Showing papers in "Computing in 1990"

Algorithms for the maximum satisfiability problem

Abstract: Old and new algorithms for the Maximum Satisfiability problem are studied We first summarize the different heuristics previously proposed, ie, the approximation algorithms of Johnson and of Lieberherr for the general Maximum Satisfiability problem, and the heuristics of Lieberherr and Specker, Poljak and Turzik for the Maximum 2-Satisfiability problem We then consider two recent local search algorithmic schemes, the Simulated Annealing method of Kirkpatrick, Gelatt and Vecchi and the Steepest Ascent Mildest Descent method, and adapt them to the Maximum Satisfiability problem The resulting algorithms, which avoid being blocked as soon as a local optimum has been found, are shown empirically to be more efficient than the heuristics previously proposed in the literature

On adaptive sampling

Abstract: We analyze the storage/accuracy trade-off of an adaptive sampling algorithm due to Wegman that makes it possible to evaluate probabilistically the number of distinct elements in a large file stored on disk.

Computational aspects of a branch and bound algorithm for quadratic zero-one programming

Abstract: In this paper we describe computational experience in solving unconstrained quadratic zero-one problems using a branch and bound algorithm. The algorithm incorporates dynamic preprocessing techniques for forcing variables and heuristics to obtain good starting points. Computational results and comparisons with previous studies on several hundred test problems with dimensions up to 200 demonstrate the efficiency of our algorithm.

Recurrence relations for rational cubic methods, part I. The Halley method

Abstract: In this paper we present a system of a priori error bounds for the Halley method in Banach spaces. Our theorem supplies sufficient conditions on the initial point to ensure the convergence of Halley iterates, by means of a system of “recurrence relations”, analogous to those given for the Newton method by Kantorovich, improving previous results by Doring [4]. The error bounds presented are optimal for second degree polynomials. Other rational cubic methods, as the Chebyshev method, will be treated in a subsequent paper.

Heuristics for Graph Coloring

Abstract: Heuristics for Graph Coloring. Some sequential coloring techniques are reviewed. A few general principles for designing heuristics are outlined and recent coloring techniques baaed on tabu search are discussed.

Yet another application of a binomial recurrence. Order statistics

Abstract: We investigate the moments of the maximum of a set of i.i.d geometric random variables. Computationally, the exact formula for the moments (which does not seem to be available in the literature) is inhibited by the presence of an alternating sum. A recursive expression for the moments is shown to be superior. However, the recursion can be both computationally intensive as well as subject to large round-off error when the set of random variables is large, due to the presence of factorial terms. To get around this difficulty we develop accurate asymptotic expressions for the moments and verify our results numerically.

Numerical integration of the one-dimensional Schro¨dinger equations

Abstract: Three Numerov-type methods with phase-lag of order eight and ten are developed for the numerical integration of the one-dimensional Schrodinger equation. One has a large interval of periodicity and the other two areP-stable. Extensive numerical testing on the resonance problem indicates that these new methods are generally more accurate than other previously developed finite difference methods for this problem.

e-optimality for bicriteria programs and its application to minimum cost flows

Abstract: A subsetS⊂X of feasible solutions of a multicriteria optimization problem is called e-optimal w.r.t. a vector-valued functionf:X→Y $$ \subseteq $$ ℝ K if for allx∈X there is a solutionz x∈S so thatf k(z x)≤(1+e)f k (x) for allk=1,...,K. For a given accuracy e>0, a pseudopolynomial approximation algorithm for bicriteria linear programming using the lower and upper approximation of the optimal value function is given. Numerical results for the bicriteria minimum cost flow problem on NETGEN-generated examples are presented.

A parallel preconditioned conjugate gradient method using domain decomposition and inexact solvers on each subdomain

Abstract: We describe a preconditioned conjugate gradient solution strategy for a multiprocessor system with message passing architecture. The preconditioner combines two techniques, a Schurcomplement preconditioning over “coupling boundaries” between the subdomains and an arbitrary choice of classic preconditioning for the inner degrees of freedom on each subdomain. All computational work on the single subdomains is carried out in parallel by distributing the subdomain data over the processor network before starting the finite element solution process (including generating the element matrices and assemblying the local subdomain stiffness matrix). The resulting spectral condition number of the entire preconditioner is estimated. For the important example of choosing MIC(0)-*-preconditioning on the subdomains, the condition number obtained is essentially the product of the two condition numbers involved.

