Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations

In this paper, we extend to the time-harmonic Maxwell equations the p-version analysis technique developed in [R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version, SIAM J. Numer. Anal., 49 (2011), 264-284] for Trefftz-discontinuous Galerkin approximations of the Helmholtz problem. While error estimates in a mesh-skeleton norm are derived parallel to the Helmholtz case, the derivation of estimates in a mesh-independent norm requires new twists in the duality argument. The particular case where the local Trefftz approximation spaces are built of vector-valued plane wave functions is considered, and convergence rates are derived.

Techniques for analysis of proteins by SDS-polyacrylamide gel electrophoresis and Western blotting

The separation of mixtures of proteins by SDS-polyacrylamide gel electrophoresis (SDS-PAGE) is a technique that is widely used—and, indeed, this technique underlies many of the assays and analyses that are described in this book. While SDS-PAGE is routine in many labs, a number of issues require consideration before embarking on it for the first time. We felt, therefore, that in the interest of completeness of this volume, a brief chapter describing the basics of SDS-PAGE would be helpful. Also included in this chapter are protocols for the staining of SDS-PAGE gels to visualize separated proteins, and for the electrotransfer of proteins to a membrane support (Western blotting) to enable immunoblotting, for example. This chapter is intended to complement the chapters in this book that require these techniques to be performed. Therefore, detailed examples of why and when these techniques could be used will not be discussed here.

Error analysis of a Monte Carlo algorithm for computing bilinear forms of matrix powers

In this paper we present error analysis for a Monte Carlo algorithm for evaluating bilinear forms of matrix powers. An almost Optimal Monte Carlo (MAO) algorithm for solving this problem is formulated. Results for the structure of the probability error are presented and the construction of robust and interpolation Monte Carlo algorithms are discussed. Results are presented comparing the performance of the Monte Carlo algorithm with that of a corresponding deterministic algorithm. The two algorithms are tested on a well balanced matrix and then the effects of perturbing this matrix, by small and large amounts, is studied.

On aposteriori error analysis of dG schemes approximating hyperbolic conservation laws

This contribution is concerned with aposteriori error analysis of discontinuous Galerkin (dG) schemes approximating hyperbolic conservation laws. In the scalar case the aposteriori analysis is based on the L1 contraction property and the doubling of variables technique. In the system case the appropriate stability framework is in L2, based on relative entropies. It is only applicable if one of the solutions, which are compared to each other, is Lipschitz. For dG schemes approximating hyperbolic conservation laws neither the entropy solution nor the numerical solution need to be Lipschitz. We explain how this obstacle can be overcome using a reconstruction approach which leads to an aposteriori error estimate.

A priori error analysis of space–time Trefftz discontinuous Galerkin methods for wave problems

We present and analyse a space–time discontinuous Galerkin method for wave propagation problems. The special feature of the scheme is that it is a Trefftz method, namely that trial and test functions are solution of the partial differential equation to be discretised in each element of the (space–time) mesh. The method considered is a modification of the discontinuous Galerkin schemes of Kretzschmar et al. (2014) and of Monk & Richter (2005). For Maxwell’s equations in one space dimension, we prove stability of the method, quasi-optimality, best approximation estimates for polynomial Trefftz spaces and (fully explicit) error bounds with high order in the meshwidth and in the polynomial degree. The analysis framework also applies to scalar wave problems and Maxwell’s equations in higher space dimensions. Some numerical experiments demonstrate the theoretical results proved and the faster convergence compared to the non-Trefftz version of the scheme.

A new linear initialization in SOM for biomolecular data

In the past decade, the amount of data in biological field has become larger and larger; Bio-techniques for analysis of biological data have been developed and new tools have been introduced. Several computational methods are based on unsupervised neural network algorithms that are widely used for multiple purposes including clustering and visualization, i.e. the Self Organizing Maps (SOM). Unfortunately, even though this method is unsupervised, the performances in terms of quality of result and learning speed are strongly dependent from the neuron weights initialization. In this paper we present a new initialization technique based on a totally connected undirected graph, that report relations among some intersting features of data input. Result of experimental tests, where the proposed algorithm is compared to the original initialization techniques, shows that our technique assures faster learning and better performance in terms of quantization error.

Past, present and future mathematical models for buildings (ii)

This article is the second part of a review of the historical evolution of mathematical models applied in the development of building technology. The first part described the current state of the art and contrasted various models with regard to the applications to conventional buildings and intelligent buildings. It concluded that mathematical techniques adopted in neural networks, expert systems, fuzzy logic and genetic models, that can be used to address model uncertainty, are well suited for modelling intelligent buildings. Despite the progress, the possible future development of intelligent buildings based on the current trends implies some potential limitations of these models. This paper attempts to uncover the fundamental limitations inherent in these models and provides some insights into future modelling directions, with special focus on the techniques of semiotics and chaos. Finally, by demonstrating an example of an intelligent building system with the mathematical models that have been developed for such a system, this review addresses the influences of mathematical models as a potential aid in developing intelligent buildings and perhaps even more advanced buildings for the future.

