Applied Mathematics and Numerical Analysis Seminar 

16/07/2015, 15:00 — 16:00 — Room P3.10, Mathematics Building
Nuno Lopes, Instituto Superior de Engenharia de Lisboa

Analytical and Numerical Methods of the type FEM-C/D for Improved Boussinesq Models

In this talk, some analytical and numerical models are developed for the generation and propagation of surface water waves. These problems are solved using asymptotic and numerical methods. Regarding the numerical methods, we consider the continuous and continuous/discontinuous Galerkin finite element methods (FEM-C/D) with penalty terms. In the first problem, the model of Zhao et al. (2004) is extended in order to include some effects like dissipation and absorption of the energy of the surface water waves. We show that this model is robust with respect to the instabilities related to steep bottom gradients of the bathymetry. A new class of nonlinear Boussinesq-type systems is derived in the second problem. A CFL type condition is obtained for the linearized problem with constant bathymetry. The consistency of the dispersion relation as well as the good stability properties of this model are verified. From the numerical tests, we can conclude that the proposed numerical model is appropriate to model surface water waves. In the third problem, a class of Korteweg, de Vries–Benjamin, Bona and Mahony (KdV-BBM) type equations is deduced. The Nwogu’s parameter is determined in order to optimise the velocity potential of the linearized KdV-BBM model. Moreover, a numerical analysis of the proposed model is performed. We conclude that the KdV-BBM model is less prone to instabilities than the KdV model. Finally, a new Boussinesq-type differential equation of sixth-order to model bidirectional waves is derived and exact travelling wave solutions are obtained. A new analytical travelling wave solution is found. This is a joint work with P. J. S. Pereira and L. Trabucho.

25/06/2015, 15:00 — 16:00 — Room P3.10, Mathematics Building
Svilen S. Valtchev, IPL & CEMAT

Extending the Range of Application of the Method of Fundamental Solutions

The method of fundamental solutions (MFS) is a meshfree boundary collocation method, originally developed for the numerical solution of homogeneous PDEs, coupled with sufficiently regular boundary conditions. In this talk we show how to extend the range of application of the classical MFS to two more general situations.

In the first case, the numerical solution of a non-homogeneous PDE is addressed. Here, fundamental solutions with different source points and test frequencies are used as shape functions in order to approximate the solution of the Cauchy-Navier equations of elastodynamics.

In the second situation, a potential problem with singular (discontinuous) boundary conditions is solved. In order to reduce the effects of the Gibbs phenomenon in the neighborhood of the singularities, the MFS approximation basis is augmented with a set of harmonic functions which possess a discontinuous inner trace at the boundary of the domain.

In both methods presented here the meshfree characteristics of the original MFS are preserved and the total approximate solution of the BVP is calculated by solving a single collocation linear system. Several 2D numerical examples will be presented in order to illustrate the performance of the methods.

27/05/2015, 14:30 — 15:30 — Room P5.18, Mathematics Building
Vincenzo Coscia, Department of Mathematics, University of Ferrara, Italy

Modeling complexity: From vascular biomechanics to social systems

In this talk I will give insight on the activity of the research center Mathematics for Technology, Medicine & Biosciences of the University of Ferrara. I'll report on the studies concerning the mechanics of the human venous system as well as on the mathematical modeling of social systems, such as pedestrian dynamics and vehicular traffic, with the common paradigm of complexity modeling.

07/05/2015, 15:00 — 16:00 — Room P3.10, Mathematics Building
Pedro Antunes, GFM-ULisboa

Numerical calculation of localized eigenfunctions of the Laplacian

It is well known that for some planar domains, the Laplacian eigenfunctions are localized in a small region of the domain and decay rapidly outside this region. We address a shape optimization problem of minimizing or maximizing the $L^2$ norm of the eigenfunctions in some sub-domains. This problem is solved by a numerical method involving the Method of Fundamental Solutions and the adjoint method for a fast calculation of the shape gradient.

