A numerical method for computing singular minimizers

Summary. A numerical method, with truncation methods as a special case, for computing singular minimizers in variational problems is described. It is proved that the method can avoid Lavrentiev phenomenon and detect singular minimizers. The convergence of the method is also established. Numerical results on a 2-D problem are given. Received September 21, 1994     

A numerical method for computing the Hamiltonian Schur form

We derive a new numerical method for computing the Hamiltonian Schur form of a Hamiltonian matrix that has no purely imaginary eigenvalues. We demonstrate the properties of the new method by showing its performance for the benchmark collection of continuous-time algebraic Riccati equations. Despite the fact that no complete error analysis for the method is yet available, the numerical results indicate that if no eigenvalues of are close to the imaginary axis then the method computes the exact Hamiltonian Schur form of a nearby Hamiltonian matrix and thus is numerically strongly backward stable. The new method is of complexity and hence it solves a long-standing open problem in numerical analysis. Volker Mehrmann was supported by Deutsche Forschungsgemeinschaft, Research Grant Me 790/11-3.     

A discrete eigenfunctions method for computing mixed hyperbolic problems based on an implicit difference scheme

In this paper numerical solutions of mixed hyperbolic problems are computed using a discrete eigenfunctions method combined with an implicit difference scheme. This new numerical technique preserves the qualitative properties of the analytic solution due to the Sturm-Liouville structure of the underlying discrete linear boundary-value problem and has computational stability advantages vs other methods. Illustrative examples are included.     

Mathematical model of an optical antenna based on the discrete sources method

We consider an optical antenna that is a linear cluster of nanoparticles inside a metal film on a substrate. Based on discrete sources method has been developed and implemented a mathematical model of the optical antenna. Numerical results demonstrate the ability to control the width of the lobe, and directionality of the antenna radiation.     

A numerical method for computing the response of composites with periodic microstructure

The stress‐strain response of a periodic structure due to an applied eigenstrain field is investigated. The periodic structure allows reduction of the problem to a representative volume element (RVE) problem. A review of Fourier series as a particular method of solving the set of differential equations is given, and an example problem is studied. Results of the latter method are compared to results obtained using discrete Fourier transforms in a different way.     

A copula-based model of speculative price dynamics in discrete time

This paper suggests a new technique to construct first order Markov processes using products of copula functions, in the spirit of Darsow et al. (1992) [10]. The approach requires the definition of (i) a sequence of distribution functions of the increments of the process, and (ii) a sequence of copula functions representing dependence between each increment of the process and the corresponding level of the process before the increment. The paper shows how to use the approach to build several kinds of processes (stable, elliptical, Farlie-Gumbel-Morgenstern, Archimedean and martingale processes), and how to extend the analysis to the multivariate setting. The technique turns out to be well suited to provide a discrete time representation of the dynamics of innovations to financial prices under the restrictions imposed by the Efficient Market Hypothesis.     

A conjugate-direction method based on a nonquadratic model

A conjugate-gradient method for unconstrained optimization, which is based on a nonquadratic model, is proposed. The technique has the same properties as the Fletcher-Reeves algorithm when applied to a quadratic function. It is shown to be efficient when tried on general functions of different dimensionality.     

A numerical method for the dynamics and stability of spiral waves

In this paper we propose a new method of investigating the change of dynamics in reaction-diffusion equations, which is based on approximating the Euclidian norm of state variables along with the introduction of phase space. Our method is simple in implementation and can be applied to study the dynamics of multiple spirals. The method is extended to study the stability of spiral waves by developing an algorithm which is applied to circular and meandering motions.     

Global dynamics of a general brucellosis model with discrete delay

For the prevention and control of brucellosis, it is important to investigate the mechanism of brucellosis transmission. Based on the characteristics of the spread of brucellosis, a susceptible-exposed-infectious-brucella (SEIB) delay dynamic model is proposed with the general incidence, elimination rate and shedding rate of pathogen. Under biologically motivated assumptions, it shows the uniqueness of the endemic equilibrium, and investigates the global asymptotically stability of the disease-free equilibrium and the endemic equilibrium. The results suggest that the global stability of equilibria depends entirely on the basic reproduction number $R_0$ and time delay is harmless for the stability of equilibria. Finally, some specific examples and numerical simulations are used to illustrate the utilization of research results and reveal the biological significance of hypothesis $(H_7)$, which implies that the dynamics of brucellosis transmission depend largely on the development of the prevention and control strategies.     

Numerical stability of a discrete model in the dynamics of ferromagnetic bodies

A discrete penalty method for the numerical study of the evolution equations for soft and undeformed magnetoelastic solids is proposed. Some results concerning the stability and the boundedness of the numerical solution are established. Since the aim of the article is also to show the numerical development of singularities (these are expected, due to the similarities with the evolution equations for liquid crystals), some numerical results on a specific test problem are reported and discussed. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 544–557, 1999     

A study on the modeling ability of the IDS method: A soft computing technique using pattern-based information processing

The ink drop spread (IDS) method is a modeling technique developed by algorithmically mimicking the information-handling processes of the human brain. This method has been proposed as a new approach to soft computing. IDS modeling is characterized by processing that uses intuitive pattern information instead of complex formulas, and it is capable of stable and fast convergences. This paper investigates the modeling ability of the IDS method based on three typical benchmarks. Experimental results demonstrated that the IDS method can handle various modeling targets, ranging from logic operations to complex nonlinear systems, and that its modeling performance is satisfactory in comparison with that of feedforward neural networks.     

A numerical procedure for computing the Moore-Penrose inverse

An algorithm for computing the Moore-Penrose inverse of an arbitraryn×m real matrixA is presented which uses a Gram-Schmidt like procedure to form anA-orthogonal set of vectors which span the subspace perpendicular to the kernel ofA. This one procedure will work for any value ofn andm, and for any value of rank (A).     

A method of solving self-similar problems in membrane dynamics

The possibility of extending the range of application of the functionally-invariant Smirnov–Sobolev solutions to plane dynamic problems is demonstrated using the solution of the Rakhmatulin problem as an example.     

A discrete optimization method based on a parameterization of a Grassmannian in multidimensional dichotomous data structuring

An approach to reducing large computational time in the problem of multidimensional dichotomous data structuring based on algebraic properties of finite geometries is proposed. A vector parameterization of the Grassmannian Gr₂(k, n) reducing memory expenditures and the number of operations required to solve this problem is introduced. A parallelization algorithm based on this parameterization and Gray coding which further reduces computational time is constructed.     

A thermodynamically consistent numerical method for a phase field model of solidification

A discretization is presented for the initial boundary value problem of solidification as described in the phase-field model developed by Penrose and Fife (1990) [1] and Wang et al. (1993) [2]. These are models that are completely derived from the laws of thermodynamics, and the algorithms that we propose are formulated to strictly preserve them. Hence, the discrete solutions obtained can be understood as discrete dynamical systems satisfying discrete versions of the first and second laws of thermodynamics. The proposed methods are based on a finite element discretization in space and a midpoint-type finite-difference discretization in time. By using so-called discrete gradient operators, the conservation/entropic character of the continuum model is inherited in the numerical solution, as well as its Lyapunov stability in pure solid/liquid equilibria.