Sector Decomposition Approach Research Articles

Loop integral evaluation and asymptotic expansion with pySecDec

The evaluation of higher-loop Feynman integrals is at the core of the quest to reduce the uncertainty of theoretical predictions and match experimental data from the LHC and future colliders. pySecDec is a program to evaluate such integrals numerically based on the sector decomposition approach; its new release version 1.5 introduces features significantly improving its performance: automatic adaptive evaluation of weighted sums of integrals (e.g. amplitudes) and asymptotic expansion in kinematic ratios. Here we briefly review both, illustrating the expected performance benefits.

FIESTA5: Numerical high-performance Feynman integral evaluation

In this paper we present a new release of the FIESTA program (Feynman Integral Evaluation by a Sector decomposiTion Approach). FIESTA5 is performance-oriented — we implemented improvements of various kinds in order to make Feynman integral evaluation faster. We plugged in two new integrators, the Quasi Monte Carlo and Tensor Train. At the same time the old code of FIESTA4 was upgraded to the C++17 standard and mostly rewritten without self-made structures such as hash tables. There are also several essential improvements which are most relevant for complex integrations — the new release is capable of producing results where previously impossible. Program summaryProgram title: FIESTA5CPC Library link to program files:https://doi.org/10.17632/kyzw4zkwsd.2Developer's repository link:https://bitbucket.org/feynmanIntegrals/fiestaLicensing provisions: GPLv3Programming language:Wolfram Mathematica 8.0 or higher, C++Supplementary material: The article, usage instructions in the program package, https://bitbucket.org/feynmanIntegrals/fiestaJournal Reference of previous version: A.V. Smirnov, FIESTA4: Optimized Feynman integral calculations with GPU support, Comput. Phys. Commun. 204 (2016) 189–199, https://doi.org/10.1016/j.cpc.2016.03.013Does the new version supersede the previous version?: YesReasons for the new version: The code was not able to produce results in some cases where the competitive code, pySecDec could; moreover we were not happy with the performance of the algorithm having multiple inefficient parts of the code.Summary of revisions: New integrators, improved sector decomposition strategies, improvement of contour transformation algorithm, code optimization especially in physical regions (might be 100 times faster or more), possibility to work in some situations where previously impossible, update of the code to modern standard.Nature of problem: Sector decomposition is a well-known approach to the numerical evaluation of Feynman integrals. Feynman integrals in 4 space-time dimension are divergent and have to be regulated. Sector decomposition is used to resolve pole singularities and consists of different stages — sector decomposition itself, contour transformation (in case of physical kinematics meaning base functions changing sign therefore leading to integration in complex numbers), pole resolution, epsilon expansion and numerical integration.Solution method: Most stages are performed in Wolfram Mathematica [1] (required version is 8.0 or higher), this part is parallelized by the use of Mathematica subkelnels in shared memory. As a result a database on hard disk is produced with the use of the KyotoCabinet [2] database engine. The integration stage is written in C++ and can be run on personal computers as well as on supercomputers via MPI. It can make use of installed graphical processor units. As default integrator we use Vegas from the Cuba library [3], but also it can be switched to QMC [4] or Tensor Train [5]. The mimalloc memory allocator [6] can be used for improved performance.Additional comments including restrictions and unusual features: The complexity of the problem is mostly restricted by CPU time required to perform the integration and to obtain the desired precision.

Full-color three-loop three-point form factors in \U0001d4a9 = 4 SYM

We present the detailed computation of full-color three-loop three-point form factors of both the stress-tensor supermultiplet and a length-three BPS operator in \U0001d4a9 = 4 SYM. The integrands are constructed based on the color-kinematics (CK) duality and generalized unitarity method. An interesting observation is that the CK-dual integrands contain a large number of free parameters. We discuss the origin of these free parameters in detail and check that they cancel in the simplified integrands. We further perform the numerical evaluation of the integrals at a special kinematics point using public packages FIESTA and pySecDec based on the sector-decomposition approach. We find that the numerical computation can be significantly simplified by expressing the integrals in terms of uniformly transcendental basis, although the final three-loop computations still require large computational resources. Having the full-color numerical results, we verify that the non-planar infrared divergences reproduce the non-dipole structures, which firstly appear at three loops. As for the finite remainder functions, we check that the numerical planar remainder for the stress-tensor supermultiplet is consistent with the known result of the bootstrap computation. We also obtain for the first time the numerical results of the three-loop non-planar remainder for the stress-tensor supermultiplet as well as the three-loop remainder for the length-three operator.

