Search

This paper, together with [Ma2], constitutes a sequel to [LoMa] where some new moduli spaces of pointed curves were introduced and studied. We start with a review of the main results of [LoMa] and then give a summary of this paper.

In this paper, we define the holomorphic Laplacian for a Frobenius manifold. We give the description of the Laplacian in terms of the prepotential. This will be used to characterize the flat coordinates for the universal unfolding of the function with a simple elliptic singularity. This characterization enables us to solve the so-called “Jacobi’s inversion problem,” i.e. the description of the flat coordinates as the automorphic functions on the period domain w.r.t. the period mapping for the primitive forms. More explicitly the Laplacian relates the flat coordinates with the theta functions on the period domain (see 10.3). About the description of flat coordinates as the automorphic functions, we shall write in a forthcoming paper. The author would like to thank Prof. Claus Hertling and Prof. Atsushi Takahashi for valuable discussions. The auther also thanks the referee for the valuable advices.

Differential linear repetitive processes are a class of 2D linear systems which can be used, for example, to model industrial processes such as long-wall coal cutting. Also they can be used to study the properties of classes of iterative learning control schemes and the convergence properties of iterative algorithms for solving nonlinear dynamic optimal control problems based on the maximum principle. The key unique feature of interest in this paper is the fact that information propagation in one of the two separate directions evolves continuously over a fixed finite interval and in the other it is, in effect, discrete. This paper describes the development of discrete approximations for these processes, resulting in 2D linear systems state space models of the well known Fornasini Marchesini form on which to base further analysis. In this context, the remainder of this paper develops formulas for computing the trajectories generated by these 2D representations which, by analogy with the standard (1D) case, can be expected to play a key role in characterising basic systems theoretic properties such as controllability and observability. Some on-going work and areas for further development in these and related areas will also be briefly discussed.

In view of Paul Fuhrmann’s many important contributions to realization theory, it seems quite appropriate to devote this lecture to the stochastic partial realization problem, when today we are honoring him on his 60th birthday. Some ten years ago Christopher I. Byrnes and I launched a joint research program on this topic, and by now we have some results which I think might interest this audience [3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Some of these results we have been obtain in collaboration with S. V. Gusev in particular, but also A. S. Matveev and H. J. Landau. This short write-up is not a paper in itself but is merely intended to interest the audience in reading the papers [8, 7, 10, 11, 12] and also [32]. The stochastic partial realization problem has important applications in speech synthesis [17], spectral estimation [22, 33], stochastic systems theory [23], systems identification [32], and several other areas of systems and control.

The efficiency of a theorem prover is more directly influenced by the total number of inferences performed before a proof is found than by the size of the final proof. In general, in the field of automated deduction for full first-order logic, there has been a great deal of attention devoted to the completeness of strategies but little to their efficiency, in the sense of the total work expended in the search for a proof. The main efficiency considerations to date have to do with the times needed by particular implementations to find proofs of particular example theorems, or with the efficiencies of decision procedures for specialized theories. Of course, there has also been work on the efficiencies of low-level operations employed by theorem provers (such as unification). It is informative (and fun) to evaluate a prover by running it on a series of examples, but this could well be supplemented by analytical results. To this end, a theoretical study would be useful. It would be nice to know something about the behaviors of proposed new strategies without having to read and understand papers about them or having to run them on examples. Theoretical measures of search space size would permit this. Such measures would also make it easier to weed out bad strategies early and would stimulate the development of good ones. There is more at issue than just a quantitative measure of performance — analytical measures reveal something about how a strategy works, and how it does subgoaling. This gives some insight into the strategy. A theoretical approach could also help to pinpoint problem areas and weaknesses in a method and lead to improvements. In general, theory does not replace experiment but it does supplement it, and provides insights that might otherwise be missed. Theory tends to make general statements and to be machine-independent, whereas experiment tends to deal in specifics and to be machine-dependent. This paper is an attempt to initiate (or further) a theory of the search efficiency of automated theorem proving.

In this paper we extend Kalman’s concept of partial realization and define generalized partial realizations of finite matrix sequences by descriptor type systems. The aim is to prove a counterpart of Kalman’s main theorem of realization theory for generalized partial realizations. The paper ends with some results concerning topological aspects of generalized partial realizations.

