main-content

## Inhaltsverzeichnis

### On the Discretization of the Coupled Heat and Electrical Diffusion Problems

We consider a nonlinear system of elliptic equations, which arises when modelling the heat diffusion problem coupled with the electrical diffusion problem. The ohmic losses which appear as a source term in the heat diffusion equation yield a nonlinear term which lies in

L

1

. A finite volume scheme is proposed for the discretization of the system; we show that the approximate solution obtained with the scheme converges, up to a subsequence, to a solution of the coupled elliptic system.

Abdallah Bradji, Raphaèle Herbin

### The Vector Analysis Grid Operators for Applied Problems

Mathematical physics problems are often formulated by means of the vector analysis differential operators: divergence, gradient and rotor. For approximate solutions of such problems it is natural to use the corresponding operator statements for the grid problems, i.e., to use the so-called VAGO (Vector

A

nalys

G

rid

O

perators) method. We discuss the possibilities of such an approach in using general irregular grids. The vector analysis difference operators are constructed using the Delaunay triangulation and the Voronoi diagrams. The truncation error and the consistency property of the difference operators constructed on two types of grids are investigated.

Petr Vabishchevich

### On Some Computational Aspects of the Variational Data Assimilation Techniques

It is important to incorporate all available observations when large-scale mathematical models arising in different fields of science and engineering are used to study various physical and chemical processes. Variational data assimilation techniques can be used in the attempts to utilize efficiently observations in a large-scale model. Variational data assimilation techniques are based on a combination of three very important components

numerical methods for solving differential equations,

splitting procedures

and

optimization algorithms.

It is crucial to select an optimal (or, at least, a good) combination of these three components, because models which are very expensive computationally become much more expensive (the computing time being often increased by a factor greater than 100) when a variational data assimilation technique is applied. Therefore, it is important to study the interplay between the three components of the variational data assimilation techniques as well as to apply powerful parallel computers in the computations. Some results obtained in the search for a good combination will be reported. Parallel techniques described in [1] are used in the runs related to this paper.

Modules from a particular large-scale mathematical model, the Unified Danish Eulerian Model (UNI-DEM), are used in the experiments. The mathematical background of UNI-DEM is discussed in [1], [24] The ideas are rather general and can easily be applied in connection with other mathematical models.

Zahari Zlatev

### Weighted Iterative Operator-Splitting Methods: Stability-Theory

In the last years the need to solve complex physical models increased. Because of this motivation to solve complex models with efficient methods, we deal with advanced operator-splitting methods. They are based on weighted iterative operator-splitting methods and decouple complicate problems in simpler problems. The stability of the weighted splitting method is discussed and the efficiency of such methods. For the stiff-problems we present the A-stability property and the choice of the weighted parameters. The theory for the semi-discretized equations is introduced with respect to the gained ODE’s. A general stability-theory for linearized operators is proposed and discussed for stiff-problems. Finally we concern the weighted operator-splitting methods for multi-dimensional and multi-physical problems.

Jürgen Geiser

### Weighted Iterative Operator-Splitting Methods and Applications

The subject of our research is to solve accurately ODEs, which appear in mathematical models arising from several physical processes. For this purpose we develop a new class of weighted iterative operator splitting methods. We present applications to systems of linear ODEs, which might contain also stiff parameters. The benefit of the proposed method is demonstrated with regard to convergence results and comparison to analytical solutions. We provide improved results and convergence rates in comparison with classical operator splitting methods.

Jürgen Geiser, Christos Kravvaritis

### Multilevel Preconditioning of 2D Rannacher-Turek FE Problems; Additive and Multiplicative Methods

In the present paper we concentrate on algebraic two-level and multilevel preconditioners for symmetric positive definite problems arising from discretization by Rannacher-Turek non-conforming rotated bilinear finite elements on quadrilaterals. An important point to make is that in this case the finite element spaces corresponding to two successive levels of mesh refinement are not nested (in general). To handle this, a proper two-level basis is required in order to fit the general framework for the construction of two-level preconditioners for conforming finite elements and to generalize the methods to the multilevel case.

The proposed variants of hierarchical two-level basis are first introduced in a rather general setting. Then, the involved parameters are studied and optimized. As will be shown, the obtained bounds – in particular – give rise to optimal order AMLI methods of additive type. The presented numerical tests fully confirm the theoretical estimates.

Ivan Georgiev, Johannes Kraus, Svetozar Margenov

### A Parallel Algorithm for Systems of Convection-Diffusion Equations

The numerical solution of systems of convection-diffusion equations is considered. The problem is described by a system of second order partial differential equations (PDEs). This system is discretized by Courant-elements. The preconditioned conjugate gradient method is used for the iterative solution of the large-scale linear algebraic systems arising after the finite element discretization of the problem. Discrete Helmholtz preconditioners are applied to obtain a mesh independent superlinear convergence of the iterative method. A parallel algorithm is derived for the proposed preconditioner. A portable parallel code using Message Passing Interface (MPI) is developed. Numerical tests well illustrate the performance of the proposed method on a parallel computer architecture.

János Karátson, Tamás Kurics, Ivan Lirkov

### Comparative Analysis of Mesh Generators and MIC(0) Preconditioning of FEM Elasticity Systems

In this study, the topics of grid generation and FEM applications are studied together following their natural synergy. We consider the following three grid generators: Triangle, NETGEN and Gmsh. The quantitative analysis is based on the number of elements/nodes needed to obtain a triangulation of a given domain, satisfying a certain minimal angle condition. After that, the performance of two displacement decomposition (DD) preconditioners that exploit modified incomplete Cholesky factorization MIC(0) is studied in the case of FEM matrices arising from the discretization of the two-dimensional equations of elasticity on non-structured grids.

Nikola Kosturski, Svetozar Margenov

### Multigrid–Based Optimal Shape and Topology Design in Magnetostatics

The paper deals with an efficient solution technique to large–scale discretized shape and topology optimization problems. The efficiency relies on multigrid preconditioning. In case of shape optimization, we apply a geometric multigrid preconditioner to eliminate the underlying state equation while the outer optimization loop is the sequential quadratic programming, which is done in the multilevel fashion as well. In case of topology optimization, we can only use the steepest–descent optimization method, since the topology Hessian is dense and large–scale. We also discuss a Newton–Lagrange technique, which leads to a sequential solution of large–scale, but sparse saddle–point systems, that are solved by an augmented Lagrangian method with a multigrid preconditioning. At the end, we present a sequential coupling of the topology and shape optimization. Numerical results are given for a geometry optimization in 2–dimensional nonlinear magnetostatics.

Dalibor Lukáš

### Generalized Aggregation-Based Multilevel Preconditioning of Crouzeix-Raviart FEM Elliptic Problems

It is well-known that iterative methods of optimal order complexity with respect to the size of the system can be set up by utilizing preconditioners based on various multilevel extensions of two-level finite element methods (FEM), as was first shown in [5]. Thereby, the constant

γ

in the so-called Cauchy-Bunyakowski-Schwarz (CBS) inequality, which is associated with the angle between the two subspaces obtained from a (recursive) two-level splitting of the finite element space, plays a key role in the derivation of optimal convergence rate estimates. In this paper a generalization of an algebraic preconditioning algorithm for second-order elliptic boundary value problems is presented, where the domain is discretized using linear Crouzeix-Raviart finite elements and the two-level splitting is defined by differentiation and aggregation (DA). It is shown that the uniform estimate on the constant

γ

(as presented in [6]) can be improved if a minimum angle condition, which is an integral part in any mesh generator, is assumed to hold in the triangulation. The improved values of

γ

can then be exploited in the set up of more problem-adapted multilevel preconditioners with faster convergence rates.

Svetozar Margenov, Josef Synka

### Solving Coupled Consolidation Equations

The iterative method to solve the equations modelling the coupled consolidation problem is presented. The algorithm is tested in the finite element package Hydro-geo for the geotechnical constructions.

