360 research outputs found
Shenfun -- automating the spectral Galerkin method
With the shenfun Python module (github.com/spectralDNS/shenfun) an effort is
made towards automating the implementation of the spectral Galerkin method for
simple tensor product domains, consisting of (currently) one non-periodic and
any number of periodic directions. The user interface to shenfun is
intentionally made very similar to FEniCS (fenicsproject.org). Partial
Differential Equations are represented through weak variational forms and
solved using efficient direct solvers where available. MPI decomposition is
achieved through the {mpi4py-fft} module (bitbucket.org/mpi4py/mpi4py-fft), and
all developed solver may, with no additional effort, be run on supercomputers
using thousands of processors. Complete solvers are shown for the linear
Poisson and biharmonic problems, as well as the nonlinear and time-dependent
Ginzburg-Landau equation.Comment: Presented at MekIT'17, the 9th National Conference on Computational
Mechanic
Fast elliptic solvers in cylindrical coordinates and the Coulomb collision operator
In this paper, we describe a new class of fast solvers for separable elliptic
partial differential equations in cylindrical coordinates with
free-space radiation conditions. By combining integral equation methods in the
radial variable with Fourier methods in and , we show that
high-order accuracy can be achieved in both the governing potential and its
derivatives. A weak singularity arises in the Fourier transform with respect to
that is handled with special purpose quadratures. We show how these solvers
can be applied to the evaluation of the Coulomb collision operator in kinetic
models of ionized gases.Comment: 20 pages, 5 figure
A fast numerical solver for local barycentric coordinates
The local barycentric coordinates (LBC), proposed in Zhang et al (2014), demonstrate good locality and can be used for local control on function value interpolation and shape deformation. However, it has no closed- form expression and must be computed by solving an optimization problem, which can be time-consuming especially for high-resolution models. In this paper, we propose a new technique to compute LBC efficiently. The new solver is developed based on two key insights. First, we prove that the non-negativity constraints in the original LBC formulation is not necessary, and can be removed without affecting the solution of the optimization problem. Furthermore, the removal of this constraint allows us to reformulate the computation of LBC as a convex constrained optimization for its gradients, followed by a fast integration to recover the coordinate values. The reformulated gradient optimization problem can be solved using ADMM, where each step is trivially parallelizable and does not involve global linear system solving, making it much more scalable and efficient than the original LBC solver. Numerical experiments verify the effectiveness of our technique on a large variety of models
A bibliography on parallel and vector numerical algorithms
This is a bibliography of numerical methods. It also includes a number of other references on machine architecture, programming language, and other topics of interest to scientific computing. Certain conference proceedings and anthologies which have been published in book form are listed also
Code generation for generally mapped finite elements
Many classical finite elements such as the Argyris and Bell elements have long been absent from high-level PDE software. Building on recent theoretical work, we describe how to implement very general finite-element transformations in FInAT and hence into the Firedrake finite-element system. Numerical results evaluate the new elements, comparing them to existing methods for classical problems. For a second-order model problem, we find that new elements give smooth solutions at a mild increase in cost over standard Lagrange elements. For fourth-order problems, however, the newly enabled methods significantly outperform interior penalty formulations. We also give some advanced use cases, solving the nonlinear Cahn-Hilliard equation and some biharmonic eigenvalue problems (including Chladni plates) using C1 discretizations
Geodesics in Heat
We introduce the heat method for computing the shortest geodesic distance to
a specified subset (e.g., point or curve) of a given domain. The heat method is
robust, efficient, and simple to implement since it is based on solving a pair
of standard linear elliptic problems. The method represents a significant
breakthrough in the practical computation of distance on a wide variety of
geometric domains, since the resulting linear systems can be prefactored once
and subsequently solved in near-linear time. In practice, distance can be
updated via the heat method an order of magnitude faster than with
state-of-the-art methods while maintaining a comparable level of accuracy. We
provide numerical evidence that the method converges to the exact geodesic
distance in the limit of refinement; we also explore smoothed approximations of
distance suitable for applications where more regularity is required
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