Nineth Workshop on Mathematical Modelling of Environmental and Life Sciences Problems

Constanta, November 1–4, 2012

ROMANIAN ACADEMY ”Gheorghe Mihoc - Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics Department of Applied Mathematics Bucharest, Romania

”OVIDIUS” UNIVERSITY Faculty of Mathematics and Informatics Research Center of Applied Mathematics Constanta, Romania

sponsored by BITDEFENDER Bucharest Scientific Prof. Dr. Dorel Homentcovschi Committee Acad. Marius Iosifescu Prof. Dr. Alexandru Morega Prof. Dr. Ulrich R¨ude Prof. Dr. Christoph Schn¨orr Prof. Dr. Harry Vereeken

Organizing Prof. Dr. Eduard Marius Craciun Committee Dr. Stelian Ion Dr. Gabriela Marinoschi Prof. Dr. Constantin Popa

Conference Dr. Aurelian Nicola Secretariat Dr. Elena Pelican General Schedule

Thursday, November 1st, 2012

16.00-18.00 Registration

Friday, November 2nd, 2012

9.00-10.00 Opening Ceremony 10.00-12.30 Communications on Diffusion Processes and Material Sciences 12.30-14.30 Lunch 14.30-16.30 Communications on Mathematical Models in Biology (I) 16.30-17.00 Coffee break 17.00-18.30 Roundtable: Bio-geo-chemical processes 19.00-23.00 Banquet

Saturday, November 3rd, 2012

9.20-10.30 Communications on Numerical Analysis 10.30-11.00 Coffee break 11.00-12.00 Communications on Image Recognition 12.00-14.00 Lunch 14.00-15.00 Communications on Mathematical Models in Biology (II)

3 Conference Lectures

Friday November 2nd, 2012

10.00-12.30 Diffusion Processes and Material Sciences Chairpersons Adrian Carabineanu, Olivian Simionescu-Panait 1 Gheorghe Juncu, Aurelian Nicola, and Constantin Popa Splitting method for ternary mass transfer equations 2 Petru Jebelean and Constantin Popa Shooting method for numerical solution of singular φ-Laplacian equations ———————-Coffee break——————— 3 Elena Stroil˘a Splitting method for multicomponent mass transfer equations 4 Eduard Marius Cr˘aciun Mathematical modeling of the multiple cracks interaction in pre-stressed elastic composites 5 Olivian Simionescu-Panait TH-wave propagation in a pre-stressed layered formation 14.30-16.30 Mathematical Models in Biology Chairpersons Alexadru Morega, Gheorghe Juncu 6 Alexei Leahu and Bogdan Munteanu Monte Carlo simulation for some reliability systems 7 Alexandru M. Morega, Alin A. Dobre and Mihaela Morega Hemodynamic - Electrical Field Interactions by Impedance Cardiography and Electro-Cardiometry 8 Anca Veronica Ion Study of behavior of density of proliferating cells in a delay differential equations model of leukemia ———————-Coffee break——————— 9 Adrian Carabineanu Free-Surface Flow Past Submerged Hydrofoils 10 Stefan-Gicu Cruceanu and Dorin Marinescu Parameter Estimation for Survival Models

4 17.00-18.30 Roundtable: Bio-geo-chemical processes Moderator Stelian Ion 11 Virgil Iordache, Bodescu F., Ion S., Scradeanu D. Open problems in modeling biogeochemical processes 12 Stelian Ion PDE and CA models of soil erosion Saturday November 3rd, 2012