Several numerical simulations illustrate the good performance of the method.

23/04/2015, 15:00 — 16:00 — Room P3.10, Mathematics Building
Pedro Serranho, Universidade Aberta

Applied mathematics to the imaging of the human retina

In this talk we will focus on mathematical problems arising in the field of medical imaging and their possible numerical solutions, namely in the imaging of the human retina. We will focus on how mathematical methods (for classification and solving partial differential equations) applied to real medical imaging data can provide additional information and simulations with the potential to be an aid to the early diagnosis of diseases affecting the retina, as for instance diabetes.

12/02/2015, 14:30 — 15:30 — Room P3.10, Mathematics Building
Paolo Falsaperla, Dipartimento di Matematica e Informatica, Università di Catania, Catania, Italy

A mathematical model of anorexia and bulimia

Time evolution of pathological and harmful behaviors (such as binge drinking or drug consumption [1, 2]) can be modeled in the context of epidemiological models [3]. In this paper we propose a mathematical model to study the dynamics of anorexic and bulimic populations inspired by the model of Gonzalez et al. [4]. The model proposed takes into account, among other things, the effects of peers' influence, media influence, and education. We prove the existence of three possible equilibria, that without media influences are disease-free, bulimic-endemic, and endemic. Neglecting media and education effects we investigate the stability of such equilibria, and we prove that under the influence of media, only one of such equilibria persists and becomes a global attractor. Which of the three equilibria becomes global attractor depends on the other parameters.

Joint work with C. Ciarcia, A. Giacobbe, G. Mulone.

References

1. Mulone G, Straughan: Modeling binge drinking, Int. J. Biomath., 5(1), 1250005 (2012).
2. Mulone G, Straughan B: A note on heroin epidemics, Math. Biosci. 218, 118-141 (2009).
3. Hethcote HW: The mathematics of infectious diseases, SIAM Rev. 42, 599-653 (2000).
4. Gonzalez B et al.: Am I too fat? Bulimia as an epidemic, J. Math. Psych. 47, 515-526 (2003).

24/11/2014, 11:00 — 12:00 — Room P3.10, Mathematics Building
Sinai Robins, Brown University

Cone theta functions and rationality of spherical volumes

20/11/2014, 15:00 — 16:00 — Room P3.10, Mathematics Building
Javad Hatami, IBB-Institute for Biotechnology and Biosciences, IST, Univ Lisbon

A Mathematical Approach To Model Human Megakaryopoiesis Process in vitro

Megakaryopoiesis is a complex process, which is commenced with the proliferation and the differentiation of hematopoietic stem cells (HSC) into megakaryocytes (Mk), followed by maturation and polyploidy of Mk and ended by platelet biogenesis. An in vitro two-stage protocol including HSC expansion and Mk lineage commitment of human umbilical cord blood cells (hUCB) were established [1]. In the first stage, hUCB CD34+-enriched cells were expanded in co-culture with bone marrow human mesenchymal stem cells (BM hMSC) in a cytokines cocktail pre-optimized for CD34+ expansion. In the second stage, expanded cells were differentiated toward Mk lineage using a cocktail containing TPO and IL-3 in a serum-free medium. Phenotypic characterization of cells was performed by Flow cytometry. In order to describe the fate of HSC during the megakaryopoiesis, a mathematical approach was used based on kinetic modeling of cell expansion and differentiation. This kind of modeling, which computes the concentration of each subset during the time, can provide significant insight into the limiting step involved in the protocol and how the interaction of different factors can affect the outcome of megakaryopoiesis process. A set of ordinary differentiation equation (ODE) were used to analyze the proliferation and differentiation of UCB CD34+ cells, as evaluated by the number of HSC (CD34+ cells), Mk (CD41+ cells) and platelets (CD42b+ cells). These ODEs were solved and a general solution for each subset was fitted to the experimental result, using least square method, to determine the unknown coefficient factors. The establishment of such reliable kinetic model will be useful for development of an efficient bioreactor system devoted for production of specific hematopoietic product.References:1. Hatami J, Andrade PZ, Bacalhau D, Cirurgião F, Ferreira FC, et al. (2014) Proliferation extent of CD34+ cells as a key parameter to maximize megakaryocytic differentiation of umbilical cord blood-derived hematopoietic stem/progenitor cells in a two-stage culture protocol. Biotechnology Reports 4: 50-55.

