3,613 research outputs found
Conforming and nonconforming virtual element methods for elliptic problems
We present in a unified framework new conforming and nonconforming Virtual
Element Methods (VEM) for general second order elliptic problems in two and
three dimensions. The differential operator is split into its symmetric and
non-symmetric parts and conditions for stability and accuracy on their discrete
counterparts are established. These conditions are shown to lead to optimal
- and -error estimates, confirmed by numerical experiments on a set
of polygonal meshes. The accuracy of the numerical approximation provided by
the two methods is shown to be comparable
Towards a mathematical understanding of learning from few examples with nonlinear feature maps
We consider the problem of data classification where the training set
consists of just a few data points. We explore this phenomenon mathematically
and reveal key relationships between the geometry of an AI model's feature
space, the structure of the underlying data distributions, and the model's
generalisation capabilities. The main thrust of our analysis is to reveal the
influence on the model's generalisation capabilities of nonlinear feature
transformations mapping the original data into high, and possibly infinite,
dimensional spaces.Comment: 18 pages, 8 figure
The Impact of Atmospheric Fluctuations on Degree-scale Imaging of the Cosmic Microwave Background
Fluctuations in the brightness of the Earth's atmosphere originating from
water vapor are an important source of noise for ground-based instruments
attempting to measure anisotropy in the Cosmic Microwave Background. This paper
presents a model for the atmospheric fluctuations and derives simple
expressions to predict the contribution of the atmosphere to experimental
measurements. Data from the South Pole and from the Atacama Desert in Chile,
two of the driest places on Earth, are used to assess the level of fluctuations
at each site.Comment: 29 pages, 7 figures, 1 table, appears in The Astrophysical Journa
Efficient High-Order Space-Angle-Energy Polytopic Discontinuous Galerkin Finite Element Methods for Linear Boltzmann Transport
We introduce an -version discontinuous Galerkin finite element method
(DGFEM) for the linear Boltzmann transport problem. A key feature of this new
method is that, while offering arbitrary order convergence rates, it may be
implemented in an almost identical form to standard multigroup discrete
ordinates methods, meaning that solutions can be computed efficiently with high
accuracy and in parallel within existing software. This method provides a
unified discretisation of the space, angle, and energy domains of the
underlying integro-differential equation and naturally incorporates both local
mesh and local polynomial degree variation within each of these computational
domains. Moreover, general polytopic elements can be handled by the method,
enabling efficient discretisations of problems posed on complicated spatial
geometries. We study the stability and -version a priori error analysis of
the proposed method, by deriving suitable -approximation estimates together
with a novel inf-sup bound. Numerical experiments highlighting the performance
of the method for both polyenergetic and monoenergetic problems are presented.Comment: 27 pages, 2 figure
The Boundaries of Verifiable Accuracy, Robustness, and Generalisation in Deep Learning
In this work, we assess the theoretical limitations of determining guaranteed
stability and accuracy of neural networks in classification tasks. We consider
classical distribution-agnostic framework and algorithms minimising empirical
risks and potentially subjected to some weights regularisation. We show that
there is a large family of tasks for which computing and verifying ideal stable
and accurate neural networks in the above settings is extremely challenging, if
at all possible, even when such ideal solutions exist within the given class of
neural architectures
Anaesthetic subspecialties and sustainable healthcare: a narrative review
Summary: The principles of environmentally sustainable healthcare as applied to anaesthesia and peri‐operative care are well documented. Associated recommendations focus on generic principles that can be applied to all areas of practice. These include reducing the use of inhalational anaesthetic agents and carbon dioxide equivalent emissions of modern peri‐operative care. However, four areas of practice have specific patient, surgical and anaesthetic factors that present barriers to the implementation of some of these principles, namely: neuroanaesthesia; obstetric; paediatric; and cardiac anaesthesia. This narrative review describes these factors and synthesises the available evidence to highlight areas of sustainable practice clinicians can address today, as well as posing several unanswered questions for the future. In neuroanaesthesia, improvements can be made by undertaking awake surgery, moving towards more reusables and embracing telemedicine in quaternary services. Obstetric anaesthesia continues to present questions regarding how services can move away from nitrous oxide use or limit its release to the environment. The focus for paediatric anaesthesia is addressing the barriers to total intravenous and regional anaesthesia. For cardiac anaesthesia, a significant emphasis is determining how to focus the substantial resources required on those who will benefit from cardiac interventions, rather than universal implementation. Whilst the landscape of evidence‐based sustainable practice is evolving, there remains an urgent need for further original evidence in healthcare sustainability targeting these four clinical areas
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