Felicja Okulicka-Dłuzewska

### Parallel Schwarz Methods for T-M Modelling

The paper deals with a finite element solution of transient thermo-elasticity problems. In this context, it is especially devoted to the parallel computing of nonstationary heat equations, when the linear systems arising in each time step are solved by the overlapping domain decomposition method. The numerical tests are performed by OpenMP and/or MPI solvers on a large benchmark problem derived from geoenvironmental model KBS.

Jiří Starý, Ondřej Jakl, Roman Kohut

### Parallel Incomplete Factorization of 3D NC FEM Elliptic Systems

A new parallel preconditioner for solution of large scale second order 3D FEM elliptic systems is presented. The problem is discretized by rotated trilinear non-conforming finite elements. The algorithm is based on application of modified incomplete Cholesky factorisation (MIC(0)) to a locally constructed modification

B

of the original stiffness matrix

A

. The matrix

B

preserves the robustness of the point-wise factorisation and has a special block structure allowing parallelization. The performed numerical tests are in agreement with the derived estimates for the parallel times.

Yavor Vutov

### Extended Object Tracking Using Mixture Kalman Filtering

This paper addresses the problem of tracking extended objects. Examples of extended objects are ships and a convoy of vehicles. Such kind of objects have particularities which pose challenges in front of methods considering the extended object as a single point. Measurements of the object extent can be used for estimating size parameters of the object, whose shape is modeled by an ellipse. This paper proposes a solution to the extended object tracking problem by mixture Kalman filtering. The system model is formulated in a conditional dynamic linear (CDL) form. Based on the specifics of the task, two latent indicator variables are proposed, characterising the mode of maneuvering and size type, respectively. The developed Mixture Kalman filter is validated and evaluated by computer simulation.

Donka Angelova, Lyudmila Mihaylova

### Exact Error Estimates and Optimal Randomized Algorithms for Integration

Exact error estimates for evaluating multi-dimensional integrals are considered. An estimate is called

exact

if the rates of convergence for the low- and upper-bound estimate coincide. The algorithm with such an exact rate is called

optimal

. Such an algorithm has an

unimprovable

rate of convergence.

The problem of existing exact estimates and optimal algorithms is discussed for some functional spaces that define the regularity of the integrand. Important for practical computations data classes are considered: classes of functions with bounded derivatives and Hölder type conditions.

The aim of the paper is to analyze the performance of two optimal classes of algorithms: deterministic and randomized for computing multi-dimensional integrals. It is also shown how the smoothness of the integrand can be exploited to construct better randomized algorithms.

Ivan T. Dimov, Emanouil Atanassov

### Parallel Monte Carlo Approach for Integration of the Rendering Equation

This paper is addressed to the numerical solving of the rendering equation in realistic image creation. The rendering equation is integral equation describing the light propagation in a scene accordingly to a given illumination model. The used illumination model determines the kernel of the equation under consideration. Nowadays, widely used are the Monte Carlo methods for solving the rendering equation in order to create photorealistic images.

In this work we consider the Monte Carlo solving of the rendering equation in the context of the parallel sampling scheme for hemisphere. Our aim is to apply this sampling scheme to stratified Monte Carlo integration method for parallel solving of the rendering equation. The domain for integration of the rendering equation is a hemisphere. We divide the hemispherical domain into a number of equal sub-domains of orthogonal spherical triangles. This domain partitioning allows to solve the rendering equation in parallel. It is known that the Neumann series represent the solution of the integral equation as a infinity sum of integrals. We approximate this sum with a desired truncation error (systematic error) receiving the fixed number of iteration. Then the rendering equation is solved iteratively using Monte Carlo approach. At each iteration we solve multi-dimensional integrals using uniform hemisphere partitioning scheme. An estimate of the rate of convergence is obtained using the stratified Monte Carlo method.

This domain partitioning allows easy parallel realization and leads to convergence improvement of the Monte Carlo method. The high performance and Grid computing of the corresponding Monte Carlo scheme are discussed.

Ivan T. Dimov, Anton A. Penzov, Stanislava S. Stoilova

### Parallel Monte Carlo Sampling Scheme for Sphere and Hemisphere

The sampling of certain solid angle is a fundamental operation in realistic image synthesis, where the rendering equation describing the light propagation in closed domains is solved. Monte Carlo methods for solving the rendering equation use sampling of the solid angle subtended by unit hemisphere or unit sphere in order to perform the numerical integration of the rendering equation.

In this work we consider the problem for generation of uniformly distributed random samples over hemisphere and sphere. Our aim is to construct and study the parallel sampling scheme for hemisphere and sphere. First we apply the symmetry property for partitioning of hemisphere and sphere. The domain of solid angle subtended by a hemisphere is divided into a number of equal sub-domains. Each sub-domain represents solid angle subtended by orthogonal spherical triangle with fixed vertices and computable parameters. Then we introduce two new algorithms for sampling of orthogonal spherical triangles.

Both algorithms are based on a transformation of the unit square. Similarly to the Arvo’s algorithm for sampling of arbitrary spherical triangle the suggested algorithms accommodate the stratified sampling. We derive the necessary transformations for the algorithms. The first sampling algorithm generates a sample by mapping of the unit square onto orthogonal spherical triangle. The second algorithm directly compute the unit radius vector of a sampling point inside to the orthogonal spherical triangle. The sampling of total hemisphere and sphere is performed in parallel for all sub-domains simultaneously by using the symmetry property of partitioning. The applicability of the corresponding parallel sampling scheme for Monte Carlo and Quasi-Monte Carlo solving of rendering equation is discussed.

Ivan T. Dimov, Anton A. Penzov, Stanislava S. Stoilova

### A Hybrid Monte Carlo Method for Simulation of Quantum Transport

In this work we propose a hybrid Monte Carlo method for solving the Levinson equation. This equation describes the electron-phonon interaction on a quantum-kinetic level in a wire. The evolution problem becomes inhomogeneous due to the spatial dependence of the initial condition. The properties of the presented algorithm, such as computational complexity and accuracy, are investigated on the Grid by mixing quasi-random numbers and pseudo-random numbers. The numerical results are obtain for a physical model with GaAs material parameters in the case of zero electrical field.

Todor Gurov, Emanouil Atanassov, Sofiya Ivanovska

### Quasi-random Walks on Balls Using C.U.D. Sequences

This paper presents work on solving elliptic BVPs problems based on quasi-random walks, by using a subset of uniformly distributed sequences—completely uniformly distributed (c.u.d.) sequences. This approach is novel for solving elliptic boundary value problems. The enhanced uniformity of c.u.d. sequences leads to faster convergence. We demonstrate that c.u.d. sequences can be a viable alternative to pseudorandom numbers when solving elliptic boundary value problems. Analysis of a simple problem in this paper showed that c.u.d. sequences achieve better numerical results than pseudorandom numbers, but also have the potential to converge faster and so reduce the computational burden.

Aneta Karaivanova, Hongmei Chi, Todor Gurov

### On the Exams of a Multi-Attribute Decision Making Electronic Course

This paper gives brief information about a new electronic course for enhanced learning in making optimal decisions using the Multi-Attribute Decision Making paradigm. Emphasis is put on the construction of the exams of this electronic course. In order to provide ready to use tests for students’ exams, an algorithm based on a Monte Carlo method was conceived. This algorithm and its benefits are presented.

Cornel Resteanu, Marius Somodi, Marin Andreica

### Random Walks for Solving Boundary-Value Problems with Flux Conditions

We consider boundary-value problems for elliptic equations with constant coefficients and apply Monte Carlo methods to solving these equations. To take into account boundary conditions involving solution’s normal derivative, we apply the new mean-value relation written down at boundary point. This integral relation is exact and provides a possibility to get rid of the bias caused by usually used finite-difference approximation. We consider Neumann and mixed boundary-value problems, and also the problem with continuity boundary conditions, which involve fluxes. Randomization of the mean-value relation makes it possible to continue simulating walk-on-spheres trajectory after it hits the boundary. We prove the convergence of the algorithm and determine its rate. In conclusion, we present the results of some model computations.

