1,280 research outputs found
A Generic Lazy Evaluation Scheme for Exact Geometric Computations
We present a generic C++ design to perform efficient and exact geometric
computations using lazy evaluations. Exact geometric computations are critical
for the robustness of geometric algorithms. Their efficiency is also critical
for most applications, hence the need for delaying the exact computations at
run time until they are actually needed. Our approach is generic and extensible
in the sense that it is possible to make it a library which users can extend to
their own geometric objects or primitives. It involves techniques such as
generic functor adaptors, dynamic polymorphism, reference counting for the
management of directed acyclic graphs and exception handling for detecting
cases where exact computations are needed. It also relies on multiple precision
arithmetic as well as interval arithmetic. We apply our approach to the whole
geometric kernel of CGAL
A Generic Lazy Evaluation Scheme for Exact Geometric Computations
International audienceWe present a generic C++ design to perform exact geometric computations efficiently using lazy evaluations. Exact geometric computations are critical for the robustness of geometric algorithms. Their efficiency is also important for many applications, hence the need for delaying the costly exact computations at run time until they are actually needed, if at all. Our approach is generic and extensible in the sense that it is possible to make it a library that users can apply to their own geometric objects and primitives. It involves techniques such as generic functor-adaptors, static and dynamic polymorphism, reference counting for the management of directed acyclic graphs, and exception handling for triggering exact computations when needed. It also relies on multi-precision arithmetic as well as interval arithmetic. We apply our approach to the whole geometry kernel of CGAL
Engineering Art Galleries
The Art Gallery Problem is one of the most well-known problems in
Computational Geometry, with a rich history in the study of algorithms,
complexity, and variants. Recently there has been a surge in experimental work
on the problem. In this survey, we describe this work, show the chronology of
developments, and compare current algorithms, including two unpublished
versions, in an exhaustive experiment. Furthermore, we show what core
algorithmic ingredients have led to recent successes
Asynchronous Stabilisation and Assembly Techniques for Additive Multigrid
Multigrid solvers are among the best solvers in the world, but once
applied in the real world there are issues they must overcome. Many multigrid
phases exhibit low concurrency. Mesh and matrix assembly are challenging to
parallelise and introduce algorithmic latency. Dynamically adaptive codes exacerbate
these issues. Multigrid codes require the computation of a cascade of matrices and
dynamic adaptivity means these matrices are recomputed throughout the solve.
Existing methods to compute the matrices are expensive and delay the solve. Non-
trivial material parameters further increase the cost of accurate equation integration.
We propose to assemble all matrix equations as stencils in a delayed element-wise
fashion. Early multigrid iterations use cheap geometric approximations and more
accurate updated stencil integrations are computed in parallel with the multigrid
cycles. New stencil integrations are evaluated lazily and asynchronously fed to the
solver once they become available. They do not delay multigrid iterations. We
deploy stencil integrations as parallel tasks that are picked up by cores that would
otherwise be idle. Coarse grid solves in multiplicative multigrid also exhibit limited
concurrency. Small coarse mesh sizes correspond to small computational workload
and require costly synchronisation steps. This acts as a bottleneck and delays
solver iterations. Additive multigrid avoids this restriction, but becomes unstable
for non-trivial material parameters as additive coarse grid levels tend to overcorrect.
This leads to oscillations. We propose a new additive variant, adAFAC-x, with a
stabilisation parameter that damps coarse grid corrections to remove oscillations.
Per-level we solve an additional equation that produces an auxiliary correction.
The auxiliary correction can be computed additively to the rest of the solve and
uses ideas similar to smoothed aggregation multigrid to anticipate overcorrections.
Pipelining techniques allow adAFAC-x to be written using single-touch semantics
on a dynamically adaptive mesh
05391 Abstracts Collection -- Algebraic and Numerical Algorithms and Computer-assisted Proofs
From 25.09.05 to 30.09.05, the Dagstuhl Seminar 05391 ``Algebraic and Numerical Algorithms and Computer-assisted Proofs\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper.
Links to extended abstracts or full papers are provided, if available
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