Explicit-in-Time Variational Formulations for Goal-Oriented Adaptivity

Goal-Oriented Adaptivity (GOA) is a powerful tool to accurately approximate physically relevant features of the solution of Partial Differential Equations (PDEs). It delivers optimal grids to solve challenging engineering problems. In time dependent problems, GOA requires to represent the error in the Quantity of Interest (QoI) as an integral over the whole space-time domain in order to reduce it via adaptive refinements. A full space-time variational formulation of the problem allows the aforementioned error representation. Thus, variational spacetime formulations for PDEs have been of great interest in the last decades, among other things, because they allow to develop mesh-adaptive algorithms. Since it is known that implicit time marching schemes have variational structure, they are often employed for GOA in time-domain problems. When coming to explicit-intime methods, these were introduced for Ordinary In this dissertation, we prove that the explicit Runge-Kutta (RK) methods can be expressed as discontinuous-in-time Petrov-Galerkin (dPG) methods for the linear advection-diffusion equation. We systematically build trial and test functions that, after exact integration in time, lead to one, two, and general stage explicit RK methods. This approach enables us to reproduce the existing time domain goal-oriented adaptive algorithms using explicit methods in time. Here, we employ the lowest order dPG formulation that we propose to recover the Forward Euler method and we derive an appropriate error representation. Then, we propose an explicit-in-time goal-oriented adaptive algorithm that performs local refinements in space. In terms of time domain adaptivity, we impose the Courant-Friedrichs-Lewy (CFL) condition to ensure the stability of the method. We provide some numerical results in one-dimensional (1D)+time for the diffusion and advection-diffusion equations to show the performance of the proposed algorithm. On the other hand, time-domain adaptive algorithms involve solving a dual problem that runs backwards in time. This process is, in general, computationally expensive in terms of memory storage. In this work, we dene a pseudo-dual problem that runs forwards in time. We also describe a forward-in-time adaptive algorithm that works for some specific problems. Although it is not possible to dene a general dual problem running forwards in time that provides information about future states, we provide numerical evidence via one-dimensional problems in space to illustrate the efficiency of our algorithm as well as its limitations. As a complementary method, we propose a hybrid algorithm that employs the classical backward-in-time dual problem once and then performs the adaptive process forwards in time. We also generalize a novel error representation for goal-oriented adaptivity using (unconventional) pseudo-dual problems in the context of frequency-domain wave-propagation problems to the time-dependent wave equation. We show via 1D+time numerical results that the upper bounds for the new error representation are sharper than the classical ones. Therefore, this new error representation can be used to design more efficient goal-oriented adaptive methodologies. Finally, as classical Galerkin methods may lead to instabilities in advection-dominated-diffusion problems and therefore, inappropriate refinements, we propose a novel stabilized discretization method, which we call Isogeometric Residual Minimization (iGRM) with direction splitting. This method combines the benefits resulting from Isogeometric Analysis (IGA), residual minimization, and Alternating Direction Implicit (ADI) methods. We employ second order ADI time integrator schemes, B-spline basis functions in space and, at each time step, we solve a stabilized mixed method based on residual minimization. We show that the resulting system of linear equations has a Kronecker product structure, which results in a linear computational cost of the direct solver, even using implicit time integration schemes together with the stabilized mixed formulation. We test our method in 2D and 3D+time advection-diffusion problems. The derivation of a time-domain goal-oriented strategy based on iGRM will be considered in future works

Photosynthesis of three dessert banana cultivars along an altitudinal gradient

Poster presented at Tropentag 2011 - Development on the Margin. Bonn (Germany), 3-7 Oct 2011

Game-based learning: using business simulators in the university classroom [Aprender jugando: la utilización de simuladores empresariales en el aula universitaria]

In recent years, technological advances have paved the way for advances in education through the deployment of innovative teaching tools designed to improve the learning experience. Such active, experiential teaching/learning tools include business simulation games which aim to boost motivation, autonomous learning and learner-centered control of the learning process. The present study provides an analysis of the implantation of a college-level course designed around a business simulation game-with a special focus on the learning process as experienced by the learners themselves. To this end, the paper seeks to identify those factors which have a positive impact on the learning experience and contribute to learner satisfaction. More specifically, learner personality traits and attitude, the role emotions may play, facilitative learning conditions and learners'' intention to use business simulation games again in the future are analyzed. Our findings demonstrate that support from teachers, a high degree of learner motivation and perceived fun will have a particularly positive impact on the opinion learners hold of this innovative teaching/learning tool. Los avances tecnológicos han permitido que la educación también evolucione introduciendo herramientas pedagógicas innovadoras que mejoran la experiencia de aprendizaje para los individuos. Una de estas herramientas para el aprendizaje activo y experiencial es el uso de juegos de simulación empresarial. Estas nuevas herramientas pedagógicas permiten mejorar la motivación, autonomía y control del estudiante en su proceso de aprendizaje. Este estudio pretende analizar el éxito en la implantación de una asignatura basada en un juego de simulación empresarial mediante el estudio de la experiencia de aprendizaje del estudiante a través de dicho juego. Para ello, el trabajo persigue identificar aquellos factores que pueden contribuir a que dicha experiencia de aprendizaje sea satisfactoria para los alumnos. En concreto, se analiza el papel que pueden jugar las emociones, la personalidad del individuo, las condiciones facilitadoras, la actitud y la intención futura de uso del juego de simulación empresarial. Los resultados ponen de manifiesto que el apoyo recibido por parte del profesorado, la diversión del estudiante y un alto nivel de entusiasmo experimentado con el juego de simulación contribuirán de forma especialmente positiva a la opinión de los estudiantes sobre esta herramienta pedagógica innovadora

