# Applied Mathematics and Numerical Analysis Seminar

## Past sessions

### Reconstruction of PDE coefficients with overprescripton of Cauchy data at the boundary

Frequently incomplete information about coefficients in partial differential equations is compensated by overprescribed Cauchy data on the boundary.

We analyse this kind of boundary value problems in an elliptic system in Lipschitz domains. Main techniques are variational formulation, boundary integral equations and the Calderon projector. To estimate those coefficients we propose a variational formulation based on internal discrepancy observed in the mixed boundary value problem, obtained by splitting the overprescribed Cauchy data. Some numerical experiments are presented.

### Models for sustainable biodiesel production

Several countries have already begun to invest in alternative energies due to smaller and smaller fossil fuel resources. In particular, for biodiesel production the Jatropha curcas appears to be a possible resource, in that it thrives even in harsh and very dry conditions. From its seeds a relevant quantity of oil can be extracted, for production of high grade biodiesel fuel.

But this plant is subject to parasitism from a mosaic virus, the Begomovirus, that is carried by white-flies Bemisia tabaci. The talk is centered on the investigation of two models for the fight of this plant infection.

In the case of large plantations we investigate the optimal insecticide spraying policy. Here the most relevant parameters of the ecosystem appear to be the infection transmission rate from vectors to plants and the vector mortality. The results indicate that spraying should be administered only after 10 days of the epidemics insurgence, relentlessly continued for about three months, after which disease eradication is obtained, [2].

At the small scale instead, we consider possible production by individuals, that cultivate this plant in small plots that would be otherwise be left wild and unproductive, [1]. We consider the effects of media campaigns that keep people aware of this plant disease, and indicate means for fighting it. The model shows that awareness campaigns should be implemented rather intensively, in order to effectively reduce or completely eradicate the infection.

References

[1] Priti Kumar Roy, Fahad Al Basir, Ezio Venturino (2017) Effects of Awareness Program for Controlling Mosaic Disease in Jatropha curcas Plantations, to appear in MMAS.

[2] Ezio Venturino, Priti Kumar Roy, Fahad Al Basir, Abhirup Datta (2016) A model for the control of the Mosaic Virus disease in Jatropha Curcas plantations, to appear in Energy, Ecology and Environment, doi:10.1007/s40974-016-0033-8.

(work in collaboration with Fahad Al Basir and Priti K. Roy, Javadpur University, India)

### A hierarchy of models for the flow of fluids through porous solids

The celebrated equations due to Fick and Darcy are approximations that can be obtained systematically on the basis of numerous assumptions within the context of Mixture Theory; these equations however not having been developed in such a manner by Fick or Darcy. Relaxing the assumptions made in deriving these equations via mixture theory, selectively, leads to a hierarchy of mathematical models and it can be shown that popular models due to Forchheimer, Brinkman, Biot and many others can be obtained via appropriate approximations to the equations governing the flow of interacting continua. It is shown that a variety of other generalizations are possible in addition to those that are currently in favor, and these might be appropriate for describing numerous interesting technological applications.

### Perturbative methods in thermal imaging

I will discuss shortly three different inverse problems in the field of thermal nondestructive testing. In all cases we have a thin metallic plate $\Omega_0$ whose top boundary $S_{top}$ is not accessible while we are able to operate on the opposite surface $S_{bot}$.

We heat the specimen by applying a heat flux of density $\phi$ on $S_{bot}$ and measure a sequence of temperature maps.

Problem 1. Recover a perturbation of the heat transfer coefficient on $S_{top}$. [Inglese, Olmi - 2017]

Problem 2. Recover a surface damage on the inaccessible side $S_{top}$. [Inglese, Olmi - in progress]

Problem 3. Recover a nonlinear heat transfer coefficient on $S_{top}$. [Clarelli, Inglese - 2016]

### Two cubic spline methods for solving Fredholm integral equations

In this lecture, we propose two methods based on the use of natural and quasi cubic spline interpolations for approximating the solution of the second kind Fredholm integral equations.

Convergence analysis is established. Some numerical examples are given to show the validity of the presented methods.

Notice the Room: Q 4.6

### Numerical investigation of the free boundary regularity for a degenerate advection-diffusion problem

In this talk I will describe the free boundary regularity of the traveling wave solutions to a degenerate advection-diffusion problem of Porous Medium type, whose existence was proved in a previous work with A. Novikov (PennState Univ, USA) and J.-M. Roquejoffre (IMT Toulouse, France). I will set up a finite-difference scheme allowing to compute approximate solutions and capture the free boundaries, and will carry out a numerical investigation of their regularity. Based on some nondegeneracy assumptions supported by solid numerical evidence, I will show the Lipschitz regularity of the free boundaries. Simulations indicate that this regularity is optimal, and the free boundaries seem to develop Lipschitz corners at least for some values of the nonlinear diffusion exponent. I will also discuss the existence of corners in the more analytical framework of viscosity solutions to certain periodic Hamilton-Jacobi equations, whose validity will be again supported by numerical evidence.

