Applied Mathematics and Numerical Analysis Seminar 

In this paper we shall study a numerical method for computing the spectrum of some elliptic operators. It will be shown that the essential spectrum of an elliptic operator in an infinite periodic waveguide may contain gaps. Moreover, we construct examples where the number of band-gaps can be made arbitrarily large. Finally, we present some numerical examples by using the finite element method. The theoretical tool to accomplish this task is provided by the Krein-Birman-Vishik theory. Using the Gelfand transform we formulate the spectral problem for a quadratic pencil of formally self-adjoint operators on the periodicity cell. For fixed Gelfand parameter the model problem in the periodicity cell admits an increasing sequence of real and non-negative eigenvalues. We approximate these eigenvalues using the mixed finite element method using the Raviart-Thomas elements for the approximation of the eigenfunctions.

T.B.A

During this work, four different segmentation methods based on clustering were implemented with the objective of identifying blood vessels using in vivo medical images. Comparison of the resulting segmentations between each method is performed. 3D geometry reconstruction and meshing leads to CFD analysis. Taking examples of the Otsu and Kittler segmentation methods, the full steps from medical images to CFD results are carried out and results are analysed.

The study of blood flow dynamics through mathematical models and numerical simulations is an important, non invasive tool to help the prediction of pathologies, as well as the consequences of surgery. In particular, the application of mathematical simplified models has proved to give useful information at fair computational costs, allowing also for the simulation of large arterial networks. In this work a reduced one-dimensional (1D) model is studied for blood flow in arteries, describing the evolution in time and one dimensional space coordinate of the mean pressure, flow rate and area, illustrating the wave propagation behaviour of blood flow in arteries. The hyperbolic system of equations is discretized in space using a second order Taylor-Galerkin method, and in time with the Lax-Wendroff scheme. The model was tested using a variety of non-physiological and physiological conditions occurring in the human arterial network. Properties and strengths of the model, regarding the numerical simulations in view of clinical applications are analyzed, and their limitations and drawbacks commented.

Computing systems have evolved rapidly and continuously for decades offering high computational potential and becoming ubiquitous. At the same time, current systems are very complex and can only be fully exploited with great knowledge of the algorithm, the architecture and the programming model. Assuming an audience with great knowledge on different kinds of algorithms, in this talk we will provide an overview of the computer architecture and some programming models, and how we may need to develop new algorithms in search for high performance. - We will introduce the memory hierarchy and the concept of locality together with some techniques to exploit them. - We will describe different levels of parallelism and clarify terms such as: latency, throughput, Pipelined processor, Super-scalar processor, Vector units, Multi-threading, [Homogeneous or Heterogeneous] Multi-core, Many-core, Shared Memory, Distributed memory or Distributed Shared Memory; and acronyms such as: SIMD, SIMT, MIMD, SSE, SMT, Hyper-threading, SMP, DM, DSM. - We will see the reasons which have driven the shift to multi-core processors and why that increases the burden on the software side. - We will comment on some available Programming Models. - We will then discuss the possibility to rethink our algorithms and/or data structures to better exploit the available hardware resources with some examples from linear algebra software: Data storage formats; Iterative refinement; Tiled algorithms.

We begin with a brief review of our previous work with autonomous and non-autonomous linear MTFDEs using collocation, least squares and finite element methods. Then we focus on the approximate solution of a nonlinear mixed type functional differential equation (MTFDE) arising from nerve conduction theory. The considered model describes the conduction of neuroelectric signals in a myelinated nerve axon (composed by a membrane and nodes) . In this case, when the membrane is depolarized at a node, the myelin tends not to depolarize the adjacent region of membrane, but instead it appears to jump to the next node to excite the membrane there, as described by the authors of [1]. As a consequence, the variation in time of the electric potential at each node depends on the electric potential of the neighbour nodes and is modeled by a first order nonlinear functional differential equation with deviated arguments. Following the approach introduced previously for linear MTFDEs, we propose and analyse a new computational method for the solution of this problem.

References

1. H. Chi, J.Bell and B. Hassard, Numerical solution of a nonlinear advance-delay-differential equation from nerve conduction theory, J.Math.Biology, 24 (1986), 583-601.