Nikolai A. Simonov

### Examining Performance Enhancement of p-Channel Strained-SiGe MOSFET Devices

We examine performance enhancement of p-channel SiGe devices using our particle-based device simulator that takes into account self-consistently the bandstructure and the quantum mechanical space-quantization and mobility enhancement effects. We find surface roughness to be the dominant factor for the bad performance of p-channel SiGe devices when compared to conventional bulk p-MOSFETs at high bias conditions. At low and moderate bias conditions, when surface-roughness does not dominate the carrier transport, we observe performance enhancement in the operation of p-channel SiGe MOSFETs versus their conventional Si counterparts.

D. Vasileska, S. Krishnan, M. Fischetti

### A Monte Carlo Model of Piezoelectric Scattering in GaN

A non-parabolic piezoelectric model of electron-phonon interaction in Gallium Nitride is discussed. The Monte Carlo aspects of the model, needed for the simulation tools which provide the characteristics of GaN-based devices are analyzed in details. The piezo-scattering rate is derived by using quantum-mechanical considerations. The angular dependence is avoided by a proper spherical averaging and the non-parabolicity of the bands is accounted for. For the selection of the after-scattering state we deploy the rejection technique. The model is implemented in a simulation software. We employ a calibrated experimentally verified set of input material parameters to obtain valuable data for the transport characteristics of GaN. The simulation results are in good agreement with experimental data available for different physical conditions.

S. Vitanov, M. Nedjalkov, V. Palankovski

### Solving the Illumination Problem with Heuristics

In this article we propose optimal and quasi optimal solutions to the problem of searching for the

maximum lighting point

inside a polygon

P

of

n

vertices. This problem is solved by using three different techniques:

random search

,

simulated annealing

and

. Our comparative study shows that simulated annealing is very competitive in this application. To accomplish the study, a new polygon generator has been implemented, which greatly helps in the general validation of our claims on the illumination problem as a new class of optimization task.

Manuel Abellanas, Enrique Alba, Santiago Canales, Gregorio Hernández

### Optimal Placement of Antennae Using Metaheuristics

In this article we solve the radio network design problem (RND). This NP-hard combinatorial problem consist of determining a set of locations for placing radio antennae in a geographical area in order to offer high radio coverage using the smallest number of antennae. This problem is originally found in mobile telecommunications (such as mobile telephony), and is also relevant in the rising area of sensor networks. In this work we propose an evolutionary algorithm called CHC as the state of the art technique for solving RND problems and determine its expected performance for different instances of the RND problem.

Enrique Alba, Guillermo Molina, Francisco Chicano

### Sparse Array Optimization by Using the Simulated Annealing Algorithm

Sparse synthetic transmit aperture (STA) imaging systems are a good alternative to the conventional phased array systems. Unfortunately, the sparse STA imaging systems suffer from some limitations, which can be overcome with a proper design. In order to do so, a simulated annealing algorithm, combined with an effective approach can used for optimization of a sparse STA ultrasound imaging system. In this paper, three two-stage algorithms for optimization of both the positions of the transmit sub-apertures and the weights of the receive elements are considered and studied. The first stage of the optimization employs a simulated annealing algorithm that optimizes the locations of the transmit sub-aperture centers for a set of weighting functions. Three optimization criteria used at this stage of optimization are studied and compared. The first two criteria are conventional. The third criterion, proposed in this paper, combines the first two criteria. At the second stage of optimization, an appropriate weighting function for the receive elements is selected.

The sparse STA system under study employs a 64-element array, where all elements are used in receive and six sub-apertures are used in transmit. Compared to a conventional phased array imaging system, this system acquires images of better quality 21 times faster than an equivalent phased array system.

Vera Behar, Milen Nikolov

### An Iterative Fixing Variable Heuristic for Solving a Combined Blending and Distribution Planning Problem

In this paper, we consider a combined blending and distribution planning problem faced by a company that manages wheat supply chain. The distribution network consists of loading ports, and customers. Products are loaded on bulk vessels of various capacity levels for delivery to overseas customers. The purpose of this model is simultaneous planning of the assignment of an appropriate type and number of vessels to each customer order, the planning of quantities blended at ports, loaded from ports, and transported from loading ports to customers. We develop a mixed integer programming (MIP) model and provide a heuristic solution procedure for this distribution planning problem. An iterative fixing variable heuristic algorithm is used to assure that acceptable solutions are obtained quickly. The effectiveness of the proposed heuristic algorithm is evaluated by computational experiment.

Bilge Bilgen

### Hybrid Heuristic Algorithm for GPS Surveying Problem

This paper introduces several approaches based on ant colony optimization for efficient scheduling the surveying activities of designing satellite surveying networks. These proposed approaches use a set of agents called ants that cooperate to iteratively construct potential observation schedules. Within the context of satellite surveying, a positioning network can be defined as a set of points which are coordinated by placing receivers on these point to determine sessions between them. The problem is to search for the best order in which these sessions can be observed to give the best possible schedule. The same problem arise in Mobile Phone Surveying networks. Several case studies have been used to experimentally assess the performance of the proposed approaches in terms of solution quality and computational effort.

Stefka Fidanova

### A Hybrid Metaheuristic for a Real Life Vehicle Routing Problem

This paper presents a solution methodology to tackle a new realistic vehicle routing problem that incorporates heterogeneous fleet, multiple commodities and multiple vehicle compartments. The objective is to find minimum cost routes for a fleet of heterogeneous vehicles without violating capacity, loading and time window constraints. The solution methodology hybridizes in a reactive fashion systematic diversification mechanisms of Greedy Randomized Adaptive Search Procedures with Variable Neighborhood Search for intensification local search. Computational results reported justify the applicability of the methodology.

Panagiotis P. Repoussis, Christos D. Tarantilis, George Ioannou

### Multipopulation Genetic Algorithms: A Tool for Parameter Optimization of Cultivation Processes Models

This paper endeavors to show that genetic algorithms, namely Multipopulation genetic algorithms (MpGA), are of great utility in cases where complex cultivation process models have to be identified and, therefore, rational choices have to be made. A system of five ordinary differential equations is proposed to model biomass growth, glucose utilization and acetate formation. Parameter optimization is carried out using experimental data set from an

E. coli

cultivation. Several conventional algorithms for parameter identification (Gauss-Newton, Simplex Search and Steepest Descent) are compared to the MpGA. A general comment on this study is that traditional optimization methods are generally not universal and the most successful optimization algorithms on any particular domain, especially for the parameter optimization considered here. They have been fairly successful at solving problems of type which exhibit bad behavior like multimodal or nondifferentiable for more conventional based techniques.

Olympia Roeva

### Design of Equiripple 2-D Linear-Phase FIR Digital Filters Using Genetic Algorithm

The paper presents a method for designing 2-D linear-phase FIR filters with an equiripple magnitude response. The filter design problem is transformed into an equivalent nonlinear optimization problem. In order to improve the speed of convergence, a two-step solution procedure of the considered problem is proposed. In the first step, a genetic algorithm is applied. The final point from the genetic algorithm is used as the starting point for a local optimization method. The proposed technique is applied to the design of 2-D FIR linear-phase filters with different symmetries. Design examples are included.

Felicja Wysocka-Schillak

### A Simple and Efficient Algorithm for Eigenvalues Computation

A simple algorithm for the computation of eigenvalues of real or complex square matrices is proposed. This algorithm is based on an additive decomposition of the matrix. A sufficient condition for convergence is proved. It is also shown that this method has many properties of the QR algorithm : it is invariant for the Hessenberg form, shifts are possible in the case of a null element on the diagonal. Some other interesting experimental properties are shown. Numerical experiments are given showing that most of the time the behavior of this method is not much different from that of the QR method, but sometimes it gives better results, particularly in the case of a bad conditioned real matrix having real eigenvalues.

