3:00pm - 5:00pmMassively parallel computations in algebraic geometry
Chair(s): Janko Böhm (TU Kaiserlautern, Germany), Anne Frühbis-Krüger (Leibniz Universität Hannover)
Massively parallel methods have been a success story in high performance numerical simulation, but so far have rarely been used in computational algebraic geometry. Recent developments like the combination of the parallelization framework GPI-Space with the computer algebra system Singular have made such approaches accessible to the mathematician without the need to deal with a multitude of technical details. The minisymposium aims at bringing together researchers pioneering this approach, discussing the current state of the art and possible future developments. We plan to address applications in classical algebraic geometry, tropical geometry, geometric invariant theory and links to high energy physics.
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
GPI-Space - Fraunhofer’s integrated solution to solve big problems on ultra scale machines
Franz-Josef Pfreundt, Mirko Rahn, Alexandra Carpen-Amarie
Fraunhofer ITWM
High Performance Computing is an essential prerequisite for numerous modern scientific discoveries, which often rely on carefully tuned software stacks to harness computing power across thousands of resources and to handle massive amounts of data. One of the main challenges in this field is to enable domain scientists to take advantage of this huge computing power, while hiding the complexity of the efficient management of such resources.
This talk will introduce GPI-Space, a workflow-management system for parallel applications, designed to automatically coordinate scalable, parallel executions in large, complex environments. A key advantage of GPI-Space is the separation it provides between the automatic management of parallel executions and the description of the problem-specific computational tasks and inter-dependencies.
The talk will highlight the features that make GPI-Space a suitable run-time environment for algebraic geometry. More specifically, it will be discussed how parallel patterns in algorithms can be exploited by GPI-Space, with a focus on its description language (based on Petri nets), which allows scientists to model domain-specific applications independently of the execution environment. Finally, it will be presented how GPI-Space was used on top of the Singular computer algebra system to speed up the execution of algorithms for topics such
as algebraic geometry, or singularity theory.
Using Petri nets for parallelizing algorithms in algebraic geometry
Lukas Ristau
TU Kaiserslautern / Fraunhofer ITWM
In theory, smoothness of an algebraic variety is checked by the classical Jacobian criterion. In many practical contexts, however, a direct application of this criterion is infeasible, in particular, if the codimension of the variety in its ambient space is large. A new hybrid smoothness test was recently suggested by Böhm and Frühbis-Krüger, which is based on the termination criterion from Hironaka's proof of resolution of singularities. This algorithm creates a sufficiently fine covering with affine charts, such that a relative version of the Jacobian criterion can be applied in each chart. In this talk, a massively parallel version of the algorithm is presented which has been implemented using Singular and GPI-Space.
This is joint work with Janko Böhm, Wolfram Decker, Anne Frühbis-Krüger, Franz-Josef Pfreundt and Mirko Rahn.
Parallel enumeration of triangulations
Lars Kastner
TU Berlin
Triangulations of point configurations are a key tool in many areas of mathematics. In tropical geometry, enumeration of all regular triangulations of certain point configurations gives rise to classification results. We will present the software mptopcom for computing triangulations of point configurations. Its core algorithm reverse search allows it to overcome the memory restraint that prevented previous software to succeed. Furthermore it is able to run in parallel.
If there is a group acting on our point configuration, then it is often sufficient to enumerate orbits of triangulations rather than all triangulations. To this end we need an effective method of comparing triangulations up to symmetry. The method introduced in mptopcom avoids computing full orbits and hence is able to exploit the group action to improve performance.
Module intersection method for multi-loop Feynman integral reduction
Yang Zhang
Max Planck Institute for Physics, Munich
We aim at a bottleneck problem of high energy physics, the linear reduction of multi-loop Feynman integrals. Our idea is to use module intersection computations in algebraic geometry to greatly simplify the Feynman integral linear relations, and then to apply a sparse linear algebra method for the reduction. With our method, the open problem of complete linear reduction of the 2-loop 5-point nonplanar hexagon-box Feynman integral was solved.