60 research outputs found
Blended numerical schemes for the advection equation and conservation laws
In this paper we propose a method to couple two or more explicit numerical
schemes approximating the same time-dependent PDE, aiming at creating new
schemes which inherit advantages of the original ones. We consider both
advection equations and nonlinear conservation laws. By coupling a macroscopic
(Eulerian) scheme with a microscopic (Lagrangian) scheme, we get a new kind of
multiscale numerical method
Can local single-pass methods solve any stationary Hamilton-Jacobi-Bellman equation?
The use of local single-pass methods (like, e.g., the Fast Marching method)
has become popular in the solution of some Hamilton-Jacobi equations. The
prototype of these equations is the eikonal equation, for which the methods can
be applied saving CPU time and possibly memory allocation. Then, some natural
questions arise: can local single-pass methods solve any Hamilton-Jacobi
equation? If not, where the limit should be set? This paper tries to answer
these questions. In order to give a complete picture, we present an overview of
some fast methods available in literature and we briefly analyze their main
features. We also introduce some numerical tools and provide several numerical
tests which are intended to exhibit the limitations of the methods. We show
that the construction of a local single-pass method for general Hamilton-Jacobi
equations is very hard, if not impossible. Nevertheless, some special classes
of problems can be actually solved, making local single-pass methods very
useful from the practical point of view.Comment: 19 page
A differential model for growing sandpiles on networks
We consider a system of differential equations of Monge-Kantorovich type
which describes the equilibrium configurations of granular material poured by a
constant source on a network. Relying on the definition of viscosity solution
for Hamilton-Jacobi equations on networks, recently introduced by P.-L. Lions
and P. E. Souganidis, we prove existence and uniqueness of the solution of the
system and we discuss its numerical approximation. Some numerical experiments
are carried out
A numerical method for Mean Field Games on networks
We propose a numerical method for stationary Mean Field Games defined on a
network. In this framework a correct approximation of the transition conditions
at the vertices plays a crucial role. We prove existence, uniqueness and
convergence of the scheme and we also propose a least squares method for the
solution of the discrete system. Numerical experiments are carried out
A level set based method for fixing overhangs in 3D printing
3D printers based on the Fused Decomposition Modeling create objects
layer-by-layer dropping fused material. As a consequence, strong overhangs
cannot be printed because the new-come material does not find a suitable
support over the last deposed layer. In these cases, one can add some support
structures (scaffolds) which make the object printable, to be removed at the
end. In this paper we propose a level set method to create object-dependent
support structures, specifically conceived to reduce both the amount of
additional material and the printing time. We also review some open problems
about 3D printing which can be of interests for the mathematical community
A measure theoretic approach to traffic flow optimization on networks
We consider a class of optimal control problems for measure-valued nonlinear
transport equations describing traffic flow problems on networks. The objective
isto minimise/maximise macroscopic quantities, such as traffic volume or
average speed,controlling few agents, for example smart traffic lights and
automated cars. The measuretheoretic approach allows to study in a same setting
local and nonlocal drivers interactionsand to consider the control variables as
additional measures interacting with the driversdistribution. We also propose a
gradient descent adjoint-based optimization method, ob-tained by deriving
first-order optimality conditions for the control problem, and we providesome
numerical experiments in the case of smart traffic lights for a 2-1 junction.Comment: 20 pages, 6 figure
Reliable optimal controls for SEIR models in epidemiology
We present and compare two different optimal control approaches applied to
SEIR models in epidemiology, which allow us to obtain some policies for
controlling the spread of an epidemic. The first approach uses Dynamic
Programming to characterise the value function of the problem as the solution
of a partial differential equation, the Hamilton-Jacobi-Bellman equation, and
derive the optimal policy in feedback form. The second is based on Pontryagin's
maximum principle and directly gives open-loop controls, via the solution of an
optimality system of ordinary differential equations. This method, however, may
not converge to the optimal solution. We propose a combination of the two
methods in order to obtain high-quality and reliable solutions. Several
simulations are presented and discussed
Approximation of the value function for optimal control problems on stratified domains
In optimal control problems defined on stratified domains, the dynamics and
the running cost may have discontinuities on a finite union of submanifolds of
RN. In [8, 5], the corresponding value function is characterized as the unique
viscosity solution of a discontinuous Hamilton-Jacobi equation satisfying
additional viscosity conditions on the submanifolds. In this paper, we consider
a semi-Lagrangian approximation scheme for the previous problem. Relying on a
classical stability argument in viscosity solution theory, we prove the
convergence of the scheme to the value function. We also present HJSD, a free
software we developed for the numerical solution of control problems on
stratified domains in two and three dimensions, showing, in various examples,
the particular phenomena that can arise with respect to the classical
continuous framework
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