René Alt

### Numerical Computations with Hausdorff Continuous Functions

Hausdorff continuous (H-continuous) functions appear naturally in many areas of mathematics such as Approximation Theory [11], Real Analysis [1], [8], Interval Analysis, [2], etc. From numerical point of view it is significant that the solutions of large classes of nonlinear partial differential equations can be assimilated through H-continuous functions [7]. In particular, discontinuous viscosity solutions are better represented through Hausdorff continuous functions [6]. Hence the need to develop numerical procedures for computations with H-continuous functions. It was shown recently, that the operations addition and multiplication by scalars of the usual continuous functions on

$\Omega\subseteq\mathbb{R}^n$

can be extended to H-continuous functions in such a way that the set ℍ(

Ω

) of all Hausdorff continuous functions is a linear space [4]. In fact ℍ(

Ω

) is the largest linear space involving interval functions. Furthermore, multiplication can also be extended [5], so that ℍ(

Ω

) is a commutative algebra. Approximation of ℍ(

Ω

) by a subspace were discussed in [3]. In the present paper we consider numerical computations with H-continuous functions using ultra-arithmetical approach [9], namely, by constructing a functoid of H-continuous functions. For simplicity we consider

$\Omega\subseteq\mathbb{R}$

. In the next section we recall the definition of the algebraic operations on ℍ(

Ω

). The concept of functoid is defined in Section 3. In Section 4 we construct a functoid comprising a finite dimensional subspace of ℍ(

Ω

) with a Fourier base extended by a set of H-continuous functions. Application of the functoid to the numerical solution of the wave equation is discussed in Section 5.

Roumen Anguelov, Svetoslav Markov

### Mixed Discretization-Optimization Methods for Nonlinear Elliptic Optimal Control Problems

An optimal control problem is considered, for systems governed by a nonlinear elliptic partial differential equation, with control and state constraints. Since this problem may have no classical solutions, it is also formulated in the relaxed form. The classical problem is discretized by using a finite element method, where the controls are approximated by elementwise constant, linear, or multilinear, controls. Our first result is that strong accumulation points in

L

2

of sequences of admissible and extremal discrete controls are admissible and weakly extremal classical for the continuous classical problem, and that relaxed accumulation points of sequences of admissible and extremal discrete controls are admissible and weakly extremal relaxed for the continuous relaxed problem. We then propose a penalized gradient projection method, applied to the discrete problem, and a corresponding discretization-optimization method, applied to the continuous classical problem, that progressively refines the discretization during the iterations, thus reducing computing time and memory. We prove that accumulation points of sequences generated by the first method are admissible and extremal for the discrete problem, and that strong classical (resp. relaxed) accumulation points of sequences of discrete controls generated by the second method are admissible and weakly extremal classical (resp. relaxed) for the continuous classical (resp. relaxed) problem. Finally, numerical examples are given.

Ion Chryssoverghi

### Stability and Bifurcation Analysis of a Nonlinear Model of Bioreactor

The stability characteristics of a nonlinear model of a continuously stirred tank bioreactor with cell recycle are studied. Assuming that some practically important model parameters are uncertain, existence of bifurcations of equilibrium points is shown. The dynamic behaviour of the system near bifurcation points is also demonstrated. Numerical simulations in the computer algebra system

Maple

are presented.

Neli Dimitrova, Plamena Zlateva

### Discrete Approximations of Singularly Perturbed Systems

In the paper we study discrete approximations of singularly perturbed system in a finite dimensional space. When the right-hand side is almost upper semicontinuous with convex compact values and one-sided Lipschitz we show that the distance between the solution set of the original and the solution set of the discrete system is

$O\left(h^{\frac{1}{2}}\right)$

.

Tzanko Donchev, Vasile Lupulescu

### A Generalized Interval LU Decomposition for the Solution of Interval Linear Systems

Generalized intervals (intervals whose bounds are not constrained to be increasingly ordered) extend classical intervals providing better algebraic properties. In particular, the generalized interval arithmetic is a group for addition and for multiplication of zero free intervals. These properties allow one constructing a LU decomposition of a generalized interval matrix

A

: the two computed generalized interval matrices

L

and

U

satisfy

A

=

LU

with equality instead of the weaker inclusion obtained in the context of classical intervals. Some potential applications of this generalized interval LU decomposition are investigated.

Alexandre Goldsztejn, Gilles Chabert

### On the Relationship Between the Sum of Roots with Positive Real Parts and Polynomial Spectral Factorization

This paper is concerned with the relationship between the sum of roots with positive real parts (SORPRP) of an even polynomial and the polynomial spectral factor of the even polynomial. The SORPRP and its relationship to Gröbner bases are firstly reviewed. Then it is shown that the system of equations satisfied by the coefficients of the polynomial spectral factor is directly related to a Gröbner basis. It is then demonstrated by means of an

${\mathcal{H}}_2$

optimal control problem that the above fact can be used to facilitate guaranteed accuracy computation.

Masaaki Kanno, Hirokazu Anai, Kazuhiro Yokoyama

### Lie Brackets and Stabilizing Feedback Controls

The relation between a class of high-order control variations and the asymptotic stabilizability of a smooth control system is briefly discussed. Assuming that there exist high-order control variations ”pointing” to a closed set at every point of some its neighborhood, an approach for constructing stabilizing feedback controls is proposed. Two illustrative examples are also presented.

Mikhail I. Krastanov

### Interval Based Morphological Colour Image Processing

In image analysis and pattern recognition fuzzy sets play the role of a good model for segmentation and classifications tasks when the regions and the classes cannot be strictly defined. Fuzzy morphology has been shown to be a very eficient tool in processing and segmentation of grey-scale images. In this work we show that using interval modelling we can apply efficiently fuzzy morphological operations to colour images. In this case intervals help us to avoid the problem of lack of total ordering in multidimensional Euclidean spaces, in particular in the three dimensional RGB colour space.

Antony T. Popov

### Solving Linear Systems Whose Input Data Are Rational Functions of Interval Parameters

The paper proposes an approach for self-verified solving of linear systems involving rational dependencies between interval parameters. A general inclusion method is combined with an interval arithmetic technique providing inner and outer bounds for the range of monotone rational functions. The arithmetic on proper and improper intervals is used as an intermediate computational tool for eliminating the dependency problem in range computation and for obtaining inner estimations by outwardly rounded interval arithmetic. Supporting software tools with result verification, developed in the environment of CAS Mathematica, are reported.

Evgenija D. Popova

### Differential Games and Optimal Tax Policy

In this paper we consider the problem of finding the solution of a differential game in relation with choosing an optimal tax policy rule. The paper extends the existing literature in two directions. First, instead of treating the tax base as given, in our formulation it is a control variable for the government. Secondly, we impose a phase constraint of mixed type for the considered problem of taxation. We present new conditions under which the solution of the differential game is found explicitly. The obtained optimal tax policy is time-consistent.

Rossen Rozenov, Mikhail I. Krastanov

### Evolutionary Optimization Method for Approximating the Solution Set Hull of Parametric Linear Systems

Systems of parametric interval equations are encountered in many practical applications. Several methods for solving such systems have been developed during last years. Most of them produce both outer and inner interval solutions, but the amount of overestimation, resp. underestimation is not exactly known. If a solution of a parametric system is monotonic and continuous on each interval parameter, then the method of combination of endpoints of parameter intervals computes its interval hull. Recently, a few polynomial methods computing the interval hull were developed. They can be applied if some monotonicity and continouity conditions are fulfilled. To get the most accurate inner approximation of the solution set hull for problems with any bounded solution set, an evolutionary optimization method is applied.

Iwona Skalna

### Formulae for Calculation of Normal Probability

Various approximate formulae for calculation of the normal probability integral

$$P(x)=\frac{1}{\sqrt{2\pi}}\int\limits_{0}^{x}e^{-t^2/2}dt,\ \ x\ge 0,$$

are given. Our formulae provide a good approximation to

P

(

x

) over the entire range 0 <

x

< ∞, hence they can be used in practice instead of the usual numerical tables. Error bounds based on the Peano kernel technique are proved.

keywordsNormal probability, error function, quadrature formulae, Peano kernels

Vesselin Gushev, Geno Nikolov

### Iterative-Collocation Method for Integral Equations of Heat Conduction Problems

The integral equations studied here play very important role in the theory of parabolic initial-boundary value problems (heat conduction problems) and in various physical, technological and biological problems (epidemiology problems). This paper is concerned with the iterative-collocation method for solving these equations. We propose an iterative method with corrections based on the interpolation polynomial of spatial variable of the Lagrange type with given collocation points. The coefficients of these corrections can be determined by a system of Volterra integral equations. The convergence of the presented algorithm is proved and an error estimate is established. The presented theory is illustrated by numerical examples and a comparison is made with other methods.