This paper presents a brief survey on the investigation of the coupled BEM and FEM applied to a nonlinear exterior boundary value problem. The aim is to find the solution of a nonlinear partial differential equation considered in an annular bounded domain and the Laplace equation outside. These equations are bound together by transmission conditions and are equipped with boundary conditions. The nonlinear problem in the interior domain is combined with an integral equation obtained with the aid of potential theory and then the whole problem is reformulated in a weak sense. The discretization is carried out by the coupled finite element — boundary element method. We discuss the existence and uniqueness of the solution of the discrete as well continuous problem, the convergence of approximate solutions to an exact one, provided the size of the grid tends to zero, and the iterative solution of the nonlinear discrete problem. Proofs are the subject of more detailed papers (see, e.g., [5]).

In this paper, we deal with a cascadic conjugate-gradient method (shortly called CCG-algorithm). This algorithm is a simpler kind of multigrid (multilevel) methods. We define it recurrently for discrete symmetric positive-definite problems on a sequence of grids. At the coarsest grid, the linear system is solved directly. At the finer grid, the system is solved iteratively by the conjugate-gradient method. A starting guess is an interpolation of the approximate solution from the previous grid. We do not implement any preconditioning or restriction onto a coarser grid. Nevertheless, CCG-algorithm has the same optimal property as the multigrid method. Namely, this algorithm converges with a rate which is independent of an amount of unknowns and a number of grids. In [6], this property was proved for a two-dimensional elliptic second-order Dirichlet problem in a convex polygon Ω, where a solution u belongs to space H 2(Ω). Here we study the problem in a non-convex polygon Ω, where u ∉ H 2(Ω) due to a strong growth of second derivatives near some angular points. We prove the rate of convergence in two cases: for uniform grids and for grids with special refinement near some angular points. The paper [1] contains impressive numerical examples both for usual and deficient regularity.

The most important impulse to the creation of the theory, is its ability to be a framework for the mathematical theory of compressible, heat conductive fluids. The theory provides proof of the existence of global (in time) solutions, what in spite of big efforts, the classical theory, based on linear Stoke’s stress-strain relation, does not make possible. The theory is compatible with principles of thermodynamics and with the principle of material frame indifference. The physical theory of multipolar fluids appeared in the paper by Necas, Silhavý [1] and follows the general ideas of Green, Rivlin [2], [3]. The mathematical theory is developed in a serie of papers by Necas, Necas, Silhavý [4], [5], [6], Necas, Novotný [7], Necas [8], Malek, Necas, Růžicka [9], [10], Bellout, Bloom, Necas [11], [12], where also limits to monopolar, in general non-newtonian fluids are studied.

In this paper we present a nontrivial set of modules which measure performance parameters of node processors and interconnection networks. With the help of these parameters we explain the mu time of the following algorithms conjugate gradient method, one-dimensional partial differential equation solver and two-dimensional partial differential equation solver on the parallel machine Ncube-2. The iPSC/860 Hypercube and the vector machine VP100 are analyzed in an other paper (see [3]). Our explanations are sometimes within 0.5% and almost always within 5% of the measured run times.

The use of adaptive mesh spatial discretisation methods, coupled spatial and temporal error control and domain decomposition methods make it possible to construct efficient automatic methods for the numerical solution of time-dependent Navier Stokes problems. This paper describes the unstructured triangular mesh spatial discretisation method being used in a prototype package for compressible flows. The scheme is a cell-centred, second-order finite volume scheme that uses a ten triangle stencil. Previous work has concentrated on algorithms and error estimates for convection dominated problems. In this paper the algorithm is extended to include a new treatment of the diffusion terms. The prototype software uses an adaptive time error control and space remeshing strategy is used to attempt to control the numerical error in the solution.

In the paper of R. Sujatha and the author [N-S], the Brauer group of a real Enriques surface was studied. Here we continue the study of Brauer group with the remark that most of the results of these paper generally valid for an arbitrary smooth projective real algebraic surface.

We present in this paper two results concerning the convergence of the teo-grid algebraic algorithm for arbitrary symmetric systems of linear equations which are also positive definite.Ue obtain these results using a special construction of the interpolation operator based on Gaussian elimination on a sub-matrix of the original system matrix.At the end of the paper we make also some remarks concerning the symmetric indefinite systems.