### An MFS-fading regularization approach to inverse BVPs in two-dimensional linear elasticity.

We investigate the numerical reconstruction of the missing displacements (Dirichlet data) and tractions (Neumann data) on an inaccessible part of the boundary in the case of a linear isotropic elastic material from the knowledge of over-prescribed noisy measurements taken on the remaining accessible boundary part. This inverse problem is solved using the fading regularization method, originally proposed by Cimetière et al. (2000, 2001) for the Laplace equation, in conjunction with a meshless method, namelythe method of fundamental solutions (MFS). The stabilization of the numerical method proposed herein is achieved by stopping the iterative procedure according to Morozov's discrepancy principle.

This is a joint work with Franck Delvare (University of Caen) and Alain Cimetière (University of Poitiers).

### Aplicações da Matemática a Simulações do Sistema Cardiovascular

Divulgação do trabalho desenvolvido no departamento de matemática do IST, relativo à modelação do sistema cardiovascular.

(Dirigido a alunos do 12º ano da Escola Secundária Seomara da Costa Primo, a frequentar o curso de Ciência e Tecnologia.)

Em colaboração com Jorge Tiago (CEMAT-IST).

### The Traceless Oldroyd viscoelastic model

In the traceless Oldroyd viscoelastic model, the viscoelastic extra stress tensor is decomposed  into its traceless (deviatoric) and spherical parts, leading to a reformulation of the classical Oldroyd model. The equivalence of the two models is established comparing model predictions for simple test cases. The new  model  is validated using several 2D benchmark problems. The structure and behaviour of the new model are discussed and the future use of the new model in envisioned, both on the theoretical and numerical perspectives.

Work in collaboration with T. Bodnár.

#### References

Bodnár, T., Pires, M. and Janela, J. (2014). Blood Flow Simulation Using Traceless Variant of Johnson-Segalman Viscoelastic Model. Mathematical Modelling of Natural Phenomena 9,(6), 117-141.

### FibReGen: Modelling Myofibre Regeneration

In young individuals, myofibres are capable of altering their profile in response to perturbation, but plasticity of ageing skeletal muscle is less clearly understood. The age-related loss of muscle mass in sarcopenia is mediated by a reduction in the total number of myofibres, a decrease in size of fast-twitch myosin heavy chain fibres and altered morphology. These maladaptations create negative metabolic and functional implications that impede healthy ageing.

Despite modern advances, Duchenne muscular dystrophy (DMD) remains fatal and incurable. Muscle is extensively replaced by adipose tissue, and heart failure often results. We propose to model for the molecular pathogenesis centred around the increased susceptibility of glycolytic fibres to degeneration in DMD and connect the histological findings of hypercontracted fibres, segmental necrosis and grouped necrosis to glycolytic fibres and investigate recent evidence from animal models suggesting that oxidative fibre type switching may ameliorate the effects of the disease.

Early physiological changes often start at the cellular or fascicular level, which is beyond the capabilities of conventional MRI. Histology, requiring invasive biopsies, is necessary to assess early treatment and training effects. Diffusion tensor imaging (DTI) provides a sensitive noninvasive readout of early physiological changes in tissue microstructure. DTI can also be applied for in vivo quantification and 3D visualisation of the macroscopic muscle fibre architecture.

The aim of FibReGen is to develop subject-specific and patient-specific computational models of skeletal and cardiac muscle entirely from MRI data. These computational models will integrate anatomical, functional, metabolic and mechanical data, and will characterise fibre type proportion and interconversion in a wide-ranging spectrum of subjects including elite athletes, those with age-related sarcopenia and patients with DMD.

Financial support:

### Optimal control of epidemiological models

We apply optimal control theory to a Tuberculosis (TB) and a TB-HIV/AIDS co-infection models. The models are given by systems of ordinary differential equations.

For the TB model, optimal control strategies are proposed to minimize the number of active infectious and persistent latent individuals, as well as the cost of interventions. A cost-effectiveness analysis is done, to compare the application of each one of the control measures, separately or in combination.

We introduce delays in the TB model, representing the time delay on the diagnosis and commencement of treatment of individuals with active TB infection. The stability of the disease free and endemic equilibriums is investigated for any time delay. Corresponding optimal control problems, with time delays in both state and control variables, are formulated and studied.

We propose a model for TB-HIV/AIDS coinfection transmission dynamics. We analyze separately the HIV-only, TB-only and TB-HIV/AIDS models. The respective basic reproduction numbers are computed, equilibria and stability are studied. Optimal control theory is applied to the TB-HIV/AIDS model and optimal treatment strategies for co-infected individuals with HIV and TB are derived. Numerical simulations to the optimal control problem show that non intuitive measures can lead to the reduction of the number of individuals with active TB and AIDS.

### Effective behavior of lattices with angular interactions

Angular interactions are of primary importance in mechanical trusses that they stabilize as well as in atomistic lattices, see Allinger and Tersoff-Brenner potentials. Graphenes are nowadays the best known example of hexagonal lattices. We will concentrate on the behavior of 2d-lattices undergoing deformations in the 3d-space, where major difficulties are already present when seeking for an equivalent behavior. We will give an example where homogenization is not required in the formulation of an equivalent continuous problem. We will show that for hexagonal lattices, on the contrary, homogenization is required even when only bond energy is taken into account. When angular interactions are added, we characterize the equivalent behavior by an alternate method. We will discuss the practical interest of the representation formulas.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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).

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