Showing papers presented at "International Conference on Numerical Analysis and Its Applications in 2009"

The Weierstrass Canonical Form of a Regular Matrix Pencil: Numerical Issues and Computational Techniques

Abstract: In the present paper, we study the derivation of the Weierstrass Canonical Form (WCF) of a regular matrix pencil In order to compute the WCF, we use two important computational tools: a) the QZ algorithm to specify the required root range of the pencil and b) the updating technique to compute the index of annihilation The proposed updating technique takes advantages of the already computed rank of the sequences of matrices that appears during our procedure reducing significantly the required floating-point operations The algorithm is implemented in a numerical stable manner, giving efficient results Error analysis and the required complexity of the algorithm are included

On Weakening Conditions for Discrete Maximum Principles for Linear Finite Element Schemes

Abstract: In this work we discuss weakening requirements on the set of sufficient conditions due to Ph. Ciarlet [4,5] for matrices associated to linear finite element schemes, which is commonly used for proving validity of discrete maximum principles (DMPs) for the second order elliptic problems.

Numerical Approximation of a Free Boundary Problem for a Predator-Prey Model

Abstract: This paper is concerned with the numerical approximation of a free boundary problem associated with a predator-prey ecological model. Taking into account the local dynamic of the system, a stable finite difference scheme is used, and numerical results are presented.

An Iterative Numerical Algorithm for a Strongly Coupled System of Singularly Perturbed Convection-Diffusion Problems

Abstract: An iterative numerical method is constructed for a coupled system of singularly perturbed convection-diffusion-reaction two-point boundary value problems. It combines a standard finite difference operator with a piecewise-uniform Shishkin mesh, and uses a Jacobi-type iteration to compute a solution. Under certain assumptions on the coefficients in the differential equations, a bound on the maximum-norm error in the computed solution is established; this bound is independent of the values of the singular perturbation parameter. Numerical results are presented to illustrate the performance of the numerical method.

Numerical Analysis of a 2d Singularly Perturbed Semilinear Reaction-Diffusion Problem

Abstract: A semilinear reaction-diffusion equation with multiple solutions is considered in a smooth two-dimensional domain. Its diffusion parameter e 2 is arbitrarily small, which induces boundary layers. We extend the numerical method and its maximum norm error analysis of the paper [N. Kopteva: Math. Comp. 76 (2007) 631–646], in which a parametrization of the boundary $\partial\Omega$ is assumed to be known, to a more practical case when the domain is defined by an ordered set of boundary points. It is shown that, using layer-adapted meshes, one gets second-order convergence in the discrete maximum norm, uniformly in e for e ≤ Ch. Here h > 0 is the maximum side length of mesh elements, while the number of mesh nodes does not exceed Ch − 2. Numerical results are presented that support our theoretical error estimates.

Question of Existence and Uniqueness of Solution for Navier-Stokes Equation with Linear Do-Nothing Type Boundary Condition on the Outflow

Abstract: The paper deals with the mathematical model of a flow of a viscous incompressible fluid through a 2D cascade of profiles. We consider a splited "do---nothing" type boundary condition on the outflow. The existence of a weak solution of a corresponding steady boundary value problem is known, see [2] and [3]. We recall the weak formulation, the theorem on existence and we study the uniqueness of the weak solution in this paper.

Model Predictive Control --- Numerical Methods for the Invariant Sets Approximation

Abstract: This paper deals with the computational issues encountered in the construction of invariant sets for LTI (Linear Time Invariant) systems subject to linear constraints. Three algorithms to compute or approximate the invariant set are presented. Two of theme are based on expansive and contractive strategy, while the third one uses the transition graph over the partition of the closed loop piecewise affine system.

