We present a class of iterative fully distributed fixed point methods to solve a system of linear equations, such that each agent in the network holds one of the equations of the system. Under a generic directed, strongly connected network, we prove a convergence result analogous to the one for fixed point methods in the classical, centralized, framework: the proposed method converges to the solution of the system of linear equations at a linear rate. We further explicitly quantify the rate in terms of the linear system and the network parameters. Next, we show that the algorithm provably works under time-varying directed networks provided that the underlying graph is connected over bounded iteration intervals, and we establish a linear convergence rate for this setting as well. A set of numerical results is presented, demonstrating practical benefits of the method over existing alternatives.
The discrete-time algebraic Riccati equation (DARE) have extensive applications in optimal control problems. We provide new theoretical supports to the stability properties of solutions to the DARE and reduce the convergence conditions under which the accelerated fixed-point iteration (AFPI) can be applied to compute the numerical solutions of DARE. In particular, we verify that the convergence of AFPI is R-superlinear when the spectral radius of the closed-loop matrix is greater than 1, which is shown by mild assumption and only using primary matrix theories. Numerical examples are shown to i…
In this paper, applying the Newton method, we transform the complex continuous-time algebraic Riccati matrix equation into a Lyapunov equation. Then, we introduce an efficient general alternating-direction implicit (GADI) method to solve the Lyapunov equation. The inexact Newton-GADI method is presented to save computational amount effectively. Moreover, we analyze the convergence of the Newton-GADI method. The convergence rate of the Newton-GADI and Newton-ADI methods is compared by analyzing their spectral radii. Furthermore, we give a way to select the quasi-optimal parameter. Corresponding…
Singular equations with rank-deficient Jacobians arise frequently in algebraic computing applications. As shown in case studies in this paper, direct and intuitive modeling of algebraic problems often results in nonisolated singular solutions. The challenges become formidable when the problems need to be solved from empirical data of limited accuracy. A newly discovered low-rank Newton's iteration emerges as an effective regularization mechanism that enables solving singular equations accurately with an error bound in the same order as the data error. This paper elaborates applications of new …
A Kaczmarz Algorithm for Solving Tree Based Distributed Systems of Equations
Electrical and Computer Engineering, Iowa State University, Ames, IA 50011（电子与计算机工程系，爱荷华州立大学，爱荷华州阿姆斯，IA 50011）；Department of Mathematics, Iowa State University, 396 Carver Hall, Ames, IA 50011（数学系，爱荷华州立大学，396号Carver大楼，爱荷华州阿姆斯，IA 50011）
Discontinuity with respect to data perturbations is common in algebraic computation where solutions are often highly sensitive. Such problems can be modeled as solving systems of equations at given data parameters. By appending auxiliary equations, the models can be formulated to satisfy four easily verifiable conditions so that the data form complex analytic manifolds on which the solutions maintain their structures and the Lipschitz continuity. When such a problem is given with empirical data, solving the system becomes a least squares problem whose solution uniquely exists and enjoys Lipsch…
Iterative solvers preconditioned with algebraic multigrid have been devised as an optimal technology to speed up the response of large sparse linear systems. In this work, this technique was implemented in the framework of the dual delineation approach. This involves a single groundwater flow solve and a pure advective transport solve with different right-hand sides. The new solver was compared with traditional preconditioned iterative methods and direct sparse solvers on several two- and three-dimensional benchmark problems spanning homogeneous and heterogeneous formations. For the groundwate…
This paper describes a massively parallel algebraic multigrid method based on non-smoothed aggregation. It is especially suited for solving heterogeneous elliptic problems as it uses a greedy heuristic algorithm for the aggregation that detects changes in the coefficients and prevents aggregation across them. Using decoupled aggregation on each process with data agglomeration onto fewer processes on the coarse level, it weakly scales well in terms of both total time to solution and time per iteration to nearly 300,000 cores. Because of simple piecewise constant interpolation between the levels…