AMS Subject Classification:

65R20.

Lechosław Ha̧cia

### Numerical Computation of the Markov Factors for the Systems of Polynomials with the Hermite and Laguerre Weights

Denote by

π

n

the set of all real algebraic polynomials of degree at most

n

, and let

$U_{n} := \{e^{-x^2} p(x) : p \in \pi_{n} \}$

,

$V_{n} := \{e^{-x} p(x) : p \in \pi_{n} \}$

. It was

$\sup \Vert {\rm I\!R} \in \not \equiv u_{*,n}^{(k)}$

proved in [9] that

$M_{k}(U_{n}) := \sup \{ \Vert u^{(k)} \Vert_{{\rm I\!R}}/ \Vert u \Vert_{{\rm I\!R}} : u \in U_{n}, u \not \equiv 0\} = \Vert u_{*,n}^{(k)} \Vert_{{\rm I\!R}},~ \forall n, k \in {\rm I\!N}$

, and

$M_{k}(V_{n}) := {\rm sup} \{ \Vert v^{(k)} \Vert_{{\rm I\!R}_{+}}/ \Vert v \Vert_{{\rm I\!R}_{+}} : v \in V_{n}, v \not \equiv 0\} = \Vert v_{*,n}^{(k)} \Vert_{{\rm I\!R}_{+}}$

, where

$\Vert \cdot \Vert_{{\rm I\!R}}$

(

$\Vert \cdot \Vert_{{\rm I\!R}_{+}}$

) is the supremum norm on IR (IR

+

: = [0, ∞ )) and

u

*,

n

(

v

*,

n

) is the Chebyshev polynomial from

U

n

(

V

n

). We prove here the convergence of an algorithm for the numerical construction of the oscillating weighted polynomial from

U

n

(

V

n

), which takes preassigned values at its extremal points. As an application, we obtain numerical values for the Markov factors

M

k

(

U

n

) and

M

k

(

V

n

) for 1 ≤

n

≤ 10 and 1 ≤

k

≤ 5.

Lozko Milev

### Connection of Semi-integer Trigonometric Orthogonal Polynomials with Szegő Polynomials

In this paper we investigate connection between semi-integer orthogonal polynomials and Szegő’s class of polynomials, orthogonal on the unit circle. We find a representation of the semi-integer orthogonal polynomials in terms of Szegő’s polynomials orthogonal on the unit circle for certain class of weight functions.

Gradimir V. Milovanović, Aleksandar S. Cvetković, Zvezdan M. Marjanović

### Trigonometric Orthogonal Systems and Quadrature Formulae with Maximal Trigonometric Degree of Exactness

Turetzkii [Uchenye Zapiski, Vypusk

1

(149) (1959), 31–55, (English translation in East J. Approx.

11

(2005) 337–359)] considered quadrature rules of interpolatory type with simple nodes, with maximal trigonometric degree of exactness. For that purpose Turetzkii made use of orthogonal trigonometric polynomials of semi–integer degree.

Ghizzeti and Ossicini [Quadrature Formulae, Academie-Verlag, Berlin, 1970], and Dryanov [Numer. Math.

67

(1994), 441–464], considered quadrature rules of interpolatory type with multiple nodes with maximal trigonometric degree of exactness. Inspired by their results, we study here

s

–orthogonal trigonometric polynomials of semi–integer degree. In particular, we consider the case of an even weight function.

Gradimir V. Milovanović, Aleksandar S. Cvetković, Marija P. Stanić

### On the Calculation of the Bernstein-Szegö Factor for Multivariate Polynomials

Let IR

d

be the Euclidean space with the usual norm |.|

2

,

${\mathcal P}_n^d$

be the set of all polynomials over IR

d

of degree

n

, and

K

⊂ IR

d

be a convex body. An algorithm for calculation of the Bernstein-Szegö factor:

$$BS(K):= \sup_{{\bf x\in{\rm int}(K)}\atop {P\in {\cal P}_n^d, n\in{\rm I\!N}}} \Bigg\{{|\rm grad P(\bf x)|_2w(K)\sqrt{1-\alpha^2(K,\bf x)} \over n \sqrt{||P||^2_{C(K)} - P^2(\bf x)} }\Bigg\}$$

is considered, where

w

(

K

) is the width of

K

and

α

(

K

,

x

) is the generalized Minkowsky functional. It is known that

$BS(K)\in [2,2\sqrt2]$

. On the basis of computer experiments, we show that the existing in the literature hypothesis, that

BS

(

K

) = 2 for any convex body

K

⊂ IR

d

, fails to hold.

Nikola Naidenov

### Quadrature Formula Based on Interpolating Polynomials: Algorithmic and Computational Aspects

The aim of this article is to obtain a quadrature formula for functions in several variables and to analyze the algorithmic and computational aspects of this formula. The known information about the integrand is

$\{\lambda_i(f)\}_{i=1}^{n}$

, where

λ

i

are linearly independent linear functionals. We find a form of the coefficients of the quadrature formula which can be easy used in numerical calculations. The main algorithm we use in order to obtain the coefficients and the remainder of the quadrature formula is based on the Gauss elimination by segments method. We obtain an expression for the exactness degree of the quadrature formula. Finally, we analyze some computational aspects of the algorithm in the particular case of the Lagrange conditions.

Dana Simian, Corina Simian

### Stability of Semi-implicit Atmospheric Models with Respect to the Reference Temperature Profile

The dependence of the linear stability of two-time-level semi-implicit schemes on choice of the reference temperature profile is studied. Analysis is made for large time steps, keeping general form of such model parameters as the number of vertical levels, their distribution, and the values of the viscosity coefficients. The obtained results reveal more restrictive conditions on the reference temperature profile than those for three-time-level schemes. Nevertheless, general conclusions are consistent with the previous analysis: instability generated by inappropriate choice of the temperature profile is absolute and the scheme stability can be recovered by setting the reference temperature to be warmer than the actual one.

Andrei Bourchtein, Ludmila Bourchtein, Maxim Naumov

### Using Singular Value Decomposition in Conjunction with Data Assimilation Procedures

In this study we apply the singular value decomposition (SVD) technique of the so-called ‘observability’ matrix to analyse the information content of observations in 4D-Var assimilation procedures. Using a simple one-dimensional transport equation, the relationship between the optimal state estimate and the right singular vectors of the observability matrix is examined. It is shown the importance of the value of the variance ratio, between the variances of the background and the observational errors, in maximizing the information that can be extracted from the observations by using Tikhonov regularization theory. Numerical results are presented.

Gabriel Dimitriu

### New Operator Splitting Methods and Their Analysis

In this paper we give a short overview of some traditional operator splitting methods. Then we introduce two new methods, namely the additive splitting and the iterated splitting. We analyze these methods and compare them to the traditional ones.

István Faragó

### On an Implementation of the TM5 Global Model on a Cluster of Workstations

TM5 is a global chemistry Transport Model. It allows two-way nested zooming which leads to possibility to run the model on relatively very fine space grid (1° ×1°) over selected regions (Europe is most often used in up to now experiments but North America, Africa and Asia can be treated separately or in combinations). The boundary conditions for the zoomed subdomains are provided consistently from the global model. The TM5 model is a good tool for studying some effects due to the grid refinement on global atmospheric chemistry issues (intercontinental transport of air pollutants, etc.).

The huge increase in the number of the multi-processor platforms and their differences leads to a need of different approaches in order to meet the requirements for the optimality of the computer runs. The paper is devoted to an implementation of a parallel version of the TM5 model on a cluster of SUN Workstations and to the developing of a new parallel algorithm. It is based on the decomposition, in some sense, of the computational domain supposing that the zoomed regions are more than one. If it is assumed that the number of zoomed regions is

N

and the number of the processors available is

p

. The processors are divided in

N/p

groups and each group is responsible for the whole computational domain and one of the zoomed regions. Some communications are needed in order to impose the inner boundary conditions. The new algorithm has better parallel feathers than the old one which is used in the inner level. Some results concerning the CPU time, speed up and efficiency can be found.