A mathematical approach to chemical equilibrium theory for gaseous systems—II: extensions and applications

Straightforward mathematical techniques are used innovatively to form a coherent theoretical system to deal with chemical equilibrium problems. For a systematic theory it is necessary to establish a system to connect different concepts. This paper shows the usefulness and consistence of the system by applications of the theorems introduced previously. Some theorems are shown somewhat unexpectedly to be mathematically correlated and relationships are obtained in a coherent manner. It has been shown that theorem 1 plays an important part in interconnecting most of the theorems. The usefulness of theorem 2 is illustrated by proving it to be consistent with theorem 3. A set of uniform mathematical expressions are associated with theorem 3. A variety of mathematical techniques based on theorems 1–3 are shown to establish the direction of equilibrium shift. The equilibrium properties expressed in initial and equilibrium conditions are shown to be connected via theorem 5. Theorem 6 is connected with theorem 4 through the mathematical representation of theorem 1.

A twenty-year independent record of sea surface temperature for climate from Along-track Scanning Radiometers

A new record of sea surface temperature (SST) for climate applications is described. This record provides independent corroboration of global variations estimated from SST measurements made in situ. Infrared imagery from Along-Track Scanning Radiometers (ATSRs) is used to create a 20 year time series of SST at 0.1° latitude-longitude resolution, in the ATSR Reprocessing for Climate (ARC) project. A very high degree of independence of in situ measurements is achieved via physics-based techniques. Skin SST and SST estimated for 20 cm depth are provided, with grid cell uncertainty estimates. Comparison with in situ data sets establishes that ARC SSTs generally have bias of order 0.1 K or smaller. The precision of the ARC SSTs is 0.14 K during 2003 to 2009, from three-way error analysis. Over the period 1994 to 2010, ARC SSTs are stable, with better than 95% confidence, to within 0.005 K yr−1(demonstrated for tropical regions). The data set appears useful for cleanly quantifying interannual variability in SST and major SST anomalies. The ARC SST global anomaly time series is compared to the in situ-based Hadley Centre SST data set version 3 (HadSST3). Within known uncertainties in bias adjustments applied to in situ measurements, the independent ARC record and HadSST3 present the same variations in global marine temperature since 1996. Since the in situ observing system evolved significantly in its mix of measurement platforms and techniques over this period, ARC SSTs provide an important corroboration that HadSST3 accurately represents recent variability and change in this essential climate variable.

Mathematical solutions for electricity networks in a low carbon future

Smart meters are becoming more ubiquitous as governments aim to reduce the risks to the energy supply as the world moves toward a low carbon economy. The data they provide could create a wealth of information to better understand customer behaviour. However at the household, and even the low voltage (LV) substation level, energy demand is extremely volatile, irregular and noisy compared to the demand at the high voltage (HV) substation level. Novel analytical methods will be required in order to optimise the use of household level data. In this paper we briefly outline some mathematical techniques which will play a key role in better understanding the customer's behaviour and create solutions for supporting the network at the LV substation level.

Efficient evaluation of highly oscillatory acoustic scattering surface integrals

We consider the approximation of some highly oscillatory weakly singular surface integrals, arising from boundary integral methods with smooth global basis functions for solving problems of high frequency acoustic scattering by three-dimensional convex obstacles, described globally in spherical coordinates. As the frequency of the incident wave increases, the performance of standard quadrature schemes deteriorates. Naive application of asymptotic schemes also fails due to the weak singularity. We propose here a new scheme based on a combination of an asymptotic approach and exact treatment of singularities in an appropriate coordinate system. For the case of a spherical scatterer we demonstrate via error analysis and numerical results that, provided the observation point is sufficiently far from the shadow boundary, a high level of accuracy can be achieved with a minimal computational cost.

How well do high resolution models reproduce tropical convection?

Cascade is a multi-institution project studying the temporal and spatial organization of tropical convective systems. While cloud resolving numerical models can reproduce the observed diurnal cycle of such systems they are sensitive to the chosen resolution. As part of this effort, we are comparing results from the Met. Office Unified Model to data from the Global Earth Radiation Budget satellite instrument over the African Monsoon Interdisciplinary Analyses region of North Africa. We use a variety of mathematical techniques to study the outgoing radiation and the evolution of properties such as the cloud size distribution. The effectiveness of various model resolutions is tested with a view to determining the optimum balance between resolution and the need to reproduce the observations.

Linear and nonlinear generalized Fourier transforms

This article presents an overview of a transform method for solving linear and integrable nonlinear partial differential equations. This new transform method, proposed by Fokas, yields a generalization and unification of various fundamental mathematical techniques and, in particular, it yields an extension of the Fourier transform method.