Parallelization of Modular Algorithms

In this paper we investigate the parallelization of two modular algorithms. In fact, we consider the modular computation of Gr\"obner bases (resp. standard bases) and the modular computation of the associated primes of a zero-dimensional ideal and describe their parallel implementation in SINGULAR. Our modular algorithms to solve problems over Q mainly consist of three parts, solving the problem modulo p for several primes p, lifting the result to Q by applying Chinese remainder resp. rational reconstruction, and a part of verification. Arnold proved using the Hilbert function that the verification part in the modular algorithm to compute Gr\"obner bases can be simplified for homogeneous ideals (cf. \cite{A03}). The idea of the proof could easily be adapted to the local case, i.e. for local orderings and not necessarily homogeneous ideals, using the Hilbert-Samuel function (cf. \cite{Pf07}). In this paper we prove the corresponding theorem for non-homogeneous ideals in case of a global ordering.Comment: 16 page

An algorithm for primary decomposition in polynomial rings over the integers

We present an algorithm to compute a primary decomposition of an ideal in a polynomial ring over the integers. For this purpose we use algorithms for primary decomposition in polynomial rings over the rationals resp. over finite fields, and the idea of Shimoyama-Yokoyama resp. Eisenbud-Hunecke-Vasconcelos to extract primary ideals from pseudo-primary ideals. A parallelized version of the algorithm is implemented in SINGULAR. Examples and timings are given at the end of the article.Comment: 8 page

Parallel algorithms for normalization

Given a reduced affine algebra A over a perfect field K, we present parallel algorithms to compute the normalization \bar{A} of A. Our starting point is the algorithm of Greuel, Laplagne, and Seelisch, which is an improvement of de Jong's algorithm. First, we propose to stratify the singular locus Sing(A) in a way which is compatible with normalization, apply a local version of the normalization algorithm at each stratum, and find \bar{A} by putting the local results together. Second, in the case where K = Q is the field of rationals, we propose modular versions of the global and local-to-global algorithms. We have implemented our algorithms in the computer algebra system SINGULAR and compare their performance with that of the algorithm of Greuel, Laplagne, and Seelisch. In the case where K = Q, we also discuss the use of modular computations of Groebner bases, radicals, and primary decompositions. We point out that in most examples, the new algorithms outperform the algorithm of Greuel, Laplagne, and Seelisch by far, even if we do not run them in parallel.Comment: 19 page

Parameter identification for soil simulation based on the discrete element method and application to small scale shallow penetration tests

The Discrete Element Method (DEM) is well-established and widely used in soil-tool interaction related applications. As for all simulation tools, a proper calibration of the model parameters is crucial. In this contribution, we present the parametrization procedure of the DEM software GRAnular Physics Engine (GRAPE), developed and implemented at Fraunhofer ITWM, and attempt to use two parametrized soil samples for the simulation of small scale shallow penetration tests. The results are compared to laboratory measurements

Triaxial compression and direct shear tests in the parametrization of soil modeled via the Discrete Element Method

Investigation in the parametrization of soil modeled with DEM based on measurements in triaxial compression and direct shear tests

Groebner bases of symmetric ideals

In this article we present two new algorithms to compute the Groebner basis of an ideal that is invariant under certain permutations of the ring variables and which are both implemented in SINGULAR (cf. [DGPS12]). The first and major algorithm is most performant over finite fields whereas the second algorithm is a probabilistic modification of the modular computation of Groebner bases based on the articles by Arnold (cf. [A03]), Idrees, Pfister, Steidel (cf. [IPS11]) and Noro, Yokoyama (cf. [NY12], [Y12]). In fact, the first algorithm that mainly uses the given symmetry, improves the necessary modular calculations in positive characteristic in the second algorithm. Particularly, we could, for the first time even though probabilistic, compute the Groebner basis of the famous ideal of cyclic 9-roots (cf. [BF91]) over the rationals with SINGULAR.Comment: 17 page

About the computation of the signature of surface singularities z^N+g(x,y)=0

In this article we describe our experiences with a parallel SINGULAR-implementation of the signature of a surface singularity defined by z^N+g(x,y)=0.Comment: 6 page

Parameter identification for soil simulation based on the discrete element method and application to small scale shallow penetration tests

The Discrete Element Method (DEM) is well-established and widely used in soil-tool interaction related applications. As for all simulation tools, a proper calibration of the model parameters is crucial. In this contribution, we present the parametrization procedure of the DEM software GRAnular Physics Engine (GRAPE), developed and implemented at Fraunhofer ITWM, and attempt to use two parametrized soil samples for the simulation of small scale shallow penetration tests. The results are compared to laboratory measurements