We present an efficient method to shorten the analytic integration-by-parts (IBP) reduction coefficients of multi-loop Feynman integrals. For our approach, we develop an improved version of Leinartas’ multivariate partial fraction algorithm, and provide a modern implementation based on the computer algebra system Singular. Furthermore, we observe that for an integral basis with uniform transcendental (UT) weights, the denominators of IBP reduction coefficients with respect to the UT basis are either symbol letters or polynomials purely in the spacetime dimension D. With a UT basis, the partial fraction algorithm is more efficient both with respect to its performance and the size reduction. We show that in complicated examples with existence of a UT basis, the IBP reduction coefficients size can be reduced by a factor of as large as ∼ 100. We observe that our algorithm also works well for settings without a UT basis.

In this paper, we show that with the state-of-art module intersection IBP reduction method and our improved Leinartas' algorithm, IBP relations for very complicated Feynman integrals can be solved and the analytic reduction coefficients can be dramatically simplified. We develop a large scale parallel implementation of our improved Leinartas' algorithm, based on the Singular/GPI-Space framework. We demonstrate our method by the reduction of two-loop five-point Feynman integrals with degree-five numerators, with a simple and sparse IBP system. The analytic reduction result is then greatly simplified by our improved Leinartas' algorithm to a usable form, with a compression ratio of two order of magnitudes. We further discover that the compression ratio increases with the complexity of the Feynman integrals.

No abstract

Die Nachfrage nach präziseren Werkzeugmaschinen steigt kontinuierlich. Aus diesem Grund müssen thermo-elastische Fehlerkompensationsmethoden unter thermischen Echtzeitbedingungen genauer werden. Die korrekte Parametrierung eines modellbasierten Korrekturansatzes für die Kompensation thermo-elastischer Maschinenfehler hat einen maßgeblichen Einfluss auf die Modellgenauigkeit. Insbesondere für die Langzeitkompensation der Maschinenfehler benötigt das Modell eine hohe Anpassungsfähigkeit aufgrund variierender Lasten. In diesem Beitrag wird eine Simulationsmodellarchitektur vorgestellt, die ein FEM-basiertes thermo-elastisches White-Box-Modell der Werkzeugmaschine mit Temperaturmessungen der Maschinenstruktur kombiniert. Die Messdaten der Temperatursensoren werden für eine prozessparallele Anpassung thermischer Modellparameter genutzt, um die Genauigkeit der Kompensationsmethode zu steigern.