Solving Minimal Residual Methods in W−1,pW^{-1,p} with large Exponents pp

We introduce a numerical scheme that approximates solutions to linear PDE's by minimizing a residual in the $W^{-1,p}(\Omega)$ norm with exponents $p> 2$. The resulting problem is solved by regularized Kacanov iterations, allowing to compute the solution to the non-linear minimization problem even for large exponents $p\gg 2$. Such large exponents remedy instabilities of finite element methods for problems like convection-dominated diffusion

A Hybrid Differential Evolution Approach to Designing Deep Convolutional Neural Networks for Image Classification

Convolutional Neural Networks (CNNs) have demonstrated their superiority in image classification, and evolutionary computation (EC) methods have recently been surging to automatically design the architectures of CNNs to save the tedious work of manually designing CNNs. In this paper, a new hybrid differential evolution (DE) algorithm with a newly added crossover operator is proposed to evolve the architectures of CNNs of any lengths, which is named DECNN. There are three new ideas in the proposed DECNN method. Firstly, an existing effective encoding scheme is refined to cater for variable-length CNN architectures; Secondly, the new mutation and crossover operators are developed for variable-length DE to optimise the hyperparameters of CNNs; Finally, the new second crossover is introduced to evolve the depth of the CNN architectures. The proposed algorithm is tested on six widely-used benchmark datasets and the results are compared to 12 state-of-the-art methods, which shows the proposed method is vigorously competitive to the state-of-the-art algorithms. Furthermore, the proposed method is also compared with a method using particle swarm optimisation with a similar encoding strategy named IPPSO, and the proposed DECNN outperforms IPPSO in terms of the accuracy.Comment: Accepted by The Australasian Joint Conference on Artificial Intelligence 201

A Feature-Based Comparison of Evolutionary Computing Techniques for Constrained Continuous Optimisation

Evolutionary algorithms have been frequently applied to constrained continuous optimisation problems. We carry out feature based comparisons of different types of evolutionary algorithms such as evolution strategies, differential evolution and particle swarm optimisation for constrained continuous optimisation. In our study, we examine how sets of constraints influence the difficulty of obtaining close to optimal solutions. Using a multi-objective approach, we evolve constrained continuous problems having a set of linear and/or quadratic constraints where the different evolutionary approaches show a significant difference in performance. Afterwards, we discuss the features of the constraints that exhibit a difference in performance of the different evolutionary approaches under consideration.Comment: 16 Pagesm 2 Figure

Interpolation Operator on negative Sobolev Spaces

We introduce a Scott--Zhang type projection operator mapping to Lagrange elements for arbitrary polynomial order. In addition to the usual properties, this operator is compatible with duals of first order Sobolev spaces. More specifically, it is stable in the corresponding negative norms and allows for optimal rates of convergence. We discuss alternative operators with similar properties. As applications of the operator we prove interpolation error estimates for parabolic problems and smoothen rough right-hand sides in a least squares finite element method

Computational statistics using the Bayesian Inference Engine

This paper introduces the Bayesian Inference Engine (BIE), a general parallel, optimised software package for parameter inference and model selection. This package is motivated by the analysis needs of modern astronomical surveys and the need to organise and reuse expensive derived data. The BIE is the first platform for computational statistics designed explicitly to enable Bayesian update and model comparison for astronomical problems. Bayesian update is based on the representation of high-dimensional posterior distributions using metric-ball-tree based kernel density estimation. Among its algorithmic offerings, the BIE emphasises hybrid tempered MCMC schemes that robustly sample multimodal posterior distributions in high-dimensional parameter spaces. Moreover, the BIE is implements a full persistence or serialisation system that stores the full byte-level image of the running inference and previously characterised posterior distributions for later use. Two new algorithms to compute the marginal likelihood from the posterior distribution, developed for and implemented in the BIE, enable model comparison for complex models and data sets. Finally, the BIE was designed to be a collaborative platform for applying Bayesian methodology to astronomy. It includes an extensible object-oriented and easily extended framework that implements every aspect of the Bayesian inference. By providing a variety of statistical algorithms for all phases of the inference problem, a scientist may explore a variety of approaches with a single model and data implementation. Additional technical details and download details are available from http://www.astro.umass.edu/bie. The BIE is distributed under the GNU GPL.Comment: Resubmitted version. Additional technical details and download details are available from http://www.astro.umass.edu/bie. The BIE is distributed under the GNU GP

The parabolic p-Laplacian with fractional differentiability

Breit D, Diening L, Storn J, Wichmann J. The parabolic p-Laplacian with fractional differentiability. 2020.We study the parabolic p-Laplacian system in a bounded domain. We deduce optimal convergence rates for the space-time discretization based on an implicit Euler scheme in time. Our estimates are expressed in terms of Nikolskii spaces and therefore cover situations when the (gradient of) the solution has only fractional derivatives in space and time. The main novelty is that, different to all previous results, we do not assume any coupling condition between the space and time resolution h and τ. The theoretical error analysis is complemented by numerical experiments

The parabolic p-Laplacian with fractional differentiability

We study the parabolic $p$-Laplacian system in a bounded domain. We deduce optimal convergence rates for the space-time discretization based on an implicit Euler scheme in time. Our estimates are expressed in terms of Nikolskii spaces and therefore cover situations when the (gradient of) the solution has only fractional derivatives in space and time. The main novelty is that, different to all previous results, we do not assume any coupling condition between the space and time resolution $h$ and $\tau$. The theoretical error analysis is complemented by numerical experiments.Comment: Source file for experiments included in submissio

DET ÄR DE SMÅ TINGEN SOM GÖR DET... Rakknivens respektive pincettens roller och dualism i den nordiska bronsålderns samhälle

ABSTRACT The aim of this essay is to analyze and discuss the razors? and tweezers? roles and dualism between them in the Nordic Bronze Age societies. This paper is an attempt to investigate what these little things can tell us about the people that were buried with them. For this purpose I have examined material of burial that consist of finds of razors respective tweezers and their connection with artefacts in the graves. I have done a comparative and contextual study together using quantitative and qualitative methods to investigate their roles and dualism. Burials from Denmark and Scania are used as examples and to illustrate roles and dualism in the study. My first conclusion is that the tweezer formed a dualism with the razor. The razor shows dualism with the tweezer, but it seems also to have been a complement for several other objects to dualism. The tweezer on other hand seems to create an identity foremost through the razor. My second conclusion is that they have more than one role, which alternate over time. The razor and the tweezer followed the construction of the Bronze Age society and were influenced by the people through their ideology and social values. I discovered that no gravecontext represents one role, instead each grave constructed several roles. Therefore my conclusion is that they have more than a practical function, and through the roles a person got a unique identity and individuality. In my third conclusion I suggest that the shaft of the razor and the neck of the tweezer enabled them to be hung, perhaps on clothing, especially during late Bronze Age. Thus they could have functioned as marks of identity-documents for example a craftsman, tradesman and/or a leader. Finally, they were only found in gravecontext which indicate an identity which was unique