Mathematical Models and Methods in Applied Sciences

This paper deals with the modeling, qualitative and numerical analysis, of welfare dynamics in societies viewed as complex evolutive systems subject to different policies of wealth distribution. A nonlinear model of wealth distribution is presented. The state of a population is modeled by a probability distribution over wealth classes and the dynamic of interaction is parameterized by a threshold, whose dynamics depends on an internal competition related to the wealth distribution. Therefore, the model is a system of equations in which the threshold is one of the dynamic variables. The approach contains the whole path from modeling to simulations, through a qualitative analysis of the initial value problem.

The system of functional differential equations considered here is motivated by a concrete class of multiscale models for tumor cell migration involving chemotaxis, haptotaxis, and subcellular dynamics proposed in [J. Kelkel and C. Surulescu, A multiscale approach to cell migration in tissue networks, Math. Models Methods Appl. Sci. 22 (2012) 1150017]. Tissue fibers, cell densities and concentrations of chemotactic signals are assumed to be just square Lebesgue integrable in space, but not necessarily essentially bounded (as in [J. Kelkel and C. Surulescu, A multiscale approach to cell migration in tissue networks, Math. Models Methods Appl. Sci. 22 (2012) 1150017] and related previous settings). The focus of interest is on sufficient conditions for the well-posedness of the underlying larger problem class.

This paper provides a review of the space–time (ST) and Arbitrary Lagrangian–Eulerian (ALE) techniques developed by the first three authors' research teams for patient-specific cardiovascular fluid mechanics modeling, including fluid–structure interaction (FSI). The core methods are the ALE-based variational multiscale (ALE-VMS) method, the Deforming-Spatial-Domain/Stabilized ST formulation, and the stabilized ST FSI technique. A good number of special techniques targeting cardiovascular fluid mechanics have been developed to be used with the core methods. These include: (i) arterial-surface extraction and boundary condition techniques, (ii) techniques for using variable arterial wall thickness, (iii) methods for calculating an estimated zero-pressure arterial geometry, (iv) techniques for prestressing of the blood vessel wall, (v) mesh generation techniques for building layers of refined fluid mechanics mesh near the arterial walls, (vi) a special mapping technique for specifying the velocity profile at an inflow boundary with non-circular shape, (vii) a scaling technique for specifying a more realistic volumetric flow rate, (viii) techniques for the projection of fluid–structure interface stresses, (ix) a recipe for pre-FSI computations that improve the convergence of the FSI computations, (x) the Sequentially-Coupled Arterial FSI technique and its multiscale versions, (xi) techniques for calculation of the wall shear stress (WSS) and oscillatory shear index (OSI), (xii) methods for stent modeling and mesh generation, (xiii) methods for calculation of the particle residence time, and (xiv) methods for an estimated element-based zero-stress state for the artery. Here we provide an overview of the special techniques for WSS and OSI calculations, stent modeling and mesh generation, and calculation of the residence time with application to pulsatile ventricular assist device (PVAD). We provide references for some of the other special techniques. With results from earlier computations, we show how these core and special techniques work.

The problem of recovering the initial temperature of a body from discrete temperature measurements made at later times is studied. While this problem has a general formulation, the results of this paper are only given in the simplest setting of a finite (one-dimensional), constant coefficient, linear rod. It is shown that with a judicious placement of a thermometer on this rod, the initial temperature profile of the rod can be completely determined by later time measurements. The paper then studies the number of measurements that are needed to recover the initial profile to a prescribed accuracy and provides an optimal reconstruction algorithm under the assumption that the initial profile is in a Sobolev class.

A hierarchy of models for pedestrian flow is numerically investigated using particle methods. It includes microscopic models based on interacting particle system coupled to an eikonal equation, hydrodynamic models using equations for density and mean velocity, nonlocal continuum equations for the density and diffusive Hughes equations. Particle methods are used on all levels of the hierarchy. Numerical test cases are investigated by comparing the above models.

This paper is devoted to the introduction and study of a photoacoustic tomography model, an imaging technique based on the reconstruction of an internal photoacoustic source distribution from measurements acquired by scanning ultrasound detectors over a surface that encloses the body containing the source under study. In a nutshell, the inverse problem consists in determining absorption and diffusion coefficients in a system coupling a hyperbolic equation (acoustic pressure wave) with a parabolic equation (diffusion of the fluence rate), from boundary measurements of the photoacoustic pressure. Since such kinds of inverse problems are known to be generically ill-posed, we propose here an optimal control approach, introducing a penalized functional with a regularizing term in order to deal with such difficulties. The coefficients we want to recover stand for the control variable. We provide a mathematical analysis of this problem, showing that this approach makes sense. We finally write necessary first-order optimality conditions and give preliminary numerical results.