Optimal propagating fronts using Hamilton-Jacobi equations

The optimal handling of level sets associated to the solution of Hamilton-Jacobi equations such as the normal flow equation is investigated. The goal is to find the normal velocity minimizing a suitable cost functional that accounts for a desired behavior of level sets over time. Sufficient conditions of optimality are derived that require the solution of a system of nonlinear Hamilton-Jacobi equations. Since finding analytic solutions is difficult in general, the use of numerical methods to obtain approximate solutions is addressed by dealing with some case studies in two and three dimensions

Moving Horizon Trend Identification Based on Switching Models for Data Driven Decomposition of Fluid Flows

Modal decomposition is pretty popular in fluid mechanics, especially for data-driven analysis. Dynamic mode decomposition (DMD) allows to identify the modes that describe complex phenomenona such as those physically modelled by the Navier-Stokes equation. The identified modes are associated with residuals, which can be used to detect a meaningful change of regime, e.g., the formation of a vortex. Toward this end, moving horizon estimation (MHE) is applied to identify the trend of the norm of the residuals that result from the application of DMD for the purpose to automatically classify the time evolution of fluid flows. The trend dynamics is modelled as a switching nonlinear system and hence an MHE problem is solved in such a way to monitor the time behavior of the fluid and quickly identify changes of regime. The stability of the estimation error given by MHE is proved. The combination of DMD and MHE provide successful results as shown by processing experimental datasets of the velocity field of fluid flows obtained by a particle image velocimetry

Evolution of cancer cell populations under cytotoxic therapy and treatment optimisation: Insight from a phenotype-structured model

We consider a phenotype-structured model of evolutionary dynamics in a population of cancer cells exposed to the action of a cytotoxic drug. The model consists of a nonlocal parabolic equation governing the evolution of the cell population density function. We develop a novel method for constructing exact solutions to the model equation, which allows for a systematic investigation of the way in which the size and the phenotypic composition of the cell population change in response to variations of the drug dose and other evolutionary parameters. Moreover, we address numerical optimal control for a calibrated version of the model based on biological data from the existing literature, in order to identify the drug delivery schedule that makes it possible to minimise either the population size at the end of the treatment or the average population size during the course of treatment. The results obtained challenge the notion that traditional high-dose therapy represents a "one-fits-all solution" in anticancer therapy by showing that the continuous administration of a relatively low dose of the cytotoxic drug performs more closely to i.e. the optimal dosing regimen to minimise the average size of the cancer cell population during the course of treatment

Eulerian approximate ray tracing and applications to grid generation

We introduce a scheme to compute the viscosity solution of the Riemannian Eikonal equation, on a regular grid or triangular mesh and which uses the order given by the Sweeping algorithm, to update the points. We also compute the bicharacteristic curves of the viscosity solution in a domain Omega in Eulerian way: instead of solving a system of ODE for every source point belongs to partial derivativeOmega, we label each ray by a parameter theta and thus we compute a function theta = theta(s, t), theta : Omega subset of R-2 --> R, whose level sets are the rays. We then present some numerical results

An homogenized hyperbolic model of multiclass traffic flow: a few examples

We introduce a new homogenized hyperbolic (multiclass) traffic flow model which allows to take into account the behaviors of different type of vehicles (cars, trucks, buses, etc.). We discretize the system with a Godunov Lagrangian scheme with oscillating initial data which describe the non homogeneity of the traffic, and we study the propagation of oscillations as time goes. We show the convergence of the scheme and the existence and uniqueness of the entropy solution to the homogenized system, which allows to completely identify the Young measure for all (x, t). Moreover, we show that this limit system is the hydrodynamic limit of the corresponding multi-class discrete system. Simulations are also presented

A multi-class homogenized hyperbolic model of traffic flow

We introduce a new homogenized hyperbolic (multiclass) traffic flow model, which allows us to take into account the behaviors of different type of vehicles (cars, trucks, buses, etc.) and drivers. We discretize the starting Lagrangian system introduced below with a Godunov scheme, and we let the mesh size h in (x, t) go to 0: the typical length (of a vehicle) and time vanish. Therefore, the variables - here (w, a) - which describe the heterogeneity of the reactions of the different car-driver pairs in the traffic, develop large oscillations when h --> 0. These (known) oscillations in (w, a) persist in time, and we describe the homogenized relations between velocity and density. We show that the velocity is the unique solution "`a la Kruzkov" of a scalar conservation law, with variable coefficients, discontinuous in x. Finally, we prove that the same macroscopic homogenized model is also the hydrodynamic limit of the corresponding multiclass Follow-the-Leader model

The eikonal equation on a manifold. Applications to grid generation or refinement

The aim of this work is the generation of anisotropic meshes which are automatically refined in some regions, typically where we want to solve numerically a PDE whose solution is singular. The basic idea is to consider an initial closed curve and to move this curve by the Hamilton-Jacobi equation on a manifold. Similar ideas could also be useful in image processing, in particular for the active contours method

State observation for Lipschitz nonlinear dynamical systems based on Lyapunov functions and functionals

State observers for systems having Lipschitz nonlinearities are considered for what concerns the stability of the estimation error by means of a decomposition of the dynamics of the error into the cascade of two systems. First, conditions are established in order to guarantee the asymptotic stability of the estimation error in a noise-free setting. Second, under the effect of system and measurement disturbances regarded as unknown inputs affecting the dynamics of the error, the proposed observers provide an estimation error that is input-to-state stable with respect to these disturbances. Lyapunov functions and functionals are adopted to prove such results. Third, simulations are shown to confirm the theoretical achievements and the effectiveness of the stability conditions we have established

On the Application of a Vortex Lattice Method to Lifting Bodies Close to a Free Surface

The interaction of the free surface with either lifting and non lifting, submerged, bodies moving beneath it is of primary interest in naval architecture. Indeed, there are many examples of possible applications such as rudders, stabilizer fins, hydrofoils among the others. The hydrodynamic problem of a submerged lifting body moving close to a free surface presents several complexities that need to be properly addressed in order to achieve a reliable solution. The problem is studied in the framework of a potential flow theory and solved by using an ad-hoc developed Vortex Lattice Method (VLM). The developed method is described and validated by comparison against available data on a flat plate. The analysis then focuses on the convergence properties of the method, especially with respect to the panel dimensions used for the free surface discretization, and on a sensitivity with respect to some peculiar operating parameters such as the depth of the body with respect to the free surface and the angle of attack with respect to the incoming flow