Hierarchical parallelisation of functional renormalisation group calculations — hp-fRG

Abstract

The functional renormalisation group (fRG) has evolved into a versatile tool in condensed matter theory for studying important aspects of correlated electron systems. Practical applications of the method often involve a high numerical effort, motivating the question in how far High Performance Computing (HPC) can leverage the approach.In this work we report on a multi-level parallelisation of the underlying computational machinery and show that this can speed up the code by several orders of magnitude. This in turn can extend the applicability of the method to otherwise inaccessible cases.We exploit three levels of parallelisation: Distributed computing by means of Message Passing (MPI), shared-memory computing using OpenMP, and vectorisation by means of SIMD units (single-instruction-multiple-data). Results are provided for two distinct High Performance Computing (HPC) platforms, namely the IBM-based BlueGene/Q system JUQUEEN and an Intel SandyBridge-based development cluster. We discuss how certain issues and obstacles were overcome in the course of adapting the code. Most importantly, we conclude that this vast improvement can actually be accomplished by introducing only moderate changes to the code, such that this strategy may serve as a guideline for other researcher to likewise improve the efficiency of their codes.

Quantitative functional renormalization group description of the two-dimensional Hubbard model

“…Minor deviations between the fRG and PA (on one side) and DQMC (on the other side) are instead observed, as expected, by increasing the interaction values. Further optimization and parallelization of the code [54] will allow us to overcome the present restriction to essentially a single s-wave form factor. This is necessary to explore broader parameter regions.…”

“…Strongly inspired by earlier channel-decomposition schemes [48] and the singular-mode fRG by Wang et al [41], the so-called truncated unity fRG was set up [53]. This formalism combines various technical improvements and allows for the development of a highly parallelizable and fast-performing code, mainly involving 2D (or 3D) integrations and matrix multiplications [54]. The name "truncated unity" comes from the insertion of unity into loop diagrams.…”

Quasi-particle functional Renormalisation Group calculations in the two-dimensional half-filled Hubbard model at finite temperatures

“…1 We consider the coarseness of the momentum discretisation as a parameter, albeit not in a fully comprehensive manner due to compute time limits. Thanks to a highly scalable parallel algorithm [38] we can treat flow equations for up to about 40 million couplings and simultaneously the two-loop contributions for the self-energy at a sufficiently accurate level. 2 Figure 4 summarises various fRG treatments we used, together with results from other methods available in the literature [14,[39][40][41][42][43].…”

“…In fact on the formal level we can write (iω − ξ 0 k − gΣ g ) −1 ∂ g (g Σ g ) = −∂ g ln(iω − ξ 0 k − gΣ g ) = ∂ g ln((iω − ξ 0 k − gΣ g ) −1 ) ≈ ∂ g ln(Z g (iω − ξ 0 k ) −1 ) = ∂ g (ln(Z g ) − ln(iω − ξ 0 k )) = ∂ g ln(Z g ) = (∂ g Z g )Z −1 g q.e.d. , (38) where we have inserted equation (15) in the third step. This provides a consistency check to some extent.…”

The High-Q Club: Experience Extreme-scaling Application Codes

“…Examination of Scalasca execution traces was key to determining optimal load-balancing of MPI and OpenMP computations, and associated workload distribution and loop scheduling strategies, to allow hp-fRG to scale effectively to use all of the JUQUEEN compute nodes with 1.8 million threads [31].…”