Showing papers on "Algebraic equation published in 1998"

Convergence of Algebraic Multigrid Based on Smoothed Aggregation

Abstract: We prove aconvergence estimate for the Algebraic Multigrid Method with prolongations defined by aggregation using zero energy modes, followed by a smoothing. The method input is the problem matrix and a matrix of the zero energy modes. The estimate depends only polylogarithmically on the mesh size, and requires only a weak approximation property for the aggregates, which can be a-priori verified computationally. Construction of the prolongator in the case of a general second order system is described, and the assumptions of the theorem are verified for a scalar problem discretized by linear conforming finite elements.

Geometry and Classificatin of Solutions of the Classical Dynamical Yang–Baxter Equation

Abstract: The classical Yang–Baxter equation(CYBE) is an algebraic equation central in the theory of integrable systems. Its nondegenerate solutions were classified by Belavin and Drinfeld. Quantization of CYBE led to the theory of quantum groups. A geometric interpretation of CYBE was given by Drinfeld and gave rise to the theory of Poisson–Lie groups.

On the Optimisation of Periodic Adsorption Processes

Abstract: A rigorous mathematical programming based approach to the optimisation of general periodic adsorption processes is presented. Detailed dynamic models taking account of the spatial variation of properties within the adsorption bed(s) are used. The resulting systems of partial differential and algebraic equations are reduced to sets of algebraic constraints by discretisation with respect to both spatial and temporal dimensions. Periodic boundary conditions are imposed to represent directly the “cyclic steady-state” of the system. Additional constraints are introduced to characterise the interactions between multiple beds in the process as well as any relevant design specifications and operating restrictions. The optimal operating and/or design decisions can be determined by solving an optimisation problem with constraints representing a single bed over a single cycle of operation, irrespective of the number of adsorption beds in the process.

Consistent Initial Condition Calculation for Differential-Algebraic Systems

Abstract: In this paper we describe a new algorithm for the calculation of consistent initial conditions for a class of systems of differential-algebraic equations which includes semi-explicit index-one systems. We consider initial condition problems of two types---one where the differential variables are specified, and one where the derivative vector is specified. The algorithm requires a minimum of additional information from the user. We outline the implementation in a general-purpose solver DASPK for differential-algebraic equations, and present some numerical experiments which illustrate its effectiveness.

Global Optimization in Parameter Estimation of Nonlinear Algebraic Models via the Error-in-Variables Approach

Abstract: The estimation of parameters in nonlinear algebraic models through the error-in-variables method has been widely studied from a computational standpoint. The method involves the minimization of a weighted sum of squared errors subject to the model equations. Due to the nonlinear nature of the models used, the resulting formulation is nonconvex and may contain several local minima in the region of interest. Current methods tailored for this formulation, although computationally efficient, can only attain convergence to a local solution. In this paper, a global optimization approach based on a branch and bound framework and convexification techniques for general twice differentiable nonlinear optimization problems is proposed for the parameter estimation of nonlinear algebraic models. The proposed convexification techniques exploit the mathematical properties of the formulation. Classical nonlinear estimation problems were solved and will be used to illustrate the various theoretical and computational aspec...

Calculation of Fuchsian groups associated to billiards in a rational triangle

Abstract: We define, following Veech, the Fuchsian group .We consider the Fuchsian groups of various rational triangles. First, we calculate explicitly the Fuchsian groups of a new sequence of triangles, and discover they are lattices. Interestingly, the lattices found are not commensurable with those previously known. We then demonstrate a class of triangles whose Fuchsian groups are {\it not\/} lattices. These are the first examples of such triangles. Finally, we end by showing how one may specify algebraically, i.e. by an explicit polynomial in two variables, the Riemann surfaces and holomorphic one-forms that are associated to a simply-connected rational polygon. Previously, these surfaces were known by their geometric description. As an example, we show a connection between the billiard in a regular polygon and the well-known Fermat curves of the algebraic equation .