Subject classifications:

65Y05.

Krassimir Georgiev

### On the Sign-Stability of the Finite Difference Solutions of One-Dimensional Parabolic Problems

In the numerical solutions of partial differential equations, the preservation of the qualitative properties of the original problem is a more and more important requirement. For 1D parabolic equations, one of this properties is the so-called sign-stability: the number of sign-changes of the solution cannot increase in time. This property is investigated for the finite difference solutions, and a sufficient condition is given to guarantee the numerical sign-stability. We prove sufficient conditions for the sign-stability and sign-unstability of tridiagonal matrices.

Róbert Horváth

### Comprehensive Modelling of PM10 and PM2.5 Scenarios for Belgium and Europe in 2010

The extended EUROS model has been used to calculate concentrations of PM

10

and PM

2.5

for Europe for the years 2002 and 2010 using a recent emission scenario. The obtained results for Belgium show decreases in PM-concentrations between 5 and 26% in this period, depending on the location. The contribution of anthropogenic sources in Flanders to annual averaged PM

10

concentrations amounts to 17% in 2002 and 15% in 2010 on average. The most important contribution to PM

10

concentrations originates from agricultural activities in Flanders, whereas the sector ”traffic” is the dominant source for anthropogenic PM

2.5

in Flanders.

C. Mensink, F. Deutsch, L. Janssen, R. Torfs, J. Vankerkom

### Parallel and GRID Implementation of a Large Scale Air Pollution Model

Large-scale environmental models are powerful tools, designed to meet the increasing demand in various environmental studies. The atmosphere is the most dynamic component of the environment, where the pollutants and other chemical species actively interact with each other, and can quickly be moved in a very long distance. Therefore the advanced modeling is usually done in a large computational domain. Moreover, all relevant physical, chemical and photochemical processes should be taken into account, which heavily depend on the meteorological conditions. All this makes the air pollution modeling a huge and rather difficult computational task, requiring a large amount of computational power. The most powerful supercomputers have been used for the development and test runs of such a model, the Danish Eulerin Model (DEM). Distributed parallel computing via MPI is one of the most efficient techniques in achieving good performance and getting results in real time. The quickly advancing GRID computing technology is another powerful tool that can be used to reach higher level of performance of such a huge model. Both techniques and their inherent problems are discussed in this paper. Results of numerical experiments are presented and analysed and some conclusions are drown, based on the experiments.

Tzvetan Ostromsky, Zahari Zlatev

### Simulation of an Extreme Air Pollution Episode in the City of Stara Zagora, Bulgaria

Several industrial hot spots exist in Bulgaria and detail study of every one is worth to be done, but often incidents with high pollution levels over usually relatively clean populated areas cause big political concern. Stara Zagora is one of the biggest towns in Bulgaria (300 000 inhabitants) located in the middle of the country. In the summer of 2004, two high level

SO

2

pollution events happened causing big public complains and even political and judicial consequences. Analogous events happened in 2005, too. All this requires appropriate measures to be taken by local authorities, first of all a suitable monitoring and forecasting system to be set. For the moment such system does not exist; only ambient air concentrations are measured in several points. This is not sufficient neither to predict nor even to explain the cases. As far as the mathematical modeling is alternative and supplementing tool according to the EU Framework Directive on Air Quality (96/62/ES) and its daughter directives (see, [5,6,7,16]), an attempt to simulate one of these events was done applying one of the most comprehensive and up-to-science modeling tools, mainly the US EPA Model-3 system.

Maria Prodanova, Juan Perez, Dimiter Syrakov, Roberto San Jose, Kostadin Ganev, Nikolai Miloshev, Stefan Roglev

### Studying the Properties of Variational Data Assimilation Methods by Applying a Set of Test-Examples

The variational data assimilation methods can successfully be used in different fields of science and engineering. An attempt to utilize available sets of observations in the efforts to improve (i) the models used to study different phenomena and/or (ii) the model results is systematically carried out when data assimilation methods are used.

The main idea, on which the variational data assimilation methods are based, is pretty general. A functional is formed by using a weighted inner product of differences of model results and measurements. The value of this functional is to be minimized. Forward and backward computations are carried out by using the model under consideration and its adjoint equations (both the model and its adjoint are defined by systems of differential equations). The major difficulty is caused by the huge increase of both the computational load (normally by a factor more than 100) and the storage needed. This is why it might be appropriate to apply some splitting procedure in the efforts to reduce the computational work.

Five test-examples have been created. Different numerical aspects of the data assimilation methods and the interplay between the major computational parts of any data assimilation method (numerical algorithms for solving differential equations, splitting procedures and optimization algorithms) have been studied by using these tests. The presentation will include results from testing carried out in the study.

Per Grove Thomsen, Zahari Zlatev

### On the Numerical Solutions of Eigenvalue Problems in Unbounded Domains

The aim of this study is to propose a procedure for coupling of finite element (FE) and infinite large element (ILE). This FE/ILE method is applied to the second-order self-adjoint eigenvalue problems in the plane. We propose a conforming method for approximation of eigenpairs in unbounded domains. Finally, some numerical results are presented.

Andrey B. Andreev, Milena R. Racheva

### Mesh Independent Superlinear Convergence of an Inner-Outer Iterative Method for Semilinear Elliptic Systems

We propose the damped inexact Newton method, coupled with inner iterations, to solve the finite element discretization of a class of nonlinear elliptic systems. The linearized equations are solved by a preconditioned conjugate gradient (PCG) method. Both the inner and the outer iterations have mesh independent superlinear convergence.

István Antal

### Solving Second Order Evolution Equations by Internal Schemes of Approximation

Numerical approximation for the solution of a second order evolution equation is proposed. An internal scheme of approximation is used. The equation is associated with a maximal monotone operator in a real Hilbert space together with bilocal boundary conditions. A numerical example is investigated.

Narcisa Apreutesei, Gabriel Dimitriu

### Target Detection and Parameter Estimation Using the Hough Transform

In recent years, the algorithms that extract information for target’s behavior through mathematical transformation of the signals reflected from a target, find ever-widening practical application. In this paper, a new two-stage algorithm for target detection and target’s radial velocity estimation that exploits the Hough transform is proposed. The effectiveness of the proposed algorithm is formulated in terms of both quality parameters - the probability of detection and the accuracy of velocity estimation. The quality parameters are estimated using the Monte Carlo simulation approach.

Vera Behar, Lyubka Doukovska, Christo Kabakchiev

### Mechanical Failure in Microstructural Heterogeneous Materials

Various heterogeneous materials with multiple scales and multiple phases in the microstructure have been produced in the recent years. We consider a mechanical failure due to the initiation and propagation of cracks in places of high pore density in the microstructures. A multi–scale method based on the asymptotic homogenization theory together with the mesh superposition method (

s

-version of FEM) is presented for modeling of cracks. The homogenization approach is used on the global domain excluding the vicinity of the crack where the periodicity of the microstructures is lost and this approach fails. The multiple scale method relies on efficient combination of both macroscopic and microscopic models. The mesh superposition method uses two independent (global and local) finite element meshes and the concept of superposing the local mesh onto the global continuous mesh in such a way that both meshes not necessarily coincide. The homogenized material model is considered on the global mesh while the crack is analyzed in the local domain (patch) which allows to have an arbitrary geometry with respect to the underlying global finite elements. Numerical experiments for biomorphic cellular ceramics with porous microstructures produced from natural wood are presented.

