Applied Mathematics and Numerical Analysis Seminar 

In this talk, we discuss the following two problems. First, we present the Auxiliary Performance Functional (API) developed by Prof. I.V. Semushin and study its role in the state and parameter estimation of linear discrete-time stochastic systems. The minimization procedure of the API with respect to parameters of the Data Model is then considered. Our approach differs from what has been done earlier in the adaptive filtering theory. We may mention that the minimization with respect to the parameters of the Adaptive Filter instead of the Data Model was considered previously. Second, we concern with robust array adaptive filters grounded in SR and UD covariance matrix factorizations used for the API gradient evaluation in identification algorithms. As a reallife example, we consider an application of the API approach to linear time-invariant statespace stochastic MIMO filter systems arising in human body temperature daily variation adaptive stochastic modeling. Simulation results and conclusions are also provided. Key words: linear stochastic system, parameter estimation, model identification, Auxiliary Performance Functional (API) approach, state sensitivity evaluation methods, stochastic modeling, homeostasis, thermoregulation.

This talk will start with an introduction, where we present the Laboratory of Mathematical Modeling which was recently established in Ulyanovsk State Pedagogical University. The report will cover its hardware and software facilities and main areas of conducted research: cosmology, molecular biology, combinatorial optimization, parameter identification. Then we will switch to the main subject of the talk. We consider the state minimization problem for nondeterministic finite automata (NFA) which is known to be computationally hard (PSPACE-complete) and introduce ReFaM – a software tool for its solution. This software tool provides a number of parallel algorithms: parallel versions of exact Kameda- Weiner and Melnikov methods as well as their hybrids with popular metaheurstics (genetic algorithm, simulated annealing, etc.). All parallel algorithms are implemented using MPI and OpenMP techniques. We discuss the implementation details and provide the results of numerical experiments

Observations of the microstructure of objects by means of a microscope are carried out in different technical fields. The state organization of iron with quenching-annealing and the human tissue by biopsy are typical examples. Since the results of such observations depend on the skills of the technician, objective methods are required for the quantification of structures. When the observed structures are very complex, the performance of pattern recognition and Fourier methods is not satisfactory. In this talk, I will introduce an algorithm based on the concept of homology. By applying this method, we obtain rigorous quantitative estimates of a structure. Several examples of its application will be presented.

Blood coagulation is an extremely complex biological process in which blood forms clots to prevent bleeding, following by their dissolution and the subsequent repair of the injured tissue. The process involves different interactions between the plasma, the vessel wall and platelets having a huge impact of the flowing blood on the thrombus growth regularization. The blood coagulation model we are working on consists of a system of reaction-advection-diffusion equations, describing the cascade of biochemical reactions, coupled with rheological models for the blood flow (Newtonian, shear-thinning and viscoelastic models). We introduce the effect of blood slip at the vessel wall emphasizing an extra supply of activated platelets to the clotting site. We expect that such contribution could be dominant, resulting in the acceleration of thrombin production and eventually of the whole clot progression. Such model will have the capacity to predict effects of specific perturbations in the hemostatic system that can’t be done by laboratory tests, and will assist in clinical diagnosis and therapies of blood coagulation diseases. A mathematical model and numerical results for thrombus development will be presented. The chain of biochemical reactions interacting with the platelets, resulting in a fibrin-platelets clot formation and the additional blood flow influence on thrombus development will be discussed.

In the present talk we will discuss some properties of Integral-Algebraic Equations (IAE) and their numerical solution. Concerning analytical methods, me will consider IAE with kernels of convolution type and weakly singular kernels. We will also discuss the regularization of IAE. The discussion of numerical methods will start with the case of systems of Volterra integral equations of the first kind. Then we will describe multistep methods for IAE.

More than 2500 years after the famous statement πάντα ῥεῖ by Heracleitos the investigation of the mechanical and dynamical behavior of fluid flow is more than ever of fundamental importance. Due to a large number of technical, experimental and computational innovations and related theoretical problems the investigation of fluid flow represents a challenging and exciting subject requiring a wide variety of profound mathematical methods, efficient numerical algorithms and complex experimental simulations. Fascinating from the mathematical point of view, of course, is the fact that the fundamental equations of Navier–Stokes, formulated the first time by the French engineer Navier in 1822, could not be solved in the general three–dimensional case up to now. So the famous American Clay Mathematics Institute created the Navier–Stokes Millennium Price Problem and offered one Million US–Dollar for its solution, stating: „Although the Navier–Stokes equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory, which will unlock the secrets hidden in the Navier–Stokes equations“.