Summation of asymptotic expansions of multiple-valued functions using algebraic approximants: Application to anharmonic oscillators

Abstract: The divergent Rayleigh-Schrodinger perturbation expansions for energy eigenvalues of cubic, quartic, sextic and octic oscillators are summed using algebraic approximants. These approximants are generalized Pade approximants that are obtained from an algebraic equation of arbitrary degree. Numerical results indicate that given enough terms in the asymptotic expansion the rate of convergence of the diagonal staircase approximant sequence increases with the degree. Different branches of the approximants converge to different branches of the function. The success of the high-degree approximants is attributed to their ability to model the function on multiple sheets of the Riemann surface and to reproduce the correct singularity structure in the limit of large perturbation parameter. An efficient recursive algorithm for computing the diagonal approximant sequence is presented.

Analysis and modification of Newton's method for algebraic Riccati equations

Abstract: When Newton's method is applied to find the maximal symmetric solution of an algebraic Riccati equation, convergence can be guaranteed under moderate conditions. In particular, the initial guess need not be close to the solution. The convergence is quadratic if the Frechet derivative is invertible at the solution. In this paper we examine the behaviour of the Newton iteration when the derivative is not invertible at the solution. We find that a simple modification can improve the performance of the Newton iteration dramatically.

An analysis of complex reaction networks in groundwater modeling

Abstract: The complex chemistry describing the biogeochemical dynamics in the natural subsurface environments gives rise to heterogeneous reaction networks, the individual segments of which can feature a wide range of timescales. This paper presents a formulation of the mass balance equations for the batch chemistry and the transport of groundwater contaminants participating in such arbitrarily complex networks of reactions. We formulate the batch problem as an initial-value differential algebraic equation (DAE) system and compute its “index” so that the ease of solvability of the system is determined. We show that when the equilibrium reactions obey the law of mass action, the index of this initial-value DAE system is always unity (thus solvable with well-developed techniques) and that the system can be decoupled into a set of linearly implicit ordinary differential equations and a set of explicit algebraic equations. The formulations for the transport of these reaction networks can take advantage of their solvability properties under batch conditions. To avoid the error associated with time splitting fast reactions from transport, we present a split-kinetics approach where the fast equilibrium reactions are combined with transport equations while only the slower kinetic reactions are time split. These results are used to formulate and solve a simplified reaction network for the biogeochemical transformation of Co(II) ethylenediaminetetraacetic acid (EDTA) in the presence of iron-coated sediments.

Regular solutions of nonlinear differential-algebraic equations and their numerical determination

Abstract: For a general class of nonlinear (possibly higher index) differential-algebraic equations we show existence and uniqueness of solutions. These solutions are regular in the sense that Newton's method will converge locally and quadratically. On the basis of the presented theoretical results, numerical methods for the determination of consistent initial values and for the computation of regular solutions are developed. Several numerical examples are included.

Algebraic Multi-grid for Discrete Elliptic Second-Order Problems

Abstract: This paper is devoted to the construction of Algebraic Multi-Grid (AMG) methods, which are especially suited for the solution of large sparse systems of algebraic equations arising from the finite element discretization of second-order elliptic boundary value problems on unstructured, fine meshes in two or three dimensions. The only information needed is recovered from the stiffness matrix. We present two types of coarsening algorithms based on the graph of the stiffness matrix. In some special cases of nested mesh refinement, we observe, that some geometrical version of the multi-grid method turns out to be a special case of our AMG algorithms. Finally, we apply our algorithms on two and three dimensional heat conduction problems in domains with complicated geometry (e.g. micro-scales), as well as to plane strain elasticity problems with jumping coefficients.

