2,005 research outputs found
Fast Ewald summation for free-space Stokes potentials
We present a spectrally accurate method for the rapid evaluation of
free-space Stokes potentials, i.e. sums involving a large number of free space
Green's functions. We consider sums involving stokeslets, stresslets and
rotlets that appear in boundary integral methods and potential methods for
solving Stokes equations. The method combines the framework of the Spectral
Ewald method for periodic problems, with a very recent approach to solving the
free-space harmonic and biharmonic equations using fast Fourier transforms
(FFTs) on a uniform grid. Convolution with a truncated Gaussian function is
used to place point sources on a grid. With precomputation of a scalar grid
quantity that does not depend on these sources, the amount of oversampling of
the grids with Gaussians can be kept at a factor of two, the minimum for
aperiodic convolutions by FFTs. The resulting algorithm has a computational
complexity of O(N log N) for problems with N sources and targets. Comparison is
made with a fast multipole method (FMM) to show that the performance of the new
method is competitive.Comment: 35 pages, 15 figure
Efficient, sparse representation of manifold distance matrices for classical scaling
Geodesic distance matrices can reveal shape properties that are largely
invariant to non-rigid deformations, and thus are often used to analyze and
represent 3-D shapes. However, these matrices grow quadratically with the
number of points. Thus for large point sets it is common to use a low-rank
approximation to the distance matrix, which fits in memory and can be
efficiently analyzed using methods such as multidimensional scaling (MDS). In
this paper we present a novel sparse method for efficiently representing
geodesic distance matrices using biharmonic interpolation. This method exploits
knowledge of the data manifold to learn a sparse interpolation operator that
approximates distances using a subset of points. We show that our method is 2x
faster and uses 20x less memory than current leading methods for solving MDS on
large point sets, with similar quality. This enables analyses of large point
sets that were previously infeasible.Comment: Conference CVPR 201
Continuation-conjugate gradient methods for the least squares solution of nonlinear boundary value problems
We discuss in this paper a new combination of methods for solving nonlinear boundary value problems containing a parameter. Methods of the continuation type are combined with least squares formulations, preconditioned conjugate gradient algorithms and finite element approximations.
We can compute branches of solutions with limit points, bifurcation points, etc.
Several numerical tests illustrate the possibilities of the methods discussed in the present paper; these include the Bratu problem in one and two dimensions, one-dimensional bifurcation and perturbed bifurcation problems, the driven cavity problem for the Navier–Stokes equations
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
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