We present a new efficient computational approach for time-dependent
first-order Hamilton-Jacobi-Bellman PDEs. Since our method is based on
a time-implicit Eulerian discretization, the numerical scheme is
unconditionally stable, but discretized equations for each
time-slice are coupled and non-linear.
We show that the same system can be re-interpreted as
a discretization of a static Hamilton-Jacobi-Bellman PDE on
the same physical domain. The latter was shown
to be "causal" in [Vladimirsky 2006], making fast (non-iterative)
methods applicable.
The implicit discretization results in higher computational cost per time
slice compared to the explicit time marching.
However, the latter is subject to a CFL-stability condition,
and the implicit approach becomes significantly more efficient whenever
the accuracy demands on the time-step are less restrictive than
the stability. We also present a hybrid
method, which aims to combine the advantages of both the explicit and
implicit discretizations.
We demonstrate the efficiency of our approach using
several examples in optimal control of isotropic fixed-horizon
processes.