9.20-10.30 Numerical Analysis Chairpersons Doina Carp 13 Doina Carp and Constantin Popa On numerical solution of inconsistent systems of linear inequalities 14 Ioana Pomparau On an accelerated version of Kaczmarz algorithm with a priori clustering 15 Cristina S¸erban and Adam Dosa Iterative Solution of Augmented Systems for Structural Mechanics 11.00-12.00 Image Recognition Chairpersons Constantin Popa 16 L˘acr˘amioaraLit¸˘aand Elena Pelican Comparative Study of NN, PCA and COD algorithms for Pattern Recog- nition 17 Ljiljana Cvetkovi´c Nekrasov matrices - generalizations and benefits 18 Stefania Petra, Constantin Popa, Christoph Schn¨orr Multiplicative Updates for Box Constrained Linear Systems 14.00-15.00 Mathematical Models in Biology Chairpersons Eduard Marius Craciun 19 N. Ray, T. von Norden, F.A. Radu, P. Knabner, W. Friess Drug release from collagen matrices including an evolving microstructure 20 Tudor Udrescu On using adaptive wavelet compression to approximate committor probabilities in metastable biochemical systems

5 Abstracts

6 Free-Surface Flow Past Submerged Hydrofoils by Adrian Carabineanu

We construct a Green function in order to obtain an integral representa- tion of the velocity field. We obtain an integral equation for the tangential velocity on the hydrofoil. The integral equation is solved numerically and the hydrodynamic coefficients are calculated.

On numerical solution of inconsistent systems of linear inequalities by Doina Carp and Constantin Popa

In the paper [1] the author described a finitely terminating algorithm for solving an inconsistent system Ax ≤ b of linear inequalities in a least squares sense. His algorithm uses a singular value decomposition of a submatrix of A om each iteration, making it impractical for all but only smaller problems. In this paper we show that a modification of Han’s algorithm allows us to introduce a controlled (possible iterative) approximation to the original singular value solution. Ref. [1] s. -P. Han, Least squares solution of linear inequalities, tech. rep. TR-2141, Mathematics Research Center, University of Wisconsin - Madison, 1980.

Mathematical modeling of the multiple cracks interaction in pre-stressed elastic composites by Eduard Marius Cr˘aciun

We consider a composite containing three equal and collinear cracks in the same plane. The upper and lower faces of the cracks are acted by sym- metrically distributed normal stresses. Our aim is to determine the elastic states in the body, cracks interaction and the cracks propagation directions. The present paper represents analitical, numerical and experimental studies of the three equal and collinear cracks propagation and interaction. In or- der to assess the displacement and strain fields in the tested materials, the

7 3-D image correlation system ARAMIS has been used. The tests were per- formed at Lublin University of Technology from Poland on the orthotropic polymer matrix composite material used for helicopter parts manufacturing, prepared by Polish Aviation Works in Poland.

Parameter Estimation for Survival Models by S¸tefan Gicu Cruceanu and Dorin Marinescu

We consider some survival models derived from the classical Gompertz model. Using real data, our purpose is to determine a set of unknown pa- rameters associated to each of these models by efficiently solving either a Least Square or a Least Absolute Value minimization problem. In order to numerically evaluate the values of these parameters, we will adapt Cellu- lar Exclusion Methods for direct solving the previous global optimization problems.

Nekrasov matrices - generalizations and benefits by Ljiljana Cvetkovi´c

While the class of strictly diagonally dominant (SDD) matrices is closely related to the Jacobi iterative method, in a similar fashion, the class of Nekrasov matrices arises from the Gauss-Seidel method. There is a con- nection between these two classes, which enables several interesting appli- cations. On the other hand, some of generalization directions for the SDD class can be applied to Nekrasov class, and produce further benefits in sev- eral areas of matrix theory.

Study of behavior of density of proliferating cells in a delay differential equations model of leukemia by Anca Veronica Ion

We consider a model of leukemia (found in the literature), model consisting of two delay differential equations, one for the evolution of the density of

8 proliferating cells P and one for the density of so called ”resting” cells N - cells that, for a certain period of time do not divide within the cancerous process. The equation for N is independent of P hence it can be studied separately. In previous works, we studied for this equation the stability of equilibrium points, the points of Hopf and Bautin bifurcation, and the stability of the periodic solutions emerged by these bifurcations. In the present work we study the equation for the proliferating cells, P .

