1,688 research outputs found
Linear Hamilton Jacobi Bellman Equations in High Dimensions
The Hamilton Jacobi Bellman Equation (HJB) provides the globally optimal
solution to large classes of control problems. Unfortunately, this generality
comes at a price, the calculation of such solutions is typically intractible
for systems with more than moderate state space size due to the curse of
dimensionality. This work combines recent results in the structure of the HJB,
and its reduction to a linear Partial Differential Equation (PDE), with methods
based on low rank tensor representations, known as a separated representations,
to address the curse of dimensionality. The result is an algorithm to solve
optimal control problems which scales linearly with the number of states in a
system, and is applicable to systems that are nonlinear with stochastic forcing
in finite-horizon, average cost, and first-exit settings. The method is
demonstrated on inverted pendulum, VTOL aircraft, and quadcopter models, with
system dimension two, six, and twelve respectively.Comment: 8 pages. Accepted to CDC 201
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
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
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
Error estimates for a tree structure algorithm solving finite horizon control problems
In the Dynamic Programming approach to optimal control problems a crucial
role is played by the value function that is characterized as the unique
viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation. It is well
known that this approach suffers of the "curse of dimensionality" and this
limitation has reduced its practical in real world applications. Here we
analyze a dynamic programming algorithm based on a tree structure. The tree is
built by the time discrete dynamics avoiding in this way the use of a fixed
space grid which is the bottleneck for high-dimensional problems, this also
drops the projection on the grid in the approximation of the value function. We
present some error estimates for a first order approximation based on the
tree-structure algorithm. Moreover, we analyze a pruning technique for the tree
to reduce the complexity and minimize the computational effort. Finally, we
present some numerical tests
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