A high frequency boundary element method for scattering by a class of nonconvex obstacles

In this paper we propose and analyse a hybrid numerical-asymptotic boundary element method for the solution of problems of high frequency acoustic scattering by a class of sound-soft nonconvex polygons. The approximation space is enriched with carefully chosen oscillatory basis functions; these are selected via a study of the high frequency asymptotic behaviour of the solution. We demonstrate via a rigorous error analysis, supported by numerical examples, that to achieve any desired accuracy it is sufficient for the number of degrees of freedom to grow only in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods. This appears to be the first such numerical analysis result for any problem of scattering by a nonconvex obstacle. Our analysis is based on new frequency-explicit bounds on the normal derivative of the solution on the boundary and on its analytic continuation into the complex plane.

Applying robust control theory to solve problems in bio-medical sciences: study of an apoptotic model

Biological models of an apoptotic process are studied using models describing a system of differential equations derived from reaction kinetics information. The mathematical model is re-formulated in a state-space robust control theory framework where parametric and dynamic uncertainty can be modelled to account for variations naturally occurring in biological processes. We propose to handle the nonlinearities using neural networks.

A Summary of the NATO ASI on Polar Cap Boundary Phenomena

The polar cap boundary is a subject of central importance to current magnetosphere-ionosphere research and its applications in “space weather” activities. The problems are that it has a number of definitions, and that the most physically meaningful definition (namely the open-closed field line boundary) is very difficult to identify in observations. New understanding of the importance of the structure and dynamics of the boundary region made the time right for a meeting reviewing our knowledge in this area. The Advanced Study Institute (ASI) on Svalbard in June 1997 discussed the boundary on both the dayside and the nightside, mapping magnetically to the dayside magnetopause and to tail plasma sheet/lobe interface, respectively. We held a “brainstorming” session, in which different ideas which arose from the presented papers were discussed and developed, and a summary session, in which session convenors gave a personal view of progress that has been made and problems which still need solving. Both were designed as ways of promoting further discussion. This paper attempts to distil some of the themes that emerged from these discussions.

Acoustic scattering : high frequency boundary element methods and unified transform methods

We describe some recent advances in the numerical solution of acoustic scattering problems. A major focus of the paper is the efficient solution of high frequency scattering problems via hybrid numerical-asymptotic boundary element methods. We also make connections to the unified transform method due to A. S. Fokas and co-authors, analysing particular instances of this method, proposed by J. A. De-Santo and co-authors, for problems of acoustic scattering by diffraction gratings.

Evolution PDEs and augmented eigenfunctions. half line.

The solution of an initial-boundary value problem for a linear evolution partial differential equation posed on the half-line can be represented in terms of an integral in the complex (spectral) plane. This representation is obtained by the unified transform introduced by Fokas in the 90's. On the other hand, it is known that many initial-boundary value problems can be solved via a classical transform pair, constructed via the spectral analysis of the associated spatial operator. For example, the Dirichlet problem for the heat equation can be solved by applying the Fourier sine transform pair. However, for many other initial-boundary value problems there is no suitable transform pair in the classical literature. Here we pose and answer two related questions: Given any well-posed initial-boundary value problem, does there exist a (non-classical) transform pair suitable for solving that problem? If so, can this transform pair be constructed via the spectral analysis of a differential operator? The answer to both of these questions is positive and given in terms of augmented eigenfunctions, a novel class of spectral functionals. These are eigenfunctions of a suitable differential operator in a certain generalised sense, they provide an effective spectral representation of the operator, and are associated with a transform pair suitable to solve the given initial-boundary value problem.