Computational Analysis of Expected Climate Change in the Carpathian Basin Using a Dynamical Climate Model

Abstract: For analyzing the possible regional climate change in the Carpathian Basin, model PRECIS has been adapted, which is the hydrostatic regional climate model HadRM3P developed at the UK Met Office, Hadley Centre, and nested in HadCM3 GCM. First, control run simulations (1961-1990) of the PRECIS model (with two different sets of boundary conditions) are analyzed. In the validation, seasonal temperature and precipitation mean values from the CRU datasets are used. According to the results, model PRECIS slightly overestimates the temperature and underestimates the precipitation. Then, model results for the periods 2071-2100 (using SRES A2 scenario) and 1961-1990 (as the reference period) are compared. The results suggest that the temperature increase expected in the Carpathian Basin may considerably exceed the global warming rate. The climate of this region is expected to become wetter in winter and drier in the other seasons.

Interpolation Method for a Function with a Singular Component

Abstract: An interpolation formula for functions with boundary layer components is proposed. It is exact on the singular boundary layer component, that leads to uniform accuracy of the interpolation. It is shown, that the proposed formula can be used for interpolation of numerical solutions of boundary value problems with exponential and power layers.

On a Class of Almost Orthogonal Polynomials

Abstract: In this paper, we define a new class of almost orthogonal polynomials which can be used successfully for modelling of electronic systems which generate orthonormal basis. Especially, for the classical weight function, they can be considered like a generalization of the classical orthogonal polynomials (Legendre, Laguerre, Hermite, ...). They are very suitable for analysis and synthesis of imperfect technical systems which are projected to generate orthogonal polynomials, but in the reality generate almost orthogonal polynomials.

On the Discretization Time-Step in the Finite Element Theta-Method of the Discrete Heat Equation

Abstract: In this paper the numerical solution of the one dimensional heat conduction equation is investigated, by applying Dirichlet boundary condition at the left hand side and Neumann boundary condition was applied at the right hand side. To the discretization in space, we apply the linear finite element method and for the time discretization the well-known theta-method. The aim of the work is to derive an adequate numerical solution for the homogenous initial condition by this approach. We theoretically analyze the possible choice of the time-discretization step-size and establish the interval where the discrete model is reliable to the original physical phenomenon. As the discrete model, we arrive at the task of the one-step iterative method. We point out that there is a need to obtain both lower and upper bounds of the time-step size to preserve the qualitative properties of the real physical solution. The main results of the work is to determine the interval for the time-step size to be used in this special finite element method and analyze the main qualitative characterstics of the model.

Memetic Simulated Annealing for the GPS Surveying Problem

Abstract: In designing Global Positioning System (GPS) surveying network, a given set of earth points must be observed consecutively (schedule). The cost of the schedule is the sum of the time needed to go from one point to another. The problem is to search for the best order in which this observation is executed. Minimizing the cost of this schedule is the goal of this work. Solving the problem for large networks to optimality requires impractical computational times. In this paper, several Simulated Annealing (SA) algorithms are developed to provide near-optimal solutions for large networks with bounded computational effort.

Weight Uniform Accuracy Estimates of Finite Difference Method for Poisson Equation, Taking into Account Boundary Effect

Abstract: Poisson equation in polyhedral domain Ω ⊂ R n , n = 2,3 with boundary Γ, when Dirichlet conditions are given on all faces or on all but one where Neimann conditions are given, is considered. Traditional difference schemes with semi-constant steps along axes precisely approximate Dirichlet conditions hence it is expected that their accuracy order increases approaching to corresponding part of boundary γ. This paper is dedicated to quantitative investigation of this boundary effect. It is also shown that analogous boundary effect in the mesh knots takes place also for finite-element method (super convergence).

On a Discrete Maximum Principle for Linear FE Solutions of Elliptic Problems with a Nondiagonal Coefficient Matrix

Abstract: In this paper we give a sufficient condition for the validity of a discrete maximum principle (DMP) for a class of elliptic problems of the second order with a nondiagonal coefficient matrix, solved by means of linear finite elements (FEs). Numerical tests are presented.

Surface Reconstruction via L1-Minimization

Abstract: A surface reconstruction technique based on the L 1- minimization of the variation of the gradient is introduced. This leads to a non-smooth convex programming problem. Well-posedness and convergence of the method is established and an interior point based algorithm is introduced. The L 1-surface reconstruction algorithm is illustrated on various test cases including natural and urban terrain data.

