arXiv subjects

Mazen Saad

Publications and source records attributed to Mazen Saad.

9 recordsLinked to original sources

arXiv2024

Bilinear optimal control for chemotaxis model: The case of two-sidedly degenerate diffusion with Volume-Filling Effect

In this paper, we study an optimal control problem for a coupled non-linear system of reaction-diffusion equations with degenerate diffusion, consisting of two partial differential equations representing the density of cells and the concentration of the chemotactic agent. By controlling the concentration of the chemical substrates, this study can guide the optimal growth of cells. The novelty of this work lies on the direct and dual models that remain in a weak setting, which is uncommon in the recent literature for solving optimal control systems. Moreover, it is known that the adjoint problems offer a powerful approach to quantifying the uncertainty associated with model inputs. However, these systems typically lack closed-form solutions, making it challenging to obtain weak solutions. For that, the well-posedness of the direct problem is first well guaranteed. Then, the existence of an optimal control and the first-order optimality conditions are established. Finally, weak solutions for the adjoint system to the non-linear degenerate direct model, are introduced and investigated.

math.OC↗

arXiv2022

Microscopic tridomain model of electrical activity in the heart with dynamical gap junctions. Part 2 -- Derivation of the macroscopic tridomain model by unfolding homogenization method

We study the homogenization of a novel microscopic tridomain system, allowing for a more detailed analysis of the properties of cardiac conduction than the classical bidomain and monodomain models. In (Acta Appl.Math. 179 (2022) 1--35), we detail this model in which gap junctions are considered as the connections between adjacent cells in cardiac muscle and could serve as alternative or supporting pathways for cell-to-cell electrical signal propagation. Departing from this microscopic cellular model, we apply the periodic unfolding method to derive the macroscopic tridomain model. Several difficulties prevent the application of unfolding homogenization results, including the degenerate temporal structure of the tridomain equations and a nonlinear dynamic boundary condition on the cellular membrane. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we use the boundary unfolding operator and a Kolmogorov--Riesz compactness's result.

math.AP↗

arXiv2022

Microscopic Tridomain Model of Electrical Activity in the Heart with Dynamical Gap Junctions. Part 1 -- Modeling and Well-Posedness

We present a novel microscopic tridomain model describing the electrical activity in cardiac tissue with dynamical gap junctions. The microscopic tridomain system consists of three PDEs modeling the tissue electrical conduction in the intra-and extra-cellular domains, supplemented by a nonlinear ODE system for the dynamics of the ion channels and the gap junctions. We establish the global existence and uniqueness of the weak solutions to our microscopic tridomain model. The global existence of solution, which constitutes the main result of this paper, is proved by means of an approximate non-degenerate system, the Faedo-Galerkin method, and an appropriate compactness argument.

math.AP↗

arXiv2022

Three Scale Unfolding Homogenization Method Applied to Cardiac Bidomain Model

In this paper, we are dealing with a rigorous homogenization result at two different levels for the bidomain model of cardiac electro-physiology. The first level associated with the mesoscopic structure such that the cardiac tissue consists of extracellular and intracellular domains separated by an interface (the sarcolemma). The second one related to the microscopic structure in such a way that the intracellular medium can only be viewed as a periodical layout of unit cells (mitochondria). At the interface between intra-and extracellular media, the fluxes are given by nonlinear functions of ionic and applied currents. A rigorous homogenization process based on unfolding operators is applied to derive the macroscopic (homogenized) model from our meso-microscopic bidomain model. We apply a three-scale unfolding method in the intracellular problem to obtain its homogenized equation at two levels. The first level upscaling of the intracellular structure yields the mesoscopic equation. The second step of the homogenization leads to obtain the intracellular homogenized equation. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we use the boundary unfolding method and a Kolmogorov-Riesz compactness's result. Next, we use the standard unfolding method to homogenize the extracellular problem. Finally, we obtain, at the limit, a reaction-diffusion system on a single domain (the superposition of the intracellular and extracellular media) which contains the homogenized equations depending on three scales. Such a model is widely used for describing the macroscopic behavior of the cardiac tissue, which is recognized to be an important messengers between the cytoplasm (intracellular) and the other extracellular inside the biological cells. Contents 2020 Mathematics Subject Classification. 65N55 and 35A01 and 35B27 and 35K57 and 65M..

math.AP↗

arXiv2022

Derivation of a new macroscopic bidomain model including three scales for the electrical activity of cardiac tissue

