Macroscopic modeling and simulations of room evacuation

We analyze numerically two macroscopic models of crowd dynamics: the classical Hughes model and the second order model being an extension to pedestrian motion of the Payne–Whitham vehicular traffic model. The desired direction of motion is determined by solving an eikonal equation with density dependent running cost, which results in minimization of the travel time and avoidance of congested areas. We apply a mixed finite volume-finite element method to solve the problems and present error analysis for the eikonal solver, gradient computation and the second order model yielding a first order convergence. We show that Hughes’ model is incapable of reproducing complex crowd dynamics such as stop-and-go waves and clogging at bottlenecks. Finally, using the second order model, we study numerically the evacuation of pedestrians from a room through a narrow exit.     

A general framework for modeling tumor-immune system competition at the mesoscopic level

We propose a class of nonlinear integro-differential equations that at the mesoscopic level models the competition between a tumor and the immune system. The model describes the evolution of a distribution function of the microscopic parameter referred to as activity of cells. The idea is somehow similar to the Enskog theory in kinetic theory. By averaging with respect to the parameter, the mesoscopic class of models reduces to the general class of macroscopic models introduced by A. d’Onofrio that may assess the effect of delays in stimulation of the immune system by tumor cells. The existence and uniqueness theory is developed.     

On the concept of generalized order optimality

The main results of this paper include a detailed analysis of the notion of generalized order optimality and some sufficient conditions for a point satisfying the necessary optimality condition of Mordukhovich (2006) [1] and [2] for being a generalized order solution of the optimization problem under consideration.     

An effective modeling and solution approach for the generalized independent set problem

The generalized independent set (GIS) problem was first introduced by Hochbaum and Pathria (Forest Sci 43(4), 544–554, 1997) and independently explored in greater detail by Hochbaum (Manage Sci 50(6), 709–123, 2004). This problem, with applications in forest management and a variety of related areas, is a generalization of the classical maximum independent set problem. In this paper we highlight a natural, nonlinear formulation for the problem that is an attractive alternative to the linear model found in the literature. The effectiveness of this alternative formulation is demonstrated by computational experience on test problems of varying size and density, disclosing a dramatic reduction in the time to obtain optimal and near optimal solutions and an ability to solve much larger problems.     

Modeling complex multi-component reactive-transport systems: towards a simulation environment based on the concept of a Knowledge Base

A modelling framework within which transport processes in the hydrosphere can be described and interfaced with relevant biogeochemical reactions is presented. Three key elements of this simulation environment are discussed: (1) a numerical engine for solving sets of coupled non-linear process equations; (2) an automated procedure for model code generation (`Automatic Code Generator'); (3) a Web-distributed Knowledge Base (KB) of processes. The Automatic Code Generator translates the information selected in the KB into computer algorithms using the principles defined in the numerical engine. The code CONTRASTE is a first attempt at developing such a modelling framework. It allows one to easily select, adapt and combine a specific set of biogeochemical processes relevant to a user-defined application. The workings of CONTRASTE are described by means of examples which demonstrate how the various components of the simulation environment are coupled and automated. Prospects for future developments towards a fully automated model generation procedure are discussed.     

Reconstruction of inhomogeneous conductivities via the concept of generalized polarization tensors

This paper extends the concept of generalized polarization tensors (GPTs), which was previously defined for inclusions with homogeneous conductivities, to inhomogeneous conductivity inclusions. We begin by giving two slightly different but equivalent definitions of the GPTs for inhomogeneous inclusions. We then show that, as in the homogeneous case, the GPTs are the basic building blocks for the far-field expansion of the voltage in the presence of the conductivity inclusion. Relating the GPTs to the Neumann-to-Dirichlet (NtD) map, it follows that the full knowledge of the GPTs allows unique determination of the conductivity distribution. Furthermore, we show important properties of the the GPTs, such as symmetry and positivity, and derive bounds satisfied by their harmonic sums. We also compute the sensitivity of the GPTs with respect to changes in the conductivity distribution and propose an algorithm for reconstructing conductivity distributions from their GPTs. This provides a new strategy for solving the highly nonlinear and ill-posed inverse conductivity problem. We demonstrate the viability of the proposed algorithm by preforming a sensitivity analysis and giving some numerical examples.     

