Showing papers in "Computers & Mathematics With Applications in 2013"

Equivalent projectors for virtual element methods

Abstract: In the original virtual element space with degree of accuracy k, projector operators in the H^1-seminorm onto polynomials of degree @?k can be easily computed. On the other hand, projections in the L^2 norm are available only on polynomials of degree @?k-2 (directly from the degrees of freedom). Here, we present a variant of the virtual element method that allows the exact computations of the L^2 projections on all polynomials of degree @?k. The interest of this construction is illustrated with some simple examples, including the construction of three-dimensional virtual elements, the treatment of lower-order terms, the treatment of the right-hand side, and the L^2 error estimates.

Stable calculation of Gaussian-based RBF-FD stencils

Abstract: Traditional finite difference (FD) methods are designed to be exact for low degree polynomials. They can be highly effective on Cartesian-type grids, but may fail for unstructured node layouts. Radial basis function-generated finite difference (RBF-FD) methods overcome this problem and, as a result, provide a much improved geometric flexibility. The calculation of RBF-FD weights involves a shape parameter @e. Small values of @e (corresponding to near-flat RBFs) often lead to particularly accurate RBF-FD formulas. However, the most straightforward way to calculate the weights (RBF-Direct) becomes then numerically highly ill-conditioned. In contrast, the present algorithm remains numerically stable all the way into the @e->0 limit. Like the RBF-QR algorithm, it uses the idea of finding a numerically well-conditioned basis function set in the same function space as is spanned by the ill-conditioned near-flat original Gaussian RBFs. By exploiting some properties of the incomplete gamma function, it transpires that the change of basis can be achieved without dealing with any infinite expansions. Its strengths and weaknesses compared with the Contour-Pade, RBF-RA, and RBF-QR algorithms are discussed.

The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator

Abstract: We analyze a nonlocal diffusion operator having as special cases the fractional Laplacian and fractional differential operators that arise in several applications. In our analysis, a nonlocal vector calculus is exploited to define a weak formulation of the nonlocal problem. We demonstrate that, when sufficient conditions on certain kernel functions hold, the solution of the nonlocal equation converges to the solution of the fractional Laplacian equation on bounded domains as the nonlocal interactions become infinite. We also introduce a continuous Galerkin finite element discretization of the nonlocal weak formulation and we derive a priori error estimates. Through several numerical examples we illustrate the theoretical results and we show that by solving the nonlocal problem it is possible to obtain accurate approximations of the solutions of fractional differential equations circumventing the problem of treating infinite-volume constraints.

Lattice Boltzmann simulations of thermal convective flows in two dimensions

Abstract: In this paper we study the lattice Boltzmann equation (LBE) with multiple-relaxation-time (MRT) collision model for incompressible thermo-hydrodynamics with the Boussinesq approximation. We use the MRT thermal LBE (TLBE) to simulate the following two flows in two dimensions: the square cavity with differentially heated vertical walls and the Rayleigh-Benard convection in a rectangle heated from below. For the square cavity, the flow parameters in this study are the Rayleigh number Ra=10^3-10^6, and the Prandtl number Pr=0.71; and for the Rayleigh-Benard convection in a rectangle, Ra=2@?10^3, 10^4 and 5@?10^4, and Pr=0.71 and 7.0.

Modeling complex systems with adaptive networks

Abstract: Adaptive networks are a novel class of dynamical networks whose topologies and states coevolve. Many real-world complex systems can be modeled as adaptive networks, including social networks, transportation networks, neural networks and biological networks. In this paper, we introduce fundamental concepts and unique properties of adaptive networks through a brief, non-comprehensive review of recent literature on mathematical/computational modeling and analysis of such networks. We also report our recent work on several applications of computational adaptive network modeling and analysis to real-world problems, including temporal development of search and rescue operational networks, automated rule discovery from empirical network evolution data, and cultural integration in corporate merger.

A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model

Abstract: Evolution equations containing fractional derivatives can provide suitable mathematical models for describing anomalous diffusion and transport dynamics in complex systems that cannot be modeled accurately by normal integer order equations. Recently, researchers have found that many physical processes exhibit fractional order behavior that varies with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. In this paper, we consider the mobile-immobile advection-dispersion model with the Coimbra variable time fractional derivative which is preferable for modeling dynamical systems and is more efficient from the numerical standpoint. A novel implicit numerical method for the equation is proposed and the stability of the approximation is investigated. As for the convergence of the numerical method, we only consider a special case, i.e., the time fractional derivative is independent of the time variable t. The case where the time fractional derivative depends on both the time variable t and the space variable x will be considered in a future work. Finally, numerical examples are provided to show that the implicit difference approximation is computationally efficient.

