TY - JOUR A1 - Diethelm, Kai T1 - Error bounds for spline-based quadrature methods for strongly singular integrals JF - Journal of Computational and Applied Mathematics N2 - For the numerical evaluation of finite-part integrals with singularities of order p ⩾ 1, we give error bounds for quadrature methods based on spline approximation. These bounds behave in the same way as the optimal ones. The ideas of the proof are also useful for methods based on other approximation processes. Y1 - 1998 U6 - https://doi.org/10.1016/S0377-0427(97)00245-8 N1 - Referred to by: Erratum to “Error bounds for spline-based quadrature methods for strongly singular integrals” VL - 89 IS - 2 SP - 257 EP - 261 ER - TY - JOUR A1 - Diethelm, Kai T1 - Erratum to “Error bounds for spline-based quadrature methods for strongly singular integrals” JF - Journal of Computational and Applied Mathematics N2 - The purpose of this note is to point out a mistake in the author's paper Y1 - 2002 U6 - https://doi.org/10.1016/S0377-0427(02)00356-4 N1 - Refers to: Error bounds for spline-based quadrature methods for strongly singular integrals VL - 142 IS - 2 SP - 449 EP - 450 ER - TY - CHAP A1 - Diethelm, Kai ED - Chui, C. K. ED - Schumaker, L. L. T1 - Fractional error constants for quadrature formulas T2 - Approximation Theory IX, Vol. 1: Theoretical Aspects Y1 - 1998 SP - 113 EP - 118 PB - Vanderbilt University Press CY - Nashville ER - TY - JOUR A1 - Diethelm, Kai T1 - Existence and construction of definite estimation functionals JF - Studia Scientiarum Mathematicarum Hungarica Y1 - 1999 VL - 35 SP - 217 EP - 236 ER - TY - CHAP A1 - Diethelm, Kai A1 - Freed, Alan D. ED - Keil, Frerich ED - Mackens, Wolfgang ED - Voß, Heinrich ED - Werther, Joachim T1 - On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity. T2 - Scientific Computing in Chemical Engineering II - Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties N2 - The authors have recently developed a mathematical model for the description of the behavior of viscoplastic materials. The model is based on a nonlinear differential equation of order β, where β is a material constant typically in the range 0 < β < 1. This equation is coupled with a first-order differential equation. In the present paper, we introduce and discuss a numerical scheme for the numerical solution of these equations. The algorithm is based on a PECE-type approach. Y1 - 1999 SN - 978-3-642-64295-1 SN - 978-3-642-60185-9 U6 - https://doi.org/10.1007/978-3-642-60185-9_24 SP - 217 EP - 224 PB - Springer CY - Heidelberg ER - TY - CHAP A1 - Diethelm, Kai ED - Lipitakis, E. A. T1 - Predictor-corrector strategies for single- and multi-term fractional differential equations T2 - Proceedings of the 5th Hellenic-European Conference on Computer Mathematics and its Applications Y1 - 2002 SP - 117 EP - 122 PB - LEA Press CY - Athen ER - TY - JOUR A1 - Diethelm, Kai T1 - Efficient solution of multi-term fractional differential equations using P(EC)mE methods JF - Computing N2 - We investigate strategies for the numerical solution of the initial value problem with initial conditions where 0<α1<α2<⋯<αν. Here y (α j ) denotes the derivative of order α j >0 (not necessarily α j ∈ ℕ) in the sense of Caputo. The methods are based on numerical integration techniques applied to an equivalent nonlinear and weakly singular Volterra integral equation. The classical approach leads to an algorithm with very high arithmetic complexity. Therefore we derive an alternative that leads to lower complexity without sacrificing too much precision. Y1 - 2003 U6 - https://doi.org/10.1007/s00607-003-0033-3 VL - 71 SP - 305 EP - 319 ER - TY - JOUR A1 - Diethelm, Kai A1 - Weilbeer, Marc T1 - A numerical approach for Joulin's model of a point source initiated flame JF - Fractional Calculus and Applied Analysis Y1 - 2004 VL - 7 SP - 191 EP - 212 ER - TY - JOUR A1 - Diethelm, Kai A1 - Ford, Neville J. A1 - Freed, Alan D. T1 - Detailed error analysis for a fractional Adams method JF - Numerical Algorithms N2 - We investigate a method for