Decomposition (computer science)

Abstract: A technique is presented for the decomposition of a linear program that permits the problem to be solved by alternate solutions of linear sub-programs representing its several parts and a coordinating program that is obtained from the parts by linear transformations. The coordinating program generates at each cycle new objective forms for each part, and each part generates in turn from its optimal basic feasible solutions new activities columns for the interconnecting program. Viewed as an instance of a “generalized programming problem” whose columns are drawn freely from given convex sets, such a problem can be studied by an appropriate generalization of the duality theorem for linear programming, which permits a sharp distinction to be made between those constraints that pertain only to a part of the problem and those that connect its parts. This leads to a generalization of the Simplex Algorithm, for which the decomposition procedure becomes a special case. Besides holding promise for the efficient computation of large-scale systems, the principle yields a certain rationale for the “decentralized decision process” in the theory of the firm. Formally the prices generated by the coordinating program cause the manager of each part to look for a “pure” sub-program analogue of pure strategy in game theory, which he proposes to the coordinator as best he can do. The coordinator finds the optimum “mix” of pure sub-programs using new proposals and earlier ones consistent with over-all demands and supply, and thereby generates new prices that again generates new proposals by each of the parts, etc. The iterative process is finite.

Abstract: In engineering design, all products and artifacts have some intended reason behind their existence: the product or artifact function. Functional modeling provides an abstract, yet direct, method for understanding and representing an overall product or artifact function. Functional modeling also strategically guides design activities such as problem decomposition, physical modeling, product architecting, concept generation, and team organization. A formal function representation is needed to support functional modeling, and a standardized set of function-related terminology leads to repeatable and meaningful results from such a representation. We refer to this representation as a functional basis; in this paper, we seek to reconcile and integrate two independent research efforts into a significantly evolved functional basis. These efforts include research from the National Institute of Standards and Technology and two US universities, and their industrial partners. The overall approach for integrating the functional representations and the final results are presented. This approach also provides a mechanism for evaluating whether future revisions are needed to the functional basis and, if so, how to proceed. The integration process is discussed relative to differences, similarities, insights into the representations, and product validation. Based on the results, a more versatile and comprehensive design vocabulary emerges. This vocabulary will greatly enhance and expand the frontiers of research in design repositories, product architecture, design synthesis, and general product modeling.

Abstract: A new method for performing a balanced reduction of a high-order linear system is presented. The technique combines the proper orthogonal decomposition and concepts from balanced realization theory. The method of snapshotsisused to obtainlow-rank,reduced-rangeapproximationsto thesystemcontrollability and observability grammiansineitherthetimeorfrequencydomain.Theapproximationsarethenusedtoobtainabalancedreducedorder model. The method is particularly effective when a small number of outputs is of interest. It is demonstrated for a linearized high-order system that models unsteady motion of a two-dimensional airfoil. Computation of the exact grammians would be impractical for such a large system. For this problem, very accurate reducedorder models are obtained that capture the required dynamics with just three states. The new models exhibit far superiorperformancethanthosederived using a conventionalproperorthogonal decomposition. Although further development is necessary, the concept also extends to nonlinear systems.