An interpretation and derivation of the lattice Boltzmann method using Strang splitting

Abstract: The lattice Boltzmann space/time discretisation, as usually derived from integration along characteristics, is shown to correspond to a Strang splitting between decoupled streaming and collision steps. Strang splitting offers a second-order accurate approximation to evolution under the combination of two non-commuting operators, here identified with the streaming and collision terms in the discrete Boltzmann partial differential equation. Strang splitting achieves second-order accuracy through a symmetric decomposition in which one operator is applied twice for half timesteps, and the other operator is applied once for a full timestep. We show that a natural definition of a half timestep of collisions leads to the same change of variables that was previously introduced using different reasoning to obtain a second-order accurate and explicit scheme from an integration of the discrete Boltzmann equation along characteristics. This approach extends easily to include general matrix collision operators, and also body forces. Finally, we show that the validity of the lattice Boltzmann discretisation for grid-scale Reynolds numbers larger than unity depends crucially on the use of a Crank-Nicolson approximation to discretise the collision operator. Replacing this approximation with the readily available exact solution for collisions uncoupled from streaming leads to a scheme that becomes much too diffusive, due to the splitting error, unless the grid-scale Reynolds number remains well below unity.

Fractional Sturm-Liouville problem

Abstract: In this paper, we define some Fractional Sturm-Liouville Operators (FSLOs) and introduce two classes of Fractional Sturm-Liouville Problems (FSLPs) namely regular and singular FSLP. The operators defined here are different from those defined in the literature in the sense that the operators defined here contain left and right Riemann-Liouville and left and right Caputo fractional derivatives. For both classes we investigate the eigenvalue and eigenfunction properties of the FSLOs. In the class of regular FSLPs, we discuss two types of FSLPs. As an operator for the class of singular FSLPs, we introduce a Fractional Legendre Equation (FLE) and discuss its solution. It is shown that the Legendre Polynomials resulting from an FLE are the same as those obtained from the integer order Legendre equation; however, the eigenvalues of the two equations differ. Using the Legendre integral transform we demonstrate some applications of our results by solving two fractional differential equations, one ordinary and the other partial. It is our hope that this paper will initiate new research in the area of FSLPs and many of its variations.

Construction of exact solutions for fractional order differential equations by the invariant subspace method

Abstract: The invariant subspace method for constructing particular solutions is modified for fractional differential equations. It allows one to reduce a fractional partial differential equation to a system of nonlinear ordinary fractional differential equations. Point symmetries of such systems are used to construct their solutions which generate solutions of the original fractional partial differential equation.

Thermal bifurcation buckling of piezoelectric carbon nanotube reinforced composite beams

Abstract: The nonlinear thermal bifurcation buckling behavior of carbon nanotube reinforced composite (CNTRC) beams with surface-bonded piezoelectric layers is studied in this paper. The governing equations of piezoelectric CNTRC beam are obtained based on the Euler-Bernoulli beam theory and von Karman geometric nonlinearity. Two kinds of carbon nanotube-reinforced composite (CNTRC) beams, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The material properties of FG-CNTRC beam are assumed to be graded in the thickness direction. The SWCNTs are assumed aligned, straight and with a uniform layout. Exact solutions are presented to study the thermal buckling behavior of beams made of a symmetric single-walled carbon nanotube reinforced composite with surface-bonded piezoelectric layers. The critical temperature load is obtained for the nonlinear problem. The effects of the applied actuator voltage, temperature, beam geometry, boundary conditions, and volume fractions of carbon nanotubes on the buckling of piezoelectric CNTRC beams are investigated.

A new approach for solving a system of fractional partial differential equations

Abstract: In this paper we propose a new method for solving systems of linear and nonlinear fractional partial differential equations. This method is a combination of the Laplace transform method and the Iterative method and here after called the Iterative Laplace transform method. This method gives solutions without any discretization and restrictive assumptions. The method is free from round-off errors and as a result the numerical computations are reduced. The fractional derivative is described in the Caputo sense. Finally, numerical examples are presented to illustrate the preciseness and effectiveness of the new technique.

On the behavior and performance of chaos driven PSO algorithm with inertia weight

Abstract: In this paper, the utilization of chaos pseudorandom number generators based on three different chaotic maps to alter the behavior and overall performance of PSO algorithm is proposed. This paper presents results of testing the performance and behavior of the proposed algorithm on typical benchmark functions that represent unimodal and multimodal problems. The promising results are analyzed and discussed.