Volterra (or Wiener) series models of non-linear systems have been studied by many authors, e.g. Rugh[1], Tomlinson[2]. By means of the Multi-dimensional Frequency Response Function (MFRF) based on the Volterra (or Wiener) kernel the non-linear dynamical systems can be analysed in the frequency domain. However, there are some crucial problems concerning the estimation and application of MFRFs[3]. The first is the expense of estimating MFRFs by direct methods, which impels us to study the parameter estimation methods. Billings[4] has developed the estimation method of MFRF by means of the NARMAX model. The polynomial approach for estimating MFRF is discussed in detail in the paper. From the point of view of the estimation of MFRF, the NARMAX model looks more convenient because the output relates directly to the input. However, from the point of view of the discussion on dynamics behaviour, there are more advantages with the parameter identification of the polynomial non-linear model. This is because it is used together with the equation of motion of the dynamic system. Natke and Zamirowski[5] discussed the method of structure identification for the class of polynomials within mechanical systems, which laid a foundation for the parameter estimation method of MFRF. The second crucial problem is that the difficulties arise from using MFRF to describe the non-linear properties of the system because the multi-frequency is without a physical meaning. In the second part of this paper an attempt is made to decrease these difficulties. While the standardized formula of the parameter estimation is given with the response and the excitation, emphasis is given to the study of the statistical equivalent 3rd order polynomial system, the spectral structure of the response expressed with MFRF is analysed in detail, and the Extended Transfer Functions (ETF) are defined, which are only functions of one-dimension frequency. In addition, the effectiveness of the statistical equivalence is analysed. The advantages of the method used here are that the MFRF of the polynomial non-linear system has the theoretical analytical expression, and by studying a non-linear system particular properties with extended transform functions avoid the difficulty of graphing the MFRF.

Elliptic boundary value problems in domains with edges were considered by many authors. Beside the general theory of Rempel/Schulze [16] and Melrose [14] there are papers of Kondrat’ev [10], Maz’ja/Rosmann [13], Grisvard [7], Dauge [4], Costabel/Dauge [3] and other authors. There the edge problem is attributed to the cone theory, established by Kondrat’ev in the fundamental paper [9] and further developed by Maz’ja/Plamenevskij [11].

The covering domain method is a tool to treat problems of static elasticity in multiply connected domains with piece-wise smooth arbitrary boundaries by the method of Fredholm integral equations. First the physical problem is decomposed mathematically into a system of coupled problems each holding in a different simply connected domain, which covers the physical solution domain. Having upgraded these simpler problems a system of nonsingular Fredholm integral equations is formed for the complete problem by superposition in the physical plane. For plane problems, which are discussed in this paper, conformal mapping is most efficient doing this preparation. The paper deals with the forming of kernel functions and influence coefficients using rational fractional functions for conformal mapping. Beyond an introductory example in order to explain the method of covering domains, applications are dealt with for a plate with an elliptic hole and a rectangular plate with a lip crack.

This paper discusses some recent progress in the field of stability of fluid and plasma equilibria. The objective is to derive explicit criteria which guarantee the nonlinear stability of specific equilibria. Most of the work described was done by H. Abarbanel, V. Arnold, R. Hazeltine, D. Holm, P. Morrison, M. Pulvirente, T. Ratiu, Y. Tang, Y.H. Wan, A. Weinstein and the author, although others have been involved in related work cited in the paper.

The paper presents an application of the discrete ordinates method to the investigation of a two-dimensional shock structure close to a wall. The region of a shock wave is investigated from kinetic theory view point. The flow of a monatomic gas in a stationary coordinates system moving with the wave is described by the Boltzmann model kinetic equations (BGK and Ellipsoidal type). The distributions of number density, velocity and temperature in the flow field are obtained as the result of the calculations which are carried out. The present paper is work in progress.

In this paper a brief and fragmentary account is made of our contribution to the GAMM workshop on Numerical Methods for the Computation of Inviscid Transonic Flow with Shock Waves, since the main objective is the comparison, during the actual workshop, of the results obtained by various methods. Emphasis however is put in the first paragraph on the basic assumptions underlying the mathematical modelling of transonic flow, using the full-potential equation. In particular the semi-conservative form of the equation, used in the method, is derived. In the second part, the discretization schemes and the solution algorithm are sketched and reference is given to a more detailed paper. A sample of results is presented in the last paragraph, and the sensitivity of the numerical solution to the space discretization is shown due to an insufficient mesh concentration of the proposed mesh in the nose region.

