561 research outputs found
A Rotating-Grid Upwind Fast Sweeping Scheme for a Class of Hamilton-Jacobi Equations
We present a fast sweeping method for a class of Hamilton-Jacobi equations
that arise from time-independent problems in optimal control theory. The basic
method in two dimensions uses a four point stencil and is extremely simple to
implement. We test our basic method against Eikonal equations in different
norms, and then suggest a general method for rotating the grid and using
additional approximations to the derivatives in different directions in order
to more accurately capture characteristic flow. We display the utility of our
method by applying it to relevant problems from engineering
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 parallel Heap-Cell Method for Eikonal equations
Numerous applications of Eikonal equations prompted the development of many
efficient numerical algorithms. The Heap-Cell Method (HCM) is a recent serial
two-scale technique that has been shown to have advantages over other serial
state-of-the-art solvers for a wide range of problems. This paper presents a
parallelization of HCM for a shared memory architecture. The numerical
experiments in show that the parallel HCM exhibits good algorithmic
behavior and scales well, resulting in a very fast and practical solver.
We further explore the influence on performance and scaling of data
precision, early termination criteria, and the hardware architecture. A shorter
version of this manuscript (omitting these more detailed tests) has been
submitted to SIAM Journal on Scientific Computing in 2012.Comment: (a minor update to address the reviewers' comments) 31 pages; 15
figures; this is an expanded version of a paper accepted by SIAM Journal on
Scientific Computin
A Model for Optimal Human Navigation with Stochastic Effects
We present a method for optimal path planning of human walking paths in
mountainous terrain, using a control theoretic formulation and a
Hamilton-Jacobi-Bellman equation. Previous models for human navigation were
entirely deterministic, assuming perfect knowledge of the ambient elevation
data and human walking velocity as a function of local slope of the terrain.
Our model includes a stochastic component which can account for uncertainty in
the problem, and thus includes a Hamilton-Jacobi-Bellman equation with
viscosity. We discuss the model in the presence and absence of stochastic
effects, and suggest numerical methods for simulating the model. We discuss two
different notions of an optimal path when there is uncertainty in the problem.
Finally, we compare the optimal paths suggested by the model at different
levels of uncertainty, and observe that as the size of the uncertainty tends to
zero (and thus the viscosity in the equation tends to zero), the optimal path
tends toward the deterministic optimal path
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