## Interner Bericht des Fachbereich Informatik

- 263
- Information Complexity of Multivariate Fredholm Equations in Sobolev Classes (1995)
- In this paper, the complexity of full solution of Fredholm integral equations of the second kind with data from the Sobolev class \(W^r_2\) is studied. The exact order of information complexity is derived. The lower bound is proved using a Gelfand number technique. The upper bound is shown by providing a concrete algorithm of optimal order, based on a specific hyperbolic cross approximation of the kernel function. Numerical experiments are included, comparing the optimal algorithm with the standard Galerkin method.

- 272
- Evaluation of a BRDF using BSP Trees (1995)
- In this paper an analytic hidden surface removal algorithm is presented which uses a combination of 2D and 3D BSP trees without involving point sampling or scan conversion. Errors like aliasing which result from sampling do not occur while using this technique. An application of this algorithm is outlined which computes the energy locally reflected from a surface having an arbitrary BRDF. A simplification for diffuse reflectors is described, which has been implemented to compute analytic form factors from diffuse light sources to differential receivers as they are needed for shading and radiosity algorithms.

- 275
- Variance reduction for Monte Carlo methods by means of deterministic numerical computation (1995)
- A new variance reduction technique for the Monte Carlo solution of integral equations is introduced. It is based on separation of the main part. A neighboring equation with exactly known solution is constructed by the help of a deterministic Galerkin scheme. The variance of the method is analyzed, and an application to the radiosity equation of computer graphics, together with numerical test results is given.

- 267
- Efficient algorithms for computing the \(L_2\) discrepancy (1995)
- The \(L_2\)-discrepancy is a quantitative measure of precision for multivariate quadrature rules. It can be computed explicitly. Previously known algorithms needed \(O(m^2\)) operations, where \(m\) is the number of nodes. In this paper we present algorithms which require \(O(m(log m)^d)\) operations.

- 269
- Visualization of Unstable Surface Regions (1995)
- Computer processing of free form surfaces forms the basis of a closed construction process starting with surface design and up to NC-production. Numerical simulation and visualization allow quality analysis before manufacture. A new aspect in surface analysis is described, the stability of surfaces versus infinitesimal bendings. The stability concept is derived from the kinetic meaning of a special vector field which is given by the deformation. Algorithms to calculate this vector field together with an appropriate visualization method give a tool able to analyze surface stability.

- 268
- Numerical Aspects of Stability Investigations on Surfaces (1995)
- The CAD/CAM-based design of free-form surfaces is the beginning of a chain of operations, which ends with the numerically controlled (NC-) production of the designed object. During this process the shape control is an important step to amount efficiency. Several surface interrogation methods already exist to analyze curvature and continuity behaviour of the shape. This paper deals with a new aspect of shape control: the stability of surfaces with respect to infnitesimal bendings. Each inEnitesimal bending of a surface determines a so called instability surface, which is used for the stability investigations. The kinematic meaning of this instability surface will be discussed and we present algorithms to calculate it.

- 271
- Complexity of Multivariate Integral Equations (1995)
- The problem of full solution of Fredholm integral equations of the second kind with data from Sobolev spaces with dominating mixed derivative is studied. Existing estimates for the univariate case are extended to arbitrary dimension.

- 276
- An Optimal Algorithm for the Local Solution of Integral Equations (1995)
- The local solution problem of multivariate Fredholm integral equations is studied. Recent research proved that for several function classes the complexity of this problem is closely related to the Gelfand numbers of some characterizing operators. The generalization of this approach to the situation of arbitrary Banach spaces is the subject of the present paper. Furthermore, an iterative algorithm is described which - under some additional conditions - realizes the optimal error rate. The way these general theorems work is demonstrated by applying them to integral equations in a Sobolev space of periodic functions with dominating mixed derivative of various order.

- 266
- The geometry of optimal degree reduction of Bezier curves (1995)
- Optimal degree reductions, i.e. best approximations of \(n\)-th degree Bezier curves by Bezier curves of degree \(n\) - 1, with respect to different norms are studied. It is shown that for any \(L_p\)-norm the euclidean degree reduction where the norm is applied to the euclidean distance function of two curves is identical to componentwise degree reduction. The Bezier points of the degree reductions are found to lie on parallel lines through the Bezier points of any Taylor expansion of degree \(n\) - 1 of the original curve. This geometric situation is shown to hold also in the case of constrained degree reduction. The Bezier points of the degree reduction are explicitly given in the unconstrained case for \(p\) = 1 and \(p\) = 2 and in the constrained case for \(p\) = 2.

- 265
- MVP-L Language Report Version 2 (1995)
- Intellectual control over software development projects requires the existence of an integrated set of explicit models of the products to be developed, the processes used to develop them, the resources needed, and the productivity and quality aspects involved. In recent years the development of languages, methods and tools for modeling software processes, analyzing and enacting them has become a major emphasis of software engineering research. The majority of current process research concentrates on prescriptive modeling of small, completely formalizable processes and their execution entirely on computers. This research direction has produced process modeling languages suitable for machine rather than human consumption. The MVP project, launched at the University of Maryland and continued at Universität Kaiserslautern, emphasizes building descriptive models of large, real-world processes and their use by humans and computers for the purpose of understanding, analyzing, guiding and improving software development projects. The language MVP-L has been developed with these purposes in mind. In this paper, we motivate the need for MVP-L, introduce the prototype language, and demonstrate its uses. We assume that further improvements to our language will be triggered by lessons learned from applications and experiments.