ACM Transactions on Mathematical Software最新文献

We present a parallelized geometric multigrid (GMG) method, based on the cell-based Vanka smoother, for higher order space-time finite element methods (STFEM) to the incompressible Navier–Stokes equations. The STFEM is implemented as a time marching scheme. The GMG solver is applied as a preconditioner for generalized minimal residual iterations. Its performance properties are demonstrated for 2D and 3D benchmarks of flow around a cylinder. The key ingredients of the GMG approach are the construction of the local Vanka smoother over all degrees of freedom in time of the respective subinterval and its efficient application. For this, data structures that store pre-computed cell inverses of the Jacobian for all hierarchical levels and require only a reasonable amount of memory overhead are generated. The GMG method is built for the deal.II finite element library. The concepts are flexible and can be transferred to similar software platforms.

The minimum distance of a linear code is a key concept in information theory. Therefore, the time required by its computation is very important to many problems in this area. In this article, we introduce a family of implementations of the Brouwer–Zimmermann algorithm for distributed-memory architectures for computing the minimum distance of a random linear code over 𝔽₂. Both current commercial and public-domain software only work on either unicore architectures or shared-memory architectures, which are limited in the number of cores/processors employed in the computation. Our implementations focus on distributed-memory architectures, thus being able to employ hundreds or even thousands of cores in the computation of the minimum distance. Our experimental results show that our implementations are much faster, even up to several orders of magnitude, than current implementations widely used nowadays.

For control nets outlining a large class of topological polyhedra, not just tensor-product grids, bi-cubic polyhedral splines form a piecewise polynomial, first-order differentiable space that associates one function with each vertex. Akin to tensor-product splines, the resulting smooth surface approximates the polyhedron. Admissible polyhedral control nets consist of quadrilateral faces in a grid-like layout, star-configuration where n ≠ 4 quadrilateral faces join around an interior vertex, n-gon configurations, where 2n quadrilaterals surround an n-gon, polar configurations where a cone of n triangles meeting at a vertex is surrounded by a ribbon of n quadrilaterals, and three types of T-junctions where two quad-strips merge into one.

The bi-cubic pieces of a polyhedral spline have matching derivatives along their break lines, possibly after a known change of variables. The pieces are represented in Bernstein-Bézier form with coefficients depending linearly on the polyhedral control net, so that evaluation, differentiation, integration, moments, and so on, are no more costly than for standard tensor-product splines. Bi-cubic polyhedral splines can be used both to model geometry and for computing functions on the geometry. Although polyhedral splines do not offer nested refinement by refinement of the control net, polyhedral splines support engineering analysis of curved smooth objects. Coarse nets typically suffice since the splines efficiently model curved features. Algorithm 1032 is a C++ library with input-output example pairs and an IGES output choice.

Floating-point numbers represent only a subset of real numbers. As such, floating-point arithmetic introduces approximations that can compound and have a significant impact on numerical simulations. We introduce Encapsulated error, a new way to estimate the numerical error of an application and provide a reference implementation, the Shaman library. Our method uses dedicated arithmetic over a type that encapsulates both the result the user would have had with the original computation and an approximation of its numerical error. We thus can measure the number of significant digits of any result or intermediate result in a simulation. We show that this approach, while simple, gives results competitive with state of the art methods. It has a smaller overhead, and it is compatible with parallelism, making it suitable for the study of large scale applications.