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In this paper we use Bernoulli polynomials to derive a new spectral method to find the numerical solutions of second order linear boundary value problems. The proposed method has been applied to several BVPs with Dirichlet, Neumann and mixed types boundary conditions. Numerical examples provide favorable comparisons with other existing methods. [ABSTRACT FROM AUTHOR]

In this paper, a stabilized mixed finite element method for a coupled steady Stokes–Darcy problem is proposed and investigated. This method is based on two local Gauss integrals for the Stokes equations. Its originality is to use a difference between a consistent mass matrix and an under-integrated mass matrix for the pressure variable of the coupled Stokes–Darcy problem by using the lowest equal-order finite element triples. This new method has several attractive computational features: parameter free, flexible, and altering the difficulties inherited in the original equations. Stability and error estimates of optimal order are obtained by using the lowest equal-order finite element triples ( P 1 − P 1 − P 1 ) and ( Q 1 − Q 1 − Q 1 ) for approximations of the velocity, pressure, and hydraulic head. Finally, a series of numerical experiments are given to show that this method has good stability and accuracy for the coupled problem with the Beavers–Joseph–Saffman–Jones and Beavers–Joseph interface conditions. [ABSTRACT FROM AUTHOR]

In this paper we propose a new iterative finite difference (IFD) scheme based on the Newton–Raphson–Kantorovich approximation method in function space to solve the classical one-dimensional Bratu’s problem. This new numerical method produces accurate solutions with low computational cost. The effectiveness and accuracy of the IFD method are confirmed through several numerical examples and compared to some existing numerical solvers. [ABSTRACT FROM AUTHOR]

Modelling has proven to be an important tool in the design of manifolds and silencers for internal combustion engines. Although simple 1D models are generally sufficiently precise in the case of manifold models, they would usually fail to predict the high frequency behaviour of modern compact manifold designs and, of course, of a complex-shaped silencing system. Complete 3D models are able to account for transversal modes and other non-1D phenomena, but at a high computational cost. A suitable alternative is provided by time-domain non-linear quasi-3D models, whose computational cost is relatively low but still providing an accurate description of the high frequency behaviour of certain elements. In this paper, a quasi-3D model which makes use of a non-linear second order time and space discretization based on finite volumes is presented. As an alternative for avoiding overshoots at discontinuities, a Flux-Corrected Transport technique has been adapted to the quasi-3D method in order to achieve convergence and avoid numerical dispersion. It is shown that the combination of dissipation via damping together with the phoenical form of the anti-diffusion term provides satisfactory results. [ABSTRACT FROM AUTHOR]

Multivariate normal stable Tweedie models are recently introduced as an extension to normal gamma and normal inverse Gaussian models. The aim of this paper is to characterize these models through their variance functions. Then, according to the power variance parameter values, the nature of polynomials associated with these models is deduced. [ABSTRACT FROM AUTHOR]

This paper uses the Newton iteration to study new identification methods for determining the parameters of dynamical systems from step responses. On the basis of the step response analysis, we present the Newton iterative algorithms. Moreover, in order to test the accuracy of the estimated parameters, the frequency and step response experiment are applied to the dynamical systems between the estimated and true models. The simulation results show that the obtained models can capture the dynamics of the systems, i.e., the estimated model’s outputs are close to the outputs of the actual systems. This confirms the effectiveness of the proposed Newton iterative identification methods. [ABSTRACT FROM AUTHOR]

The convergence of the block cyclic projection for compressed sensing based tomography (BCPCS) algorithm had been proven recently in the case of underrelaxation parameter λ ∈ ( 0 , 1 ] . In this paper, we prove its convergence with overrelaxation parameter λ ∈ ( 1 , 2 ) . As a result, the convergence of the other two algorithms (BCAVCS and BDROPCS) with overrelaxation parameter λ ∈ ( 1 , 2 ) in a special case is derived. Experiments are given to demonstrate the convergence behavior of the BCPCS algorithm with different values of λ . [ABSTRACT FROM AUTHOR]