A multi-model evaluation of aerosols over South Asia: common problems and possible causes

Atmospheric pollution over South Asia attracts special attention due to its effects on regional climate, water cycle and human health. These effects are potentially growing owing to rising trends of anthropogenic aerosol emissions. In this study, the spatio-temporal aerosol distributions over South Asia from seven global aerosol models are evaluated against aerosol retrievals from NASA satellite sensors and ground-based measurements for the period of 2000–2007. Overall, substantial underestimations of aerosol loading over South Asia are found systematically in most model simulations. Averaged over the entire South Asia, the annual mean aerosol optical depth (AOD) is underestimated by a range 15 to 44% across models compared to MISR (Multi-angle Imaging SpectroRadiometer), which is the lowest bound among various satellite AOD retrievals (from MISR, SeaWiFS (Sea-Viewing Wide Field-of-View Sensor), MODIS (Moderate Resolution Imaging Spectroradiometer) Aqua and Terra). In particular during the post-monsoon and wintertime periods (i.e., October–January), when agricultural waste burning and anthropogenic emissions dominate, models fail to capture AOD and aerosol absorption optical depth (AAOD) over the Indo–Gangetic Plain (IGP) compared to ground-based Aerosol Robotic Network (AERONET) sunphotometer measurements. The underestimations of aerosol loading in models generally occur in the lower troposphere (below 2 km) based on the comparisons of aerosol extinction profiles calculated by the models with those from Cloud–Aerosol Lidar with Orthogonal Polarization (CALIOP) data. Furthermore, surface concentrations of all aerosol components (sulfate, nitrate, organic aerosol (OA) and black carbon (BC)) from the models are found much lower than in situ measurements in winter. Several possible causes for these common problems of underestimating aerosols in models during the post-monsoon and wintertime periods are identified: the aerosol hygroscopic growth and formation of secondary inorganic aerosol are suppressed in the models because relative humidity (RH) is biased far too low in the boundary layer and thus foggy conditions are poorly represented in current models, the nitrate aerosol is either missing or inadequately accounted for, and emissions from agricultural waste burning and biofuel usage are too low in the emission inventories. These common problems and possible causes found in multiple models point out directions for future model improvements in this important region.

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.

Sparse space-time boundary element methods for the heat equation

The goal of this work is the efficient solution of the heat equation with Dirichlet or Neumann boundary conditions using the Boundary Elements Method (BEM). Efficiently solving the heat equation is useful, as it is a simple model problem for other types of parabolic problems. In complicated spatial domains as often found in engineering, BEM can be beneficial since only the boundary of the domain has to be discretised. This makes BEM easier than domain methods such as finite elements and finite differences, conventionally combined with time-stepping schemes to solve this problem. The contribution of this work is to further decrease the complexity of solving the heat equation, leading both to speed gains (in CPU time) as well as requiring smaller amounts of memory to solve the same problem. To do this we will combine the complexity gains of boundary reduction by integral equation formulations with a discretisation using wavelet bases. This reduces the total work to O(h

A genome-wide test of the differential susceptibility hypothesis reveals a genetic predictor of differential response to psychological treatments for child anxiety disorders

Background: The differential susceptibly hypothesis suggests that certain genetic variants moderate the effects of both negative and positive environments on mental health and may therefore be important predictors of response to psychological treatments. Nevertheless, the identification of such variants has so far been limited to preselected candidate genes. In this study we extended the differential susceptibility hypothesis from a candidate gene to a genome-wide approach to test whether a polygenic score of environmental sensitivity predicted response to Cognitive Behavioural Therapy (CBT) in children with anxiety disorders. Methods: We identified variants associated with environmental sensitivity using a novel method in which within-pair variability in emotional problems in 1026 monozygotic (MZ) twin pairs was examined as a function of the pairs’ genotype. We created a polygenic score of environmental sensitivity based on the whole-genome findings and tested the score as a moderator of parenting on emotional problems in 1,406 children and response to individual, group and brief parent-led CBT in 973 children with anxiety disorders. Results: The polygenic score significantly moderated the effects of parenting on emotional problems and the effects of treatment. Individuals with a high score responded significantly better to individual CBT than group CBT or brief parent-led CBT (remission rates: 70.9%, 55.5% and 41.6% respectively). Conclusions: Pending successful replication, our results should be considered exploratory. Nevertheless, if replicated, they suggest that individuals with the greatest environmental sensitivity may be more likely to develop emotional problems in adverse environments, but also benefit more from the most intensive types of treatment.