The lecture introduces the Navier–Stokes equations from an historical and physical point of view, touches some fundamental mathematical problems of viscous incompressible fluid flow and ends up with new regularity results.

Proteins are large molecules composed of linear chains of connected aminoacids, being largely responsible for the processes taking place in living organisms. The conformation (spatial structural arrangment) adopted by those chains ensures the chemical and physical properties required for proper protein functioning, with a loss of the normal conformational features leading to malfunction or disease. Therefore, a detailed characterization of the distribution of conformations is determinant to understand and rationalize the behavior of proteins. This talk discusses some aspects and open issues in the study of protein conformational behavior in the field of Molecular Modelling. Although deterministic or stochastic computational methods can be used to sample protein conformations, the sampling is often partial and many alternative dissimilarity measures and classification algorithms exist. Thus, questions remain about the suitability of available measures, conformational spaces and clustering methods for properly reflecting the underlying energetics of these molecules and, ultimately, their thermodynamic and kinetic properties. Overall, the viewpoint adopted in the talk is that of a physical biochemist trying to take advantage of mathematical methods (and hoping to get helpful suggestions from the audience).

Quadrature rules are usually classified in terms of their degree of exactness. We show that the vector of weights of a rule, as the least-squares or minimax solution of a fundamental linear system, leads to a new parameter which we call the angle of the rule. Numerical evidence for some standard quadrature rules shows that the angle of a rule can be more interesting than its degree of exactness, providing an explanation why low degree rules can be almost performant as rules of much higher degree.

We aim to provide an accurate representation of industrial CFD activities in Portugal. For this purpose we review some work undertaken at blueCAPE. The presentation is thus comprised of three sections: basic description of physical models and numerical discretization techniques; application examples from industrial applications; hardware developments and their implications in CFD codes. The first section sets the context and the nomenclature for the examples that follow in broad technical and scientific terms. The second section puts forward some examples taken from a number of industrial problems that blueCAPE has addressed through CFD analysis. The third and final section discusses some of the implications that the widening gap between fine and coarse-grained parallelism imposed by hardware developments have on the current generation of commercial CFD codes. Questions will be taken.

We consider the Navier-Stokes equations in a 2D-bounded domain with general non-homogeneous Navier slip boundary conditions prescribed on permeable boundaries, and study the vanishing viscosity limit. We prove that solutions of the Navier-Stokes equations converge to solutions of the Euler equations satisfying the same Navier slip boundary condition on the inflow region of the boundary.

The communication is concerned with maximum regularity estimates in anisotropic Sobolev spaces $W_p^{2;1}$ for solutions of linear problems arising in hydrodynamics and magnetohydrodynamics. The proof is based on the Marcinkiewicz-Mikhlin-Lizorkin theorem on multipliers in the Fourier integrals.

In the recent years mathematical models and numerical simulations applied to the vascular system have been increasing due to their applicability in different physiological cases and pathologies such as aneurysms, thrombosis, atherosclerosis, etc. The presentation focuses on the modelling of the coupled fluid-structure interaction (FSI) problem which arises in haemodynamics using different constitutive laws to describe the vessel wall dynamics. The mechanical behaviour of the tissues composing the vessel wall is highly nonlinear. Moreover, it is known that it shows a non-homogeneous composition and anisotropic behaviour. In this work the arterial tissue has been described as an homogeneous isotropic non-linear material. Numerical simulations of the coupled fluid-structure interaction problem have been carried out using the different constitutive laws and analysing the effects of the different arterial modelling on the numerical results. In particular the attention of this work is referred to a anatomically realistic geometry of cerebral aneurysm that developed on the internal carotid artery.

We develop a computational method for the simulation of the flow of blood through cerebral aneurysms, which may occur in the human brain. These aneurysms are weak regions in the vessel, presenting a serious risk of rupture to the patient. The goal is to understand the flow in these diseased parts of the vessel system. An immersed boundary method, based on volume penalization, is developed to compute the pulsatile flow. The raw medical imagery representing the patient-specific geometry is processed to extract the ‘masking function’, which is needed to simulate flow patterns and obtain wall shear stresses under realistic physiological conditions. We illustrate the numerical method for several model and realistic aneurysms. In each case we observe a transition to complex time-dependent flow in case the flow speed and/or the aneurysm size become sufficiently large. High frequency variations appear in the flow, which may be an easy method for monitoring the progress of a developing aneurysm and the risk it represents.

