Adaptive spline interpolation for Hamilton-Jacobi-Bellman equations

L. Grüne, F. Bauer, W. Semmler: Adaptive spline interpolation for Hamilton-Jacobi-Bellman equations
Applied Numerical Mathematics 56, 1196 - 1210, 2006

DOI: 10.1016/j.apnum.2006.03.011
Keywords: viscosity solution; optimal control; adaptive discretization; spline interpolation; adaptive grids; fixed point equation; numerical example; convergence; numerical stability
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Abstract:

We study the performace of adaptive spline interpolation in semi--Lagrangian discretization schemes for Hamilton--Jacobi--Bellman equations. We investigate the local approximation properties of cubic splines on locally refined grids by a theoretical analysis. Numerical examples show how this method performs in practice. Using those examples we also illustrate numerical stability issues.

Animation

The following animation shows a numerically unstable iteration (cf. Figure 3 in the paper):
[animated GIF for numerically unstable iteration]

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