In probabilistic model analysis and optimization, expected values of a model output f ( x ) in face of continuous random inputs x are estimated through n -dimensional integrals, where n = d i m ( x ) . Cubature formulae are approximations of these integrals by a weighted sum of function evaluations at carefully chosen points. When each function evaluation corresponds to a heavy computational simulation, and particularly in optimization problems, one needs very efficient formulae with few integration points, even though only having modest accuracy. In this paper, we evaluate the performance of several cubature formulae with few points, including Smolyak type formulae, also known as sparse grid integration, and recently proposed thinned cubatures, constructed using orthogonal arrays. Tests are made for a wide family of smooth and non-oscillatory functions f ( x ) , possibly with significant anisotropy, and covering both normal and uniform input probability distributions. Two practical case studies are also presented, one of analysis of a large scale mass transfer model with uncertain parameters and a second one of optimal production planning under uncertain market conditions. Results clearly indicate that cubatures with large negative weights, including Smolyak type formulae, are not reliable, contrary to positive thinned cubatures that produce very reasonable estimates up to dimension 24. These thinned cubatures may also surpass quasi-Monte Carlo methods also up to dimension 24. [ABSTRACT FROM AUTHOR]

In this paper we consider the numerical upscaling of the Brinkman equation in the presence of high-contrast permeability fields. We develop and analyze a robust and efficient Generalized Multiscale Finite Element Method (GMsFEM) for the Brinkman model in two dimensions. In the fine grid, we use mixed finite element method with the velocity and pressure being continuous piecewise quadratic and piecewise constant finite element spaces, respectively. Using the GMsFEM framework we construct suitable coarse-scale spaces for the velocity and pressure that yield a robust mixed GMsFEM. We develop a novel approach to construct a coarse approximation for the velocity snapshot space and a robust small offline space for the velocity space. The stability of the mixed GMsFEM and a priori error estimates are derived. A variety of two-dimensional numerical examples are presented to illustrate the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we present a characterization for the Hausdorff distance between two given algebraic curves in the n -dimensional space (parametrically or implicitly defined) to be finite. The characterization is related with the asymptotic behavior of the two curves and it can be easily checked. More precisely, the Hausdorff distance between two curves C and C ¯ is finite if and only if for each infinity branch of C there exists an infinity branch of C ¯ such that the terms with positive exponent in the corresponding series are the same, and reciprocally. [ABSTRACT FROM AUTHOR]

This paper deals with the asymptotic stability of numerical solutions for differential equations with piecewise continuous arguments (EPCAs). The necessary and sufficient condition is given for non-confluent Runge–Kutta methods to preserve the stability of nonlinear scalar EPCAs. As for systems, we prove that some algebraically stable methods can preserve the asymptotic stability. [ABSTRACT FROM AUTHOR]

Many physics problems can only be studied by coupling various numerical codes, each modeling a subaspect of the physics problem that is addressed. In most cases, the “brute force” technique of running the codes one after the other in a loop until convergence is reached requires excessive CPU time. The present paper illustrates that re-writing the coupling as a root-finding problem, to which a quasi-Newton method–here the (Inverse) Column Updating Method–can be applied, is useful to push down the computation time, at the expense of a very modest amount of supplementary programming. A simplified version of the set of codes commonly used to describe plasma heating by radio frequency waves in a tokamak plasma is adopted for illustrating the potential of the speed-up method. It consists of a wave equation as well as a Fokker–Planck velocity space diffusion and a radial energy diffusion model. It is shown that with this approach a substantial reduction in CPU time needed for convergence can be obtained. [ABSTRACT FROM AUTHOR]

This paper investigates the pricing and hedging of variance swaps under a 3/2 volatility model using explicit formulae. Pricing and hedging is performed under the benchmark approach, which only requires the existence of the numéraire portfolio. The growth optimal portfolio is used as numéraire together with the real world probability measure as pricing measure. This real world pricing concept provides minimal prices for variance swaps even when an equivalent risk neutral probability measure does not exist. [ABSTRACT FROM AUTHOR]

In this paper a general drift-flux model describing a subsonic and isentropic multi-phase fluid in connected pipes is considered. Each phase is assumed to be isentropic with its own sonic speed. The components are gamma-law gases with γ > 1 . For such, a computational challenge at a junction is the computation of rarefaction waves which do not have a readily available analytical form. Firstly, the well-posedness of the Riemann problem at the junction is discussed. It is suggested that rarefaction waves should be linearized in order to obtain a more efficient numerical method for coupling such multi-component flow. Some computational results on the dynamics of the multi-phase gas in the pipes demonstrate the qualitative behavior of this approach. [ABSTRACT FROM AUTHOR]

In this paper, we consider the mixed Stokes–Darcy problem which describes a fluid flow coupled with a porous media. We present a modified two-grid method for decoupling this mixed model. Stability is proved and optimal error estimates are derived. The numerical results show that the modified two-grid method is effective and has the same accuracy as the coupling scheme when we choose h = H 2 . [ABSTRACT FROM AUTHOR]