Numerical Experiments for Reaction-Diffusion Equations Using Exponential Integrators

Abstract: In this study we focus on a comparative numerical approach of two reaction-diffusion models arising in biochemistry by using exponential integrators. The goal of exponential integrators is to treat exactly the linear part of the differential model and allow the remaining part of the integration to be integrated numerically using an explicit scheme. Numerical simulations including both the global error as a function of time step and error as a function of computational time are shown.

Numerical Modelling of Cellular Immune Response to Virus

Abstract: We present a mathematical model of cellular immune response to viral infection. The model is a bilinear system of integro-differential equations of Boltzmann type. Results of numerical experiments are presented.

Qualitative Analysis of the Crank-Nicolson Method for the Heat Conduction Equation

Abstract: The preservation of the basic qualitative properties --- besides the convergence --- is a basic requirement in the numerical solution process. For solving the heat conduction equation, the finite difference/linear finite element Crank-Nicolson type full discretization process is a widely used approach. In this paper we formulate the discrete qualitative properties and we also analyze the condition w.r.t. the discretization step sizes under which the different qualitative properties are preserved. We give exact conditions for the discretization of the one-dimensional heat conduction problem under which the basic qualitative properties are preserved.

A Two-Grid Approximation of an Interface Problem for the Nonlinear Poisson-Boltzmann Equation

Abstract: We present a robust and efficient numerical method for solution of an interface problem for a generalization of the Poisson-Boltzmann equation, arising in molecular biophysics. The differential problem is solved by FEM (finite element method) technique on two (coarse and fine) subspaces. The solution of the nonlinear system of algebraic equations on the fine mesh is reduced to the solution on two small (one linear and one nonlinear) systems on the coarse grid and a large linear one on the fine grid. It is shown, both theoretically and numerically, that the coarse space can be extremely coarse and still achieve asymptotically optimal approximation.

A New Method for Solving Transient Lossy Transmission Line Problem

Abstract: This paper presents a new technique for the transient analysis of lossy transmission lines. The proposed method is based on discretization of Telegrapher's equation via the auxiliary problem equations to which well known numerical methods can be applied easily. The new method also lets simple and well structured algorithm be developed. A SPICE model is used to verify the results obtained from the new method.

A Two-Grid Algorithm for Solution of the Difference Equations of a System of Singularly Perturbed Semilinear Equations

Abstract: We propose a two-grid algorithm for implementation of a generalized A.M.Il'in's scheme to a system of semilinear diffusion convection-dominated equations. To solve the nonlinear algebraic system of difference equations we use Newton method. We derive the difference scheme on a coarse mesh and, then using uniform interpolation, taking into account the boundary layers, we obtain the initial guess for an iterative method on a fine mesh. Estimates of the accuracy and the computational work are obtained. The main advantage of the proposed algorithm is the computational cost.

Numerical Solution of the Discrete-Time Coupled Algebraic Riccati Equations

Abstract: We consider the numerical solution of a set of discrete-time coupled algebraic Riccati equations that arise in quadratic optimal control Several iterations for computing a symmetric solution of this system are investigated and compared New iterations are based on the properties of a Stein equation It is necessary to solve a Stein equation at each step of considered algorithms We will compare the corresponding solvers in regard of accuracy, number of iterations and time of executing Several sets of test examples are used to demonstrate the performance

Multilevel Splitting of Weighted Graph-Laplacian Arising in Non-conforming Mixed FEM Elliptic Problems