06/11/2014, 15:00 — 16:00 — Room P3.10, Mathematics Building
Anca-Maria Toader, CMAF and Faculdade de Ciências da Universidade de Lisboa

The Adjoint Method in Optimization of Eigenvalues and Eigenmodes

The Adjoint Method goes back to the works of Pontryagin in the framework of Ordinary Differential Equations. In the eighties, J. Cea employed the Adjoint Method in a practical way, from the perspective of Lagrange multipliers. Since then, applications of the Adjoint Method were successfully used in Shape Optimization, Topology Optimization and very recently to optimize eigenvalues and eigenmodes (eigenvectors).

The main contribution of this study is to show how the Adjoint Method is applied to the optimization of eigenvalues and eigenmodes.  The general case of an arbitrary cost function will be detailed. In this framework, the direct problem does not involve a bilinear form and a linear form as usual in other applications. However, it is possible to follow the spirit of the method and deduce $N$ adjoint problems and obtain $N$ adjoint states, where $N$ is the number of eigenmodes taken into account for optimization.

An optimization algorithm based on the derivative of the cost function is developed. This derivative depends on the derivatives of the eigenmodes and the Adjoint Method allows one to express it in terms of the the adjoint states and of the solutions of the direct eigenvalue problem.

This method was applied in [1] for material identification purposes in the framework of free material design. In [2] this study is applied to optimization of microstructures, modeled by Bloch wave techniques.

References

1. S. Oliveira, A.-M. Toader, P. Vieira, Damage identification in a concrete dam by fitting measured modal parameters. Nonlinear Analysis: Real World Applications, 13, Issue 6, 2888-2899, 2012.
2. C. Barbarosie, A.-M. Toader, The Adjoint Method in the framework of Bloch Waves (in preparation).

09/10/2014, 14:30 — 15:30 — Room P3.10, Mathematics Building
Sílvia Barbeiro, CMUC, Department of Mathematics, University of Coimbra

Modeling electromagnetic wave’s propagation in human eye’s structure

In this talk we will discuss the a mathematical model that describes the electromagnetic wave’s propagation through the eye’s structures in order to create a virtual OCT scan. Our model is based on time-dependent Maxwell’s equations. We use the discontinuous Galerkin method for the integration in space and a low-storage Runge-Kutta method for the integration in time. In the model we consider anisotropic permittivity tensors which arise naturally in our application of interest. We illustrate the performance of the method with some numerical experiments.

09/09/2014, 11:30 — 12:30 — Room P3.10, Mathematics Building
Maria Specovius-Neugebauer, University of Kassel, Germany

The time periodic Stokes system in a layer: asymptotic behavior at infinity

While there are numerous papers on the time decay of solutions to the Stokes and Navier-Stokes initial boundary value in various types of domains only few results are devoted to the spatial decay. In this talk we consider the solutions to the time periodic Stokes problem in a layer where the data are also time periodic and smooth with bounded support for simplicity. The results were obtained in a joint work with Konstantin Pileckas,Vilnius.

23/07/2014, 11:00 — 13:00 — Room P3.10, Mathematics Building
Willi Jäger, Maria Neuss-Radu, University of Heidelberg, University of Erlangen-Nuremberg (resp.)

Interactions of the fluid and solid phases in complex media — coupling reactive flows, transport and mechanics, and applications to medical processes.