In his paper of 1929 [56], C.L. Siegel, after defining G-functions and giving some examples, announced some results which one could obtain by the techniques he found (and described in the same paper) for studying the diophantine approximation properties of values of what he called E-functions. However no proof had appeared, and the first attempt in the direction of Siegel’s statements was in M.S. Numagomedov’s work, more than fourty years later. The successive work of A.I. Galockin [30], Y. Flicker, E. Bombieri [7] and D.V. & G.V. Chudnovsky [18], finally completed the proof of a G-function theorem that Siegel could have envisioned; roughly speaking, this is a quantitative result on the non-existence of too many algebraic relations among the values at some algebraic point ξ of certain G-functions, when ξ is “arithmetically” small enough (depending on the degree of the relations). Therefore, the diophantine theory of values of G-functions belongs to irrationality rather than transcendence theory. Nevertheless, its typical feature (and strength), discovered by Bombieri, is the possibility of a local-to-global setting.

This note, written in January 19971, grew out of an attempt to understand references [Be], [BeFa] and [LiTi]. In these papers two related but different methods are presented for the construction of a certain Chow class on moduli spaces of stable (parametrized) curves in a projective manifold V, called virtual fundamental class. This class replaces the usual fundamental class of these spaces in the definition of basic enumerative invariants of V involving curves, called Gromov-Witten (GW-) invariants. They are invariant under smooth deformations of V.

The concept of a Frobenius manifold was introduced by B. Dubrovin [9] to capture in an axiomatic form the properties of correlators found by physicists (see [8]) in two-dimensional topological field theories “coupled to gravity at the tree level”. The purpose of these notes is to reiterate and expand the viewpoint, outlined in the paper [7] of T. Coates and the author, which recasts this concept in terms of linear symplectic geometry and exposes the role of the twisted loop group L (2) GL N of hidden svmmetries.

This paper is a sequel to [9]. Its goal is to verify that the virtual top Chern class c 1/r in the Chow group of the moduli space of higher spin curves \(\overline M _{{g,n}}^{{1/r}} \) constructed in [9], satisfies all the axioms of spin virtual class formulated in [5]. Hence, according to [5], it gives rise to a cohomological field theory in the sense of Kontsevich-Manin [7]. As was observed in [9], the only non-trivial axioms that have to be checked for the class c 1/r are two axioms that we call Vanishing axiom and Ramond factorization axiom. The first of them requires c 1/r to vanish on all the components of the moduli space \(\overline M _{{g,n}}^{{1/r}} \) where one of the markings is equal to r - 1. The second demands vanishing of the push-forward of c 1/r restricted to the components of the moduli space corresponding to the so called Ramond sector, under some natural finite maps.

The moduli space A g of principally polarized abelian varieties of dimension g is a locally symmetric variety. Denote the closure in A g of the locus of jacobians by J g . In this paper we make a preliminary investigation of locally symmetric subvarieties X of A g that are contained in J 9 and contain the moduli point of the jacobian of a smooth curve. Under certain hypotheses (X is “simple”, the corresponding family of abelian varieties can be lifted to a family of curves and a rank condition), we prove that such an X has to be a ball quotient. Our main tools are group cohomology and naive geometric considerations.

In this paper we study the moduli space A of principally polarized abelian varieties of dimension g defined over a field of characteristic p. For moduli spaces one can try to obtain a stratification, by defining a discrete invariant for the objects to be classified, and by taking as strata those loci where the invariant considered is constant. Here we use the observation that finite group schemes annihilated by p geometrically “have no moduli”. For every abelian variety X we consider the finite group scheme X [ p ] (the kernel of multiplication by p). This can be encoded conveniently in the notion of an “elementary sequence”. Raynaud proved that the largest stratum, the ordinary locus, is quasi-affine. It is the generalization of that method which makes everything work. In particular we show that every stratum in the EO-stratification is quasi-affine. A careful study of the way strata attach to each other gives a connectedness result. This generalizes a result by Faltings and by Chai which says that A is irreducible. This is joint work with T. Ekedahl.

