A local convergence proof for the iterative aggregation method
TL;DR: In this article, the iterative aggregation method for the solution of a system of linear algebraic equations x = Ax + b, where A ≥ 0, b ≥ 0 and s > 0, and s < s, is proved to be locally convergent.
About: This article is published in Linear Algebra and its Applications.The article was published on 1983-06-01 and is currently open access. It has received 42 citations till now. The article focuses on the topics: Local convergence & Iterative method.
Citations
More filters
TL;DR: This paper develops a general framework for aggregation and disaggregation methodology, survey previous work regarding aggregation and aggregating techniques for optimization problems, illuminate the appropriate role of aggregation and segregating methodology for optimization applications, and proposes future research directions.
Abstract: A fundamental issue in the use of optimization models is the tradeoff between the level of detail and the ease of using and solving the model. Aggregation and disaggregation techniques have proven to be valuable tools for manipulating data and determining the appropriate policies to employ for this tradeoff. Furthermore, aggregation and disaggregation techniques offer promise for solving large-scale optimization models, supply a set of promising methodologies for studying the underlying structure of both univariate and multivariate data sets, and provide a set of tools for manipulating data for different levels of decision makers. In this paper, we develop a general framework for aggregation and disaggregation methodology, survey previous work regarding aggregation and disaggregation techniques for optimization problems, illuminate the appropriate role of aggregation and disaggregation methodology for optimization applications, and propose future research directions.
239 citations
01 Jan 1984
TL;DR: In this paper, an introductory survey of the defect correction approach is presented, which may serve as a unifying frame of reference for the subsequent papers on special subjects, such as special subjects.
Abstract: This is an introductory survey of the defect correction approach which may serve as a unifying frame of reference for the subsequent papers on special subjects.
66 citations
TL;DR: The construction of auxiliary problems as well as applications to elasto-plastic model and linear programming are described and the auxiliary problem for the dual of a perturbed linear program is interpreted as a dual of perturbed aggregated linear program.
Abstract: This paper is concerned with multilevel iterative methods which combine a descent scheme with a hierarchy of auxiliary problems in lower dimensional subspaces. The construction of auxiliary problems as well as applications to elasto-plastic model and linear programming are described. The auxiliary problem for the dual of a perturbed linear program is interpreted as a dual of perturbed aggregated linear program. Coercivity of the objective function over the feasible set is sufficient for the boundedness of the iterates. Equivalents of this condition are presented in special cases.
52 citations
Cites methods from "A local convergence proof for the i..."
...Liesegang [17] for solution of large-scale LP. Closely related iterative methods for the solution of large-scale linear systems accelerated by aggregation were proposed and studied in Vakhutinsky et al. [28], Miranker and Pan [26], Chatelin and Miranker [9] and Mandel and Sekerka [ 21 ]....
[...]
TL;DR: In this paper, an iterative aggregation-disaggregation procedure that alternates between solving an aggregated problem and disaggregating the variables, one block at a time, in terms of the aggregate variables of the other blocks is proposed.
Abstract: The equation v = q + Mv, where M is a matrix with nonnegative elements and spectral radius less than one, arises in Markovian decision processes and input-output models. In this paper, we solve the equation using an iterative aggregation-disaggregation procedure that alternates between solving an aggregated problem and disaggregating the variables, one block at a time, in terms of the aggregate variables of the other blocks. The disaggregated variables are then used to guide the choice of weights in the subsequent aggregation. Computational experiments on randomly generated and inventory problems indicate that this algorithm is significantly faster than successive approximations when the spectral radius of M is near one, and is slower in unstructured problems with spectral radii in the neighborhood of 0.8. The algorithm appears promising for large structured problems, where it can often reduce computational time and main memory storage requirements and offer greater robustness to initial values.
45 citations
TL;DR: A black-box parallel iterative method suitable for solving both elliptic and certain non-elliptic problems discretized on unstructured meshes is proposed and the validity of the proved convegence estimate is confirmed.
Abstract: We propose a black-box parallel iterative method suitable for solving both elliptic and certain non-elliptic problems discretized on unstructured meshes. The method is analyzed in the case of the second order elliptic problems discretized on quasiuniform P1 and Q1 finite element meshes. The numerical experiments confirm the validity of the proved convegence estimate and show that the method can successfully be used for more difficult problems (e.g. plates, shells and Helmholtz equation in high-frequency domain.)
44 citations
Cites background from "A local convergence proof for the i..."
...It was introduced in the early 1950’s by Leontief [6] and frequently used in the problems of economic modeling (cf. Mandel and Sekerka [7] and the references therein.)...
[...]
...Mandel and Sekerka [7] and the references therein....
[...]
References
More filters
01 Jan 1980
98 citations
TL;DR: In this article, the authors give convergence criteria for general difference schemes for boundary value problems in Lipschitzian regions, and prove convergence for the multi-grid algorithm with Gauss-Seidel's iteration as smoothing procedure.
Abstract: Convergence proofs for the multi-grid iteration are known for the case of finite element equations and for the case of some difference schemes discretizing boundary value problems in a rectangular region. In the present paper we give criteria of convergence that apply to general difference schemes for boundary value problems in Lipschitzian regions. Furthermore, convergence is proved for the multi-grid algorithm with Gauss-Seidel's iteration as smoothing procedure.
93 citations
TL;DR: Methods of successive approximation for solving linear systems or minimization problems are accelerated by aggregation-disaggregation processes, characterized by means of Galerkin approximations, and this in turn permits analysis of the method.
81 citations
IBM1
TL;DR: A class of methods for accelerating the convergence of iterative methods for solving linear systems by replacing the given linear system with a derived one of smaller size, the aggregated system is studied.
71 citations