Stéphane Bordas, Ronald H. W. Hoppe, Svetozara I. Petrova

### Round-Trip Operator Technique Applied for Optical Resonators with Dispersion Elements

The round-trip operator technique is widely used for dispersionless optical resonators beginning from pioneering studies of Fox and Li. The resonator modes are determined as eigenfunctions of the round-trip operator and may be calculated by means of numerical linear algebra. Corresponding complex eigenvalues determine the wavelength shifts relative to reference value and threshold gains. Dispersion elements, for example, Bragg mirrors in a vertical cavity surface emitting laser (VCSEL) cause a dependence of the propagation operator on the wavelength and threshold gain. We can determine the round-trip operator in this case also, but the unknown values of the wavelength and threshold gain enter into the operator in a complicated manner. Trial-and-error method for determination of the wavelength shifts and the threshold gains is possible but it is rather time consuming method. The proposed approximate numerical method for calculation of resonator modes is based on the solution of linear eigenvalue problem for the round-trip operator with reference wavelength and zero attenuation. The wavelength shifts and threshold gains can be calculated by simple formulae using the eigenvalues obtained and the computed effective length of the resonator. Calculations for a cylindrical antiresonant-reflecting optical waveguide (ARROW) VCSEL are performed for verification of the model.

Nikolay N. Elkin, Anatoly P. Napartovich, Dmitry V. Vysotsky, Vera N. Troshchieva

### On the Impact of Tangential Grid Refinement on Subgrid-Scale Modelling in Large Eddy Simulation

The paper presents Large Eddy Simulations of plane channel flow at a friction Reynolds number of 180 and 395 with a block-structured Finite Volume method. Local grid refinement near the solid wall is employed in order to reduce the computational cost of such simulations or other simulations of wall-bounded flows. Different subgrid-scale models are employed and different expressions for the length scale in these models are investigated. It turns out that the numerical discretization has an non-negligible impact on the computed fluctuations.

Jochen Fröhlich, Jordan A. Denev, Christof Hinterberger, Henning Bockhorn

### Detection Acceleration in Hough Parameter Space by K-Stage Detector

The sequential algorithm for target detection in impulse environment by K-stage processing in polar Hough (PH) space is investigated. This algorithm permits the search radar to minimize the time of target detection, because the sequential detector used minimizes the number of the necessary radar antenna scans, while the conventional detector uses a fix scan number. The aim of our study is to minimize the detection time by minimizing the number of radar scans, when the detector’s parameters and characteristics are fixed. The average estimates of the minimal number of scans are obtained using the Monte Carlo simulation approach when the detector’s parameters are fixed. The proposed algorithm is simulated in MATLAB environment. The efficiency factor of the investigated detector for radar signal detection and track determination in conditions of binomial distributed impulse interference is obtained by using Monte-Carlo approach. The prepositional detector, compared to the conventional detector, accelerates the detection procedure several times.

Ivan Garvanov, Christo Kabakchiev, Hermann Rohling

### A Method for Calculating Active Feedback System to Control Vertical Position of Plasma in a Tokamak

In designing tokamaks, the maintenance of vertical stability of plasma is one of the most important problems. For this purpose systems of passive and active feedbacks are applied. The role of passive system consisting of a vacuum vessel and passive stabilizer plates is to suppress fast MHD (magnetohydrodynamic) instabilities. The active feedback system is applied to control slow motions of plasma. The objective of this paper is to investigate three successive problems the solution of which will allow to determine the possibility to control plasma motions. The first problem is the vertical stability problem under the assumption of ideal conductivity of plasma and passive stabilizing elements. The problem is solved analytically and on the basis of the obtained solution a criterion of MHD-stability is formulated.

The second problem is the vertical stability when finite conductivity of stabilizing elements is taken into account. The dispersion equation relative to instability growth rate is obtained and analyzed. For practical values of the parameters it is shown that there is a unique root with positive real part, which presents the growth rate of only unstable mode.

The third problem is connected with the control of plasma vertical position with application of active feedback system. The problem of calculation of feedback control parameters is formulated as an optimization problem and its approximate solving method is suggested.

Nizami Gasilov

### Numerical Algorithm for Simultaneously Determining of Unknown Coefficients in a Parabolic Equation

A numerical algorithm for an inverse problem of simultaneously determining unknown coefficients in a linear parabolic equation subject to the specifications of the solution at internal points along with the usual initial and boundary conditions is proposed. The approach based on TTF (Trace Type Functional) formulation of the problem is used. To avoid instability in this approach the Tikhonov regularization method is applied. Some numerical examples using the proposed algorithm are presented.

Nizami Gasilov, Afet Golayoglu Fatullayev, Krassimir Iankov

### Data Compression of Color Images Using a Probabilistic Linear Transform Approach

In this work, we design an efficient algorithm for color image compression using a model for the rate-distortion connection. This model allows the derivation of an optimal color components transform, which can be used to transform the RGB primaries or matrices into a new color space more suitable for compression. Sub-optimal solutions are also proposed and examined. The model can also be used to derive optimal bits allocation for the transformed subbands. An iterative algorithm for the calculation of optimal quantization steps is introduced using the subband rates (entropies). We show that the rates can be approximated based on a probabilistic model for subband transform coefficients to reduce the algorithm’s complexity. This is demonstrated for the Discrete Cosine Transform (DCT) as the operator for the subband transform and the Laplacian distribution assumption for its coefficients. The distortion measure considered is the MSE (Mean Square Error) with possible generalization to WMSE (Weighted MSE). Experimental results of compressed images are presented and discussed for two versions of the new compression algorithm.

Evgeny Gershikov, Moshe Porat

### On a Local Refinement Solver for Coupled Flow in Plain and Porous Media

A local refinement algorithm for computer simulation of flow through oil filters is presented. The mathematical model is based on laminar incompressible Navier-Stokes-Brinkman equations for flow in pure liquid and in porous zones. A finite volume method based discretization on cell-centered, collocated, locally refined grids is used. Special attention is paid to the conservation of the mass on the interface between the coarse and the fine grid. A projection method, SIMPLEC, is used to decouple momentum and continuity equations. The corresponding software is implemented in a flexible way for arbitrary 3D geometries, approximated by an union of parallelepipeds with different sizes. Results from numerical experiments show that the solver could be successfully used for simulation of coupled flow in plain and in porous media.

Oleg Iliev, Daniela Vasileva

### Properties of the Lyapunov Iteration for Coupled Riccati Equations in Jump Linear Systems

We analyze the solution of the system of coupled algebraic Riccati equations of the optimal control problem of jump linear system. We prove that the Lyapunov iterations converge to a positive semidefinite stabilizing solution under mild conditions.

Ivan Ganchev Ivanov

### Numerical Analysis of Blow-Up Weak Solutions to Semilinear Hyperbolic Equations

We study numerical approximations of weak solutions of hyperbolic problems with discontinuous coefficients and nonlinear source terms in the equation. By a semidiscretization of a Dirichlet problem in the space variable we obtain a system of ordinary differential equations (SODEs), which is expected to be an approximation of the original problem. We show at conditions similar to those for the hyperbolic problem, that the solution of the SODEs blows up. Under certain assumptions, we also prove that the numerical blow-up time converges to the real blow-up time when the mesh size goes to zero. Numerical experiments are analyzed.

Bośko S. Jovanovic, Miglena N. Koleva, Lubin G. Vulkov

### A Monotone Iterative Method for Numerical Solution of Diffusion Equations with Nonlinear Localized Chemical Reactions

We study the numerical solution of a model two-dimensional problem, where the nonlinear reaction takes place only at some interface curves, due to the present of catalyst. A finite difference algorithm, based on a monotone iterative method and the immersed interface method (IIM), is proposed and analyzed. Our method is efficient with respect to flexibility in dealing with the geometry of the interface curve. The numerical results indicate first order of accuracy.

Juri D. Kandilarov

### Numerical Solution of an Elliptic Problem with a Non-classical Boundary Condition

We investigate an elliptic problem with a boundary condition given by a sum of normal derivative and an elliptic operator in tangential variables (also known as ”Venttsel” boundary condition). The differential problem is discretized by a specific finite difference method. Error estimates of the numerical method in the discrete Sobolev space

$W_2 ^1$

are obtained. The rate of convergence in this space is optimal, i.e. it is

m

− 1 for solutions from

$W_2 ^{m}$

, 1 <

m

< 2.5.