In this paper diffusion through a viscoelastic biodegradable material is studied. The phenomenon is described by a set of three coupled partial differential equations that take into account passive diffusion, stress driven diffusion and the degradation of the material. The stability properties of the model are studied. Erodible viscoelastic materials, as biodegradable polymers, have a huge range of applications in medicine to make drug eluting implants. Using the mathematical model the behavior of a particular ocular drug eluting implant which describes drug delivery into the vitreous chamber of the eye is presented. The model consists of coupled systems of partial differential equations linked by interface conditions. The chemical structure, the viscoelastic properties and the diffusion in the implant as well as the transport in the vitreous are taken into account to simulate the evolution in vivo of released drug. The dependence of the delivery profile on the properties of the material is addressed. Numerical simulations that illustrate the interplay between these phenomena are included. [ABSTRACT FROM AUTHOR]

In this paper the numerical simulation of aeroelastic interactions of flexibly supported two-degrees of freedom (2-DOF) airfoil in two-dimensional (2D) incompressible viscous turbulent flow subjected to a gust (sudden change of flow conditions) is considered. The flow is modeled by Reynolds averaged Navier-Stokes equations (RANS), and by k - ω turbulence model. The considered flow problem is discretized in space using the fully stabilized finite element (FE) method implemented in the developed in-house program, which allows to solve interaction problems. In order to treat the time dependent inlet boundary condition the standard stabilization procedure was modified. Further, the under relaxation procedure was introduced in order to overcome the artificial instability of the coupling algorithm. The aeroelastic response to a sudden gust is numerically analyzed with the aid of the developed FE code. [ABSTRACT FROM AUTHOR]

In this paper, a new nonmonotone trust region algorithm with simple quadratic models is proposed. Unlike traditional nonmonotone trust region method, our trust region subproblem is very simple by using a new scale approximation of the minimizing function's Hessian. The global convergence of the proposed algorithm is established under some reasonable conditions. Numerical tests on a set of large scale standard test problems are presented and show that the new algorithm is efficient and robust. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider splitting methods for nonlinear ordinary differential equations in which one of the (partial) flows that results from the splitting procedure cannot be computed exactly. Instead, we insert a well-chosen state into the corresponding nonlinearity , which results in a linear term whose exact flow can be determined efficiently. Therefore, in the spirit of splitting methods, it is still possible for the numerical simulation to satisfy certain properties of the exact flow. However, Strang splitting is no longer symmetric (even though it is still a second order method) and thus high order composition methods are not easily attainable. We will show that an iterated Strang splitting scheme can be constructed which yields a method that is symmetric up to a given order. This method can then be used to attain high order composition schemes. We will illustrate our theoretical results, up to order six, by conducting numerical experiments for a charged particle in an inhomogeneous electric field, a post-Newtonian computation in celestial mechanics, and a nonlinear population model and show that the methods constructed yield superior efficiency as compared to Strang splitting. For the first example we also perform a comparison with the standard fourth order Runge–Kutta methods and find significant gains in efficiency as well better conservation properties. [Copyright &y& Elsevier]

Abstract: In this paper we propose a method for the accurate calculation of output quantities resulting from a parameter-dependent, single-phase flow model. In particular, given a small-dimensional set of inputs (as compared to the fine model), we treat the problem using a combined local–global model reduction technique. The local model reduction is achieved through the use of the Generalized Multiscale Finite Element Method (GMsFEM) where a set of independently calculated basis functions are used in order to construct a suitable coarse approximation space. The multiscale basis function computations are localized to specified coarse subdomains, and follow an offline–online procedure in which a set of eigenvalue problems are used to capture the underlying behavior of the system. Because the offline stage accounts for a one-time preprocessing step, the online coarse space may be cheaply constructed for a given input state. We then apply balanced truncation (BT) to the online coarse system in order to obtain a global reduced-order approximation of the output state. BT recasts the model equation into a systems framework where the input–output mapping may be approximated through the spectral construction of a reduced-order model, and requires the solution of a set of Lyapunov equations. As the Lyapunov equations represent an expensive computation, the efficiency of the proposed method depends on the size of the online coarse space. The combined approach is shown to be flexible with respect to the online space and reduced dimensions, and may be readily modified in order to ensure that the resulting output errors are comparable. [Copyright &y& Elsevier]