the numerical solution of the nonlinear fractional differential equation D * α y(t)=f(t,y(t)), equipped with initial conditions y (k)(0)=y 0 (k), k=0,1,...,⌈α⌉−1. Here α may be an arbitrary positive real number, and the differential operator is the Caputo derivative. The numerical method can be seen as a generalization of the classical one-step Adams–Bashforth–Moulton scheme for first-order equations. We give a detailed error analysis for this algorithm. This includes, in particular, error bounds under various types of assumptions on the equation. Asymptotic expansions for the error are also mentioned briefly. The latter may be used in connection with Richardson's extrapolation principle to obtain modified versions of the algorithm that exhibit faster convergence behaviour. Y1 - 2004 U6 - https://doi.org/10.1023/B:NUMA.0000027736.85078.be VL - 36 SP - 31 EP - 52 ER - TY - JOUR A1 - Diethelm, Kai T1 - Monotonicity results for a compound quadrature method for finite-part integrals JF - Journal of Inequalities in Pure and Applied Mathematics N2 - Given a function f ∈ C^3[0, 1] and some q ∈ (0, 1), we look at the approximation for the Hadamard finite-part integral = R1/0x^−q−1f(x)dx based on a piecewise linear interpolant for f at n equispaced nodes (i.e., the product trapezoidal rule). The main purpose of this paper is to give sufficient conditions for the sequence of approximations to converge against the correct value of the integral in a monotonic way. An application of the results yields detailed information on the error term of a backward differentiation formula for a fractional differential equation. Y1 - 2004 UR - http://emis.icm.edu.pl/journals/JIPAM/article400.html?sid=400 VL - 5 ER - TY - JOUR A1 - Diethelm, Kai T1 - Non-optimality of certain quadrature rules for Cauchy principal value integrals JF - Zeitschrift für angewandte Mathematik und Mechanik Y1 - 1994 VL - 74 IS - 6 SP - T689 EP - T690 ER - TY - JOUR A1 - Diethelm, Kai T1 - Uniform convergence of optimal order quadrature rules for Cauchy principal value integrals JF - Journal of Computational and Applied Mathematics N2 - For the numerical evaluation of Cauchy principal value integrals of the form , λε(−1, 1), f εCs[− 1, 1], we consider a quadrature method based on spline interpolation of odd degree 2k + 1,k ∈N0. We show that these rules converge uniformly for λ ∈ (− 1, 1). In particular, we calculate the exact order of magnitude of the error and show that it is equal to the order of the optimal remainder in the class of functions with bounded sth derivative if s ε s;;2k + 1, 2k + 2};. Finally, we compare the rule to the well-known quadrature rule of Elliott and Paget which only converges pointwise. Y1 - 1994 U6 - https://doi.org/10.1016/0377-0427(94)90086-8 VL - 56 IS - 3 SP - 321 EP - 329 ER - TY - JOUR A1 - Diethelm, Kai T1 - Gaussian quadrature formulae of the third kind for Cauchy principal value integrals: Basic properties and error estimates JF - Journal of Computational and Applied Mathematics Y1 - 1995 U6 - https://doi.org/10.1016/0377-0427(95)00103-4 VL - 65 IS - 1-3 SP - 97 EP - 114 ER - TY - JOUR A1 - Diethelm, Kai T1 - Asymptotically sharp error bounds for a quadrature rule for Cauchy principal value integrals based on piecewise linear interpolation JF - Approximation Theory and its Applications Y1 - 1995 UR - https://link.springer.com/article/10.1007/BF02836832 U6 - https://doi.org/10.1007/BF02836832 VL - 11 IS - 4 SP - 78 EP - 89 ER - TY - JOUR A1 - Diethelm, Kai T1 - A definiteness criterion for linear functionals and its application to Cauchy principal value quadrature JF - Journal of Computational and Applied Mathematics Y1 - 1996 U6 - https://doi.org/10.1016/0377-0427(95)00175-1 VL - 66 IS - 1-2 SP - 167 EP - 176 ER - TY - CHAP A1 - Diethelm, Kai ED - Alefeld, G. ED - Herzberger, J. T1 - Error bounds for compound quadratures for Hadamard-type finite-part integrals T2 - Numerical Methods and Error Bounds Y1 - 1996 SP - 58 EP - 63 PB - Akademie- Verlag CY - Berlin ER - TY - JOUR A1 - Diethelm, Kai T1 - Definite