Solving parabolic equations on the unit sphere via Laplace transforms and radial basis functions

We propose a method to construct numerical solutions of parabolic equations on the unit sphere. The time discretization uses Laplace transforms and quadrature. The spatial approximation of the solution employs radial basis functions restricted to the sphere. The method allows us to construct high accuracy numerical solutions in parallel. We establish $L_2$ error estimates for smooth and nonsmooth initial data, and describe some numerical experiments.Comment: 26 pages, 1 figur

Can disordered mobile phone use be considered a behavioral addiction? An update on current evidence and a comprehensive model for future research

Despite the many positive outcomes, excessive mobile phone use is now often associated with potentially harmful and/or disturbing behaviors (e.g., symptoms of deregulated use, negative impact on various aspects of daily life such as relationship problems, and work intrusion). Problematic mobile phone use (PMPU) has generally been considered as a behavioral addiction that shares many features with more established drug addictions. In light of the most recent data, the current paper reviews the validity of the behavioral addiction model when applied to PMPU. On the whole, it is argued that the evidence supporting PMPU as an addictive behavior is scarce. In particular, it lacks studies that definitively show behavioral and neurobiological similarities between mobile phone addiction and other types of legitimate addictive behaviors. Given this context, an integrative pathway model is proposed that aims to provide a theoretical framework to guide future research in the field of PMPU. This model highlights that PMPU is a heterogeneous and multi-faceted condition

Smoothness-Increasing Accuracy-Conserving (SIAC) filtering and quasi interpolation: A unified view

Filtering plays a crucial role in postprocessing and analyzing data in scientific and engineering applications. Various application-specific filtering schemes have been proposed based on particular design criteria. In this paper, we focus on establishing the theoretical connection between quasi-interpolation and a class of kernels (based on B-splines) that are specifically designed for the postprocessing of the discontinuous Galerkin (DG) method called Smoothness-Increasing Accuracy-Conserving (SIAC) filtering. SIAC filtering, as the name suggests, aims to increase the smoothness of the DG approximation while conserving the inherent accuracy of the DG solution (superconvergence). Superconvergence properties of SIAC filtering has been studied in the literature. In this paper, we present the theoretical results that establish the connection between SIAC filtering to long-standing concepts in approximation theory such as quasi-interpolation and polynomial reproduction. This connection bridges the gap between the two related disciplines and provides a decisive advancement in designing new filters and mathematical analysis of their properties. In particular, we derive a closed formulation for convolution of SIAC kernels with polynomials. We also compare and contrast cardinal spline functions as an example of filters designed for image processing applications with SIAC filters of the same order, and study their properties

Computers in Secondary Schools: Educational Games

This entry introduces educational games in secondary schools. Educational games include three main types of educational activities with a playful learning intention supported by digital technologies: educational serious games, educational gamification, and learning through game creation. Educational serious games are digital games that support learning objectives. Gamification is defined as the use of "game design elements and game thinking in a non-gaming context" (Deterding et al. 2011, p. 13). Educational gamification is not developed through a digital game but includes game elements for supporting the learning objectives. Learning through game creation is focused on the process of designing and creating a prototype of a game to support a learning process related to the game creation process or the knowledge mobilized through the game creation process. Four modalities of educational games in secondary education are introduced in this entry to describe educational games in secondary education: educational purpose of entertainment games, serious games, gamification, and game design

One-Sided Position-Dependent Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering Over Uniform and Non-Uniform Meshes

In this paper, we introduce a new position-dependent Smoothness-Increasing Accuracy-Conserving (SIAC) filter that retains the benefits of position dependence while ameliorating some of its shortcomings. As in the previous position-dependent filter, our new filter can be applied near domain boundaries, near a discontinuity in the solution, or at the interface of different mesh sizes; and as before, in general, it numerically enhances the accuracy and increases the smoothness of approximations obtained using the discontinuous Galerkin (dG) method. However, the previously proposed position-dependent one-sided filter had two significant disadvantages: (1) increased computational cost (in terms of function evaluations), brought about by the use of $4k+1$ central B-splines near a boundary (leading to increased kernel support) and (2) increased numerical conditioning issues that necessitated the use of quadruple precision for polynomial degrees of $k\ge 3$ for the reported accuracy benefits to be realizable numerically. Our new filter addresses both of these issues --- maintaining the same support size and with similar function evaluation characteristicsas the symmetric filter in a way that has better numerical conditioning --- making it, unlike its predecessor, amenable for GPU computing. Our new filter was conceived by revisiting the original error analysis for superconvergence of SIAC filters and by examining the role of the B-splines and their weights in the SIAC filtering kernel. We demonstrate, in the uniform mesh case, that our new filter is globally superconvergent for $k=1$ and superconvergent in the interior (e.g., region excluding the boundary) for $k\ge2$. Furthermore, we present the first theoretical proof of superconvergence for postprocessing over smoothly varying meshes, and explain the accuracy-order conserving nature of this new filter when applied to certain non-uniform meshes cases. We provide numerical examples supporting our theoretical results and demonstrating that our new filter, in general, enhances the smoothness and accuracy of the solution. Numerical results are presented for solutions of both linear and nonlinear equation solved on both uniform and non-uniform one- and two-dimensional meshes