In the present paper, a new three-scale asymptotic homogenization method is proposed to study the electrical behavior of the cardiac tissue structure with multiple heterogeneities at two different levels. The first level is associated with the mesoscopic structure such that the cardiac tissue is composed of extracellular and intracellular domains. The second level is associated with the microscopic structure in such a way the intracellular medium can only be viewed as a periodical layout of unit cells (mitochondria). Then, we define two kinds of local cells that are obtained by upscaling methods. The homogenization method is based on a power series expansion which allows determining the macroscopic (homogenized) bidomain model from the microscopic bidomain problem at each structural level. First, we use the two-scale asymptotic expansion to homogenize the extracellular problem. Then, we apply a three-scale asymptotic expansion in the intracellular problem to obtain its homogenized equation at two levels. The first upscaling level of the intracellular structure yields the mesoscopic equation and the second step of the homogenization leads to obtain the intracellular homogenized equation. Both the mesoscopic and microscopic information is obtained by homogenization to capture local characteristics inside the cardiac tissue structure. Finally, we obtain the macroscopic bidomain model and the heart domain coincides with the intracellular medium and extracellular one, which are two inter-penetrating and superimposed continua connected at each point by the cardiac cellular membrane. The interest of the proposed method comes from the fact that it combines microscopic and mesoscopic characteristics to obtain a macroscopic description of the electrical behavior of the heart.

math.AP↗

arXiv2013

A combined finite volume--nonconforming finite element scheme for compressible two phase flow in porous media

We propose and analyze a combined finite volume--nonconforming finite element scheme on general meshes to simulate the two compressible phase flow in porous media. The diffusion term, which can be anisotropic and heterogeneous, is discretized by piecewise linear nonconforming triangular finite elements. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh, where the dual volumes are constructed around the sides of the original mesh. The relative permeability of each phase is decentred according the sign of the velocity at the dual interface. This technique also ensures the validity of the discrete maximum principle for the saturation under a non restrictive shape regularity of the space mesh and the positiveness of all transmissibilities. Next, a priori estimates on the pressures and a function of the saturation that denote capillary terms are established. These stabilities results lead to some compactness arguments based on the use of the Kolmogorov compactness theorem, and allow us to derive the convergence of a subsequence of the sequence of approximate solutions to a weak solution of the continuous equations, provided the mesh size tends to zero. The proof is given for the complete system when the density of the each phase depends on the own pressure.

math.NA↗

arXiv2012

Study of full implicit petroleum engineering finite volume scheme for compressible two phase flow in porous media

An industrial scheme, to simulate the two compressible phase flow in porous media, consists in a finite volume method together with a phase-by-phase upstream scheme. The implicit finite volume scheme satisfies industrial constraints of robustness. We show that the proposed scheme satisfy the maximum principle for the saturation, a discrete energy estimate on the pressures and a function of the saturation that denote capillary terms. These stabilities results allow us to derive the convergence of a subsequence to a weak solution of the continuous equations as the size of the discretization tends to zero. The proof is given for the complete system when the density of the each phase depends on the own pressure.

math.NA↗

arXiv2012

Study of degenerate parabolic system modeling the hydrogen displacement in a nuclear waste repository

Our goal is the mathematical analysis of a two phase (liquid and gas) two components (water and hydrogen) system modeling the hydrogen displacement in a storage site for radioactive waste. We suppose that the water is only in the liquid phase and is incompressible. The hydrogen in the gas phase is supposed compressible and could be dissolved into the water with the Henry's law. The flow is described by the conservation of the mass of each components. The model is treated without simplified assumptions on the gas density. This model is degenerated due to vanishing terms. We establish an existence result for the nonlinear degenerate parabolic system based on new energy estimate on pressures.

math.AP↗

arXiv2011

Convergence of a Finite Volume Scheme for Gas Water Flow in a Multi-Dimensional Porous Media

A classical model for water-gas flows in porous media is considered. The degenerate coupled system of equations obtained by mass conservation is usually approximated by finite volume schemes in the oil reservoir simulations. The convergence properties of these schemes are only known for incompressible fluids. This chapter deals with construction and convergence analysis of a finite volume scheme for compressible and immiscible flow in porous media. In comparison with incompressible fluid, compressible fluids requires more powerful techniques. We present a new result of convergence in a two or three dimensional porous medium and under the only modification that the density of gas depends on global pressure.

math.NA↗