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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 }
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\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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