PDE and CA models of soil erosion by Stelian Ion

The soil erosion is a very complex phenomenon that needs an interdisci- plinary treatment to quantify it or to make predictions about its evolution. In a rough classification the mathematical models cast in the regressive models and physically based models. Among the physically based models the distributed models gain attention of the modelers and practitioners due, mainly, to the progress in data acquisition technics and the development of mathematical tools and calculus power. In the paper we discuss two types of physically based and distributed model, the models that use the partial differential equations in their formulation and the cellular automata based models. We focus on their common points and what makes them different. Some numerical simulations illustrate our talk.

Shooting method for numerical solution of singular φ-Laplacian equations by Petru Jebelean and Constantin Popa

In this paper we are concerned with numerical solutions to Dirichlet problem [φ(u0)]0 = f(x) in [α, β]; u(α) = A, u(β) = B, where φ :(−η, η) → R (η < +∞) is an increasing diffeomorphism with φ0(y) ≥ d > 0 for all y ∈ (−η, η). The obtained algorithm combines the shooting method with Euler’s method and it is convergent whenever the problem is solvable. We provide numerical experiments confirming the the- oretical aspects.

9 Splitting method for ternary mass transfer equations by Gheorghe Juncu, Aurelian Nicola, and Constantin Popa

In a multi-component system the diffusion of a certain species is dictated not only by its own concentration gradient but also by the concentration gradient of the other species. In this case, the mathematical model is a system of strongly coupled second - order elliptic/parabolic partial differen- tial equations. In this paper we propose a splitting method for numerical solving of multicomponent mass transfer equations. The case analyzed in detail is the linear ternary system. Theoretical results about the stability of the method are also presented.

Monte Carlo simulation for some reliability systems by Alexei Leahu and Bogdan Munteanu

In this work will be presented nuumerical results of Monte Carlo simulation for Life Time of some reliability systems, where this main characteristic is a Pascal convolution of independent, identically distributed random variables.

Comparative Study of NN, PCA and COD algorithms for Pattern Recognition by L˘acr˘amioaraLit¸˘aand Elena Pelican

Pattern recognition is a current issue of great interest in real-life problems. Over the years, different types of algorithms have been developed to solve this problem. In this paper we present a comparative study for three al- gorithms: the nearest neighbour algorithm (NN), the eigenfaces technique (PCA), and our COD proposed algorithm. The exeperiments are made on three datasets: the well known ORL dataset (also known as AT&T dataset), an own face dataset CTOVF, and an own digit dataset CTOVD. The study includes the use of different metrics, different initialization (for the COD algorithm) and different levels of truncation (for the PCA and COD algo- rithm).

10 Hemodynamic - Electrical Field Interactions by Impedance Cardiography and Electro-Cardiometry by Alexandru M. Morega, Alin A. Dobre and Mihaela Morega

The Electro Cardiometry (ECM) and the Impedance Cardiography (ICM) are investigation methods of the thoracic electrical impedance (TEB) group, concerned with monitoring the hemodynamic flow, through electrical impedance measurements. In this paper we present a mathematical model for the hemo- dynamic flow in the aorta, the dynamics of the electrical conductivity of the blood as revealed by electrical measurements on the thorax, and the electrical field problem equivalent to the ICM and ECM procedures. The mathematical models are solved by finite element (FEM) numerical simula- tion. We use 3D computational domains built out of medical scans through reconstruction techniques used the medical planning and diagnosis. The an- alytic formulae for the electrical conductivity of the blood that are available pose certain difficulties in the attempt to approach anatomically realistic computational domains. To overpass this difficulty we use equivalent values for the electrical conductivity of the blood, obtained by averaging techniques that prove to keep the sensitivity of TEB to the aorta blood flow dynam- ics. The solution to these direct problems opens the path to the inverse IMC and EMC problems, aiming to understand the flow dynamics out of experimental data.

Multiplicative Updates for Box Constrained Linear Systems by Stefania Petra, Constantin Popa, Christoph Schn¨orr

We derive multiplicative updates for solving box constrained linear systems of equations based on the minimization of the Bregman distance Df corre- sponding to the Bregman function f(x) = (x − a) log(x − a) + (b − x) log(b − x), x ∈ [a, b]. The updates have a simple closed form. We prove conver- gence and derive convergence rates for the considered iteration by extending existing results for the exponential gradient descent method. In case of non- negative constraints only, the proposed method reduces to the well known simultaneous multiplicative algebraic reconstruction technique (SMART). Thus, our work answers open questions concerning its convergence behavior.

