When multicomponent, multistage separation problems are solved on parallel computers by successive linearization methods, the solution of a large sparse linear equation system becomes a computational bottleneck, since other parts of the calculation are more easily parallelized. When the standard problem formulation is used, this system has a block-tridiagonal form. It is shown how this structure can be used in parallelizing the sparse matrix computation. By reformulating the problem so that it has a bordered-block-bidiagonal superstructure, it can be made even more amenable to parallelization. These strategies permit the use of a two-level hierarchy of parallelism that provides substantial improvements in computationalperformance on parallel machines. IntroductionThe need to solve multicomponent equilibrium-stage separation calculations arises frequently in chemical process analysis and design. The rigorous simulation of separation columns is a computationally intense problem that often represents the largest computational effort in solving flowsheet simulation and optimization problems. Thus, the solution of these problems is an attractive application for parallel computers, since these offer the potential of much higher computational speed than conventional sequential machines. Since, in principle, they involve no upper limit on speed, parallel computing architectures are the inevitable future of high-speed scientific and engineering computing. However, since current methods for solving multicomponent separation problems were developed for use on conventional serial computers, they usually are not capable of taking much advantage of the potential power of parallel machines. Thus, the strategies used to solve these problems must be rethought. We address here the development of new problem-solving strategies for exploiting the power of parallel computing in solving multicomponent equilibrium-stage separation problems.There are a wide variety of methods available for solving multicomponent separation problems (Henley and Seader, 1981), including equation-tearing methods and simultaneous correction procedures. The latter are desirable since they are more robust for difficult problems, and they allow more flex- ibility in problem specifications. Furthermore, they are analogous to the equation-based approach for solving the more general process flowsheeting problem, which Vegeais and Stadtherr (1992) have found to have the best potential for exploiting parallel computing. Thus, we concentrate on the use of simultaneous correction methods. In this case, the problem is formulated as a large set of nonlinear equations to be solved by successive linearization, usually using Newton-Raphson or some variation thereof. This involves the solution of a large sparse system of linear equations, which often represents a large fraction of the overall computing time. As noted by Vegeais and Stadtherr (1992), the sparse matrix problem becomes even more of a computational bottleneck on a parallel machine because other parts of the ca...
We formulate an Algebraic-Coding Equivalence to the Maximum Distance Separable Conjecture. Specifically, we present novel proofs of the following equivalent statements. Let (q, k) be a fixed pair of integers satisfying q is a prime power and 2 ≤ k ≤ q. We denote by P q the vector space of functions from a finite field F q to itself, which can be represented as the space P q := F q [x]/(x q − x) of polynomial functions. We denote by O n ⊂ P q the set of polynomials that are either the zero polynomial, or have at most n distinct roots in F q . Given two subspaces Y, Z of P q , we denote by Y, Z their span. We prove that the following are equivalent.A Suppose that either:(a) q is odd (b) q is even and k ∈ {3, q − 1}.Then there do not exist distinct subspaces Y and Z of P q such that:B Suppose q is odd, or, if q is even, k ∈ {3, q − 1}. There is no integer s with q ≥ s > k such that the Reed-Solomon code R over F q of dimension s can have s − k + 2 columns B = {b 1 , . . . , b s−k+2 } added to it, such that:(a) Any s × s submatrix of R ∪ B containing the first s − k columns of B is independent. (b) B ∪ {[0, 0, . . . , 0, 1]} is independent.C The MDS conjecture is true for the given (q, k).
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.