Sparse initial data indentification for parabolic pde and its finite element approximations

We address the problem of inverse source identification for parabolic equations from the optimal control viewpoint employing measures of minimal norm as initial data. We adopt the point of view of approximate controllability so that the target is not required to be achieved exactly but only in an approximate sense. We prove an approximate inversion result and derive a characterization of the optimal initial measures by means of duality and the minimization of a suitable quadratic functional on the solutions of the adjoint system. We prove the sparsity of the optimal initial measures showing that they are supported in sets of null Lebesgue measure. As a consequence, approximate controllability can be achieved efficiently by means of controls that are activated in a finite number of pointwise locations. Moreover, we discuss the finite element numerical approximation of the control problem providing a convergence result of the corresponding optimal measures and states as the discretization parameters tend to zero.The first author was supported by Spanish Ministerio de Economía y Competitividad under project MTM2011-22711. The third author was supported by the Advanced Grant NUMERIWAVES/FP7-246775 of the European Research Council Executive Agency, FA9550-14-1-0214 of the EOARD-AFOSR, FA9550-15-1-0027 of AFOSR, the BERC 2014-2017 program of the Basque Government, the MTM2011-29306 and SEV-2013-0323 Grants of the MINECO, the CIMI-Toulouse Excellence Chair in PDEs, Control and Numerics and a Humboldt Award at the University of Erlangen-Nürnberg

Evolving surface finite element method for the Cahn-Hilliard equation

We use the evolving surface finite element method to solve a Cahn- Hilliard equation on an evolving surface with prescribed velocity. We start by deriving the equation using a conservation law and appropriate transport for- mulae and provide the necessary functional analytic setting. The finite element method relies on evolving an initial triangulation by moving the nodes according to the prescribed velocity. We go on to show a rigorous well-posedness result for the continuous equations by showing convergence, along a subse- quence, of the finite element scheme. We conclude the paper by deriving error estimates and present various numerical examples

Perceived connections between information and communication technology use and mental symptoms among young adults - a qualitative study

<p>Abstract</p> <p>Background</p> <p>Prospective associations have been found between high use of information and communication technology (ICT) and reported mental symptoms among young adult university students, but the causal mechanisms are unclear. Our aim was to explore possible explanations for associations between high ICT use and symptoms of depression, sleep disorders, and stress among young adults in order to propose a model of possible pathways to mental health effects that can be tested epidemiologically.</p> <p>Methods</p> <p>We conducted a qualitative interview study with 16 women and 16 men (21-28 years), recruited from a cohort of university students on the basis of reporting high computer (n = 28) or mobile phone (n = 20) use at baseline and reporting mental symptoms at the one-year follow-up. Semi-structured interviews were performed, with open-ended questions about possible connections between the use of computers and mobile phones, and stress, depression, and sleep disturbances. The interview data were analyzed with qualitative content analysis and summarized in a model.</p> <p>Results</p> <p>Central factors appearing to explain high quantitative ICT use were personal dependency, and demands for achievement and availability originating from the domains of work, study, social life, and individual aspirations. Consequences included mental overload, neglect of other activities and personal needs, time pressure, role conflicts, guilt feelings, social isolation, physical symptoms, worry about electromagnetic radiation, and economic problems. Qualitative aspects (destructive communication and information) were also reported, with consequences including vulnerability, misunderstandings, altered values, and feelings of inadequacy. User problems were a source of frustration. Altered ICT use as an effect of mental symptoms was reported, as well as possible positive effects of ICT on mental health.</p> <p>Conclusions</p> <p>The concepts and ideas of the young adults with high ICT use and mental symptoms generated a model of possible paths for associations between ICT exposure and mental symptoms. Demands for achievement and availability as well as personal dependency were major causes of high ICT exposure but also direct sources of stress and mental symptoms. The proposed model shows that factors in different domains may have an impact and should be considered in epidemiological and intervention studies.</p

Error estimates for the discretization of the velocity tracking problem

In this paper we are continuing our work (Casas and Chrysafinos, SIAM J Numer Anal 50(5):2281–2306, 2012), concerning a priori error estimates for the velocity tracking of two-dimensional evolutionary Navier–Stokes flows. The controls are of distributed type, and subject to point-wise control constraints. The discretization scheme of the state and adjoint equations is based on a discontinuous time-stepping scheme (in time) combined with conforming finite elements (in space) for the velocity and pressure. Provided that the time and space discretization parameters, t and h respectively, satisfy t = Ch2, error estimates of order O(h2) and O(h 3/2 – 2/p ) with p > 3 depending on the regularity of the target and the initial velocity, are proved for the difference between the locally optimal controls and their discrete approximations, when the controls are discretized by the variational discretization approach and by using piecewise-linear functions in space respectively. Both results are based on new duality arguments for the evolutionary Navier–Stokes equations