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\centerline{\bf Adaptive vs.\ high order methods for}
\centerline{\bf HJB equations in optimal control}
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\centerline{Lars Gr\"une }
\vspace{0.2truecm}
\centerline{Mathematical Institute, University of Bayreuth }
\centerline{Bayreuth, Germany, e-mail: {\tt lars.gruene@uni-bayreuth.de }} 
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\par\noindent
In this talk we consider the global numerical approach to deterministic 
and stochastic optimal control problems via the solution of the associated 
Hamilton--Jacobi--Bellman (HJB) equation by semi--Lagrangian schemes.
\par\noindent
This method is appealing because it gives the full global information 
(including approximately optimal feedback laws) for the problem 
considered, but it is well known that this approach is subject to the 
``curse of dimensionality''. Hence it is of utmost importance to use 
efficient numerical methods for its implementation, in particular for the 
spatial discretization.   \par\noindent For this purpose, two different 
approaches have been developed during the last several years: low order 
approximations (i.e., piecewise linear or multilinear) with adaptive 
discretization of the state space based on suitable a posteriori error 
estimates and high order methods using polynomial or spline interpolation 
on fixed grids. While the first method is preferable for nonsmooth or 
steep solutions the second is particularly efficient in the smooth case. 
\par\noindent In this talk we will first give an introduction to both 
methods and show how they perform for different HJB equations with smooth 
and nonsmooth viscosity solutions. In the second part of the talk we will 
present a combination of both methods, discuss theoretical convergence 
results and illustrate the performance of this method with several 
examples.
\par\noindent
Joint work with F. Bauer (University of Bayreuth).

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