Stable normal forms for polynomial system solving

This paper describes and analyzes a method for computing border bases of a zero-dimensional ideal $I$. The criterion used in the computation involves specific commutation polynomials and leads to an algorithm and an implementation extending the one provided in [MT'05]. This general border basis algorithm weakens the monomial ordering requirement for \grob bases computations. It is up to date the most general setting for representing quotient algebras, embedding into a single formalism Gr\"obner bases, Macaulay bases and new representation that do not fit into the previous categories. With this formalism we show how the syzygies of the border basis are generated by commutation relations. We also show that our construction of normal form is stable under small perturbations of the ideal, if the number of solutions remains constant. This new feature for a symbolic algorithm has a huge impact on the practical efficiency as it is illustrated by the experiments on classical benchmark polynomial systems, at the end of the paper

The Semantics of Graph Programs

GP (for Graph Programs) is a rule-based, nondeterministic programming language for solving graph problems at a high level of abstraction, freeing programmers from handling low-level data structures. The core of GP consists of four constructs: single-step application of a set of conditional graph-transformation rules, sequential composition, branching and iteration. We present a formal semantics for GP in the style of structural operational semantics. A special feature of our semantics is the use of finitely failing programs to define GP's powerful branching and iteration commands

High-level programming of stencil computations on multi-GPU systems using the SkelCL library

The implementation of stencil computations on modern, massively parallel systems with GPUs and other accelerators currently relies on manually-tuned coding using low-level approaches like OpenCL and CUDA. This makes development of stencil applications a complex, time-consuming, and error-prone task. We describe how stencil computations can be programmed in our SkelCL approach that combines high-level programming abstractions with competitive performance on multi-GPU systems. SkelCL extends the OpenCL standard by three high-level features: 1) pre-implemented parallel patterns (a.k.a. skeletons); 2) container data types for vectors and matrices; 3) automatic data (re)distribution mechanism. We introduce two new SkelCL skeletons which specifically target stencil computations – MapOverlap and Stencil – and we describe their use for particular application examples, discuss their efficient parallel implementation, and report experimental results on systems with multiple GPUs. Our evaluation of three real-world applications shows that stencil code written with SkelCL is considerably shorter and offers competitive performance to hand-tuned OpenCL code

GPU-accelerated simulation of colloidal suspensions with direct hydrodynamic interactions

Solvent-mediated hydrodynamic interactions between colloidal particles can significantly alter their dynamics. We discuss the implementation of Stokesian dynamics in leading approximation for streaming processors as provided by the compute unified device architecture (CUDA) of recent graphics processors (GPUs). Thereby, the simulation of explicit solvent particles is avoided and hydrodynamic interactions can easily be accounted for in already available, highly accelerated molecular dynamics simulations. Special emphasis is put on efficient memory access and numerical stability. The algorithm is applied to the periodic sedimentation of a cluster of four suspended particles. Finally, we investigate the runtime performance of generic memory access patterns of complexity $O(N^2)$ for various GPU algorithms relying on either hardware cache or shared memory.Comment: to appear in a special issue of Eur. Phys. J. Special Topics on "Computer Simulations on GPUs