Applied Mathematics and Numerical Analysis Seminar 

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.

Recently, Kulikov [1] presented the idea of double quasi-consistency, which facilitates global error estimation and control, considerably. More precisely, a local error control implemented in such methods plays a part of global error control at the same time. However, Kulikov studied only Nordsieck formulas and proved that there exists no doubly quasi-consistent scheme among those methods. In this paper, we prove that the class of doubly quasi-consistent formulas is not empty and present the first example of such sort. This scheme belongs to the family of superconvergent explicit two-step peer methods constructed by Weiner et al. [2]. We present a sample of s-stage fixed-stepsize doubly quasi-consistent parallel explicit peer methods of order $s-1$ when $s = 3$. The notion of embedded formulas is utilized to evaluate efficiently the local error of the constructed doubly quasi-consistent peer method and, hence, its global error at the same time. Numerical examples confirm clearly that the usual local error control implemented in doubly quasi-consistent numerical integration techniques is capable of producing numerical solutions for user-supplied accuracy conditions in automatic mode. Then, we discuss variable-stepsize explicit parallel peer methods grounded in the interpolation idea. Approximation, stability and convergence are studied in detail. In particular, we prove that some interpolation-type peer methods are stable on any variable mesh in practice. Double quasi-consistency is utilized to introduce an efficient global error estimation formula in the numerical methods under discussion. The main advantage of these new adaptive schemes is the capacity of producing numerical solutions for user-supplied accuracy conditions in automatic mode and almost at no extra cost. This means that a usual local error control mechanism monitors and regulates the global error at the same time because the true error of any doubly quasi-consistent numerical method is asymptotically equal to its local error. Numerical experiments support theoretical results of this paper and illustrate how the new global error control concept works in practice. We also conduct a comparison with explicit ODE solvers in MatLab.

In this talk the computational bio-mechanics (CBM) lines of BSC will be described. They go in the direction of the "Homo Computationalis", where complex CBM problems are solved using efficient parallel solvers, capable of running in supercomputing facilities, in order to get closer to realistic simulations of human organs treated as Physical systems. Three projects will be described in detail: Cerebral Artherial System, Superior Respiratory Airways and Electromechanical Model of the Heart.

We build a family of nonempty closed convex sets using the data of the Generalized Nash equilibrium problem (GNEP), and select the sets iteratively such that the intersection of the selected sets contains solutions of the GNEP. We adapt the algorithm introduced by Iusem-Sosa (2003) to obtain solutions of the GNEP. Finally we give some numerical experiments to illustrate the numerical behavior of the algorithm.

Bifurcations in stochastic delay differential equations are quite difficult to detect. In this talk, we consider how numerical methods can be used to detect changes in the underlying behaviour of the exact solution of these problems. The talk brings together ideas from deterministic delay differential equations, from stochastic ordinary differential equations and from statistical curve-fitting to provide some interesting new insights into the problem.

Modelling and simulation of fluid-structure interaction problems (FSI) has many applications in engineering, biomechanics and medical sciences. For example, modelling elasto-plastic material deformations covered with lubricant or simulation of blood flow in veins and diseases of the cardiovascular system. The mathematical approach for investigating monolithic FSI problems via the 'arbitrary Lagrangian-Eulerian' method (ALE) will be presented. The resulting equations are nonlinear and have been solved by the Newton's method. After that, some results derived so far in my PhD project will be discussed. The focus is on stationary problems (e.g. an obstacle problem) and Benchmark configurations verifying the programming code.

The general thinking is that ocean eddies are Gaussian-like instead of Rankine-like. We will show however that in several oceanic regions, such as around the Canary Islands or in the Gulf of Tanhuentepec (Mexico), eddies are initially Rankine-like and evolve towards Gaussian type. Gaussian-like vortices have a Gaussian distribution of vorticity, smooth shear zones at their periphery, their azimuthal velocity does not vary linearly and anticyclones are unstable. On the contrary, Rankine-like vortices are on solid body rotation, have strong shears at their periphery, the azimuthal velocity field varies linearly and anticyclones are stable. A simple way to detect both vortices is through hydrographic field anomalies. In Rankine-like vortices, anomalies do not have a well-defined center and gradients increase towards the periphery. In Gaussian type vortices, anomalies have a clear center with gradients increasing toward the vortex center. In Rankine vortices there is a strong diapycnal mixing at their periphery that enhances primary production.

In this talk a finite-volume discretisation of a Lattice Boltzmann equation over unstructured grids is presented. The new scheme is based on placing the unknown fields at the nodes of the mesh and evolve them based on the fluxes crossing the surfaces of the corresponding control volumes. The method, named unstructured Lattice Boltzmann equation (ULBE) is applied here to the problem of blood flow over the endothelium in small arteries. The study shows a significant variation of wall shear stress to the corrugation degree of the endothelium.

I will discuss Mathematical Immunology, concentrating on T cells. In the first part of the talk, I will consider how a diverse repertoire of T-cell clonotypes is maintained. Understanding the dynamics of "homeostasis" leads to a continuous-time Markov chain model. In the second part of the talk I will discuss timescales for the interaction of T cells with Dendritic cells in a lymph node, which leads to a continuous-space Brownian motion model

The unsteady Poiseuille flow, describing the motion of a viscous incompressible fluid in an infinite straight pipe of constant cross-section $\sigma$, is defined as a solution of the inverse problem for the heat equation on $\sigma$. The existence and uniqueness of such flow with the prescribed flow rate $F(t)$ is proved for Newtonian and second grade fluids. It is shown that the flow rate $F(t)$ and the axial pressure drop $q(t)$ are related, at each time $t$, by a linear Volterra integral equation of the second type, where the kernel depends only upon $t$ and $\sigma$. One significant consequence of this result is that it allows us to prove that the inverse parabolic problem of finding a Poiseuille flow corresponding to a given $F(t)$ is equivalent to the resolution of the classical initial-boundary value problem for the heat equation. The behavior as $t\to\infty$ of this unsteady Poiseuille solution is studied. In particular, it is proved that in the case, where the flow rate $F(t)$ exponentially tends to a constant $F_*$, the non-stationary Poiseuille solution tends as $t\to\infty$ to the steady Poiseuille flow corresponding to the flow rate $F_*$. The unsteady Navier-Stokes system is studied in a two-dimensional domain with strip-like outlets to infinity in weighted Sobolev function spaces. It is proved that under natural compatibility conditions there exists a solution with prescribed flow rates over cross-sections of outlets to infinity and that this solution tends in each outlet to the corresponding unsteady Poiseuille flow. The decay rate of the solution is conditioned only by the decay rate of an external force and initial data. The obtained results are true for arbitrary values of norms of the data (in particular, for arbitrary fluxes) and globally in time. For the three-dimensional domain with cylindrical outlets to infinity the analogous results are obtained either for small data or for a small time interval.