3,478 research outputs found
Theory and implementation of -matrix based iterative and direct solvers for Helmholtz and elastodynamic oscillatory kernels
In this work, we study the accuracy and efficiency of hierarchical matrix
(-matrix) based fast methods for solving dense linear systems
arising from the discretization of the 3D elastodynamic Green's tensors. It is
well known in the literature that standard -matrix based methods,
although very efficient tools for asymptotically smooth kernels, are not
optimal for oscillatory kernels. -matrix and directional
approaches have been proposed to overcome this problem. However the
implementation of such methods is much more involved than the standard
-matrix representation. The central questions we address are
twofold. (i) What is the frequency-range in which the -matrix
format is an efficient representation for 3D elastodynamic problems? (ii) What
can be expected of such an approach to model problems in mechanical
engineering? We show that even though the method is not optimal (in the sense
that more involved representations can lead to faster algorithms) an efficient
solver can be easily developed. The capabilities of the method are illustrated
on numerical examples using the Boundary Element Method
Deep AutoRegressive Networks
We introduce a deep, generative autoencoder capable of learning hierarchies
of distributed representations from data. Successive deep stochastic hidden
layers are equipped with autoregressive connections, which enable the model to
be sampled from quickly and exactly via ancestral sampling. We derive an
efficient approximate parameter estimation method based on the minimum
description length (MDL) principle, which can be seen as maximising a
variational lower bound on the log-likelihood, with a feedforward neural
network implementing approximate inference. We demonstrate state-of-the-art
generative performance on a number of classic data sets: several UCI data sets,
MNIST and Atari 2600 games.Comment: Appears in Proceedings of the 31st International Conference on
Machine Learning (ICML), Beijing, China, 201
Adaptive FEM with coarse initial mesh guarantees optimal convergence rates for compactly perturbed elliptic problems
We prove that for compactly perturbed elliptic problems, where the
corresponding bilinear form satisfies a Garding inequality, adaptive
mesh-refinement is capable of overcoming the preasymptotic behavior and
eventually leads to convergence with optimal algebraic rates. As an important
consequence of our analysis, one does not have to deal with the a-priori
assumption that the underlying meshes are sufficiently fine. Hence, the overall
conclusion of our results is that adaptivity has stabilizing effects and can
overcome possibly pessimistic restrictions on the meshes. In particular, our
analysis covers adaptive mesh-refinement for the finite element discretization
of the Helmholtz equation from where our interest originated
A Posteriori Error Estimation for the p-curl Problem
We derive a posteriori error estimates for a semi-discrete finite element
approximation of a nonlinear eddy current problem arising from applied
superconductivity, known as the -curl problem. In particular, we show the
reliability for non-conforming N\'{e}d\'{e}lec elements based on a residual
type argument and a Helmholtz-Weyl decomposition of
. As a consequence, we are also able to derive an a
posteriori error estimate for a quantity of interest called the AC loss. The
nonlinearity for this form of Maxwell's equation is an analogue of the one
found in the -Laplacian. It is handled without linearizing around the
approximate solution. The non-conformity is dealt by adapting error
decomposition techniques of Carstensen, Hu and Orlando. Geometric
non-conformities also appear because the continuous problem is defined over a
bounded domain while the discrete problem is formulated over a weaker
polyhedral domain. The semi-discrete formulation studied in this paper is often
encountered in commercial codes and is shown to be well-posed. The paper
concludes with numerical results confirming the reliability of the a posteriori
error estimate.Comment: 32 page
SPHS: Smoothed Particle Hydrodynamics with a higher order dissipation switch
We present a novel implementation of Smoothed Particle Hydrodynamics (SPHS)
that uses the spatial derivative of the velocity divergence as a higher order
dissipation switch. Our switch -- which is second order accurate -- detects
flow convergence before it occurs. If particle trajectories are going to cross,
we switch on the usual SPH artificial viscosity, as well as conservative
dissipation in all advected fluid quantities (for example, the entropy). The
viscosity and dissipation terms (that are numerical errors) are designed to
ensure that all fluid quantities remain single-valued as particles approach one
another, to respect conservation laws, and to vanish on a given physical scale
as the resolution is increased. SPHS alleviates a number of known problems with
`classic' SPH, successfully resolving mixing, and recovering numerical
convergence with increasing resolution. An additional key advantage is that --
treating the particle mass similarly to the entropy -- we are able to use
multimass particles, giving significantly improved control over the refinement
strategy. We present a wide range of code tests including the Sod shock tube,
Sedov-Taylor blast wave, Kelvin-Helmholtz Instability, the `blob test', and
some convergence tests. Our method performs well on all tests, giving good
agreement with analytic expectations.Comment: 21 pages; 15 Figures. Submitted to MNRAS. Comments welcom
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