Exploiting Kronecker structure in exponential integrators: Fast approximation of the action of phi-functions of matrices via quadrature

In this article, we propose an algorithm for approximating the action of  $\varphi$-functions of matrices against vectors, which is a key operation in exponential time integrators. In particular, we consider matrices with Kronecker sum structure, which arise from problems admitting a tensor product representation. The method is based on quadrature approximations of the integral form of the  $\varphi$-functions combined with a scaling and modified squaring method. Owing to the Kronecker sum representation, only actions of 1D matrix exponentials are needed at each quadrature node and assembly of the full matrix can be avoided. Additionally, we derive a priori bounds for the quadrature error, which show that, as expected by classical theory, the rate of convergence of our method is supergeometric. Guided by our analysis, we construct a fast and robust method for estimating the optimal scaling factor and number of quadrature nodes that minimizes the total cost for a prescribed error tolerance. We investigate the performance of our algorithm by solving several linear and semilinear time-dependent problems in 2D and 3D. The results show that our method is accurate and orders of magnitude faster than the current state-of-the-art

Green marketing strategy and the firm's performance: The moderating role of environmental culture

Following the natural-resource-based view of the company, this study analyses how green marketing strategy influences different dimensions of organizational performance. In this task, it also studies how the integration of the environmental values within the firm's internal culture determines the effect of green strategies on performance. To meet these aims, the present research collects data from 361 manufacturing firms of a European country. Structural equation modelling with EQS software was the method applied to analyse the information. The findings indicate that green marketing strategy led firms to improve their profitability by optimizing marketing performance and reducing costs. However, dimensions of organizational results, like process performance, are not positively related to economic success. They also reveal that environmentally oriented firms are more likely to achieve a superior operational and marketing performance from environmental practices

Understanding online business simulation games: the role of flow experience, perceived enjoyment and personal innovativeness

This study aimed to empirically predict the degree of acceptance of an online business simulation game in an educational context. To do so, this study proposed an extended technology acceptance model that includes variables such as perceived enjoyment and flow. In addition, the moderating role of students'' personal imiovativeness in the technology field was analysed. The framework was empirically tested applying partial least squares to a sample of 266 students. Results reveal that perceived enjoyment is a key variable in explaining students'' perceived ease of use, usefulness and attitudes towards the simulation. Perceived ease of use is not significantly related to flow experience. However, this relationship is moderated by personal innovativeness. Indeed, results indicate that the higher the students'' personal innovativeness, the more attenuated the effect of perceived ease of use on the attitude towards the game and on flow experience. The study offers relevant insights for the pedagogical use of competitive digital technologies in university contexts

Equivalence between the DPG method and the Exponential Integrators for linear parabolic problems

The Discontinuous Petrov-Galerkin (DPG) method and the exponential integrators are two well established numerical methods for solving Partial Di fferential Equations (PDEs) and sti ff systems of Ordinary Di fferential Equations (ODEs), respectively. In this work, weapply the DPG method in the time variable for linear parabolic problems and we calculate the optimal test functions analytically. We show that the DPG method in time is equivalent to exponential integrators for the trace variables, which are decoupled from the interior variables. In addition, the DPG optimal test functions allow us to compute the approximated solutions in the time element interiors. This DPG method in time allows to construct a posteriori error estimations in order to perform adaptivity. We generalize this novel DPG-based time-marching scheme to general fi rst order linear systems of ODEs. We show the performance of the proposed method for 1D and 2D + time linear parabolic PDEs after discretizing in space by the nite element method

A DPG-based time-marching scheme for linear hyperbolic problems

The Discontinuous Petrov-Galerkin (DPG) method is a widely employed discretization method for Partial Di fferential Equations (PDEs). In a recent work, we applied the DPG method with optimal test functions for the time integration of transient parabolic PDEs. We showed that the resulting DPG-based time-marching scheme is equivalent to exponential integrators for the trace variables. In this work, we extend the aforementioned method to time-dependent hyperbolic PDEs. For that, we reduce the second order system in time to first order and we calculate the optimal testing analytically. We also relate our method with exponential integrators of Gautschi-type. Finally, we validate our method for 1D/2D + time linear wave equation after semidiscretization in space with a standard Bubnov-Galerkin method. The presented DPG-based time integrator provides expressions for the solution in the element interiors in addition to those on the traces. This allows to design di fferent error estimators to perform adaptivity

The DPG Method for the Convection-Reaction Problem, Revisited

We study both conforming and non-conforming versions of the practical DPG method for the convection-reaction problem. We determine that the most common approach for DPG stability analysis - construction of a local Fortin operator - is infeasible for the convection-reaction problem. We then develop a line of argument based on a direct proof of discrete stability; we find that employing a polynomial enrichment for the test space does not suffice for this purpose, motivating the introduction of a (two-element) subgrid mesh. The argument combines mathematical analysis with numerical experiments

Error representation of the time-marching DPG scheme

In this article, we introduce an error representation function to perform adaptivity in time of the recently developed time-marching Discontinuous Petrov–Galerkin (DPG) scheme. We first provide an analytical expression for the error that is the Riesz representation of the residual. Then, we approximate the error by enriching the test space in such a way that it contains the optimal test functions. The local error contributions can be efficiently computed by adding a few equations to the time-marching scheme. We analyze the quality of such approximation by constructing a Fortin operator and providing an a posteriori error estimate. The time-marching scheme proposed in this article provides an optimal solution along with a set of efficient and reliable local error contributions to perform adaptivity. We validate our method for both parabolic and hyperbolic problems