A Sector-Based Energy-Efficient Lightweight Clustering Algorithm

Wireless sensor networks consist of a large number of nodes with constrained energy supply. Energy efficiency is hot and challenging issue in wireless sensor networks. Existing studies have shown that clustering is one of the efficient techniques to improve energy efficiency. An easy-to-implement and flexible method for even clustering and cluster head selection is beneficial to optimize network stability and energy efficiency. In this study, a sector-based lightweight and flexible clustering algorithm is developed to reduce node energy dissipation and optimize energy efficiency. Our proposal divides the area into virtual sectors. In the meanwhile, an even sector cluster is created using the sector decomposition approach. Based on the total communication distance, residual energy, and local node density, each node in the cluster calculates its own priority. The node with the highest priority is selected as the cluster head. Our proposal is compared with TSC,MH-TSC,SEECP,DREEP and LEACH. Experimental results show that the proposed algorithm outperforms these algorithms in terms of network stability and network lifetime.

FIESTA4: Optimized Feynman integral calculations with GPU support

This paper presents a new major release of the program FIESTA (Feynman Integral Evaluation by a Sector decomposiTion Approach). The new release is mainly aimed at optimal performance at large scales when one is increasing the number of sampling points in order to reduce the uncertainty estimates. The release now supports graphical processor units (GPUs) for the numerical integration, methods to optimize cluster-usage, as well as other speed, memory, and stability improvements. Program summaryProgram title: FIESTA4Catalogue identifier: AECP_v4_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AECP_v4_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 3No. of lines in distributed program, including test data, etc.: 575847No. of bytes in distributed program, including test data, etc.: 43008976Distribution format: tar.gzProgramming language: Wolfram Mathematica 7.0 or higher, c++.Computer: From a desktop PC to a supercomputer.Operating system: Unix, Linux, Mac OS X.Has the code been vectorized or parallelized?: YesRAM: depends on the complexity of the problemCatalogue identifier of previous version: AECP_v3_0Journal reference of previous version: Comput. Phys. Comm. 185(2014)2090Classification: 4.4, 4.12, 5, 6.5.External routines: Wolfram Mathematica [1], KyotoCabinet [2], Cuba [3], QHull [4]Does the new version supersede the previous version?: YesNature of problem:The sector decomposition approach to evaluating Feynman integrals falls apart into the sector decomposition itself, where one has to minimize the number of sectors; the pole resolution and epsilon expansion; and the numerical integration of the resulting expression. Moreover, in cases where the integrand is complex, one has to perform a contour deformation.Solution method:The program has a number of sector decomposition strategies. Everything except the integration is performed in Wolfram Mathematica [1] (required version is 7.0 or higher). This part of the calculation is parallelized with the use of shared memory. The database is stored on hard disk with the use of the KyotoCabinet [2] database engine.The integration part of the algorithm can be performed on a cluster. It is written in c++ and does not need Wolfram Mathematica. For integration we use the Cuba library package [3].The sampling point evaluation has been vectorized. It can also use graphical processor units for the parallelization of sampling point evaluation.Reasons for new version:The main reason for the new version is the possibility to use GPUs for faster calculations.Summary of revisions:The main improvements are performance-related. The integration in the new version works 2–4 times faster, and another 2–4 times faster if one uses a GPU. Also there are multiple stability improvements.Restrictions:The complexity of the problem is mostly restricted by CPU time required to perform the integration and obtain a proper precision.Additional comments:For additional information see, http://science.sander.su, https://bitbucket.org/fiestaIntegrator/fiesta/overviewRunning time:Depends on the complexity of the problem.