Biharmonic Problem in a Rectangle

Abstract: This paper addresses the fascinating long history of the classical two-dimensional biharmonic problem for a rectangular domain. Among various mathematical and engineering approaches, the method of superposition is effective for solving mechanical problems concerning creeping flow of viscous fluid set up in a rectangular cavity by tangential velocities applied along its walls, an equilibrium of an elastic rectangle, and bending of a clamped thin rectangular elastic plate by a normal load. The object of this paper is both to clarify some purely mathematical questions connected with the solution of the infinite systems of linear algebraic equations and to provide a considerable simplification of the numerical algorithm. The method is illustrated by several examples of steady Stokes flow in a square cavity.

Control of Nonlinear Differential Algebraic Equation Systems : An Overview

Abstract: Chemical processes are inherently nonlinear and their dynamics are naturally described by systems of coupled differential and algebraic equations (DAEs); the differential equations arise from the standard dynamic balances of mass, energy and momentum, while the algebraic equations typically include thermodynamic relations, empirical correlations, quasi-steady-state relations etc. In many cases, the algebraic equations in the DAE model can be readily eliminated to obtain an equivalent ordinary differential equation (ODE) model, which can be used as the basis for controller design. On the other hand, there is a broad class of chemical processes for which the algebraic equations in the DAE models are “singular” in nature, and thus, inhibit a direct reduction of the DAE model into an ODE system. Such DAE systems with singular algebraic equations are said to have a high “index” and they are fundamentally different from ODE systems.

Rational solutions of first-order differential equations

Abstract: We prove that degrees of rational solutions of an algebraic differential equation F(dw/dz,w,z) = 0 are bounded. For given F an upper bound for degrees can be determined explicitly. This implies that one can find all rational solutions by solving algebraic equations.

Two-component formulation of the Wheeler–DeWitt equation

Abstract: The Wheeler–DeWitt equation for the minimally coupled Friedman–Robertson–Walker-massive-scalar-field minisuperspace is written as a two-component Schrodinger equation with an explicitly “time”-dependent Hamiltonian. This reduces the solution of the Wheeler–DeWitt equation to the eigenvalue problem for a nonrelativistic one-dimensional harmonic oscillator and an infinite series of trivial algebraic equations whose iterative solution is easily found. The solution of these equations yields a mode expansion of the solution of the original Wheeler–DeWitt equation. Further analysis of the mode expansion shows that in general the solutions of the Wheeler–DeWitt equation for this model are doubly graded, i.e., every solution is a superposition of two definite-parity solutions. Moreover, it is shown that the mode expansion of both even- and odd-parity solutions is always infinite. It may be terminated artificially to construct approximate solutions. This is demonstrated by working out an explicit example which turns out to satisfy DeWitt’s boundary condition at initial singularity.

Elimination Methods in Polynomial Computer Algebra

Abstract: From the Publisher: This book presents a modified method, based on multidimensional residue theory, for the elimination of unknowns from a system of nonlinear algebraic equations. An algorithm is given for constructing the resultant of the system, and a computer implementation making use of formula manipulation software is carried out. Programmes in MAPLE are available The algorithms and programmes are then applied to questions from the theory of chemical kinetics, such as the search for all stationary solutions of kinetic equations and the construction of kinetic polynolynomials. The subject of this book is closely connected with a wide range of current problems in the analysis of nonlinear systems.

On Asymptotics in Case of Linear Index-2 Differential-Algebraic Equations

Abstract: Asymptotic properties of solutions of general linear differential-algebraic equations (DAEs) and those of their numerical counterparts are discussed. New results on the asymptotic stability in the sense of Lyapunov as well as on contractive index-2 DAEs are given. The behavior of the backward differentiation formula (BDF), implicit Runge--Kutta (IRK), and projected implicit Runge--Kutta (PIRK) methods applied to such systems is investigated. In particular, we clarify the significance of certain subspaces closely related to the geometry of the DAE. Asymptotic properties like A-stability and L-stability are shown to be preserved if these subspaces are constant. Moreover, algebraically stable IRK(DAE) are B-stable under this condition. The general results are specialized to the case of index-2 Hessenberg systems.