On Approximating Discontinuous Solutions of PDEs by Adaptive Finite Elements

For singularly perturbed problems with a small diffusion, when the transient layer is very sharp and the computational mesh is relatively coarse, the solution can be viewed as discontinuous. For both linear and nonlinear hyperbolic partial differential equations, the solution can be discontinuous. When finite element methods with piecewise polynomials are used to approximate these discontinuous solutions, numerical solutions often overshoot near a discontinuity. Can this be resolved by adaptive mesh refinements? In this paper, for a simple discontinuous function, we explicitly compute its continuous and discontinuous piecewise constant or linear projections on discontinuity matched or non-matched meshes. For the simple discontinuity-aligned mesh case, piecewise discontinuous approximations are always good. For the general non-matched case, we explain that the piecewise discontinuous constant approximation combined with adaptive mesh refinements is the best choice to achieve accuracy without overshooting. For discontinuous piecewise linear approximations, non-trivial overshootings will be observed unless the mesh is matched with discontinuity. For continuous piecewise linear approximations, the computation is based on a "far away assumption", and non-trivial overshootings will always be observed under regular meshes. We calculate the explicit overshooting values for several typical cases. Several numerical tests are preformed for a singularly-perturbed reaction-diffusion equation and linear hyperbolic equations to verify our findings in the paper.Comment: 23 page

An accurate FIC-FEM formulation for the 1D advection-diffusion-reaction equation

In this paper we present an accurate stabilized FIC-FEM formulation for the 1D advection-diffusion-reaction equation in the exponential and propagation regimes using two stabilization parameters. Both the steady-state and transient solutions are considered. The stabilized formulation is based on the standard Galerkin FEM solution of the governing differential equations derived via the Finite Increment Calculus (FIC) method. The steady-state problem is considered first. The optimal value of the two stabilization parameters ensuring an exact (nodal) FEM solution using uniform meshes of linear 2-noded elements is obtained. In the absence of the absorption term the formulation simplifies to the standard one-parameter Petrov-Galerkin method for the advection-diffusion problem. For the diffusion-reaction case one stabilization parameter is just needed and the diffusion-type stabilization term is identical to that obtained by Felippa and O˜ nate [16] using a variational FIC approach. A procedure for computing the stabilization parameters for the transient problem is proposed. The accuracy of the new FIC-FEM formulation is demonstrated in the solution of steady-state and transient 1D advection-diffusion-radiation problems for a the range of physical parameters and boundary conditions. Finally we outline the procedure to extend the 1D FIC-FEM formulation to multidimensions

Through a Model, Darkly: An Investigation of Modellers’ Conceptualisation of Uncertainty in Climate and Energy Systems Modelling and an Application to Epidemiology

Policy responses to climate change require the use of complex computer models to understand the physical dynamics driving change, to evaluate its impacts and to evaluate the efficacy and costs of different mitigation and adaptation options. These models are often complex and built by large teams of dedicated researchers. All modelling requires assumptions, approximations and analytic conveniences to be employed. No model is without uncertainty. Authors have attempted to understand these uncertainties over the years and have developed detailed typologies to deal with them. However, it remains unknown how modellers themselves conceptualise the uncertainty inherent in their work. The core of this thesis involves the interviews of 38 modellers from climate science, energy systems modelling and integrated assessment to understand how they conceptualise the uncertainty in their work. This study finds that there is diversity in how uncertainty is understood and that various concepts from the literature are selectively employed to organise uncertainties. Uncertainty analysis is conceived as consisting of different phases in the model development process. The interplay between the complexity of the model and the capacities of modellers to manipulate these models shapes the ways in which uncertainty can be conceptualised. How we can attempt to wrangle with uncertainty in the present is determined by the path-dependent decisions made in the past; decisions that are influenced by a variety of factors within the context of the model’s creation. Furthermore, this thesis examines the application of these concepts to another field, epidemiology, to examine their generalisability in other contexts. This thesis concludes that in a situation such as climate change, where the nature of the problem changes in a dynamic way, emphasis should be placed on reducing the grip of these path dependencies and the resource costs of adapting models to face new challenges and answer new policy questions

Midwife of An-arché: Toward a Poetics of Becoming-with-Woman

This project explores the connections between midwifery and the ethical demands attendant to poetic practice. Through verse and prose, I unfold a figuration of the midwife that traverses the boundaries between Levinasian heteronomy and Deleuzian heteromorphism, and is a constitutive factor in sites of resistance to the biomedical territorialisation of the creative body. Chief archival and methodological components that inform the thesis include: a historiography of childbirth - tracing the development of ‘holistic’ and ‘interventionist’ paradigms, and the ideological underpinnings of the phallocratic takeover of the birthing room in certain Western countries; idiographic insights gathered from dialogues with maternal practitioners and mothers, including residents of The Farm in Tennessee - where I participated in a midwifery workshop week; an experiential inquiry into Holotropic Breathwork - to facilitate access to non-ordinary states of consciousness; and a negotiation between Marxist-feminist and poststructuralist articulations of ethico-political agency. Subject matter ranges from a consideration of the ethical import of the placental economy to the bio-intelligent tissue of the psoas, the banishment of Anne Hutchinson from Massachusetts Bay to the legacy of the ‘Twilight Sleep’ movement. Sustained critical attention is devoted to Mina Loy’s “Parturition”, and contemporary poets that have acknowledged Loy as an influence, such as Lara Glenum. I suggest that, despite the absence of a birth attendant on the symbolic level, Loy’s poem resonates with the investments of midwifery, instating a ‘subjectin- process’ that woks through and against abstruse and instrumental discourses, defying both the technocratic erasure of maternal knowing and the fetishistic reduction of labour to an end-product. Art’s capacity for opening up a corporeallycharged zone of between-ness is further elaborated in an essay on Andrei Tarkovsky’s Stalker - through which the treatment of spatiotemporality is aligned with the imperatives of midwifery guardianship.

Evolutionary Computation

This book presents several recent advances on Evolutionary Computation, specially evolution-based optimization methods and hybrid algorithms for several applications, from optimization and learning to pattern recognition and bioinformatics. This book also presents new algorithms based on several analogies and metafores, where one of them is based on philosophy, specifically on the philosophy of praxis and dialectics. In this book it is also presented interesting applications on bioinformatics, specially the use of particle swarms to discover gene expression patterns in DNA microarrays. Therefore, this book features representative work on the field of evolutionary computation and applied sciences. The intended audience is graduate, undergraduate, researchers, and anyone who wishes to become familiar with the latest research work on this field