FIESTA 3: Cluster-parallelizable multiloop numerical calculations in physical regions

The goal of this paper is to present a new major release of the program FIESTA (Feynman Integral Evaluation by a Sector decomposiTion Approach). This version presents features like cluster-parallelization, new asymptotic expansion algorithms, calculations in physical regions, new sector-decomposition strategies, as well as multiple speed, memory, and stability improvements. Program summaryProgram title: FIESTA 3Catalogue identifier: AECP_v3_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AECP_v3_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 2No. of lines in distributed program, including test data, etc.: 109631No. of bytes in distributed program, including test data, etc.: 6777017Distribution format: tar.gzProgramming language: Wolfram Mathematica 7.0 or higher, C++.Computer: From a desktop PC to a supercomputer.Operating system: Unix, Linux, Mac OS X.Has the code been vectorized or parallelized?: Yes. Number of processors used: from 1 processor up to loading a supercomputer (tests were performed up to 1024 cores) RAM: Depends on the complexity of the problemCatalogue identifier of previous version: AECP_v2_0Journal reference of previous version: Comput. Phys. Comm. 182 (2011) 790Classification: 4.4, 4.12, 5, 6.5.External routines: Wolfram Mathematica [1], KyotoCabinet [2], Cuba [3], QHull [4]Does the new version supercede the previous version?: Yes. Some obsolete options were removed as being superseded by better approaches.Nature of problem:The sector decomposition approach to evaluating Feynman integrals falls apart into the sector decomposition itself, where one has to minimize the number of sectors; the pole resolution and epsilon expansion; and the numerical integration of the resulting expression. Moreover, in cases where the integrand is complex, one has to perform a contour deformation.Solution method:The program has a number of sector decomposition strategies. One of the most important features is the ability to perform a contour deformation, as well as the so-called pre-resolution in the case of integrals at the threshold. Everything except the integration is performed in Wolfram Mathematica [1] (required version is 7.0 or higher). This part of the calculation is parallelizable with the use of shared memory. The database is stored on hard disk with the use of the KyotoCabinet [2] database engine. The integration part of the algorithm, which can be performed on a cluster, is written in c++ and does not need Wolfram Mathematica. For integration we use the Cuba library package [3].Reasons for new version:Application of FIESTA to physical values (complex integration) as well as the cluster-parallelization.Summary of revisions:FIESTA 3 is an absolutely new version with multiple new features such as •integration in physical regions•new asymptotic expansion methods•MPI-parallelization•multiple speed and memory improvements•total reconstruction: the integrator is completely separated and can be used by other programsRestrictions:The complexity of the problem is mostly restricted by the CPU time required to perform the integration and obtain a proper precision.Running time: Depends on the complexity of the problem.