Following the work of Stolcis [5], the present paper describes a computational method able to predict the viscous transonic flow development around single and multi-element aerofoils. The Reynolds-averaged Navier-Stokes equations applicable to compressible, two-dimensional turbulent flow are solved using a cell-centred, finite volume spatial discretisation. A multi-stage, explicit, time-marching scheme is used to advance the unsteady flow equations in time to a steady-state solution. Turbulence closure is achieved by using a differential Reynolds stress model (DRSM), which solves modelled transport equations for the Reynolds stress components themselves. Also, the method makes use of unstructured grids in order to be able to deal routinely with complex geometries such as multi-element aerofoil configurations. Results are presented for the RAE 2822 and MBB A3 transonic single-element aerofoils, comparing predictions using a differential Reynolds stress model and a two-equation k-∈ turbulence model with experiment.

In this paper, a new algebraic closure model formulation is presented, which is independent of wall distance. Based on comparisons with experiments of zero pressure gradient flow, all the constants involved in the model are determined. To extend the model into flows with pressure gradient, two extra correlations are derived from the experimental data processed which cover a wide range of flows. The performance of this model will be illustrated through comparison with experiments and DNS data for the three cases of the ETMA project.

The problem of technical diagnostics of the real object is reduced to the problem of identification additional external action on some subsystem of initial system which is modeling the behaviour of real object.This additional external action is related to the mathematical description of control parameter of real object. Tikhonov’s regularization method is used for solution of ill-posed (instable) problem of action identification. Here the method of choise of the optimal model was used for improvement of accuracy of regularized solution. The paper deals with two practical problems: the operative evaluation of rotor unbalance characteristics and the Krilov’s problem.

In this paper a discretisation for two-dimensional steady state transport processes in porous media is presented. To guarantee stability and good accuracy of the discrete solution even in the convection dominated case two mean concepts are used: The convective part of the equation is discretised by using the method of characteristics. Mixed finite element / finite volume techniques are applied to the resulting modified equation. The discretisation can be applied also to the hyperbolic limit problem of transport without dispersion. Some examples are presented. They demonstrate the stability of the discretisation and the sensitivity with respect to the choice of the steplength parameter for the discrete characteristics. For convection dominated problems the linear systems achieved by this discretisation are singular perturbations of subdiagonal systems, if a ‘flow oriented’ numbering of the finite element nodes is used. Gauss-Seidel and SOR-algorithm with overrelaxation can be used as efficient solvers.

Pairs of linear transformations on a finite dimensional vector space are of great relevance in the analysis of two-dimensional (2D) systems evolutions. In this paper, special properties of matrix pairs, such as finite memory, separability, property L and property P, as well as their dynamical interpretations, are investigated. Practical criteria for testing property L and property P in a finite number of steps are also presented.

Inviscid flow models remain, in the author conception, the most important for naval architecture despite the recently increased application of viscous flow tools. In this paper a physical, mathematical and numerical model for free surface potential flow around ship model hulls using finite element method is presented. Non-linear effects on the free surface are taken into account by an iterative procedure. From all application for inviscid flows, the work is focusing on the wave resistance problem only.

In this paper we examine the problem of optimal approximation in the 2-norm, of a finite square Hankel matrix by a Hankel matrix of rank one, and provide a necessary and sufficient condition for its solvability.

This paper presents fast iterative solvers for coupled non-linear Finite Element and Boundary Element problems using a damped inexact Newton method a la Axellson and Kaporin. This method converges globally even if the second Gateaux-dervitative does not exist. The used solvers for the linear saddle point problems occuring in the modified Newton algorithm are optimal in the sense, that they are independent of the number of unknowns. These linear solvers are based either on preconditioned conjugate residual like methods, where no Schur Complement construction is required, or on an inner-outer iteration of Axelsson and Vassilevski. Both methods use multigrid of seperate positive semi-definite and negative definite parts of the coupled operator.

In the present paper we give an overview on multiscale algorithms for the solution of boundary integral equations which are based on the use of wavelets. These methods have been introduced first by Beylkin, Coifman, and Rokhlin [5]. They have been developed and thoroughly investigated in the work of Alpert [1], Dahmen, Proessdorf, Schneider [16-19], Harten, Yad-Shalom [25], v.Petersdorff, Schwab [33-35], and Rathsfeld [39-40]. We describe the wavelet algorithm and the theoretical results on its stability, convergence, and complexity. Moreover, we discuss the application of the method to the solution of a two-dimensional scattering problem of acoustic or electromagnetic waves and to the solution of a fixed geodetic boundary value problem for the gravity field of the earth. The computational tests confirm the high compression rates and the saving of computation time predicted by the theory.