quadrature formulae for Cauchy principal value integrals JF - Bolyai Society Mathematical Studies Y1 - 1996 VL - 5 SP - 175 EP - 186 ER - TY - CHAP A1 - Diethelm, Kai T1 - Tools for assessing and optimizing the energy requirements of high performance scientific computing software T2 - Proceedings in Applied Mathematics and Mechanics N2 - Score-P is a measurement infrastructure originally designed for the analysis and optimization of the performance of HPC codes. Recent extensions of Score-P and its associated tools now also allow the investigation of energy-related properties and support the user in the implementation of corresponding improvements. Since it would be counterproductive to completely ignore performance issues in this connection, the focus should not be laid exclusively on energy. We therefore aim to optimize software with respect to an objective function that takes into account energy and run time. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) Y1 - 2016 U6 - https://doi.org/10.1002/pamm.201610407 VL - 16 IS - 1 SP - 837 EP - 838 ER - TY - JOUR A1 - Schuchart, Joseph A1 - Gerndt, Michael A1 - Kjeldsberg, Per Gunnar A1 - Diethelm, Kai A1 - Lysaght, Michael A1 - Horák, David A1 - Říha, Lubomír A1 - Gocht, Andreas A1 - Sourouri, Mohammed A1 - Kumaraswamy, Madhura A1 - Chowdhury, Anamika A1 - Jahre, Magnus A1 - Bouizi, Othman A1 - Sabir Mian, Umbreen A1 - Kružík, Jakub A1 - Sojka, Radim A1 - Beseda, Martin A1 - Kannan, Venkatesh A1 - Bendifallah, Zakaria A1 - Hackenberg, Daniel A1 - Nagel, Wolfgang E. T1 - The READEX formalism for automatic tuning for energy efficiency JF - Computing N2 - Energy efficiency is an important aspect of future exascale systems, mainly due to rising energy cost. Although High performance computing (HPC) applications are compute centric, they still exhibit varying computational characteristics in different regions of the program, such as compute-, memory-, and I/O-bound code regions. Some of today’s clusters already offer mechanisms to adjust the system to the resource requirements of an application, e.g., by controlling the CPU frequency. However, manually tuning for improved energy efficiency is a tedious and painstaking task that is often neglected by application developers. The European Union’s Horizon 2020 project READEX (Runtime Exploitation of Application Dynamism for Energy-efficient eXascale computing) aims at developing a tools-aided approach for improved energy efficiency of current and future HPC applications. To reach this goal, the READEX project combines technologies from two ends of the compute spectrum, embedded systems and HPC, constituting a split design-time/runtime methodology. From the HPC domain, the Periscope Tuning Framework (PTF) is extended to perform dynamic auto-tuning of fine-grained application regions using the systems scenario methodology, which was originally developed for improving the energy efficiency in embedded systems. This paper introduces the concepts of the READEX project, its envisioned implementation, and preliminary results that demonstrate the feasibility of this approach. Y1 - 2017 U6 - https://doi.org/10.1007/s00607-016-0532-7 VL - 99 SP - 727 EP - 745 ER - TY - JOUR A1 - Diethelm, Kai A1 - Siegmund, Stefan A1 - Tuan, Hoang The T1 - Asymptotic behavior of solutions of linear multi-order fractional differential systems JF - Fractional Calculus and Applied Analysis N2 - In this paper, we investigate some aspects of the qualitative theory for multi-order fractional differential equation systems. First, we obtain a fundamental result on the existence and uniqueness for multi-order fractional differential equation systems. Next, a representation of solutions of homogeneous linear multi-order fractional differential equation systems in series form is provided. Finally, we give characteristics regarding the asymptotic behavior of solutions to some classes of linear multi-order fractional differential equation systems. Y1 - 2017 U6 - https://doi.org/10.1515/fca-2017-0062 VL - 20 SP - 1165 EP - 1195 ER -