For a class of singular boundary value arising in Physiology, a finite difference scheme is proposed. Based on the asymptotic expansion of the solution in the neighborhood of the singular points, smoothing variable substitutions are introduced in order to improve the convergence order of the finite diference methods. Numerical results are presented and discussed.

In this work are obtained some criteria which guarantee the oscillatory behavior of the differential system of mixed type $$x'(t)=\int_{-1}^0 d[\nu(\theta)] x(t-r(\theta)) + \int_{-1}^0 d[\eta(\theta)]x(t+\tau(\theta))$$ where \(x(t)\in\mathbb{R}^n\), \(r(\theta)\) and \(\tau(\theta)\) are real \(n\times n\) matrix valued functions of bounded variation on \([-1,0]\).

This presentation will focus on our numerical investigations relating to mixed-type functional differential equations (MTFDEs) of the form \(x'(t)=a(t)x(t) + b(t)x(t+1) + c(t)x(t+1)\). Following a brief introduction to MTFDEs we include a review of our progress to date. Alongside our aim of developing new approaches to the numerical solution of MTFDEs we are particularly interested in using our approach in modelling applications. We will discuss some aspects of, and issues relating to, our current investigations, and will present examples to illustrate our numerical approach.

In this talk we are concerned with oscillatory functional differential equations(that is, those equations where all solutions oscillate) under numerical approximation. Our interest is in the preservation of qualitative properties of solutions under numerical discretisation. We give conditions under which an equation is oscillatory, and consider whether the discrete schemes derived using linear methods will also be oscillatory. We conclude with some general theory.

We investigate the steady state Navier-Stokes equations considered in the full space $\mathbb{R}^2$. We suplement the system with a condition at infinity which requires the solution (the velocity) to tend to a prescribed constant vector field. This problem is strictly connected with an open problem of a flow past an obstacle on the plane. The main difficulty there is to assure the convergence of a solution to a prescribed velocity at infinity. We propose a new method to deal with this problem. The class of functions, where we look for a solution, is different from standard Sobolev spaces. This is due to the fact that our analysis is carried through in a Fourier space only in one direction. In these spaces we show existence of solutions together with their basic asymptotic structure.

This talk is concerned with nonlinear stability properties of periodic traveling wave solutions for some classical equations. Periodic traveling wave solutions will be constructed by using the Jacobi elliptic functions. It will be shown that these solutions are nonlinearly stable in the respective energy space by periodic disturbances with period $L$.

We introduce a general class of mixture models suitable to describe water-dependent degradation and erosion of biodegradable polymers in conjunction with drug release from such matrices. The ability to predict and quantify degradation and erosion has direct impact in a variety of biomedical applications and is a useful design tool for biodegradable implants and tissue engineering scaffolds. The model is based on a finite number of constituents describing the polydisperse polymeric system, each representing chains of an average size, and two additional constituents, water and drug. Hydrolytic degradation of individual chains occurs at the molecular level and mixture constituents diffuse individually accordingly to Fick's 1st law at the bulk level – such analysis confers a multiscale aspect to the resulting reaction-diffusion system. A shift between two different types of behavior, each identified to surface or bulk erosion, is observed with the variation of a single nondimensional parameter measuring the relative importance of the mechanisms of reaction and diffusion. Mass loss follows a sigmoid decrease in bulk eroding polymers, whereas decreases linearly in surface eroding polymers. Polydispersity influences degradation and erosion of bulk eroding polymers and drug release from unstable surface eroding matrices is dramatically enhanced in an erosion controlled release.

Adaptive optics is a technology which allows to manipulate the characteristic of a light beam. The talk will introduce adaptive optics principles and main applications. In many scientific experiments adaptive optics has been successfully used as key components for the generation of few optical cycles laser pulses. The talk will explore some techniques for pulse compression and shaping using both modeling of deformable mirror deformation or genetic algorithms.

We introduce the class of so-called cordial Volterra integral operators $V$ (they are non-compact!) and cordial integral equations $u = V u + f$. For example, Diogo's, Lighthill's and Abel's integral equations are cordial under certain conditions. We discuss the relations between cordial equations, Mellin convolution equations and Wiener-Hopf integral equations. We determine the spectra of a cordial operator in different spaces and, in this way, we obtain results about the smoothness of a solution to a cordial integral equation. Our final purpose is to examine the convergence and convergence speed of polynomial and piecewise polynomial (spline) collocation methods for cordial integral equations.