Abstract: This paper extends the moment matching market implied calibration procedure (Guillaume and Schoutens 2012) to Markov models with piecewise constant parameters between successive quoted option maturities. The Markov property allows us to determine the parameter set of each subprocess by a bootstrapping moment matching calibration. This sequential calibration arises naturally due to the additive property of cumulants of independent random variables and consists in solving independent moment matching systems of equations, where and denote the number of quoted maturities and the number of parameters, respectively. As shown in Guillaume and Schoutens (2012), for popular Lévy processes, these systems can be transformed into systems of algebraic equations which give directly the model parameters of each subprocess in terms of the second to the th standardized moments of the log asset return process between successive maturity times. For the numerical study, we work out the bootstrapping moment matching calibration under two popular Lévy models with piecewise constant parameters, namely the VG and Meixner models and compare its performance with existing calibration procedures for term structure models. [Copyright &y& Elsevier]

Abstract: In this paper we consider the problem of deriving correlation estimates from observed option data. An implied correlation estimate arises when we match the observed index option price with a corresponding model price. The underlying model assumes that stock prices can be described using a lognormal distribution, while a Gaussian copula describes the dependence structure. Within this multivariate stock price model, the index option price is not given in a closed form and has to be approximated. Different methods exist and each choice leads to another implied correlation estimate. We show that the traditional approach for determining implied correlations is a member of our more general framework. It turns out that the traditional implied correlation underestimates the real correlation. This error is more pronounced when some stock volatilities are large compared to the other volatility levels. We propose a new approach to measure implied correlation which does not have this drawback. However, our numerical illustrations show that determining implied correlations with the traditional approach may be justified for strike prices which are close to the at-the-money strike price. We also show that implied correlation estimates can be used to define an index, called the Implied Correlation Index (ICX), which reflects the market’s perception about future (short-term) co-movement between stock prices. Using a volatility index together with the ICX gives an accurate description of the current level of market fear. [Copyright &y& Elsevier]

Abstract: Commonly in container terminals, the containers are stored in yards on top of each other using yard cranes. The split-platform storage/retrieval system (SP-AS/RS) has been invented to store containers more efficiently and to access them more quickly. The integrated scheduling of quay cranes, automated guided vehicles and handling platforms in SP-AS/RS has been formulated and solved using the simulated annealing algorithm in previous literatures. This paper presents a genetic algorithm (GA) to solve this problem more accurately and precisely. The GA includes a new operator to make a random string of tasks observing the precedence relations between the tasks. For evaluating the performance of the GA, 10 small size test cases were solved by using the proposed GA and the results were compared to those from the literature. Results show that the proposed GA is able to find fairly near optimal solutions similar to the existing simulated annealing algorithm. Moreover, it is shown that the proposed GA outperforms the existing algorithm when the number of tasks in the scheduling horizon increases (e.g. 30 to 100). [Copyright &y& Elsevier]

Abstract: The time evolution of dynamic triopoly games is modeled by a discrete dynamical system obtained by the iteration of a three-dimensional map. We present in this paper four games: a rational Cournot triopoly, a rational Bertrand triopoly, a Puu triopoly with quantity competition, and a cooperative Cournot triopoly game. For each game, the Nash equilibrium of the game is computed and complete analytical and numerical studies of the stability conditions for the fixed points, which are the Nash equilibria, are given. The analysis of bifurcations which cause qualitative changes in the behavior of games and cause loss of stability of Nash equilibrium is investigated through numerical explorations. [Copyright &y& Elsevier]

Abstract: Model updating for the quadratic eigenvalue problem aims to update the model by given eigendata. In this paper, we consider the -weighted model updating problem which can not only preserve the symmetry and definiteness of the original model but also express our confidence in the original model through assigning different confidence weights. We propose an augmented Lagrangian dual method for the -weighted model updating problem. Under some mild assumptions, our method is shown to converge at least linearly. Numerical results illustrate the effectiveness of our method. In addition, we compare our method with the semi-definite programming (SDP) method. Numerical results illustrate that when the scale of the model becomes large our method still works but the SDP method failed to converge. [Copyright &y& Elsevier]