Abstract: We consider a second-order elliptic problem in mixed form that has to be solved as a part of a projection algorithm for unsteady Navier-Stokes equations. The use of Crouzeix-Raviart non-conforming elements for the velocities and piece-wise constants for the pressure provides a locally mass-conservative algorithm. Then, the Crouzeix-Raviart mass matrix is diagonal, and the velocity unknowns can be eliminated exactly. The reduced matrix for the pressure is referred to as weighted graph-Laplacian. In this paper we study the construction of optimal order preconditioners based on algebraic multilevel iterations (AMLI). The weighted graph-Laplacian for the model 2-D problem is considered. We assume that the finest triangulation is obtained after recursive uniform refinement of a given coarse mesh. The introduced hierarchical splitting is the first important contribution of this article. The proposed construction allows for a local analysis of the constant in the strengthened Cauchy-Bunyakowski-Schwarz (CBS) inequality. This is an important characteristic of the splitting and is associated with the angle between the two hierarchical FEM subspaces. The estimates of the convergence rate and the computational cost at each iteration show that the related AMLI algorithm with acceleration polynomial of degree two or three is of optimal complexity.

On Superlinear PCG Methods for FDM Discretizations of Convection-Diffusion Equations

Abstract: The numerical solution of linear convection-diffusion equations is considered. Finite difference discretization leads to an algebraic system solved by a suitable preconditioned CG method, where the preconditioning approach is based on equivalent operators. Our goal is to study the superlinear convergence of the preconditioned CG iteration and to find mesh independent behaviour on a model problem. This is an analogue of previous results where FEM discretization was used.

On the Sign-Stability of Finite Difference Solutions of Semilinear Parabolic Problems

Abstract: The sign-stability property is one of the important qualitative properties of the one-dimensional heat conduction equation, or more generally, of one-dimensional parabolic problems. This property means that the number of the spatial sign-changes of the solution function cannot increase in time. In this paper, sufficient conditions will be given that guarantee the fulfillment of a numerical analogue of the sign-stability for the finite difference solution of a semilinear parabolic problem. The results are demonstrated on a numerical test problem.

Minimal Simplex for IFS Fractal Sets

Abstract: Fractal sets manipulation and modeling is a difficult task due to their complexity and unpredictability. One of the basic problems is to determine bounds of a fractal set given by some recursive definitions, for example by an Iterated Function System (IFS). Here we propose a method of bounding an IFS-generated fractal set by a minimal simplex that is affinely identical to the standard simplex. First, it will be proved that for a given IFS attractor, such simplex exists and it is unique. Such simplex is then used for definition of an Affine invariant Iterated Function System (AIFS) that then can be used for affine transformation of a given fractal set and for its modeling.

On the Numerical Solution of a Transmission Eigenvalue Problem

Abstract: A transmission eigenvalue problem in disjoint intervals is examined. Distribution of the eigenvalues is obtained. The corresponding difference scheme is proposed and tested on few numerical examples.

Tensor Product q -Bernstein Bézier Patches

Abstract: In this work we define a new de Casteljau type algorithm, whichis in barycentric form, for the q -Bernstein Beziercurves. We express the intermediate points of the algorithmexplicitly in two ways. Furthermore we define tensor productpatches, based on this algorithm, depending on two parameters.Degree elevation procedure for the tensor product patch is studied.Finally, the matrix representation of tensor product patch is givenand we find the transformation matrix between classical tensorproduct Bezier patch and tensor product q -Bernstein Bezier patch.

Optimal Order FEM for a Coupled Eigenvalue Problem on 2D Overlapping Domains

Abstract: In this paper we present a numerical approach to a nonstandard second-order elliptic eigenvalue problem defined on two overlapping rectangular domains with a nonlocal (integral) boundary condition. Usually, for this class of problems error estimates are suboptimal. By introducing suitable degrees of freedom and a corresponding interpolation operator we derive optimal order finite element approximation. Numerical results illustrate the efficiency of the proposed method.

Numerical Quadrature for Bessel Transformations with High Oscillations

Abstract: We explore higher order numerical quadrature for the integration of systems containing Bessel functions such as $\int_a^b f(x)J_{ u}(rx)dx$ and $\int_a^b f(x)\cos(r_1x)J_{ u}(rx)dx$. The decay of the error of the these methods drastically improves as frequency grows.