In this paper we present cubature methods for the approximation of surface integrals arising from Galerkin discretizations of 3-d boundary integral equations. This numerical integrator is fully implicit in the sense that the form of the kernel function, the surface parametrization, the trial and test space, and the order of the singularity of the kernel function is not used explicitly. Different kernels can be treated by just replacing the subroutine which evalutes the kernel function in certain surface points.

In the following paper we consider nonlinear dynamical systems depending on a parameter λ ∈ ℝ. They are described by autonomous systems of ODEs $$\frac{{dx}}{{dt}}{\text{ }} = {\text{ }}f(x,\lambda )\,,\,f\,:\,{R^n}x\,R{\text{ }} \to {\text{ }}{R^n}{\text{ }}$$ (1.1) where f ∈ C r , r ≥ 1. The widespread periodically forced nonautonomous sytems $$\frac{{dx}}{{dt}}{\text{ }} = {\text{ }}f(t,x,\lambda )\,,\,f\,:(t + T,x,\lambda ){\text{ }} = {\text{ }}f(t,x,\lambda ){\text{ }}$$ (1.2) with known period T can be rewritten as autonomous systems in the phase space S 1 x ℝ n and dealt with like equation (1.1) in principle.

Fleeces are materials which have gained in the last time increasing applicability. The large variety of fibres used for the fleece production offers a wide spectrum of different absorbtion velocities and capacities of liquids. These materials can be used for the production of baby napkins, too. One of the first papers about mathematical modelling of baby napkins is written by J.Weickert [1]. The aim of his work was to provide optimizing the distribution of the granules of a superabsorbent in the ultra-napkin. Mathematical modelling of the liquid transport in fleece should be helpful to design of washable napkins consisting of different fleece layers.

A method is presented for generating three-dimensional surface data given two-dimensional section data. The application on which this paper concentrates is that of producing wing surfaces through a set of airfoil sections. It is an extension of a new method for the the efficient parametrization of complex three-dimensional shapes, called the PDE Method. The method views surface generation as a boundary-value problem, and produces surfaces as the solutions to elliptic partial differential equations.

In this paper the development and application of a finite volume method for the prediction of compressible and incompressible, laminar and turbulent, steady and unsteady flows in complex geometries is presented. The method uses quadrilateral (2D) or hexaedral (3D) control volumes, block-structured grids (which may not fit at block interfaces) and a colocated (cell-centered) arrangement of variables on the grid. The conservation equations for mass, momentum, energy, turbulent kinetic energy and its dissipation rate are solved iteratively in a sequential manner. Discretization methods up to fourth order were tested, but second order centered approximations, together with local grid refinement, were found to be the best compromise between accuracy, efficiency and ease of implementation. The efficiency is increased by using multigrid methods and parallel computing. Results of several example calculations are presented to demonstrate the efficiency and accuracy of the method.

The paper deals with a comparative study of LES and RANS (k-e model) results for a typical bluff-body flow, namely the flow around a surface mounted cubical obstacle placed in a plane channel. For this test case detailed experimental data (Re=40,000) are available [11]. Two slightly different numerical solution procedures based on a 3-D finite-volume method are used in this investigation. The Reynolds-averaged equations for incompressible flow are solved implicitly [10], whereas in the LES code [1, 2, 3, 4, 5] an explicit second order Adams-Bashforth scheme is applied. Different formulations of the k-e turbulence model are used in the RANS simulations, the standard version with wall functions, a RNG version, a modified version proposed by Kato and Launder [7], and a two-layer approach. For modeling the non-resolvable subgrid-scale motion in the LES two different models are applied, namely the well known Smagorinsky model [16] as well as the dynamic model originally proposed by Germano et al. [6]. The capability of the different methods is demonstrated by comparison with the measurements.