Lattice Boltzmann simulation of turbulent flow laden with finite-size particles

Abstract: The paper describes a particle-resolved simulation method for turbulent flow laden with finite size particles. The method is based on the multiple-relaxation-time lattice Boltzmann equation. The no-slip boundary condition on the moving particle boundaries is handled by a second-order interpolated bounce-back scheme. The populations at a newly converted fluid lattice node are constructed by the equilibrium distribution with non-equilibrium corrections. MPI implementation details are described and the resulting code is found to be computationally efficient with a good scalability. The method is first validated using unsteady sedimentation of a single particle and sedimentation of a random suspension. It is then applied to a decaying isotropic turbulence laden with particles of Kolmogorov to Taylor microscale sizes. At a given particle volume fraction, the dynamics of the particle-laden flow is found to depend mainly on the effective particle surface area and particle Stokes number. The presence of finite-size inertial particles enhances dissipation at small scales while reducing kinetic energy at large scales. This is in accordance with related studies. The normalized pivot wavenumber is found to not only depend on the particle size, but also on the ratio of particle size to flow scales and particle-to-fluid density ratio.

Multi-GPU implementation of the lattice Boltzmann method

Abstract: The lattice Boltzmann method (LBM) is an increasingly popular approach for solving fluid flows in a wide range of applications. The LBM yields regular, data-parallel computations; hence, it is especially well fitted to massively parallel hardware such as graphics processing units (GPU). Up to now, though, single-GPU implementations of the LBM are of moderate practical interest since the on-board memory of GPU-based computing devices is too scarce for large scale simulations. In this paper, we present a multi-GPU LBM solver based on the well-known D3Q19 MRT model. Using appropriate hardware, we managed to run our program on six Tesla C1060 computing devices in parallel. We observed up to 2.15x10^9 node updates per second for the lid-driven cubic cavity test case. It is worth mentioning that such a performance is comparable to the one obtained with large high performance clusters or massively parallel supercomputers. Our solver enabled us to perform high resolution simulations for large Reynolds numbers without facing numerical instabilities. Though, we could observe symmetry breaking effects for long-extended simulations of unsteady flows. We describe the different levels of precision we implemented, showing that these effects are due to round off errors, and we discuss their relative impact on performance.

Is the k-NN classifier in high dimensions affected by the curse of dimensionality?

Abstract: There is an increasing body of evidence suggesting that exact nearest neighbour search in high-dimensional spaces is affected by the curse of dimensionality at a fundamental level. Does it necessarily mean that the same is true for k nearest neighbours based learning algorithms such as the k -NN classifier? We analyse this question at a number of levels and show that the answer is different at each of them. As our first main observation, we show the consistency of a k approximate nearest neighbour classifier. However, the performance of the classifier in very high dimensions is provably unstable. As our second main observation, we point out that the existing model for statistical learning is oblivious of dimension of the domain and so every learning problem admits a universally consistent deterministic reduction to the one-dimensional case by means of a Borel isomorphism.

Global stability for reaction-diffusion equations in biology

Abstract: The aim of this work is to study the global stability for some diffusion equations in biology by constructing Lyapunov functionals. These Lyapunov functionals are obtained from those for ordinary differential equations. Several examples from virology and epidemiology are given to illustrate our method.

A comparison study of multi-component Lattice Boltzmann models for flow in porous media applications

Abstract: A comparison study of three different multi-component Lattice Boltzmann models is carried out to explore their capability of describing binary immiscible fluid systems. The Shan-Chen pseudo potential model, the Oxford free energy model and the colour gradient model are investigated using the multi-relaxation time scheme (MRT) algorithm to study the flow of binary immiscible fluids. We investigate Poiseuille flow of layered immiscible binary fluids and capillary fingering phenomena and evaluate the results against analytical solutions. In addition, we examine the capability of the various models to simulate fluids with significant viscosity and density contrast and suitable interface thickness. This is of great importance for large scale flow in porous media applications, where it is important to minimise the interfacial thickness from a computational point of view. We find that the Shan-Chen model can simulate high density ratios up to 800 for binary fluids with the same viscosity. Imposing a viscosity contrast will lead to highly diffusive interfaces in the Shan-Chen model and therefore this will affect significantly the numerical stability. The Free Energy model and the colour gradient model have similar capabilities on this point: they can simulate binary fluids having the same density but with significant viscosity contrast. This is of great importance to study the flow of water, supercritical CO"2 and oil in porous media, for CO"2 storage and Enhanced Oil Recovery (EOR) operations.

