Canonical Formulation of Gravitational Teleparallelism in 2+1 Dimensions in Schwinger's Time Gauge

We consider the most general class of teleparallel gravitational {}{}theories quadratic in the torsion tensor, in three space-time dimensions, and carry out a detailed investigation of its Hamiltonian formulation in Schwinger's time gauge. This general class is given by a family of three-parameter theories. A consistent implementation of the Legendre transform reduces the original theory to a one-parameter family of theories. By calculating Poisson brackets we show explicitly that the constraints of the theory constitute a first-class set. Therefore the resulting theory is well defined with regard to time evolution. The structure of the Hamiltonian theory rules out the existence of the Newtonian limit.Comment: 17 pages, Latex file, no figures; a numerical coefficient has been corrected and a different result is achieve

Kolmogorov widths and low-rank approximations of parametric elliptic PDEs

Kolmogorov $n$-widths and low-rank approximations are studied for families of elliptic diffusion PDEs parametrized by the diffusion coefficients. The decay of the $n$-widths can be controlled by that of the error achieved by best $n$-term approximations using polynomials in the parametric variable. However, we prove that in certain relevant instances where the diffusion coefficients are piecewise constant over a partition of the physical domain, the $n$-widths exhibit significantly faster decay. This, in turn, yields a theoretical justification of the fast convergence of reduced basis or POD methods when treating such parametric PDEs. Our results are confirmed by numerical experiments, which also reveal the influence of the partition geometry on the decay of the $n$-widths.Comment: 27 pages, 6 figure

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

A pseudospectral matrix method for time-dependent tensor fields on a spherical shell

We construct a pseudospectral method for the solution of time-dependent, non-linear partial differential equations on a three-dimensional spherical shell. The problem we address is the treatment of tensor fields on the sphere. As a test case we consider the evolution of a single black hole in numerical general relativity. A natural strategy would be the expansion in tensor spherical harmonics in spherical coordinates. Instead, we consider the simpler and potentially more efficient possibility of a double Fourier expansion on the sphere for tensors in Cartesian coordinates. As usual for the double Fourier method, we employ a filter to address time-step limitations and certain stability issues. We find that a tensor filter based on spin-weighted spherical harmonics is successful, while two simplified, non-spin-weighted filters do not lead to stable evolutions. The derivatives and the filter are implemented by matrix multiplication for efficiency. A key technical point is the construction of a matrix multiplication method for the spin-weighted spherical harmonic filter. As example for the efficient parallelization of the double Fourier, spin-weighted filter method we discuss an implementation on a GPU, which achieves a speed-up of up to a factor of 20 compared to a single core CPU implementation.Comment: 33 pages, 9 figure

4-Dimensional General Relativity from the instrinsic spatial geometry of SO(3) Yang--Mills theory

In this paper we derive 4-dimensional General Relativity from three dimensions, using the intrinsic spatial geometry inherent in Yang--Mills theory which has been exposed by previous authors as well as as some properties of the Ashtekar variables. We provide various interesting relations, including the fact that General Relativity can be written as a Yang--Mills theory where the antiself-dual Weyl curvature replaces the Yang--Mills coupling constant. We have generalized the results of some previous authors, covering Einsteins spaces, to include more general spacetime geometries.Comment: 16 pages. Background material for revised journal articl