We study a finite{dimensional approach to the Heath Jarrow Morton model for interest rate and introduce a notion of approximate consistency for a family of functions in a deterministic and stochastic framework. This amounts to asking the decrease of the minimum distance in least squares sense. We start from a general linearly parameterized set of functions and extend the theory to a nonlinear Nelson Siegel family. Necessary and sufficient condition to have approximately consistency are given as well as a criterion of stability for the approximation.

Cancer immunotherapy aims at eliciting an immune system response against the tumor. However, it is often characterized by toxic side-effects. Limiting the tumor growth and, concurrently, avoiding the toxicity of a drug, is the problem of protocol design. We formulate this question as an optimization problem and derive an algorithm for its solution. Unlike the standard optimal control approach, the algorithm simulates impulse-like drug administrations. It relies on an exact computation of the gradient of the cost function with respect to any protocol by means of the variational equations, that can be solved in parallel with the system. In comparison with previous versions of this method [F. Castiglione, B. Piccoli, Optimal control in a model of dendritic cell transfection cancer immunotherapy, Bull. Math. Biol. 68 (2006) 255–274; B. Piccoli, F. Castiglione, Optimal vaccine scheduling in cancer immunotherapy, Physica A. 370 (2) (2007) 672–680], we optimize both the timing and the dosage of each administration and introduce a penalty term to avoid clustering of subsequent injections, a requirement consistent with the clinical practice. In addition, we implement the optimization scheme to simulate the case of multitherapies. The procedure works for any ODE system describing the pharmacokinetics and pharmacodynamics of an arbitrary number of therapeutic agents. In this work, it was tested for a well known model of the tumor–immune system interaction [D. Kirschner, J.C. Panetta, Modeling immunotherapy of tumor–immune interaction, J. Math. Biol. 37 (1998) 235–252]. Exploring three immunotherapeutic scenarios (CTL therapy, IL-2 therapy and combined therapy), we display the stability and efficacy of the optimization method, obtaining protocols that are successful compromises between various clinical requirements.

We introduce a simulation algorithm based on a fluid-dynamic model for traffic flows on road networks, which are considered as graphs composed by arcs that meet at some junctions. The approximation of scalar conservation laws along arcs is made by three velocities Kinetic schemes with suitable boundary conditions at junctions. Here we describe the algorithm and we give an example. © 2006 Elsevier B.V. All rights reserved.

We consider a model for supply chains governed by partial di®eren- tial equations. The mathematical properties of a continuous model are discussed and existence and uniqueness is proven. Moreover, Lipschitz continuous dependence on the initial data is proven. We make use of the front tracking method to construct approximate solutions.

A mathematical model for the diffusion-transport of a substance between two porous homogeneous media of different properties and dimensions is presented. A strong analogy with the one-dimensional transient heat conduction process across two-layered slabs is shown and a similar methodology of solution is proposed. Separation of variables leads to a Sturm-Liouville problem with discontinuous coefficients and an exact analytical solution is given in the form of an infinite series expansion. The model points out the role of four non-dimensional parameters which control the diffusion mechanism across the two porous layers. The drug-eluting stent constitutes the main application of the present model. Drug concentration profiles at various times are given and analyzed. Also, qualitative considerations and a quantitative description to evaluate feasibility of new drug delivery strategies are provided, and some indicators, such as the emptying time, useful to optimize the drug-eluting stent design are discussed.

We consider a hyperbolic conservation law with discontinuous flux. Such a partial differential equation arises in different applications, in particu- lar we are motivated by a model of traffic flow. We provide a new formulation in terms of Riemann Solvers. Moreover, we determine the class of Riemann Solvers which provide existence and uniqueness of the corresponding weak en- tropic solutions.

This paper is focused on continuum-discrete models for supply chains. In particular, we consider the model introduced in [ ], where a system of conservation laws describe the evolution of the supply chain status on sub-chains, while at some nodes solutions are determined by Riemann solvers. Fixing the rule of flux maximization, two new Riemann Solvers are defined. We study the equilibria of the resulting dynamics, moreover some numerical experiments on sample supply chains are reported. We provide also a comparison, both of equilibria and experiments, with the model of [ ].

We are concerned with the numerical study of a simple one-dimensional Schr\"odinger operator $-\frac 1 2 \Dxx + \alpha q(x)$ with $\alpha \in \Re$, $q(x)=\cos(x)+\eps \cos(kx)$, $\eps >0$ and $k$ being irrational. This governs the quantum wave function of an independent electron within a crystalline lattice perturbed by some impurities whose dissemination induces long-range order only, which is rendered by means of the quasi-periodic potential $q$. We study numerically what happens for various values of $k$ and $\eps$; it turns out that for $k > 1$ and $\eps\ll 1$, that is to say, in case more than one impurity shows up inside an elementary cell of the original lattice, ``impurity bands" appear and seem to be $k$-periodic. When $\eps$ grows bigger than one, the opposite case occurs.

We consider the approximation of a microelectronic device corresponding to a $n^+-n-n^+$ diode consisting in a channel flanked on both sides by two highly doped regions. This is modelled through a system of equations: ballistic for the channel and drift-diffusion elsewhere. The overall coupling stems from the Poisson equation for the self-consistent potential. We propose an original numerical method for its processing, being realizable, explicit in time and nonnegativity preserving on the density. In particular, the boundary conditions at the junctions express the continuity of the current and don't destabilize the general scheme. At last, efficiency is shown by presenting results on test-cases of some practical interest.

The aim of this text is to develop on the asymptotics of some 1-D nonlinear Schr\"odinger equations from both the theoretical and the numerical perspectives, when a caustic is formed. We review rigorous results in the field and give some heuristics in cases where justification is still needed. The scattering operator theory is recalled. Numerical experiments are carried out on the focus point singularity for which several results have been proven rigorously. Furthermore, the scattering operator is numerically studied. Finally, experiments on the cusp caustic are displayed, and similarities with the focus point are discussed. Several shortcomings of spectral time-splitting schemes are investigated.

Colorization refers to an image processing task which recovers color in grayscale images when only small regions with color are given. We propose a couple of variational models using chromaticity color components to colorize black and white images. We first consider total variation minimizing (TV) colorization which is an extension from TV inpainting to color using chromaticity model. Second, we further modify our model to weighted harmonic maps for colorization. This model adds edge information from the brightness data, while it reconstructs smooth color values for each homogeneous region. We introduce penalized versions of the variational models, we analyze their convergence properties, and we present numerical results including extension to texture colorization.