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A novel approach to integration by parts reduction
Andreas von Manteuffel, Robert M. Schabinger
TL;DR
Existing Laporta-based integration-by-parts reduction becomes difficult for demanding multi-loop calculations because of expression swell, auxiliary computations, and limited parallel scaling. The paper proposes reconstructing algebraic identities from finite-field reduction samples, expecting lower memory use, stronger parallelization, and faster runtimes.
Problem
Demanding multi-loop calculations require improvements because conventional Laporta-based reduction can involve huge symbolic systems, severe intermediate expression swell, unnecessary auxiliary identities, and limited use of available cores.
Method
The method solves many sampled systems over distinct prime fields and reconstructs the symbolic rational coefficients of the needed identities from those samples.
Results
The authors expect the strategy to avoid intermediate expression swell, target only needed relations, use massive parallelization, and improve computational efficiency over conventional Laporta-based approaches.
Takeaways & Limitations
The approach is designed to make modern computational resources more effective for integration-by-parts reduction through parallel numerical sampling and dedicated symbolic reconstruction.
Takeaways & Limitations
Conventional reduction remains constrained by high-degree intermediate coefficients, auxiliary-integral computations, and limited speed-up beyond a few tens of cores.
Abstract
from arXiv · showhide
Integration by parts reduction is a standard component of most modern multi-loop calculations in quantum field theory. We present a novel strategy constructed to overcome the limitations of currently available reduction programs based on Laporta's algorithm. The key idea is to construct algebraic identities from numerical samples obtained from reductions over finite fields. We expect the method to be highly amenable to parallelization, show a low memory footprint during the reduction step, and allow for significantly better run-times.
Abstract
The paper motivates a finite-field and reconstruction-based approach to IBP reduction that targets expression swell, unnecessary auxiliary reductions, and limited parallel scalability in Laporta-based solvers. It develops rational and rational-function reconstruction techniques intended to reduce memory demands, improve linear-system solving, and enable massive parallelization.
- Motivation: Laporta-based IBP reductions can require weeks on a desktop for demanding multi-loop calculations, motivating faster methods for future problems with more scales or loops.The paper highlights two-loop pp → t¯t virtual corrections as an example of a computationally intensive application.
- Limitations of conventional reduction: Intermediate expression swell makes coefficients expensive to manipulate and store, while increasing coefficient complexity also makes greatest-common-divisor computations costly.Masking a single integral relation produced reduction results more than an order of magnitude larger in observed cases.
- Limitations of conventional reduction: Conventional solvers compute many identities for auxiliary integrals that do not occur in the target calculation, despite those quantities being unnecessary for the specific result.The paper identifies dedicated treatment of the required subset of IBP relations as a route to avoiding such computations.
- Limitations of conventional reduction: Cluster-based parallelization of existing solvers often yields significant speed-up only up to a few tens of cores, leaving larger modern clusters underused.The optimal core count is problem specific, but the paper identifies limited scalability as a major shortcoming.
- Method: The proposed strategy samples calculations over distinct prime fields, combines samples with the Chinese remainder algorithm, and reconstructs symbolic rational coefficients at the end.The approach uses machine-sized prime fields and reconstructs rational numbers and univariate rational functions from finite-field data.
- Method: The extended Euclidean algorithm underlies finite-field inverses, rational reconstruction, and the combination of solutions obtained modulo distinct primes.The paper notes that classical reconstruction procedures can be replaced by optimized variants, including modern rational-function reconstruction strategies.
- Method and expected impact: The resulting linear-system solver avoids intermediate expression swell in the univariate polynomial case and is designed for massively parallel reduction and reconstruction steps.The reduction requires N × M copies of the same system to be solved, while reconstruction is also massively parallelizable.
- Method and expected impact: The authors expect the strategy to outperform conventional Laporta approaches by avoiding symbolic swell, reducing auxiliary manipulations, reconstructing only needed relations, and using computational resources more effectively.The paper presents these as expected efficiency advantages rather than a reported benchmark result.