Sparse numerical solution

Lanczos Method for the Numerical Solution of Large Sparse Generalized Symmetric Eigenvalue Problems. 

The solution of a sparse system of equations can be carried out in three stages 1. Partitioning, 2. Reordering or "tearing", and 3. Numerical solution. Stages 1 and 2 contain only logical operations and their objective is to obtain a system which can be solved faster and/or with smaller round-off error propagated. 

P. Concus, G. H. Golub, and D. P. O Leary, in Sparse Matrix Computations, J. R. Bunch and D. J. Rose, Eds., Academic Press, New York, 1976, pp. 309—332. A Generalized Conjugate Gradient Method for the Numerical Solution of Elliptic Partial Differential Equations. 

A Graph-Theoretic Study of the Numerical Solution of Sparse Positive-Definite Systems of Linear Equations. 

In 32 the authors consider the solution of the two-dimensional time-independent Schrodinger equation by partial discretization. The discretized problem is treated as a problem of the numerical solution of system of ordinary differential equations and has been solved numerically by symplectic methods. The problem is then transformed into an algebraic eigenvalue problem involving real, symmetric, large sparse matrices. As numerical illustrations the authors have found the eigenvalues of the two-dimensional harmonic oscillator... 

Although the inverse of J0 appears in Eq. (5-31), it should be noted that the explicit expression of Jo 1 need never be developed only the LU factorization is required. If J0 is sparse, its inverse Jo 1 is not necessarily sparse, but its factorization L0U0 is sparse. Thus, throughout the remainder of the development, inverses are shown but the actual numerical solutions are to be found by use of the LU factorizations rather than the inverses of the jacobian matrices. 

T. Ericsson and A. Ruhe Lanczos method for the numerical solution of large sparse generalized Math, of Comp. 35, 1251 (1980). 

Detailed experimental data on the rate constants associated adsorption/desorption kinetics or conformational interconversion of different forms of a protein chromatographed on -alkylsilicas are currently very sparse. The kinetics of de-naturation of several proteins on n-butyl-bonded silica surfaces have been reported. Fig. 18 for example, shows the dependence of peak area on the incubation time of lysozyme on the bonded phase surface, from which rate constants for interconversion on the stationary phase, i.e. were derived [63]. The graphical representations derived from quantitative numerical solutions of the probabihty distributions... 

Numerical values for solid diffusivities D,j in adsorbents are sparse and disperse. Moreover, they may be strongly dependent on the adsorbed phase concentration of solute. Hence, locally conducted experiments and interpretation must be used to a great extent. Summaries of available data for surface diffusivities in activated carbon and other adsorbent materials and for micropore diffusivities in zeolites are given in Ruthven, Yang, Suzuki, and Karger and Ruthven (gen. refs.). 

Warner [176] has given a comprehensive discussion of the principal approaches to the solution of stiff differential equations, including a hundred references among the most pertinent books, papers and application packages directed at simulating kinetic models. Emphasis has been put not only on numerical and software problems such as robustness, improving the linear equation solvers, using sparse matrix techniques, etc., but also on the availability of a chemical compiler, i.e. a powerful interface between kineticist and computer. 

As we noted in the discussion of solution techniques, the matrices encountered in the numerical formulations are very sparse i.e., they contain a large number of zeros. In solving a problem with a large number of nodes it may be quite time-consuming to enter all these zeros and the simple form of the Gauss-Seidel equation may be preferable. 

C. C. Paige and M. A. Saunders, Solution of sparse indefinite systems of linear equations, Siam J. Numer. Anal. 12 611 (1975). 

When investigating a model of a chemical plant or process, one of the most important tasks is to determine the influence of model parameters like operation conditions or geometric dimensions on performance and dynamics. Because in most cases a large number of parameters has to be examined, an efficient tool for the determination of parameter dependencies is required. Continuation methods in conjunction with the concepts of bifurcation theory have proved to be useful for the analysis of nonlinear systems and are increasingly used in chemical engineering science. They offer the possibility to compute steady states or periodic solutions directly as a function of one or several parameters and to detect changes in the qualitative behaviour of a system like the appearance or disappearance of multiple steady states. In this paper, numerical methods for the continuation of steady states and periodic solutions for large sparse systems with arbitrary structural properties are presented. The application of this methods to models of chemical processes and the problems which arise in this context are discussed for the example of a special type of catalytic fixed bed reactor, the so-called circulation loop reactor. 

Table 4.1. Numerical results for the computation of the solution of (4.1) on regular sparse grids.

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