On polynomial collocation for Cauchy singular integral equations with fixed singularities

In this paper we consider a polynomial collocation method for the numerical solution of Cauchy singular integral equations with fixed singularities over the interval, where the fixed singularities are supposed to be of Mellin convolution type. For the stability and convergence of this method in weighted L2 spaces, we derive necessary and sufficient conditions

Properties and numerical solution of an integral equation system to minimize airplane drag for a multiwing system

We consider an open multiwing system, composed of N 2 disjoint openplane curves, not necessarily symmetric, and examine the corresponding (con-strained) induced drag minimization problem. To this end, we first derive theassociated Euler-Lagrange system of equations, which is then reduced to anequivalent system of Cauchy singular integral equations. By generalizing a pre-vious approach of ours for the case of a single open wing, we obtain existenceand uniqueness results for the problem solution in a product of weighted Sobolevtype spaces. This system is then solved by applying to it a collocation-quadraturemethod. For this, we prove stability and derive corresponding error estimates.Finally, to test the efficiency of the proposed numerical method, we apply it tosome multiwing system

A fast algorithm for Prandtl's integro-differential equation

AbstractCollocation and quadrature methods for singular integro-differential equations of Prandtl's type are studied in weighted Sobolev spaces. A fast algorithm basing on the quadrature method is proposed. Convergence results and error estimates are given

Developing a Scalable Benchmark for Assessing Large Language Models in Knowledge Graph Engineering

As the field of Large Language Models (LLMs) evolves at an accelerated pace, the critical need to assess and monitor their performance emerges. We introduce a benchmarking framework focused on knowledge graph engineering (KGE) accompanied by three challenges addressing syntax and error correction, facts extraction and dataset generation. We show that while being a useful tool, LLMs are yet unfit to assist in knowledge graph generation with zero-shot prompting. Consequently, our LLM-KG-Bench framework provides automatic evaluation and storage of LLM responses as well as statistical data and visualization tools to support tracking of prompt engineering and model performance.Comment: To be published in SEMANTICS 2023 poster track proceedings. SEMANTICS 2023 EU: 19th International Conference on Semantic Systems, September 20-22, 2023, Leipzig, German

Quadrature Methods for Singular Integral Equations of Mellin Type Based on the Zeros of Classical Jacobi Polynomials

In this paper we formulate necessary conditions for the stability of certain quadrature methods for Mellin type singular integral equations on an interval. These methods are based on the zeros of classical Jacobi polynomials, not only on the Chebyshev nodes. The method is considered as an element of a special C*-algebra such that the stability of this method can be reformulated as an invertibility problem of this element. At the end, the mentioned necessary conditions are invertibility properties of certain linear operators in Hilbert spaces. Moreover, for the proofs we need deep results on the zero distribution of the Jacobi polynomials

On polynomial collocation for Cauchy singular integral equations with fixed singularities

In this paper we consider a polynomial collocation method for the numerical solution of Cauchy singular integral equations with fixed singularities over the interval, where the fixed singularities are supposed to be of Mellin convolution type. For the stability and convergence of this method in weighted L2 spaces, we derive necessary and sufficient conditions

Numerical analysis for one-dimensional Cauchy singular integral equations

AbstractThe paper presents a selection of modern results concerning the numerical analysis of one-dimensional Cauchy singular integral equations, in particular the stability of operator sequences associated with different projection methods. The aim of the paper is to show the main ideas and approaches, such as the concept of transforming the question of the stability of an operator sequence into an invertibility problem in a certain Banach algebra or the concept of certain scales of weighted Besov spaces to prove convergence rates of the sequence of the approximate solutions. Moreover, computational aspects, in particular the construction of fast algorithms, are discussed