Aneurysm, embolisms and atherosclerosis are, among different diseases affecting the cardiovascular system, the most studied. These pathologies include a variety of disorders and conditions that affect the heart and the blood and are usually associated with factors like biochemistry, haemodynamics and genetic predisposition. These factors are specific to each individual and it is important to represent accurately patient-specific information to evaluate correctly clinical state either at diagnosis and prognosis stages. Taking an example of a configuration of the Aorto-Iliac bifurcation, we examine the effects of image filtering and contrast enhancement on the computational reconstructed geometry. Methods to quantify the differences resulting in the images from the different filtering methods are based on the Signal Noise Ration, pixel intensity variance. Finally all the methods are applied to a synthetic image to assure the most accurate sequence of images. Comparison of the images and reconstructed geometries after different pre-processing methods identify a possible uncertainty range for this patient specific study that should be considered when discussing prognosis and diagnosis in a clinically relevant context, mainly when studying the measures of wall shear stress, wall shear stress gradient, and oscillatory shear index which have been largely used in the literature to correlate to disease. In this study we focus on the effects of uncertainty in clinically acquired medical imaging to variability in the reconstructed vessel geometry.

Blood coagulation is a biological process of fundamental importance and extreme complexity. It consists on the formation of blood clots at the site of vascular injury, preventing the blood loss. This process involves complex interactions among multiple molecular and cellular components in the blood and vessel wall, and it is also influenced by the flow of blood.

Mathematical modeling of the blood coagulation and fibrinolysis processes is a way of conceptualizing and understanding this complicated system, helping to optimize design of artificial devices and also to identify the regions of the arterial tree susceptible to the formation of thrombotic plaques and possible rupture in stenosed arteries. A good model should be simple enough in order to be applied in numerical simulations, and at the same time should be able to capture the process complexity, so to allow its better understanding.

The blood coagulation model we are working on consists of a system of convection-reaction-diffusion equations, describing the cascade of biochemical reactions, coupled with rheological models for the blood flow (Newtonian, shear-thinning and viscoelastic models). We introduce the effect of blood slip at the vessel wall emphasizing an extra supply of activated platelets to the clotting site. We expect that such contribution could be dominant, resulting in the acceleration of thrombin production and eventually of the whole clot progression. Such model will have the capacity to predict effects of specific perturbations in the hemostatic system that can't be done by laboratory tests, and will assist in clinical diagnosis and therapies of blood coagulation diseases.

Numerical results for 1D case will be presented, based on the solution of a system of reaction-diffusion equations, using the Finite Element Method. Evolution of concentration of biochemical species and clot formation and growth will be investigated in the injury site of the vessel wall.

We study the Nehari manifold $N$ associated to the boundary value problem \[-\Delta u=f(u)\,,\quad u\in H^1_0(\Omega)\] where $\Omega$ is a bounded regular domain in $\mathbb{R}^n$. Using elementary tools from Differential Geometry, we provide a local description of $N$ as an hypersurface of the Sobolev space $H^1_0(\Omega)$. We prove that, at any point $u\in N$, there exists an exterior tangent sphere whose curvature is the limit of the increasing sequence of principal curvatures of $N$. Also, the $H^1$-norm of $u\in N$ depends on the number of principal negative curvatures. Finally, we study basic properties of an angle decreasing flow on the Nehari manifold associated to homogeneous non-linearities.

Polynomial sequences generated by integral powers of first and second order differential operators conveniently chosen will be the issue. More precisely, the focus will lie on their connection with well known orthogonal polynomial sequences along with their foremost structural properties. This talk will be split in two parts. We will start by analysing the cases in which the aforementioned differential operator is of first order, bringing into analysis polynomial sequences associated to the classical linear functionals of Hermite, Laguerre, Bessel and Jacobi. Afterwards, the discussion will proceed towards the analysis of polynomial sequences generated by second order differential operators, which brings up the open problem of characterizing orthogonal polynomial sequences with respect to certain positive definite linear functionals. The Kontorovich-Lebedev transform and the central factorial numbers will be an asset to attain our goals.

References

1. Ana F. Loureiro, New results on the Bochner condition about classical orthogonal polynomials, J. Math An. Appl., 364 (2010) 307-323.
2. Ana F. Loureiro, P. Maroni, S. Yakubovich, On a nonorthogonal polynomial sequence associated with Bessel operator, Pre-Print CMUP 2011-10 (ArXiv:1104.4055v1)
3. Ana F. Loureiro, S. Yakubovich, On a polynomial sequence related to the Ditkin-Prudnikov problem, Pre-Print CMUP 2011-23 (arXiv:1110.6015v1)