11 We apply our method to binary compressed sensing problems, in particular to binary tomographic image reconstruction.

On an accelerated version of Kaczmarz algorithm with a priori clustering by Ioana Pomparau

When adding sufficient directions for projection such that Kaczmarz algo- rithm becomes direct solver of the linear least squares problem, the fill-in percentage of these directions is higher than that of the system matrix. We study some numerical results concerning a method of accelerating the Kacz- marz iterative scheme using supplementary directions for projection while preserving the sparsity pattern of the original problem via clustering.

The Effect of the Internal Liquid Viscozity on the Pulsatory Lipid Vesicle Dynamics by Dumitru Popescu1 and Alin Gabriel Popescu

In this paper, we have considered a lipid vesicle under positive osmotic stress. In certain conditions, its evolution will be a cyclic process. We have named it as a pulsatory lipid vesicle. We have realised a mathematical mod- elling of both the vesicle and the pore dynamics. The viscosity of the internal solution is an important physical parameter which influences the pulsatory vesicle running. From this reason the aim of the present paper is to analyze the effect of the internal solute viscosity on the characteristic parameters of the pulsatory lipid vesicle (swelling time, pore lifetime, number of cycles, the length time of the vesicle activity, internal liquid amount leaked out during a cycle).

Drug release from collagen matrices including an evolving microstructure by N. Ray, T. von Norden, F.A. Radu, P. Knabner, W. Friess

12 Biodegradable collagen matrices have become a promising alternative to tra- ditional drug delivery systems. The relevant mechanism in controlled drug release are the penetration of the collagen matrix by water, the swelling of the matrix where drug is released by diffusion and enzymatic degradation of the matrix with simultaneous drug release. These phenomena have been studied experimentally, via numerical simulations and also analytically ex- tensively in the past. However, a rigorous derivation of macroscopic model description is still lacking which includes the the evolving microstructure due to degradation processes. Since this can lead to the release of physically entrapped active agent and therefore an increase of drug release, a good understanding of these phenomena is very important. We present such a derivation using formal two-scale asymptotic expansion in a level set frame- work and complete our results with numerical simulations in comparison with experimental data.

TH-wave propagation in a pre-stressed layered formation by Olivian Simionescu-Panait

In this lecture we present new results concerning TH-wave propagation in pre-stressed layered materials. These results are similar to those from seis- mology concerning Love wave propagation. We obtain and analyze the dis- persion relation, into a parallel-sided plate, resp. into a layer on a substrate structure, for various classes of anisotropy, which generalize the classical re- sults from the case without initial fields. We derive here the energy density and the energy flux of the TH-wave, in order to satisfy the energy balance equation. We calculate the mean energy density, resp. the mean energy flux, over a period of time, in the layer and in the substrate. Finally, we obtain the total mean energy densities and total mean energy fluxes in the layer and in the substrate.

Splitting method for multicomponent mass transfer equations by Elena Stroil˘a

In the paper [1] the authors analyse the splitting method for a ternary mass transfer problem. In this case they establish necessary and sufficient

13 conditions for the stability of the method, according to the considerations in [2]. Based on these ideas, in this paper we propose a splitting method for numerical solution of multicomponent mass transfer equations with more than 3 species. We establish necessary conditions for the stability of the appropriate splitting technique, for 3 and 4 species. Ref. [1] Juncu Gh., Nicola A., Popa C., Splitting method for ternary mass transfer equations, in this conference. Ref. [2] G.I. Marchuk G. I., Methods of Numerical Mathematics, Springer, New York, 1975.

Iterative Solution of Augmented Systems for Structural Mechanics by Cristina S¸erban and Adam Dosa

Using the finite element discretization, the calculation of strength of ma- terials from structural mechanics leads to large, sparse and ill-conditioned linear systems of equations. Moreover, these systems have a special type of structure (2 x 2 blocks) which is why we can include them in the aug- mented systems class. In this paper, we present a comparative study of some preconditioned iterative methods for solving such systems.