Modelling reactive flows, diffusion, transport and mechanical interactions in media consisting of multiple phases, e.g. of a fluid and a solid phase in a porous medium, is giving rise to many open problems for (multi-scale) analysis and simulation. In this lecture, the following processes are studied:

• diffusion, transport, and reaction of substances in the fluid and the solid phases,
• mechanical interactions of the fluid and solid phases,
• change of the mechanical properties of the solid phase by chemical reactions,
• volume changes (“growth”) of the solid phase.

These processes occur for instance in soil and in porous materials, but also in biological membranes, tissues and in bones. The model equations consist of systems of nonlinear partial differential equations, with initial-boundary conditions and transmission conditions on fixed or free boundaries, mainly in complex domains. The coupling of processes on different scales is posing challenges to the mathematical analysis as well as to computing. In order to reduce the complexity, effective macroscopic equations have to be derived, including the relevant information from the micro-scale.

In case of processes in tissues, a homogenization limit leads to an effective, mechanical system, containing a pressure gradient, which satisfies a generalized, time-dependent Darcy law, a Biot-law, where the chemical substances satisfy diffusion-transport-reaction equations and are influencing the mechanical parameters.

The interaction of the fluid and the material transported in a vessel with its flexible wall, incorporating material and changing its structure and mechanical behavior, is a process important e.g. in the vascular system (plaque-formation) or in porous media. 

The modeling and analytic aspects addressed in our talk are also highly relevant for the study of inflammatory processes. 

The lecture is based on recent results obtained in cooperation with A. Mikelic, F. Weller and Y. Yang. 

05/06/2014, 16:00 — 17:00 — Room P3.10, Mathematics Building
Alexander G. Ramm, Department of Mathematics, Kansas State University

Wave scattering by many small particles and creating materials with desired refraction coefficients

Many-body wave scattering problems are solved asymptotically, as the size \(a\) of the particles tends to zero and the number of the particles tends to infinity.

Acoustic, quantum-mechanical, and electromagnetic wave scattering by many small particles is studied. This theory allows one to give a recipe for creating materials with a desired refraction coefficient.

One can create material with negative refraction, that is, the group velocity in this material is directed opposite to the phase velocity. One can create material with some desired wave-focusing properties. For example, one can create a new material which scatters plane wave mostly in a fixed given solid angle.

15/05/2014, 15:00 — 16:00 — Room P3.10, Mathematics Building
Jahed Naghipoor, CMUC, Universidade de Coimbra

A non-Fickian reaction diffusion equation for polymeric stent embedded in the arterial wall

In recent years, mathematical modeling of cardiovascular drug delivery systems has become an effective tool to gain deeper insights in the cardiovascular diseases like atherosclerosis.. In the case of coronary biodegradable stent which is a tiny expandable biocompatible metallic mesh tube covered by biodegradable polymer, it leads to a deeper understanding of the drug release mechanisms from polymeric stent into the arterial wall. A coupled non-Fickian model of a cardiovascular drug delivery system using a biodegradable drug eluting stent is proposed in this talk. Energy estimates are used to study the qualitative behavior of the model. The numerical results are obtained using an IMEX finite element method. The influence of arterial stiffness in the sorption of drug eluted from the stent is analyzed. The results presented in this talk open new perspectives to adapt the drug delivery profile to the needs of the patient.

27/03/2014, 15:00 — 16:00 — Room P3.10, Mathematics Building
Arvet Pedas, Institute of Applied Mathematics, Tartu University, Estonia

High-order methods for Volterra integral equations with weak singularities

We consider the numerical solution of some classes of linear Volterra integral equations with singularities. We apply to them a smoothing transformation so that the exact solution of the resulting equation does not contain any singularities in its derivatives up to a certain order. After that the regularized equation is solved by a piecewise polynomial collocation method on a mildly graded or uniform grid. Global convergence estimates are derived and some superconvergence results are given.

