The exponentially convergent trapezoidal rule

It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators

VI Workshop on Computational Data Analysis and Numerical Methods: Book of Abstracts

The VI Workshop on Computational Data Analysis and Numerical Methods (WCDANM) is going to be held on June 27-29, 2019, in the Department of Mathematics of the University of Beira Interior (UBI), Covilhã, Portugal and it is a unique opportunity to disseminate scientific research related to the areas of Mathematics in general, with particular relevance to the areas of Computational Data Analysis and Numerical Methods in theoretical and/or practical field, using new techniques, giving especial emphasis to applications in Medicine, Biology, Biotechnology, Engineering, Industry, Environmental Sciences, Finance, Insurance, Management and Administration. The meeting will provide a forum for discussion and debate of ideas with interest to the scientific community in general. With this meeting new scientific collaborations among colleagues, namely new collaborations in Masters and PhD projects are expected. The event is open to the entire scientific community (with or without communication/poster)

Curvature Aligned Simplex Gradient: Principled Sample Set Construction For Numerical Differentiation

The simplex gradient, a popular numerical differentiation method due to its flexibility, lacks a principled method by which to construct the sample set, specifically the location of function evaluations. Such evaluations, especially from real-world systems, are often noisy and expensive to obtain, making it essential that each evaluation is carefully chosen to reduce cost and increase accuracy. This paper introduces the curvature aligned simplex gradient (CASG), which provably selects the optimal sample set under a mean squared error objective. As CASG requires function-dependent information often not available in practice, we additionally introduce a framework which exploits a history of function evaluations often present in practical applications. Our numerical results, focusing on applications in sensitivity analysis and derivative free optimization, show that our methodology significantly outperforms or matches the performance of the benchmark gradient estimator given by forward differences (FD) which is given exact function-dependent information that is not available in practice. Furthermore, our methodology is comparable to the performance of central differences (CD) that requires twice the number of function evaluations.Comment: 31 Pages, 5 Figures, Submitted to IMA Numerical Analysi

SciCADE 95: International conference on scientific computation and differential equations

This report consists of abstracts from the conference. Topics include algorithms, computer codes, and numerical solutions for differential equations. Linear and nonlinear as well as boundary-value and initial-value problems are covered. Various applications of these problems are also included

Transition control based on grey, neural states

+hlm.;c

Multi-level stochastic collocation methods for parabolic and Schrödinger equations

In this thesis, we propose, analyse and implement numerical methods for time-dependent non-linear parabolic and Schrödinger-type equations with uncertain parameters. The discretisation of the parameter space which incorporates the uncertainty of the problem is performed via single- and multi-level collocation strategies. To deal with the possibly large dimension of the parameter space, sparse grid collocation techniques are used to alleviate the curse of dimensionality to a certain extent. We prove that the multi-level method is capable of reducing the overall computational costs significantly. In the parabolic case, the time discretisation is performed via an implicit-explicit splitting strategy of order two which consists shortly speaking of a combination of an implicit trapezoidal rule for the stiff linear part and Heun\u27s method for the non-linear part. In the Schrödinger case, time is discretised via the famous second-order Strang splitting method. For both problem classes we review known error bounds for both discretizations and prove new error bounds for the time discretisations which take the regularity in the parameter space into account. In the parabolic case, a new error bound for the "implicit-explicit trapezoidal method" (IMEXT) method is proved. To our knowledge, this error bound stating second-order convergence of the IMEXT method closes a current gap in the literature. Utilising the aforementioned new error bounds for both problem classes, we can rigorously prove convergence of the single- and multi-level methods. Additionally, cost savings of the multi-level methods compared to the single-level approach are predicted and verifed by numerical examples. The results mentioned above are novel contributions in two areas of mathematics. The first one is (analysis of) numerical methods for uncertainty quantification and the second one is numerical analysis of time-integration schemes for PDEs

Heavy Quark Mass Effects in Deep Inelastic Scattering and Global QCD Analysis

A new implementation of the general PQCD formalism of Collins, including heavy quark mass effects, is described. Important features that contribute to the accuracy and efficiency of the calculation of both neutral current (NC) and charged current (CC) processess are explicitly discussed. This new implementation is applied to the global analysis of the full HERA I data sets on NC and CC cross sections, with correlated systematic errors, in conjunction with the usual fixed-target and hadron collider data sets. By using a variety of parametrizations to explore the parton parameter space, robust new parton distribution function (PDF) sets (CTEQ6.5) are obtained. The new quark distributions are consistently higher in the region x ~ 10^{-3} than previous ones, with important implications on hadron collider phenomenology, especially at the LHC. The uncertainties of the parton distributions are reassessed and are compared to the previous ones. A new set of CTEQ6.5 eigenvector PDFs that encapsulates these uncertainties is also presented.Comment: 32 pages, 12 figures; updated, Publication Versio