Natalia T. Kolkovska

### Contour Determination in Ultrasound Medical Images Using Interacting Multiple Model Probabilistic Data Association Filter

The Probabilistic Data Association Filter (PDAF) with Interacting Multiple Model (IMM) approach is applied for contour determination in ultrasound images. The contour of interest is assumed to be a target trajectory which is tracked using IMMPDA filtering. The target movement is assumed to be along a circle and controlled by equally spaced radii from an arbitrary seed point inside the assumed contour. The generalized scores of the candidate points along current radius are determined on the base of two components - the Gaussian probability density function, associated with the assignment of the current point to the trajectory and the edge magnitude. A method for modeling complex contours with known true positions and method for error evaluation are proposed. These methods are used to generate Field II images and to estimate errors of contour determination using IMMPDA algorithm incorporating edge magnitude.

Pavlina Konstantinova, Dan Adam, Donka Angelova, Vera Behar

### Computational Model of 1D Continuum Motion. Case of Textile Yarn Unwinding Without Air Resistance

The textile fibers and yarns are usually modeled as one dimensional continuum of beams or as a system of particles, connected by springs. The governing equation of motion of such systems has many solutions, which switch in between depending on the available energy of the system. Effective modeling of problems with bifurcations like these, applicable for more general geometries is rarely reported. In the present work we demonstrate, that using a system of particles and an appropriate change of variables, a successful time step integration either by the Leap-Frog or the Verlet algorithms is possible.

Yordan Kyosev, Michail Todorov

### A Numerical Approach to the Dynamic Unilateral Contact Problem of Soil-Pile Interaction Under Instabilizing and Environmental Effects

The paper deals with a numerical approach for the dynamic soil-pile interaction, considered as an inequality problem of structural engineering. So, the unilateral contact conditions due to tensionless and elastoplastic softening/fracturing behavior of the soil as well as due to gapping caused by earthquake excitations are taken into account. Moreover, second-order geometric effects for the pile behavior due to preexisting compressive loads and environmental soil effects causing instabilization are taken also into account. The numerical approach is based on a double discretization and on mathematical programming. First, in space the finite element method (FEM) is used for the simulation of the pipeline and the unilateral contact interface, in combination with the boundary element method (BEM) for the soil simulation. Next, with the aid of Laplace transform, the problem conditions are transformed to convolutional ones involving as unknowns the unilateral quantities only. So the number of unknowns is significantly reduced. Then a marching-time approach is applied and finally a nonconvex linear complementarity problem is solved in each time-step.

Asterios Liolios, Konstantina Iossifidou, Konstantinos Liolios, Khairedin Abdalla, Stefan Radev

### Analysis of Soil-Structure Interaction Considering Complicated Soil Profile

The effect of soil-structure interaction (SSI) is an important consideration and cannot be neglected in the seismic design of structures on soft soil. Various methods have been developed to consider SSI effects and are currently being used. However, most of the approaches including a general finite element method cannot appropriately consider the properties and characteristics of the sites with complicated soil profiles. To overcome these difficulties, this paper presents soil-structure interaction analysis method, which can consider precisely complicated soil profiles by adopting an unaligned mesh generation approach. This approach has the advantages of rapid generation of structured internal meshes and leads to regular and precise stiffness matrix. The applicability of the proposed method is validated through several numerical examples and the influence of various properties and characteristics of soil sites on the response is investigated.

Jang Ho Park, Jaegyun Park, Kwan-Soon Park, Seung-Yong Ok

### Numerical Method for Regional Pole Assignment of Linear Control Systems

A new class of regional pole assignment problems for linear control systems is considered, in which each closed-loop system pole is placed in a desired separate region of the complex plane. A numerically stable method for regional pole assignment is proposed, in which the design freedom is parameterized directly by specific eigenvector (or principal vector) elements and pole location variables that can be chosen arbitrarily. Combined with an appropriate optimization procedure, the proposed method can be used to solve a wide range of optimization problems with pole location constraints, arising in the multi-input control systems design (

H

2

/

H

∞

optimization with pole assignment, robust pole assignment, pole assignment with maximum stability radius, etc.).

Petko Hr. Petkov, Nicolai D. Christov, Michail M. Konstantinov

### Solution Limiters and Flux Limiters for High Order Discontinuous Galerkin Schemes

We analyze a general concept of limiters for a high order DG scheme written for a 1-D problem. The limiters, which are local and do not require extended stencils, are incorporated into the solution reconstruction in order to meet the requirement of monotonicity and avoid spurious solution overshoots. A limiter

β

will be defined based on the solution jumps at grid interfaces. It will be shown that

β

should be 0 <

β

< 1 for a monotone approximate solution.

Natalia Petrovskaya

### Numerical Analysis of the Sinuous Instability of a Viscous Capillary Jet Flowing Down an Immiscible Nonviscous Fluid

The sinuous instability of a viscous jet flowing down an inviscid fluid is studied. On the basis of the 3D Navier-Sokes equations for the jet the full dispersion equation of the small disturbances is derived. Numerical results are shown, illustrating both the effect of viscosity and ambient density.

Stefan Radev, Kalin Nachev, Fabrice Onofri, Lounes Tadrist, Asterios Liolios

### Order Adaptive Quadrature Rule for Real Time Holography Applications

Order adaptive algorithm for real time holography applications is presented in this paper. The algorithm is based on Master-Worker parallel computation paradigm. Definite integrals required for visualization of fringes are computed using a novel order adaptive quadrature rule with an external detector defining the order of integration in real time mode. The proposed integration technique can be effectively applied in hybrid numerical-experimental techniques for analysis of micro-mechanical components.

Minvydas Ragulskis, Loreta Saunoriene

### FEM Modelling of Resonant Frequencies of In–Plane Parallel Piezoelectric Resonator

In the contribution, we introduce an application of finite element model of the piezoelectric resonator. The model is based on the physical description of piezoelectric materials, using linear piezoelectric state equations. Weak formulation and discretization of the problem lead to a large and sparse linear algebraic system, which results in a generalized eigenvalue problem. Resonant frequencies, the most important parameters of the resonator, are subsequently found by solving this algebraic problem. Depending on the discretization parameters, this problem may become large.

Typically, we are not interested in all eigenvalues (resonant frequencies). For determination of several of them it seems therefore appropriate to consider the Krylov subspace methods (namely the implicitly restarted Arnoldi method implemented in the ARPACK library). For coarser meshes, we compute the complete spectra and we find the frequencies of dominant oscillation modes (the selection is made according to their electromechanical coupling coefficients). Then we focus on the part of the spectra near to the chosen dominant frequency and repeat the computation for refined meshes. From the results, we can also find out intervals between the dominant resonant frequencies (which is other important parameter describing the behavior of the resonator).

The model was tested on the problem of thickness-shear vibration of the in–plane parallel quartz resonator. The results, compared with the measurement, will be given in the contribution.

Petr Rálek

### Numerical Algorithm for Non-linear Systems Identification Based on Wavelet Transform

This paper propose a method for identification of complex engineering systems using wavelet transform. This transform is chosen because it can provide a well localization both in time and in frequency. The method is applied to an electrohydraulic system that drives a shaking table. The identification is made using real signals obtained from experimental tests.

Elena Şerban

### Simulation of Turbulent Thermal Convection Using Finite Volumes

To simulate turbulent Rayleigh–Bénard convection in cylindrical domains an explicit/semi-implicit finite volume method with fourth order approximations in space was developed. Using this method and cylindrical staggered grids of about 11 million nodes clustered in vicinity of the boundary we performed simulations of turbulent Rayleigh–Bénard convection in wide cylindrical containers of the aspect ratios

Γ

= 5 and 10 and the Rayleigh number from 10

5

to 10

8

. In the present paper the method, its numerical stability and mesh generation algorithm are discussed.

Olga Shishkina, Claus Wagner

### Phase-Field Versus Level Set Method for 2D Dendritic Growth

The goal of the paper is to review and compare two of the most popular methods for modeling the dendritic solidification in 2D, that tracks the interface between phases implicitly, e.g. the phase-field method and the level set method. We apply these methods to simulate the dendritic crystallization of a pure melt. Numerical experiments for different anisotropic strengths are presented. The two methods compare favorably and the obtained tip velocities and tip shapes are in good agreement with the microscopic solvability theory.

Vladimir Slavov, Stefka Dimova

### Backmatter

Weitere Informationen