Runge-Kutta interpolants based on values from two successive integration steps

Abstract: New interpolants of the explicit Runge-Kutta method for the initial value problem are proposed. These interpolants are based on values of the solution and its derivative from two successive integration steps. In this paper, three interpolants withO(h6) local error (l.e.), for the fifth order solution, of the methods Fehlberg 4(5) (RKF 4(5)), Dormand and Prince 5(4) (RKDP 5(4)) and Verner 5(6) (RKV 5(6)) without extra cost are derived. An interpolant withO(h7) (l.e.) for the sixth order solution of the Verner's method with only one extra function evaluation per integration step is also constructed. The above advantages are obtained without any cost in the magnitude of the error.

Finite difference schemes on grids with local refinement in time and space for parabolic problems. I. Derivation, stability, and error analysis

Abstract: Finite difference schemes for parabolic initial value problems on cell-centered grids in space (rectangular for two space dimensions) with regular local refinement in space as in time are derived and their stability and convergence properties are studied. The construction of the finite difference schemes is based on the finite volume approach by approximation of the balance equation. Thus the derived schemes preserve the mass (or the heat).

Rosenbrock methods for differential-algebraic systems with solution-dependent singular matrix multiplying the derivative

Abstract: We study convergence and order conditions of Rosenbrock type methods applied to differential-algebraic systems of the formB(y)y′=a(y), with singular matrixB. An embedded pair of methods of order 3(2) is constructed.

Automatic mesh generation for finite element analysis

Abstract: The finite element analysis in engineering applications comprises three phases: domain discretization, equation solving and error analysis. The domain discretization or mesh generation is the pre-processing phase which plays an important role in the achievement of accurate solutions. In this paper, the improvement of one particularly promising technique for generating two-dimensional meshes is presented. Our technique shows advantages and efficiency over some currently available mesh generators.

Computing the external geodesic diameter of a simple polygon

Abstract: Given a simple polygonP ofn vertices, we present an algorithm that finds the pair of points on the boundary ofP that maximizes theexternal shortest path between them. This path is defined as theexternal geodesic diameter ofP. The algorithm takes0(n2) time and requires0(n) space. Surprisingly, this problem is quite different from that of computing theinternal geodesic diameter ofP. While the internal diameter is determined by a pair of vertices ofP, this is not the case for the external diameter. Finally, we show how this algorithm can be extended to solve theall external geodesic furthest neighbours problem.

Finite element approximation of electrostatic potential in one dimensional multilayer structures with quantized electronic charge

Abstract: The problem of computing the electrostatic potential in a one dimensional multilayer semiconductor device with quantized electrons density is analysed using results of monotone operator theory and perturbation calculus. An error estimate is proved for the discretization with Lagrange finite elements of degree one. A practical implementation of the method, using a quasi-Newton algorithm is presented together with some numerical results.

Implementation issues in solving nonlinear equations for two-point boundary value problems

Abstract: Complex numerical methods often contain subproblems that are easy to state in mathematical form, but difficult to translate into software. Several algorithmic isues of this nature arise in implementing a Newton iteration scheme as part of a finite-difference method for two-point boundary value problems. We describe the practical as well as theoretical considerations behind the decisions included in the final code, with special emphasis on two “watchdog” strategies designed to improve reliability and allow early termination of the Newton iterates.

Numerical solution for the Cauchy problem in nonlinear 1-D thermoelasticity

Abstract: We approximate the Cauchy problem by a problem in a bounded domain Ω R =(−R,R) withR>0 sufficiently large, and the boundary conditions on ∂Ω R are imposed in terms of the far field behavior of solutions to the Cauchy problem. Then we solve this approximate problem by the finite element method for the spatial variable and the difference method for the time variable. Moreover a coupled numerical scheme for the Cauchy problem is presented. The error estimates are established.

Shape preserving histopolation using rational quadratic splines

Abstract: In this paper the area true approximation of histograms by rational quadraticC1-splines is considered under constraints like convexity or monotonicity. For the existence of convex or monotone histosplines sufficient and necessary conditions are derived, which always can be satisfied by choosing the rationality parameters appropriately. Since the mentioned problems are in general not uniquely solvable histo-splines with minimal mean curvature areconstructed.

A note on convergence concepts for stiff problems

Abstract: Most convergence concepts for discretizations of nonlinear stiff initial value problems are based on one-sided Lipschitz continuity. Therefore only those stiff problems that admit moderately sized one-sided Lipschitz constants are covered in a satisfactory way by the respective theory. In the present note we show that the assumption of moderately sized one-sided Lipschitz constants is violated for many stiff problems. We recall some convergence results that are not based on one-sided Lipschitz constants; the concept of singular perturbations is one of the key issues. Numerical experience with stiff problems that are not covered by available convergence results is reported.

