1,353 research outputs found
Some non monotone schemes for Hamilton-Jacobi-Bellman equations
We extend the theory of Barles Jakobsen to develop numerical schemes for
Hamilton Jacobi Bellman equations. We show that the monotonicity of the schemes
can be relaxed still leading to the convergence to the viscosity solution of
the equation. We give some examples of such numerical schemes and show that the
bounds obtained by the framework developed are not tight. At last we test some
numerical schemes.Comment: 24 page
High-order filtered schemes for time-dependent second order HJB equations
In this paper, we present and analyse a class of "filtered" numerical schemes
for second order Hamilton-Jacobi-Bellman equations. Our approach follows the
ideas introduced in B.D. Froese and A.M. Oberman, Convergent filtered schemes
for the Monge-Amp\`ere partial differential equation, SIAM J. Numer. Anal.,
51(1):423--444, 2013, and more recently applied by other authors to stationary
or time-dependent first order Hamilton-Jacobi equations. For high order
approximation schemes (where "high" stands for greater than one), the
inevitable loss of monotonicity prevents the use of the classical theoretical
results for convergence to viscosity solutions. The work introduces a suitable
local modification of these schemes by "filtering" them with a monotone scheme,
such that they can be proven convergent and still show an overall high order
behaviour for smooth enough solutions. We give theoretical proofs of these
claims and illustrate the behaviour with numerical tests from mathematical
finance, focussing also on the use of backward difference formulae (BDF) for
constructing the high order schemes.Comment: 27 pages, 16 figures, 4 table
High-order filtered schemes for the Hamilton-Jacobi continuum limit of nondominated sorting
We investigate high-order finite difference schemes for the Hamilton-Jacobi
equation continuum limit of nondominated sorting. Nondominated sorting is an
algorithm for sorting points in Euclidean space into layers by repeatedly
removing minimal elements. It is widely used in multi-objective optimization,
which finds applications in many scientific and engineering contexts, including
machine learning. In this paper, we show how to construct filtered schemes,
which combine high order possibly unstable schemes with first order monotone
schemes in a way that guarantees stability and convergence while enjoying the
additional accuracy of the higher order scheme in regions where the solution is
smooth. We prove that our filtered schemes are stable and converge to the
viscosity solution of the Hamilton-Jacobi equation, and we provide numerical
simulations to investigate the rate of convergence of the new schemes
Convergent finite difference methods for one-dimensional fully nonlinear second order partial differential equations
This paper develops a new framework for designing and analyzing convergent
finite difference methods for approximating both classical and viscosity
solutions of second order fully nonlinear partial differential equations (PDEs)
in 1-D. The goal of the paper is to extend the successful framework of
monotone, consistent, and stable finite difference methods for first order
fully nonlinear Hamilton-Jacobi equations to second order fully nonlinear PDEs
such as Monge-Amp\`ere and Bellman type equations. New concepts of consistency,
generalized monotonicity, and stability are introduced; among them, the
generalized monotonicity and consistency, which are easier to verify in
practice, are natural extensions of the corresponding notions of finite
difference methods for first order fully nonlinear Hamilton-Jacobi equations.
The main component of the proposed framework is the concept of "numerical
operator", and the main idea used to design consistent, monotone and stable
finite difference methods is the concept of "numerical moment". These two new
concepts play the same roles as the "numerical Hamiltonian" and the "numerical
viscosity" play in the finite difference framework for first order fully
nonlinear Hamilton-Jacobi equations. In the paper, two classes of consistent
and monotone finite difference methods are proposed for second order fully
nonlinear PDEs. The first class contains Lax-Friedrichs-like methods which also
are proved to be stable and the second class contains Godunov-like methods.
Numerical results are also presented to gauge the performance of the proposed
finite difference methods and to validate the theoretical results of the paper.Comment: 23 pages, 8 figues, 11 table
A High-Order Scheme for Image Segmentation via a modified Level-Set method
In this paper we propose a high-order accurate scheme for image segmentation
based on the level-set method. In this approach, the curve evolution is
described as the 0-level set of a representation function but we modify the
velocity that drives the curve to the boundary of the object in order to obtain
a new velocity with additional properties that are extremely useful to develop
a more stable high-order approximation with a small additional cost. The
approximation scheme proposed here is the first 2D version of an adaptive
"filtered" scheme recently introduced and analyzed by the authors in 1D. This
approach is interesting since the implementation of the filtered scheme is
rather efficient and easy. The scheme combines two building blocks (a monotone
scheme and a high-order scheme) via a filter function and smoothness indicators
that allow to detect the regularity of the approximate solution adapting the
scheme in an automatic way. Some numerical tests on synthetic and real images
confirm the accuracy of the proposed method and the advantages given by the new
velocity.Comment: Accepted version for publication in SIAM Journal on Imaging Sciences,
86 figure
Boundary Treatment and Multigrid Preconditioning for Semi-Lagrangian Schemes Applied to Hamilton-Jacobi-Bellman Equations
We analyse two practical aspects that arise in the numerical solution of
Hamilton-Jacobi-Bellman (HJB) equations by a particular class of monotone
approximation schemes known as semi-Lagrangian schemes. These schemes make use
of a wide stencil to achieve convergence and result in discretization matrices
that are less sparse and less local than those coming from standard finite
difference schemes. This leads to computational difficulties not encountered
there. In particular, we consider the overstepping of the domain boundary and
analyse the accuracy and stability of stencil truncation. This truncation
imposes a stricter CFL condition for explicit schemes in the vicinity of
boundaries than in the interior, such that implicit schemes become attractive.
We then study the use of geometric, algebraic and aggregation-based multigrid
preconditioners to solve the resulting discretised systems from implicit time
stepping schemes efficiently. Finally, we illustrate the performance of these
techniques numerically for benchmark test cases from the literature
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