Abstract: When modelling the cardiovascular system, the effect of the vessel wall on the blood flow has great relevance. Arterial vessels are complex living tissues and three-dimensional specific models have been proposed to represent their behaviour. The numerical simulation of the 3D–3D Fluid–Structure Interaction (FSI) coupled problem has high computational costs in terms of required time and memory storage. Even if many possible solutions have been explored to speed up the resolution of such problem, we are far from having a 3D–3D FSI model that can be solved quickly. In 3D–3D FSI models two of the main sources of complexity are represented by the domain motion and the coupling between the fluid and the structural part. Nevertheless, in many cases, we are interested in the blood flow dynamics in compliant vessels, whereas the displacement of the domain is small and the structure dynamics is less relevant. In these situations, techniques to reduce the complexity of the problem can be used. One consists in using transpiration conditions for the fluid model as surrogate for the wall displacement, thus allowing problem’s solution on a fixed domain. Another strategy consists in modelling the arterial wall as a thin membrane under specific assumptions (Figueroa et al., 2006, Nobile and Vergara, 2008) instead of using a more realistic (but more computationally intensive) 3D elastodynamic model. Using this strategy the dynamics of the vessel motion is embedded in the equation for the blood flow. Combining the transpiration conditions with the membrane model assumption, we obtain an attractive formulation, in fact, instead of solving two different models on two moving physical domains, we solve only a Navier–Stokes system in a fixed fluid domain where the structure model is integrated as a generalized Robin condition. In this paper, we present a general formulation in the boundary conditions which is independent of the time discretization scheme choice and on the stress–strain constitutive relation adopted for the vessel wall structure. Our aim is, first, to write a formulation of a reduced order model with zero order transpiration conditions for a generic time discretization scheme, then to compare a 3D–3D FSI model and a reduced FSI one in two realistic patient-specific cases: a femoropopliteal bypass and an aorta. In particular, we are interested in comparing the wall shear stresses, in fact this quantity can be used as a risk factor for some pathologies such as atherosclerosis or thrombogenesis. More in general we want to assess the accuracy and the computational convenience to use simpler formulations based on reduced order models. In particular, we show that, in the case of small displacements, using a 3D–3D FSI linear elastic model or the correspondent reduced order one yields many similar results. [Copyright &y& Elsevier]

Abstract: We develop efficient algorithms for solving the compressed sensing problem. We modify the standard regularization model for compressed sensing by adding a quadratic term to its objective function so that the objective function of the dual formulation of the modified model is Lipschitz continuous. In this way, we can apply the well-known Nesterov algorithm to solve the dual formulation and the resulting algorithms have a quadratic convergence. Numerical results presented in this paper show that the proposed algorithms outperform significantly the state-of-the-art algorithm NESTA in accuracy. [Copyright &y& Elsevier]

This paper investigates an inverse problem of simultaneously reconstructing the initial value and source coefficient in a degenerate parabolic equation. Problems of this type have important applications in several fields of applied science. Being different from other inverse coefficient problems in classical parabolic equations (non-degenerate), the principal coefficient in the mathematical model discussed in the paper may vanish on both extremities of the domain. On the basis of Carleman estimate, the uniqueness and conditional stability of solution for the original problem are established. Then an iteration algorithm of Landweber type is designed to obtain the numerical solution and some typical numerical experiments are also performed. Numerical results show that the proposed method is stable and the unknown coefficients are recovered quite well. [ABSTRACT FROM AUTHOR]

A fundamental topic in numerical simulation of two-phase displacement is discussed in this paper. The mathematical model for the compressible problem is defined mainly by two nonlinear partial differential equations: a parabolic equation for the pressure and a convection–diffusion equation for the saturation. The Darcy velocity is determined by the pressure, and affects the whole physical process. The system is solved by a composite numerical scheme. The conservative mixed volume element is used for the first equation. The computational accuracy is improved for the Darcy velocity. The second equation is solved by a conservative upwind mixed volume element, where the mixed volume element and upwind approximation treat the diffusion and convection, respectively. The upwind method preserves the high computational accuracy, and numerical dispersion and nonphysical oscillation are eliminated. The saturation and its adjoint vector function are obtained simultaneously. An important feature in numerical scheme, the conservation of mass, is proved. By the traditional theoretical work of numerical analysis such as a priori estimates of differential equations, the optimal order error estimate is obtained. Finally, numerical tests show the effectiveness and practicability, then the present method possibly solves the challenging problems as a powerful tool. • The physical nature of conservation is preserved. • The compressibility of flow, molecular diffusion and mechanical diffusion are considered in the numerical model. • Strong stability and high accuracy are shown by experimental tests. [ABSTRACT FROM AUTHOR]