Fractional model for malaria transmission under control strategies

Abstract: We study a fractional model for malaria transmission under control strategies. We consider the integer order model proposed by Chiyaka et al. (2008) in [15] and modify it to become a fractional order model. We study numerically the model for variation of the values of the fractional derivative and of the parameter that models personal protection, b. From observation of the figures we conclude that as b is increased from 0 to 1 there is a corresponding decrease in the number of infectious humans and infectious mosquitoes, for all values of @a. This means that this result is invariant for variation of fractional derivative, in the values tested. These results are in agreement with those obtained in Chiyaka et al. (2008) [15] for @a=1.0 and suggest that our fractional model is epidemiologically well-posed.

Comparison of different propagation steps for lattice Boltzmann methods

Abstract: Several possibilities exist to implement the propagation step of lattice Boltzmann methods. This paper describes common implementations and compares the number of memory transfer operations they require per lattice node update. A performance model based on the memory bandwidth is then used to obtain an estimation of the maximum achievable performance on different machines. A subset of the discussed implementations of the propagation step are benchmarked on different Intel- and AMD-based compute nodes using the framework of an existing flow solver that is specially adapted to simulate flow in porous media, and the model is validated against the measurements. Advanced approaches for the propagation step like ''A-A pattern'' or ''Esoteric Twist'' require more programming effort but often sustain significantly better performance than non-naive but straightforward implementations.

Online game bot detection based on party-play log analysis

Abstract: As online games become popular and the boundary between virtual and real economies blurs, cheating in games has proliferated in volume and method. In this paper, we propose a framework for user behavior analysis for bot detection in online games. Specifically, we focus on party play which reflects the social activities among gamers: in a Massively Multi-user Online Role Playing Game (MMORPG), party play is a major activity that game bots exploit to keep their characters safe and facilitate the acquisition of cyber assets in a fashion very different from that of normal humans. Through a comprehensive statistical analysis of user behaviors in game activity logs, we establish threshold levels for the activities that allow us to identify game bots. Based on this, we also build a knowledge base of detection rules, which are generic. We apply our rule reasoner to AION, a popular online game serviced by NCsoft, Inc., a leading online game company based in Korea.

Imperialist competitive algorithm for solving systems of nonlinear equations

Abstract: Solving systems of nonlinear equations is a relatively complicated problem in which arise a diverse range of sciences. There are a number of different approaches that have been proposed. In this paper, we employ the imperialist competitive algorithm (ICA) for solving systems of nonlinear equations. Some well-known problems are presented to demonstrate the efficiency of this new robust optimization method in comparison to other known methods.

Fractional-order PI λDµ controller design

Abstract: This paper introduces a new design method of fractional-order proportional-derivative (FOPD) and fractional-order proportional-integral-derivative (FOPID) controllers. A biquadratic approximation of a fractional-order differential operator is used to introduce a new structure of finite-order FOPID controllers. Using the new FOPD controllers, the controlled systems can achieve the desired phase margins without migrating the gain crossover frequency of the uncontrolled system. This may not be guaranteed when using FOPID controllers. The proposed FOPID controller has a smaller number of parameters to tune than its existing counterparts. A systematic design procedure is identified in terms of the desired phase and the gain margins of the controlled systems. The viability of the design methods is verified using a simple numerical example.

Impact of environmental factors on mosquito dispersal in the prospect of sterile insect technique control

Abstract: The aim of this paper is to develop a mathematical model to simulate mosquito dispersal and its control taking into account environmental parameters, like wind, temperature, or landscape elements. We particularly focus on the Aedes albopictus mosquito which is now recognized as a major vector of human arboviruses, like chikungunya, dengue, or yellow fever. One way to prevent those epidemics is to control the vector population. Biological control tools, like the Sterile Insect Technique (SIT), are of great interest as an alternative to chemical control tools which are very detrimental to the environment. The success of SIT is based not only on a good knowledge of the biology of the insect, but also on an accurate modeling of the insect's distribution. We consider a compartmental approach and derive temporal and spatio-temporal models, using Advection-Diffusion-Reaction equations to model mosquito dispersal. Periodic releases of sterilized males are modeled with an impulse differential equation. Finally, using the splitting operator approach, and well-suited numerical methods for each operator, we provide numerical simulations for mosquito spreading, and test different vector control scenarios. We show that environmental parameters, like vegetation, can have a strong influence on mosquito distribution and in the efficiency of vector control tools, like SIT.

Fractional differential equations and related exact mechanical models

Abstract: The aim of the paper is the description of fractional-order differential equations in terms of exact mechanical models. This result will be archived, in the paper, for the case of linear multiphase fractional hereditariness involving linear combinations of power-laws in relaxation/creep functions. The mechanical model corresponding to fractional-order differential equations is the extension of a recently introduced exact mechanical representation (Di Paola and Zingales (2012) [33] and Di Paola et al. (2012) [34]) of fractional-order integrals and derivatives. Some numerical applications have been reported in the paper to assess the capabilities of the model in terms of a peculiar arrangement of linear springs and dashpots.