Coupling of atomistic and mesoscopic scales: Multiscale modeling of DNA translocation through nanopores

The basic principles of a novel computational method to model nano-sized systems, such as a polymer embedded in a fluctuating hydrodynamic solvent, are described. The approach is based on a concurrent multi-scale technique rellying on Molecular Dynamics and Lattice Boltzmann in stochastic formulations. We undertook a numerical study of a biopolymer translocating across a nanopore mimicking a device for automatic DNA sequencing. The study revealed that the translocation time obeys a power law dependence on the polymer length with an exponent in agreement with experimental measurements.The scaling behavior is captured by a mean-field model taking into account both the hydrodynamic drag and thermodynamic effects. Results about multi-file translocation of DNA in large pores are briefly described. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An approach to the modeling and analysis of multiobjective generalized networks

Generalized networks can often provide substantial advantages in both the modeling and solution of integer programming problems. In this paper we present a straightforward approach which combines generalized networks with goal programming so as to achieve a modeling and solution methodology for multiobjective generalized networks. Such an approach also encompasses the solution to weighted integer foal programming as well as lexicographic integer goal programming problems. In ongoing research, the resulting hybrid algorithms have indicated superior performance, for a number of problems, over that obtained by more conventional approaches. A particularly attractive feature of the methodology is its relative simplicity and robustness.     

An algorithm based on the generalized D-gap function for equilibrium problems

The equilibrium problem (EP) can be reformulated as an unconstrained minimization problem through the generalized D-gap function. In this paper, we propose an algorithm for minimizing the problem and analyze some convergence properties of the proposed algorithm. Under some reasonable conditions, we show that the iteration sequence generated by the algorithm is globally convergent and converges to a solution to the EP and the generalized D-gap function provides a global error bound for the algorithm.     

Inversion algorithm based on the generalized objective functional for compressed sensing

Owing to providing a novel insight for signal and image processing, compressed sensing (CS) has attracted increasing attention. The accuracy of the reconstruction algorithms plays an important role in real applications of the CS theory. In this paper, a generalized reconstruction model that simultaneously considers the inaccuracies on the measurement matrix and the measurement data is proposed for CS reconstruction. A generalized objective functional, which integrates the advantages of the least squares (LSs) estimation and the combinational M-estimation, is proposed. An iterative scheme that integrates the merits of the homotopy method and the artificial physics optimization (APO) algorithm is developed for solving the proposed objective functional. Numerical simulations are implemented to evaluate the feasibility and effectiveness of the proposed algorithm. For the cases simulated in this paper, the reconstruction accuracy is improved, which indicates that the proposed algorithm is successful in solving CS inverse problems.     

Finite elements based on consistently assumed stresses and displacements

Finite element stiffness matrices are derived using an extende Hellinger-Reissner principle in which internal displacements are added to serve as Lagrange multipliers to introducethe equilibrium constraint in each element. In a consistent formulation the assumed stresses are initially unconstrained and complete polynomials and the total displacements are also complete such that the corresponding strains are complete in the same order as the stressesSeveral examples indicate that resulting properties for elements constructed by this consistent formulation are ideal and are less sensitive to distortions of element geometries. The method has been used to find the optimal stress terms for plane elements, 3-D solids, axisymmetric solids, and plate bending elements.     

Mathematical modeling of heat-distribution processes using generalized equations for thermal conductivity

Problems in the mathematical modeling of heat-distribution processes on the basis of more general equations than parabolic equations are considered. We study the general structure of the relations between solutions of various approximations to the generalized heat-conductivity equations. We introduce a notion of singularly perturbed dissipative structures and analyze singularly perturbed blow-up regimes.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 70, pp. 54–60, 1990.     

Mathematical modeling of interrelationships among cutting angles,setting angles and working angles of single-point cutting tools

The angles of a cutting tool play a key role in determining its cutting efficiency. However, once the tool has been mounted on a machine tool, setting errors inevitably cause the working angles of the tool to deviate slightly from the designed cutting angles. Accordingly, this study develops mathematical models to analyze the interrelationships among the tool angles, the setting angles and the working angles. In the proposed approach, a process of homogeneous coordinate transformation is employed to develop a kinematic model of the cutting tool such that the working angles can be derived given the tool angles and setting angles. Mathematical formulae are then developed to inversely derive the tool angles required to generate the specified working angles given prior knowledge of the setting angles. An illustrative numerical example is presented to demonstrate the validity of the proposed approach. Overall, the numerical results confirm that the methodology presented in this study provides a comprehensive, simple and versatile means of modeling a variety of conventional single-point cutting tools.     

Numerical simulation and analysis of generalized difference method on triangular networks for electrical impedance tomography

Electrical impedance tomography is an inverse problem of elliptic differential equations. Numerical methods based on combining generalized difference method and Levenberg–Marquardt iteration on a planar domain are proposed. Positive semi-definiteness and existence of solution of the generalized difference scheme are proved. Element geometry matrix is introduced to shortcut calculation and standardize computer program. A series of numerical experiments verify the reliability of its mathematical model and the feasibility of the algorithm. A class of electrical current patterns is proposed to minimize the number of direct problems to be solved in each iteration. These methods have been applied successfully in practical simulation of electrical impedance tomography.