14,040 research outputs found
AlSub: Fully Parallel and Modular Subdivision
In recent years, mesh subdivision---the process of forging smooth free-form
surfaces from coarse polygonal meshes---has become an indispensable production
instrument. Although subdivision performance is crucial during simulation,
animation and rendering, state-of-the-art approaches still rely on serial
implementations for complex parts of the subdivision process. Therefore, they
often fail to harness the power of modern parallel devices, like the graphics
processing unit (GPU), for large parts of the algorithm and must resort to
time-consuming serial preprocessing. In this paper, we show that a complete
parallelization of the subdivision process for modern architectures is
possible. Building on sparse matrix linear algebra, we show how to structure
the complete subdivision process into a sequence of algebra operations. By
restructuring and grouping these operations, we adapt the process for different
use cases, such as regular subdivision of dynamic meshes, uniform subdivision
for immutable topology, and feature-adaptive subdivision for efficient
rendering of animated models. As the same machinery is used for all use cases,
identical subdivision results are achieved in all parts of the production
pipeline. As a second contribution, we show how these linear algebra
formulations can effectively be translated into efficient GPU kernels. Applying
our strategies to , Loop and Catmull-Clark subdivision shows
significant speedups of our approach compared to state-of-the-art solutions,
while we completely avoid serial preprocessing.Comment: Changed structure Added content Improved description
Kira - A Feynman Integral Reduction Program
In this article, we present a new implementation of the Laporta algorithm to
reduce scalar multi-loop integrals---appearing in quantum field theoretic
calculations---to a set of master integrals. We extend existing approaches by
using an additional algorithm based on modular arithmetic to remove linearly
dependent equations from the system of equations arising from
integration-by-parts and Lorentz identities. Furthermore, the algebraic
manipulations required in the back substitution are optimized. We describe in
detail the implementation as well as the usage of the program. In addition, we
show benchmarks for concrete examples and compare the performance to Reduze 2
and FIRE 5.
In our benchmarks we find that Kira is highly competitive with these existing
tools.Comment: 37 pages, 3 figure
An Elimination Method for Solving Bivariate Polynomial Systems: Eliminating the Usual Drawbacks
We present an exact and complete algorithm to isolate the real solutions of a
zero-dimensional bivariate polynomial system. The proposed algorithm
constitutes an elimination method which improves upon existing approaches in a
number of points. First, the amount of purely symbolic operations is
significantly reduced, that is, only resultant computation and square-free
factorization is still needed. Second, our algorithm neither assumes generic
position of the input system nor demands for any change of the coordinate
system. The latter is due to a novel inclusion predicate to certify that a
certain region is isolating for a solution. Our implementation exploits
graphics hardware to expedite the resultant computation. Furthermore, we
integrate a number of filtering techniques to improve the overall performance.
Efficiency of the proposed method is proven by a comparison of our
implementation with two state-of-the-art implementations, that is, LPG and
Maple's isolate. For a series of challenging benchmark instances, experiments
show that our implementation outperforms both contestants.Comment: 16 pages with appendix, 1 figure, submitted to ALENEX 201
Universal Constructions for (Co)Relations: categories, monoidal categories, and props
Calculi of string diagrams are increasingly used to present the syntax and
algebraic structure of various families of circuits, including signal flow
graphs, electrical circuits and quantum processes. In many such approaches, the
semantic interpretation for diagrams is given in terms of relations or
corelations (generalised equivalence relations) of some kind. In this paper we
show how semantic categories of both relations and corelations can be
characterised as colimits of simpler categories. This modular perspective is
important as it simplifies the task of giving a complete axiomatisation for
semantic equivalence of string diagrams. Moreover, our general result unifies
various theorems that are independently found in literature and are relevant
for program semantics, quantum computation and control theory.Comment: 22 pages + 3 page appendix, extended version of arXiv:1703.0824
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
Gr\"obner Bases over Algebraic Number Fields
Although Buchberger's algorithm, in theory, allows us to compute Gr\"obner
bases over any field, in practice, however, the computational efficiency
depends on the arithmetic of the ground field. Consider a field , a simple extension of , where is an
algebraic number, and let be the minimal polynomial of
. In this paper we present a new efficient method to compute Gr\"obner
bases in polynomial rings over the algebraic number field . Starting from
the ideas of Noro [Noro, 2006], we proceed by joining to the ideal to be
considered, adding as an extra variable. But instead of avoiding
superfluous S-pair reductions by inverting algebraic numbers, we achieve the
same goal by applying modular methods as in [Arnold, 2003; B\"ohm et al., 2015;
Idrees et al., 2011], that is, by inferring information in characteristic zero
from information in characteristic . For suitable primes , the
minimal polynomial is reducible over . This allows us to
apply modular methods once again, on a second level, with respect to the
factors of . The algorithm thus resembles a divide and conquer strategy and
is in particular easily parallelizable. At current state, the algorithm is
probabilistic in the sense that, as for other modular Gr\"obner basis
computations, an effective final verification test is only known for
homogeneous ideals or for local monomial orderings. The presented timings show
that for most examples, our algorithm, which has been implemented in SINGULAR,
outperforms other known methods by far.Comment: 16 pages, 1 figure, 1 tabl
Modular elliptic curves over real abelian fields and the generalized Fermat equation
Using a combination of several powerful modularity theorems and class field
theory we derive a new modularity theorem for semistable elliptic curves over
certain real abelian fields. We deduce that if is a real abelian field of
conductor , with and , , , then every
semistable elliptic curve over is modular.
Let , , be prime, with , and .To a
putative non-trivial primitive solution of the generalized Fermat
we associate a Frey elliptic curve defined over
, and study its mod representation with the help
of level lowering and our modularity result. We deduce the non-existence of
non-trivial primitive solutions if , or if and , .Comment: Introduction rewritten to emphasise the new modularity theorem. Paper
revised in the light of referees' comment
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