Privacy-preserving max/min query in two-tiered wireless sensor networks

Abstract: In a two-tiered wireless sensor network, resource-limited sensor nodes act as the lower layer for sensing data, and resource-rich storage nodes act as the upper layer for storing data and processing queries from the sink. This architecture has been widely adopted because it can save power and storage consumptions for sensors and improve the efficiency of query processing. However, storage nodes may be compromised in a hostile environment and breach privacy of sensor data. Although privacy-preserving range query and Top- k query have been studied, query for maximum or minimum has not been well addressed. In this paper, we propose a privacy-preserving protocol specializing for MAX/MIN query that prevents adversaries from gaining sensitive information from sensor collected data. To preserve privacy, Prefix Membership Verification approach is employed to encode sensor data such that a storage node can correctly process max/min queries over encoded data without knowing their actual values. Detailed theoretical and quantitative results confirm the high efficacy and efficiency of the proposed schemes.

Extension of the VIKOR method to dynamic intuitionistic fuzzy multiple attribute decision making

Abstract: In this paper, we extend the VIKOR method for dynamic intuitionistic fuzzy multiple attribute decision making (DIF-MADM). Two new aggregation operators called dynamic intuitionistic fuzzy weighted geometric (DIFWG) operator and uncertain dynamic intuitionistic fuzzy weighted geometric (UDIFWG) operator are presented. Based on the DIFWA and UDIFWA operators respectively, we develop two procedures to solve the DIF-MADM problems where all attribute values are expressed in intuitionistic fuzzy numbers or interval-valued intuitionistic fuzzy numbers, which are collected at different periods. Finally, a numerical example is used to illustrate the applicability of the proposed approach.

The multi-attribute decision making method based on interval-valued intuitionistic fuzzy Einstein hybrid weighted geometric operator

Abstract: This article proposes an approach to multi-attribute decision making (MADM) where individual assessments are provided as interval-valued intuitionistic fuzzy numbers (IVIFNs). Firstly, some Einstein geometric operators on interval-valued intuitionistic fuzzy sets, such as Einstein product, Einstein exponentiation etc., and their characteristics are introduced. Secondly, some Einstein geometric operators, such as the interval-valued intuitionistic fuzzy Einstein weighted geometric operator, interval-valued intuitionistic fuzzy Einstein ordered weighted geometric operator and interval-valued intuitionistic fuzzy Einstein hybrid weighted geometric (IV IFHWG^@e) operator, are developed for aggregating the IVIFNs. Moreover, various properties of these operators are established. Finally, an IV IFHWG^@e operator based approach to MADM under interval-valued intuitionistic fuzzy environments is proposed. An illustrative propulsion/manoeuvring system selection problem is employed to demonstrate how to apply the proposed procedure and verify the feasibility and effectiveness of the developed method.

Inversion of circular means and the wave equation on convex planar domains

Abstract: We study the problem of recovering the initial data of the two dimensional wave equation from values of its solution on the boundary @?@W of a smooth convex bounded domain @W@?R^2. As a main result we establish back-projection type inversion formulas that recover any initial data with support in @W modulo an explicitly computed smoothing integral operator K"@W. For circular and elliptical domains the operator K"@W is shown to vanish identically and hence we establish exact inversion formulas of the back-projection type in these cases. Similar results are obtained for recovering a function from its mean values over circles with centers on @?@W. Both reconstruction problems are, amongst others, essential for the hybrid imaging modalities photoacoustic and thermoacoustic tomography.

Propagation speed of the maximum of the fundamental solution to the fractional diffusion-wave equation

Abstract: In this paper, the one-dimensional time-fractional diffusion-wave equation with the fractional derivative of order @a,1<@a<2, is revisited. This equation interpolates between the diffusion and the wave equations that behave quite differently regarding their response to a localized disturbance: whereas the diffusion equation describes a process, where a disturbance spreads infinitely fast, the propagation speed of the disturbance is a constant for the wave equation. For the time-fractional diffusion-wave equation, the propagation speed of a disturbance is infinite, but its fundamental solution possesses a maximum that disperses with a finite speed. In this paper, the fundamental solution of the Cauchy problem for the time-fractional diffusion-wave equation, its maximum location, maximum value, and other important characteristics are investigated in detail. To illustrate analytical formulas, results of numerical calculations and plots are presented. Numerical algorithms and programs used to produce plots are discussed.