The Domain Decomposition Method (DD) is a powerful tool for deriving boundary element equations approximating boundary value problems with piecewise homogeneous material properties and for constructing the corresponding parallel solvers. Although the method allows the coupling of different discretization techniques, i.e., Boundary Element Methods (BEM) and Finite Element Methods (FEM), as it is desired in various applications, we discuss pure BEM formulations in this paper. We introduce the Adaptive Domain Decomposition Preprocessor ADDPre which realizes an automatic decomposition of the domain under consideration into p subdomains, where p is the number of processors to be used. Furthermore, the parallel algorithm and the preconditioners being involved are discussed. Numerical examples, including potential and linear elasticity problems, which demonstrate the high efficiency of the algorithm are presented.

This paper addresses the problem of reconstructing a free-form surface from measurement data. While the usual methods subdivide the point cloud and fit individual surfaces to these parts we fit a single integral tensor product B-spline surface to the entire point cloud. Holes in the point set, varying point densities, and free boundaries are handled. An effective algorithm is presented, which calculates a smooth approximation surface to a prescribed error tolerance with the help of energy terms.

Because of the relatively bad properties of the boundary element matrices (they are generally neither self-adjoint nor sparse) the computational cost of the Boundary Integral Equation Method is often unnecessarily high. Moreover, in case of mixed boundary conditions, the corresponding boundary integral equation is not of the second kind, so that the traditional well-known iterative methods can hardly be applied. In this paper we present a special iterative method which converts the original mixed boundary value problem to a sequence of pure Dirichlet and pure Neumann subproblems converging rapidly to the solution of the original problem. In the solution of these subproblems, standard multi-grid tools can be used, so that a significant reduction of the computational cost can be achieved. We also derive a multipole-based technique to evaluate the appearing boundary integrals in an economic way, which can further reduce the overall computational cost.

The boundary element method requires discretization only on the boundary. However, O(N 3 ) work and O(N 2 ) memory is usually required, where N is the number of boundary elements. To avoid these problems, Rokhlin [6] proposed the multipole method for the potential problem. This paper extends the method to the two dimensional elastostatic problem. Unlike the biharmonic equation formulation by Greenbaum et al. [3], a direct formulation in terms of displacement and traction variables is presented. The method requires O(N log N) work and memory. Numerical examples demonstrate the efficiency of the proposed method compared to the standard techniques.

In this paper we discuss several sparse matrix based preconditioners suitable for preconditioning boundary element equations. All preconditioners involve only O(n) nonzeros. We provide a framework for constructing operator splitting based preconditioners and use it to analyze a class of sparse preconditioners. For singular integral equations, a more efficient preconditioner is proposed that has a band-2 structure.

An algorithm for approximation of arbitrary clouds of points with integral tensor product B-spline surfaces is presented. The clouds may be scattered may have holes and may have arbitrary boundaries. The usual methods in Reverse Engineering subdivide the given cloud into rectangular parts and approximate these parts individually. In the presented paper an overall algorithm for tensor product B-spline approximation with free boundary curves is introduced.

All convex interpolants to convex bivariate Hermite scattered data are bounded above and below by two piecewise linear functions u and l respectively. This paper discusses numerical algorithms for constructing u and l and how, in certain cases, they form the basis for constructing a Cl convex interpolant using Powell-Sabin elements.

Fuzzy rule-based approximate reasoning is attracting more and more interest from researchers and practitioners nowadays. Zadeh (1973, 1975, 1979a) provided the basic machinery for fuzzy set-based approximate reasoning more than fifteen years ago. In his approach, each granule of knowledge is represented by a fuzzy set or a fuzzy relation on the appropriate universe. Then the fuzzy set representations of the different granules are combined and the result of this combination is projected onto the universe(s) of interest. A well-known particular case of this method is the pattern of inference, named ‘generalized modus ponens’, which enables us to deduce a fuzzy conclusion from a fuzzy rule and a fuzzy fact pertaining to the universe of discourse associated with the condition part of the rule. However, this framework is rather general, and a proper application of it requires a correct understanding of the intended meaning of the pieces of knowledge to be represented by fuzzy sets. In particular, fuzzy rules may have very different semantics, which lead to different choices concerning the multiple-valued connective to be used to model the rule. In this paper we distinguish between purely gradual rules and rules with uncertain conclusion parts. Purely gradual rules are of the form ‘the more X is A, the more Y is B’, which qualitatively describes a relation between the values of X and Y, but which is not pervaded with uncertainty. Gradual rules of the form ‘the closer X is to..., the closer Y is to...’ express pieces of knowledge which are of interest for interpolative reasoning purposes.