The fully Lagrangian Smoothed Particle Hydrodynamics (SPH) method was originally invented to deal with non axisymmetric problems in astrophysics (Lucy 1977, Gingold & Monaghan 1977). Since then the use of SPH has expanded in many areas of solid and fluid dynamics (involving large deformations, impacts, free-surface and multiphase flows). For example, collision of rubber cylinders (Swegle et al., 1995) in solid mechanics, dam breaking and free-surface waves (Monaghan, 1994) and two-phase flows such as Rayleigh-Benard instability (Violeau, 1999) in fluid mechanics. A major advantage of SPH over Eulerian methods is that the method does not need a grid to calculate spatial derivatives. Instead, they are found by summation of analytical differentiated interpolation formulae (Monaghan, 1992). The momentum and energy equations become sets of ordinary differential equations which are easy to understand in mechanical and thermodynamical terms. For example, the pressure gradient becomes a force between pairs of particles. While Eulerian methods have difficulties to construct a mesh for the simulation domain when it has very complex interfaces, SPH is able to do it without any special front tracking treatment. Nevertheless, despite good agreements in general, some limitations are found in the SPH method such as very small time step, which lead to very expensive CPU cost, and pressure fluctuation.

Neste trabalho propõe-se uma nova metodologia baseada em técnicas de procura directa (DMS - Direct MultiSearch), que não agrega nenhuma das funções objectivo do problema. A estrutura algorítmica baseia-se no paradigma dos passos de procura/sondagem dos métodos de procura directa direccional, recorrendo ao conceito de dominância de Pareto para manter uma lista de pontos não dominados (de onde são seleccionadas as novas iteradas a utilizar no passo de sondagem). Um dos objectivos do método é gerar o maior námero possível de pontos na frente de Pareto unicamente a partir do passo de sondagem. Pretende-se, também, manter a estrutura algorítmica o mais geral possível, possibilitando, em particular, a incorporação de estratégias de disseminação no passo de procura (que, como se sabe, é opcional). A metodologia DMS é uma generalização para optimização multiobjectivo (OMO) de todos os métodos do tipo procura directa direccional. Dois outros subprodutos desta contribuição são (i) o desenvolvimento de uma colecção de problemas para a OMO; (ii) a extensão dos perfis de desempenho e de dados para OMO, permitindo a comparação de diferentes solvers num conjunto grande de problemas teste, em termos de eficiência e robustez na determinação da frente de Pareto. Relativamente à análise de convergência desta classe de algoritmos, supondo válidas as hipóteses habitualmente consideradas na análise de convergência dos métodos de procura directa direccional em optimização uni-objectivo e recorrendo à análise não suave de Clarke, foi demonstrado que uma subsucessão da sucessão de iteradas gerada pelo DMS converge para um ponto crítico de Pareto-Clarke.

d’Angelo, Zunino and Quarteroni have developed a multiscale methodology to handle numerically complex problems that arise naturally in microcirculation and perfusion of biological tissues through networks of vascular beds. The problem is composed by two scales: (i) at the macroscale, the tissue is treated as a homogenized porous media, usually saturated with plasma where relevant chemical species diffuse, advect and react; and (ii) the microscale is a finite and discrete network of small vessels (arterioles and capillaries), from which plasma permeates, and are naturally considered as a network of one dimensional models of fluid flow in circular tubes. Starling’s filtration law, a widely established description of the permeation of plasma from the arteriole into the tissue, couples both scales in a nonlinear fashion, i.e. the amount of fluid delivered from the vessel into the tissue (or vice versa) generally depends locally on the dependent variables of both scales. Currently, the mathematical modeling of angiogenesis is generally treated with two concurrent modeling approaches: (i) on one hand, continuum models of angiogenesis are based on reaction-diffusion-advection equations that describe endothelial cell conservation and quantify their motility accordingly to the distribution of other important fields, such as the concentration of angiogenic factor or presence of fibronectin, and fall short in the discrete identification and description of the vascular network; and (ii) stochastic discrete models based on lattice descriptions of the tissue together with a set of phenomenologically based cellular automata rules of network behavior and evolution have provided interesting qualitative results to describe a multitude of different patterns of angiogenesis. The multiscale methodology developed by d’Angelo and co-workers seems to be a promising strategy to develop quite a fundamental model of angiogenesis, based on evolution of the microscale (i.e. with an evolving network in response to biological stimuli provided by the tissue) or even on the evolution of both scales (with a growing tissue in response to improved conditions provided by the enhancement of the microscale). Several challenges naturally arise with the mathematical description of this complex problem and are currently a work in progress, such as: (i) the ability of the method to handle very complex arterial networks downstream after multiple rounds of capillary sprouting, (ii) anastomosis (when two capillary tips join and form a loop, changing the topology of the microscale domain), and among others (iii) the choice of proper one dimensional models for fluid flow in one-ended tubes and the treatment of anisotropic (directional) plasma sources at the macroscale.