Adaptive Integration and Singular Boundary Transformations

We apply and compare results of transformations used to annihilate boundary singularities for multivariate integration over hyper-rectangular and simplicial domains. While classically these transformations are applied with a product trapezoidal rule, we use adaptive methods in the PARINT software package, based on rules of higher polynomial degree for the integration over subdomains. PARINT is layered over the MPI environment (Message Passing Interface) and deploys advanced parallel computation techniques such as load balancing among processes that are distributed over a network of nodes. The message passing is performed in a non-blocking and asynchronous manner, and permits overlapping of computation and communication. Comparisons of computation times using long double vs. double precision confirm that the extended format does not considerably increase the time for long doubles. We further apply the proposed methods to problems arising from self-energy Feynman loop diagrams with massless internal lines, in particular where the corresponding integrand has singularities on the boundaries of the integration domain.

[1]  Elise de Doncker,et al.  α-Load balancing for parallel adaptive task partitioning , 2004 .

[2]  Terje O. Espelid,et al.  An adaptive algorithm for the approximate calculation of multiple integrals , 1991, TOMS.

[3]  Elise de Doncker,et al.  Algorithm 45. Automatic computation of improper integrals over a bounded or unbounded planar region , 2005, Computing.

[4]  Paul Van Dooren,et al.  An adaptive algorithm for numerical integration over the n-cube , 1976 .

[5]  J. Fujimoto,et al.  High precision numerical computations A case for an HAPPY design , 2005 .

[6]  George Szekeres,et al.  Numerical evaluation of high-dimensional integrals , 1964 .

[7]  Elise de Doncker,et al.  Quadpack computation of Feynman loop integrals , 2012, J. Comput. Sci..

[8]  D. Shanks.  Non‐linear Transformations of Divergent and Slowly Convergent Sequences , 1955 .

[9]  Elise de Doncker,et al.  Performance and Irregular Behavior of Adaptive Task Partitioning , 2001, International Conference on Computational Science.

[10]  F. Yuasa,et al.  Acceleration of Feynman loop integrals in high-energy physics on many core GPUs , 2013 .

[11]  Fukuko Yuasa,et al.  Application of GRAPE9-MPX for High Precision Calculation in Particle Physics and Performance Results , 2015, ICCS.

[12]  K. Chetyrkin,et al.  Four Loop Massless Propagators: an Algebraic Evaluation of All Master Integrals , 2010, 1004.1153.

[13]  Elise de Doncker,et al.  New euler-maclaurin expansions and their application to quadrature over the s-dimensional simplex , 1979 .

[14]  R. Kass,et al.  Subregion-Adaptive Integration of Functions Having a Dominant Peak , 1997 .

[15]  H. M. Möller,et al.  Invariant Integration Formulas for the n-Simplex by Combinatorial Methods , 1978 .

[16]  A. Genz,et al.  An Imbedded Family of Fully Symmetric Numerical Integration Rules , 1983 .

[17]  Alan Genz,et al.  An Adaptive Numerical Integration Algorithm for Simplices , 1989, Great Lakes Computer Science Conference.

[18]  F. Yuasa,et al.  Toward Automatic Regularization for Feynman Loop Integrals in Perturbative Quantum Field Theory , 2012 .

[19]  P. Wynn,et al.  On a device for computing the _{}(_{}) tranformation , 1956 .

[20]  Terje O. Espelid,et al.  Algorithm 698: DCUHRE: an adaptive multidemensional integration routine for a vector of integrals , 1991, TOMS.

[21]  Avram Sidi,et al.  Extension and Completion of Wynn's Theory on Convergence of Columns of the Epsilon Table , 1996 .

[22]  Robert Piessens,et al.  Quadpack: A Subroutine Package for Automatic Integration , 2011 .

[23]  Automatic integration over a sphere , 1981 .

[24]  Atsushi Kawai,et al.  A development of an accelerator board dedicated for multi-precision arithmetic operations and its application to Feynman loop integrals II , 2014, Journal of Physics: Conference Series.

[25]  Elise de Doncker,et al.  Numerical computation of two-loop box diagrams with masses , 2011, Comput. Phys. Commun..

[26]  P. Wynn,et al.  On a Device for Computing the e m (S n ) Transformation , 1956 .

[27]  Masatake Mori,et al.  Double Exponential Formulas for Numerical Integration , 1973 .

[28]  Elise de Doncker,et al.  Two Methods for Load Balanced Distributed Adaptive Integration , 1996, HPCN Europe.

[29]  Elise de Doncker,et al.  Load Balancing in Distributed Adaptive Task Partitioning , 2003 .

[30]  Avram Sidi,et al.  Acceleration of convergence of general linear sequences by the Shanks transformation , 2011, Numerische Mathematik.

[31]  Alan Genz,et al.  An adaptive algorithm for numerical integration over an n-dimensional rectangular region , 1980 .

[32]  Elise de Doncker,et al.  Adaptive Integration for 3-loop Feynman Diagrams with Massless Propagators , 2015, ICCS.

[33]  Philip Rabinowitz,et al.  Methods of Numerical Integration , 1985 .

[34]  B. Burrows,et al.  Lower bounds for quartic anharmonic and double‐well potentials , 1993 .

[35]  Elise de Doncker,et al.  Algorithm 32 automatic computation of integrals with singular integrand, over a finite or an infinite range , 2005, Computing.

[36]  Elise de Doncker,et al.  Computation of loop integrals using extrapolation , 2004 .