FIESTA 2: Parallelizeable multiloop numerical calculations

The program FIESTA has been completely rewritten. Now it can be used not only as a tool to evaluate Feynman integrals numerically, but also to expand Feynman integrals automatically in limits of momenta and masses with the use of sector decompositions and Mellin–Barnes representations. Other important improvements to the code are complete parallelization (even to multiple computers), high-precision arithmetics (allowing to calculate integrals which were undoable before), new integrators, Speer sectors as a strategy, the possibility to evaluate more general parametric integrals. Program summary Program title: FIESTA 2 Catalogue identifier: AECP_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AECP_v2_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU GPL version 2 No. of lines in distributed program, including test data, etc.: 39 783 No. of bytes in distributed program, including test data, etc.: 6 154 515 Distribution format: tar.gz Programming language: Wolfram Mathematica 6.0 (or higher) and C Computer: From a desktop PC to a supercomputer Operating system: Unix, Linux, Windows, Mac OS X Has the code been vectorised or parallelized?: Yes, the code has been parallelized for use on multi-kernel computers as well as clusters via Mathlink over the TCP/IP protocol. The program can work successfully with a single processor, however, it is ready to work in a parallel environment and the use of multi-kernel processor and multi-processor computers significantly speeds up the calculation; on clusters the calculation speed can be improved even further. RAM: Depends on the complexity of the problem Classification: 4.4, 4.12, 5, 6.5 Catalogue identifier of previous version: AECP_v1_0 Journal reference of previous version: Comput. Phys. Comm. 180 (2009) 735 External routines: QLink [1], Cuba library [2], MPFR [3] Does the new version supersede the previous version?: Yes Nature of problem: The sector decomposition approach to evaluating Feynman integrals falls apart into the sector decomposition itself, where one has to minimize the number of sectors; the pole resolution and epsilon expansion; and the numerical integration of the resulting expression. Solution method: The sector decomposition is based on a new strategy as well as on classical strategies such as Speer sectors. The sector decomposition, pole resolution and epsilon-expansion are performed in Wolfram Mathematica 6.0 or, preferably, 7.0 (enabling parallelization) [4]. The data is stored on hard disk via a special program, QLink [1]. The expression for integration is passed to the C-part of the code, that parses the string and performs the integration by one of the algorithms in the Cuba library package [2]. This part of the evaluation is perfectly parallelized on multi-kernel computers. Reasons for new version: 1. The first version of FIESTA had problems related to numerical instability, so for some classes of integrals it could not produce a result. 2. The sector decomposition method can be applied not only for integral calculation. Summary of revisions: 1. New integrator library is used. 2. New methods to deal with numerical instability (MPFR library). 3. Parallelization in Mathematica. 4. Parallelization on multiple computers via TCP-IP. 5. New sector decomposition strategy (Speer sectors). 6. Possibility of using FIESTA to for integral expansion. 7. Possibility of using FIESTA to discover poles in d. 8. New negative terms resolution strategies. Restrictions: The complexity of the problem is mostly restricted by CPU time required to perform the evaluation of the integral Running time: Depends on the complexity of the problem

Feynman Integral Evaluation by a Sector decomposiTion Approach (FIESTA)

We present a new program performing the sector decomposition and integrating the expression afterwards. The program takes a set of propagators and a set of indices as input and returns the epsilon-expansion of the corresponding integral. Program summary Program title: FIESTA Catalogue identifier: AECP_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AECP_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GPL v2 No. of lines in distributed program, including test data, etc.: 88 281 No. of bytes in distributed program, including test data, etc.: 6 153 480 Distribution format: tar.gz Programming language: Wolfram Mathematica 6.0 [3] and C Computer: from a desktop PC to supercomputer Operating system: Unix, Linux, Windows RAM: depends on the complexity of the problem Classification: 4.4, 4.12, 5, 6.5 External routines: QLink [1], Vegas [2] Nature of problem: The sector decomposition approach to evaluating Feynman integrals falls apart into the sector decomposition itself, where one has to minimize the number of sectors; the pole resolution and epsilon expansion; and the numerical integration of the resulting expression. Solution method: The sector decomposition is based on a new strategy. The sector decomposition, pole resolution and epsilon-expansion are performed in Wolfram Mathematica 6.0 [3]. The data is stored on hard disk via a special program, QLink [1]. The expression for integration is passed to the C-part of the code, that parses the string and performs the integration by the Vegas algorithm [2]. This part of the evaluation is perfectly parallelized on multi-kernel computers. Restrictions: The complexity of the problem is mostly restricted by the CPU time required to perform the evaluation of the integral, however there is currently a limit of maximum 11 positive indices in the integral; this restriction is to be removed in future versions of the code. Running time: Depends on the complexity of the problem.

A numerical approach to the double real radiation part of jets at NNLO

We report on the sector decomposition approach to the double real radiation part of e^+e^- to 3 jets at NNLO.

The sector decomposition approach to real radiation at NNLO

A method based on sector decomposition is briefly described which is suitable for the calculation of the double real radiation part of e^+e^- to 3 jets at next-to-next-to-leading order in a fully differential way.