil Upstream Research Company（埃克森美孚上游研究公司）；Academy of Mathematics and System Sciences（中国科学院数学与系统科学研究院）；Sichuan University（四川大学）
Spectral coarse spaces based on indefinite operators: the $H_k$-GenEO method
Inria（法国国家信息与自动化研究所）；Univ. Lille（里尔大学）；Laboratoire Paul Painlevé（保罗·潘勒韦实验室）；Eindhoven University of Technology（埃因霍温理工大学）；University of Strathclyde（斯特拉斯克莱德大学）；University of Bath（巴斯大学）
Preconditioners are generally essential for fast convergence in the iterative solution of linear systems of equations. However, the computation of a good preconditioner can be expensive. So, while solving a sequence of many linear systems, it is advantageous to recycle preconditioners, that is, update a previous preconditioner and reuse the updated version. In this paper, we introduce a simple and effective method for doing this. Although our approach can be used for matrices changing slowly in any way, we focus on the important case of sequences of the type $(s_k\textbf{E}(\textbf{p}) + \text…
Application of gradient descent algorithms based on geodesic distances
School of Science, Dalian Jiaotong University（大连交通大学理学院）；School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）；Waseda Institute for Advanced Study, Waseda University（早稻田大学高等研究院）