27/03/2014, 14:00 — 15:00 — Room P3.10, Mathematics Building
Ruediger Weiner, Institut für Mathematik, University of Halle, Germany

Global error control with explicit peer methods

Step size control in the numerical solution of initial value problems $$y\prime =f(t,y),y(t_0)=y_0$$ is usually based on the control of the local error. We present numerical tests showing that this may lead to high global errors, i.e. the real error is much larger than the prescribed tolerance. There is a tolerance proportionality, with more stringent tolerances also the global error is reduced. However, tolerance and achieved global error may differ by several magnitudes. A very simple idea to overcome this problem is to use two methods of different orders with same step size sequences and local error control for the lower order method. Then the difference of the numerical approximations of both methods is an estimate of the global error of the lower order method. This strategy was implemented for pairs of explicit peer methods in Matlab . Numerical tests show the reliability of this approach. The numerical costs are comparable with those of ode45, but in contrast to ode45 the required accuracy is achieved.

13/03/2014, 14:30 — 15:30 — Room P3.10, Mathematics Building
Elias Gudino, CMUC, Universidade de Coimbra

A 3D model for mechanistic control of drug release

A 3D mathematical model for sorption/desorption by a cylindrical polymeric matrix with dispersed drug is proposed. The model is based on a system of partial differential equations coupled with boundary conditions over a moving boundary. We assume that the penetrant diffuses into a swelling matrix and causes a deformation which induces a stress driven diffusion and consequently a non-Fickian mass flux. A physically sound non linear dependence between strain and penetrant concentration is considered and introduced in a Boltzmann integral with a kernel computed from a Maxwell-Wiechert model. Numerical simulations show how the mechanistic behavior can have a role in drug delivery design.

11/12/2013, 14:00 — 16:00 — Room V1.08, Civil Engineering Building, IST
Marco Leite, UCL Institute of Neurology and Instituto de Sistemas e Robótica, IST

Modelling populations of integrate and fire neurons: a Fokker-Planck approach to population density dynamics

Much of the phenomenology of interest in the field of neuroscience arises from the interaction of large populations of densely interconnected neurons (~\(10^5\) neurons per mm³ of mammal cortex, averaging \(10^4\) connections per neuron). Different levels of abstraction may be adopted when modelling such systems, and these need to be well suited with regards to the phenomena one is interested in studying. Here we aim at the study of the (sparse) synchronization of neurons observed during electrophysiologically recorded fast oscillatory behavior of networks of large populations. For that we use a ubiquitous simplified neuronal model - the conductance based leaky integrate and fire neuron. This model may be described by a one dimensional stochastic differential equation. Under mean field assumptions we may describe, using a linear Fokker-Planck equation, the behavior of a single population of uncoupled neurons with a PDE. The coupling of different populations will render this Fokker-Planck equation strongly non-linear. In this presentation I will also explore some details of such modelling approaches, namely: the non-natural boundary conditions generated by the neuronal firing mechanism and the numerical scheme used to deal with the brittleness from there ensued. I will also present results on the types of behavior, data, and statistics that such modelling approach is able to predict, e.g. neuronal (a)synchrony, neuronal input currents, firing rates, inter spike intervals, etc... This type of approach allows for a computationally tractable and scalable study of networks of populations of neurons. In the future we plan to implement parameter estimation algorithms to this family of models.

26/11/2013, 15:00 — 16:00 — Room P3.10, Mathematics Building
G. E. Chatzarakis, School of Pedagogical and Technological Education, Athens, Greece

Oscillations of Difference Equations of Several Deviating Arguments

Sufficient conditions are given for the oscillation of all solutions of a certain retarded linear difference equation and for its (dual) advanced difference equation.

Examples illustrating the results are also given.

See also

https://www.math.tecnico.ulisboa.pt/~plima/Chatzar.pdf

26/11/2013, 14:00 — 15:00 — Room P3.10, Mathematics Building
I. P. Stavroulakis, University of Ioannina, Greece

Oscillations of Delay and Difference Equations

Results are presented about the oscillation of all solutions of a certain class of first order linear retarded differential equations and their discrete analogues.

See also

https://www.math.tecnico.ulisboa.pt/~plima/Stavrou.pdf