On using adaptive wavelet compression to approximate committor probabilities in metastable biochemical systems by Tudor Udrescu

The dynamics of biochemical systems are often described by using a Markov jump process on a high-dimensional discrete state space, with the corre- sponding probability distribution obtained by solving the chemical master equation (CME). The solution of the CME delivers an accurate picture of the dynamics, which can be further enhanced by investigating the qualita- tive behavior of the underlying Markov jump process at equilibrium. Such information about the specific transition mechanisms between certain states is particularly relevant for systems exhibiting metastable dynamics, where rare events induce a switching behavior between subsets of the state space. However, approximating the solution of the CME or computing committor probabilities - statistical objects that characterize the progress of transi- tions between two arbitrarily chosen subsets of the state space, are both

14 non-trivial problems. Here, we present a wavelet based approach for the efficient computation of the committor probabilities, which is illustrated on multi-dimensional models of genetic toggle switches with large state spaces.

15 Authors

16 Florian Bodescu Department of Systems Ecology and Sustain- able Development ”Dan Manoleli” Research Center for Ecological Services - CESEC Fac- ulty of Biology University of Bucharest Spl In- dependentei 91-95 050089 Sector 5 Bucuresti Romania e-mail: Adrian Carabineanu Faculty of Mathematics and Informatics, Uni- versity of Bucharest, Romania, ”Gheorghe Mihoc–Caius Iacob” Institute of Statistical Mathematics and Applied Math- ematics, Calea 13 Septembrie nr.13, 050711 Bucharest, Romania e-mail: [email protected] Doina Carp Constanta Maritime University, Mircea cel Batran str., nr. 104, Constanta, Romania e-mail: [email protected] Andrei C˘ald˘aru¸s Department of Systems Ecology and Sustain- able Development ”Dan Manoleli” Research Center for Ecological Services - CESEC Fac- ulty of Biology University of Bucharest Spl Indep. 91-95 050089, Bucuresti, Romania e-mail: Eduard Marius Faculty of Mathematics and Informatics, Cr˘aciun ”Ovidius” University of Constanta e-mail: [email protected] Stefan Gicu-Cruceanu ”Gheorghe Mihoc–Caius Iacob” Institute of Statistical Mathematics and Applied Math- ematics, Calea 13 Septembrie nr.13, 050711 Bucharest, Romania e-mail: [email protected] A.A. Dobre POLITEHNICA University Bucharest, Bucharest 6, Romania e-mail: Ljiljana Cvetkovi´c Department of Mathematics and Informatics, University of Novi Sad, Serbia e-mail: [email protected] Adam Dosa Transilvania University of Brasov, Eroilor Blvd. 29, Brasov, Romania e-mail: [email protected]

17 W. Friess Department Pharmazie, Ludwig- Maximilians-University Muenchen, Bute- nandtstr. 5-13, D-81377, Germany e-mail: Anca Veronica Ion ”Gheorghe Mihoc–Caius Iacob” Institute of Statistical Mathematics and Applied Math- ematics, Calea 13 Septembrie nr.13, 050711 Bucharest, Romania e-mail: [email protected] Stelian Ion ”Gheorghe Mihoc–Caius Iacob” Institute of Statistical Mathematics and Applied Math- ematics, Calea 13 Septembrie nr.13, 050711 Bucharest, Romania e-mail: ro [email protected] Virgil Iordache Department of Systems Ecology and Sustain- able Development ”Dan Manoleli” Research Center for Ecological Services - CESEC Fac- ulty of Biology University of Bucharest Spl Independentei 91-95 050089, Bucuresti, Ro- mania e-mail: [email protected] Petru Jebelean West University of Timisoara, Blvd. V. Par- van 4, 300223 Timisoara, Romania e-mail: [email protected] Gheorghe Juncu Faculty of Chemical Engineering, PO- LITEHNICA University of Bucharest, Polizu 1, 78126 Bucharest, Romania e-mail: [email protected] P. Knabner Department of Mathematics, University of Erlangen-Nuremberg, Cauerstr. 11, D-91058 Erlangen, Germany e-mail: Alexei Leahu “Ovidius” University of Constanta e-mail: [email protected] Dorin Marinescu ”Gheorghe Mihoc–Caius Iacob” Institute of Statistical Mathematics and Applied Math- ematics, Calea 13 Septembrie nr.13, 050711 Bucharest, Romania e-mail: [email protected]