Continuity of the dynamics in a localized large diffusion problem with nonlinear boundary conditions

This paper is concerned with singular perturbations in parabolic problems subjected to nonlinear Neumann boundary conditions. We consider the case for which the diffusion coefficient blows up in a subregion Omega(0) which is interior to the physical domain Omega subset of R(n). We prove, under natural assumptions, that the associated attractors behave continuously as the diffusion coefficient blows up locally uniformly in Omega(0) and converges uniformly to a continuous and positive function in Omega(1) = (Omega) over bar\Omega(0). (C) 2009 Elsevier Inc. All rights reserved.

Coincidence theorems and its applications to equilibrium problems

Given two maps h : X x K -> R and g : X -> K such that, for all x is an element of X, h(x, g(x)) = 0, we consider the equilibrium problem of finding (x) over tilde is an element of X such that h((x) over tilde, g(x)) >= 0 for every x is an element of X. This question is related to a coincidence problem.

Strongly damped wave problems: Bootstrapping and regularity of solutions

The aim of the article is to present a unified approach to the existence, uniqueness and regularity of solutions to problems belonging to a class of second order in time semilinear partial differential equations in Banach spaces. Our results are applied next to a number of examples appearing in literature, which fall into the class of strongly damped semilinear wave equations. The present work essentially extends the results on the existence and regularity of solutions to such problems. Previously, these problems have been considered mostly within the Hilbert space setting and with the main part operators being selfadjoint. In this article we present a more general approach, involving sectorial operators in reflexive Banach spaces. (C) 2008 Elsevier Inc. All rights reserved.

Continuity of attractors for parabolic problems with localized large diffusion

In this paper we study the continuity of asymptotics of semilinear parabolic problems of the form u(t) - div(p(x)del u) + lambda u =f(u) in a bounded smooth domain ohm subset of R `` with Dirichlet boundary conditions when the diffusion coefficient p becomes large in a subregion ohm(0) which is interior to the physical domain ohm. We prove, under suitable assumptions, that the family of attractors behave upper and lower semicontinuously as the diffusion blows up in ohm(0). (c) 2006 Elsevier Ltd. All rights reserved.

A singular integro-differential equation model for dryout in LMFBR boiler tubes

A 2D steady model for the annular two-phase flow of water and steam in the steam-generating boiler pipes of a liquid metal fast breeder reactor is proposed The model is based on thin-layer lubrication theory and thin aerofoil theory. The exchange of mass between the vapour core and the liquid film due to evaporation of the liquid film is accounted for using some simple thermodynamics models, and the resultant change of phase is modelled by proposing a suitable Stefan problem Appropriate boundary conditions for the now are discussed The resulting non-lineal singular integro-differential equation for the shape of the liquid film free surface is solved both asymptotically and numerically (using some regularization techniques) Predictions for the length to the dryout point from the entry of the annular regime are made The influence of both the traction tau provided by the fast-flowing vapour core on the liquid layer and the mass transfer parameter eta on the dryout length is investigated

A Mass-Conserving Algorithm for Dynamical Lubrication Problems With Cavitation

A numerical algorithm for fully dynamical lubrication problems based on the Elrod-Adams formulation of the Reynolds equation with mass-conserving boundary conditions is described. A simple but effective relaxation scheme is used to update the solution maintaining the complementarity conditions on the variables that represent the pressure and fluid fraction. The equations of motion are discretized in time using Newmark`s scheme, and the dynamical variables are updated within the same relaxation process just mentioned. The good behavior of the proposed algorithm is illustrated in two examples: an oscillatory squeeze flow (for which the exact solution is available) and a dynamically loaded journal bearing. This article is accompanied by the ready-to-compile source code with the implementation of the proposed algorithm. [DOI: 10.1115/1.3142903]