Overview of the BiNoPe-HJ project
The HJB Solver aims at computing approximate solution to Hamilton-Jacobi equations of the following form :
together with some boundary conditions on the boundary ∂Ω of Ω, and where typically :
- H(x,∇u) ≡ maxα(-f(x,α).∇u - ℓ(x,α))
- Ω : "box" domain of ℝd with d ≥ 2
- f : ℝ+ × ℝd × ℝm → ℝd : dynamics of the system
- ℓ : ℝ+ × ℝd × ℝm → ℝ : distributed cost function
- U : commands set and m := |U| number of commands (if they are discretized).
The d-dimensional solver is implemented with mainly two schemes : the Semi-Lagrangian scheme and the Finite Difference scheme. The general structure of is the following :
Also, sequentials version of the Fast Marching Method have been implemented in dimension d, adapted to solving the eikonal equation of the form c(x,u(x))∥∇u(x)∥ = 1 for x ∈ Ω, with u(x) = 0 for x ∈ Ω0 ⊂ Ω.
2 Numerical methods
- Semi Lagrangien Scheme.
- It is implemented with P1 interpolations of the caracteristics which are computed using EDO discretization method such as Euler (RK1), Heun or midpoint rule (RK2).
- Finite Differences Scheme.
- first order Lax-Friedrichs scheme and second order ENO2 (Essentially Non Oscillatory) scheme are implemented, and are coupled with Runge-Kutta methods on order 1, 2 or 3 in time.
- Fast Marching Method (FMM)
- for Eikonal equations, 3 different cases :
- classical FMM for problems with a constant sign, time-independent velocity : c(x)∥∇u(x)∥ = 1 and with c(x) > 0
- time-dependent FMM for problems with constant sign, time-dependent velocity c = c(x,t) and with c(x,t) ≥ 0
- Generalized FMM for problems with a changing-sign, time-dependant velocity : c(x,u(x)))∥∇u(x)∥ = 1 and with c(x) > 0