Systolic computation of interpolating polynomials

Abstract: Several time-optimal and spacetime-optimal systolic arrays are presented for computing a process dependence graph corresponding to the Aitken algorithm. It is shown that these arrays also can be used to compute the generalized divided differences, i.e., the coefficients of the Hermite interpolating polynomial. Multivariate generalized divided differences are shown to be efficiently computed on a 2-dimensional systolic array. The techniques also are applied to the Neville algorithm, producing similar results.

On the evaluation of one-dimensional Cauchy principal value integrals by rules based on cubic spline interpolation

Abstract: In this note we consider the numerical evaluation of one dimensional Cauchy principal value integrals of the form $$\rlap{--} \smallint _a^b \frac{{k(x)f(x)}}{{x - \lambda }}dx, a< \lambda< b,$$ by rules obtained by “subtracting out” the singularity and then applying product quadratures based on cubic spline interpolation at equally spaced nodes. Convergence results are established for Holder continuous functions of order, μ, 0<μ≤1, and asymptotic rates are obtained for functionsf≠C k [a, b],k=1, 2, 3 or 4. Some comparisons with other methods and numerical examples are also given.

Solving nonlinear equation systems via global partition and search: some experimental results

Abstract: A general framework is proposed for the solution of nonlinear equation systems, applying multiextremal (global) optimization methodology. Following a brief presentation of the background needed, numerical results are reported for two classes of easily reproducible pseudo-random test problems.

Order, stepsize and stiffness switching

Abstract: The question discussed in this paper is “When should a code switch between stiff and non-stiff options, when should it switch between one order and another and how should it adjust its stepsize from one step to the next.” Criteria are proposed for switching between the options that become available as the integration progresses and these are presented in algorithmic form.

Continuous extensions of Rosenbrock-type methods

Abstract: Solving an initial value problem by a Rosenbrock method produces, in general, the numerical solution at (in advance unknown) gridpoints. Applications requiring frequent output (as graphics, delay differential equations, problems with driving equations) normally restrict the stepsize control of these codes and increase the computational overhead considerably. In this paper we introduce a class of continuous extensions for Rosenbrock-type methods. These extensions furnish a continuous numerical solution without affecting the efficiency of the codes. — Author's Abstract

A mesh-independence principle for nonlinear operator equations and their discretizations under mild differentiability conditions

Abstract: In this note we extend the validity of the mesh-independence principle for nonlinear operator equations and their discretizations to include operators whose derivatives are only Holder continuous.

On some ways of approximating inverses of banded matrices in connection with deriving preconditioners based on incomplete block factorization

Abstract: A unified approach of deriving band approximate inverses of band symmetric positive definite matrices is considered. Such band approximations to the inverses of successive Schur complements are required throughout incomplete block factorizations of block-tridiagonal matrices. Such block-tridiagonal matrices arise, for example, in finite element solution of second order elliptic differential equations. A sharp decay rate estimate for inverses of blocktridiagonal symmetric positive definite matrices is given in addition. Numerical tests on a number of model elliptic boundary value problems are presented comparing thus derived preconditioning matrices.

Solving N + m nonlinear equations with only m nonlinear variables

Abstract: We derive a method for solvingN+m nonlinear algebraic equations inN+m unknownsy≠R m andz≠R N of the formA(y)z+b(y)=0, where the(N+m) × N matrixA(y) and vectorb(y) are continuously differentiable functions ofy alone. By exploiting properties of an orthonormal basis for null(A T (y)) the problem is reduced to solvingm nonlinear equations iny only. These equations are solved by Newton's method inm variables. Details of computational implementation and results are provided.

Multicriterion optimization using interval analysis

Abstract: Interval Analysis methods have been applied for obtaining the global optimum of the multimodal multivariable functions. We discuss here the multicriterion optimization problem, where several objective functions must be optimized in conflicting situations.

Truncated nonlinear ABS algorithm and its convergence property

Abstract: This paper shows that the nonlinear ABS algorithm [1] can be viewed as inexact Newton method [3]. From this point of view, the truncated nonlinear ABS algorithm is established and its convergence property is discussed.

A programmed algorithm for existence proofs for two-point boundary value problems

Abstract: For (scalar) nonlinear two-point boundary value problems of the second order, we present a programmed algorithm for proving the existence of a solution within calculated bounds. This algorithm is based on the theory of (functional-) differential inequalities applied to certain transformed problems.