18 Alexandru Morega POLITEHNICA University Bucharest, De- partment of Bioengineering and Biotechnol- ogy& Department of Electrical Engineering, Bucharest 6, Romania, ”Gheorghe Mihoc–Caius Iacob” Institute of Statistical Mathematics and Applied Math- ematics, Calea 13 Septembrie nr.13, 050711 Bucharest, Romania e-mail: [email protected] L˘acr˘amioaraLit¸˘a ”Ovidius” University of Constanta, Faculty of Mathematics and Informatics, Bd. Mamaia 124, 900527, Constanta, Romania e-mail: [email protected] Mihaela Morega POLITEHNICA University Bucharest, De- partment of Bioengineering and Biotechnol- ogy& Department of Electrical Engineering, Bucharest 6, Romania e-mail: [email protected] Bogdan Munteanu Air Force Academy ”H. Coanda”, Brasov, Ro- mania e-mail: Aurelian Nicola ”Ovidius” University of Constanta, Faculty of Mathematics and Informatics, Bd. Mamaia 124, 900527, Constanta, Romania e-mail: [email protected] T. von Norden Department of Mathematics, University of Erlangen-Nuremberg, Cauerstr. 11, D-91058 Erlangen, Germany e-mail: Elena Pelican “Ovidius” University of Constanta, Faculty of Mathematics and Informatics, Bd. Mamaia 124, 900527, Constanta, Romania e-mail: [email protected] Stefania Petra Image and Pattern Analysis Group, Depart- ment of Mathematics and Computer Science, University of Heidelberg, Germany e-mail: [email protected] Ioana Pomparau “Ovidius” University of Constanta, Faculty of Mathematics and Informatics, Bd. Mamaia 124, 900527 Constanta, Romania e-mail: [email protected]

19 Constantin Popa “Ovidius” University of Constanta, Faculty of Mathematics and Informatics, Bd. Mamaia 124, 900527 Constanta, Romania ”Gheorghe Mihoc–Caius Iacob” Institute of Statistical Mathematics and Applied Math- ematics, Calea 13 Septembrie nr.13, 050711 Bucharest, Romania e-mail: [email protected] Dumitru Popescu ”Gheorghe Mihoc–Caius Iacob” Institute of Statistical Mathematics and Applied Math- ematics, Calea 13 Septembrie nr.13, 050711 Bucharest, Romania e-mail: [email protected] Alin Gabriel Popescu IT CORE Bucharest, Romania e-mail: alin [email protected] F.A. Radu Department of Mathematics, University of Bergen, P. O. Box 7800, N-5020 Bergen, Nor- way e-mail: N. Ray Department of Mathematics, University of Erlangen-Nuremberg, Cauerstr. 11, D-91058 Erlangen, Germany e-mail: Christoph Schn¨orr University of Heidelberg, Image and Pattern Analysis Group, Heidelberg, Germany e-mail: [email protected] Olivian University of Bucharest, Faculty of Mathe- Simionescu-Panait matics and Informatics e-mail: [email protected] Elena Stroil˘a Research Center for Navy, Str. Stefanita Voda, nr. 4, 900402, Constanta, Romania e-mail: [email protected] Cristina Serban “Ovidius” University of Constanta, Faculty of Civil Engineering, Constanta, Romania e-mail: [email protected] Tudor Udrescu Karlsruhe Institute of Technology (KIT), In- stitute for Applied and Numerical Mathemat- ics, Karlsruhe, Germany e-mail: [email protected]

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