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Convergence Rate foraCurse-of-Dimensionality-Free Method for aClass of HJB PDEs William M. McEneaneyL. Jonathan Kluberg2009 год

Convergence Rate for a Curse-of-Dimensionality-Free Method for a Class of HJB PDEs
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Аннотация: In previous work of the first author and others, max-plus methods have been explored for solution of first-order, nonlinear Hamilton–Jacobi–Bellman partial differential equations (HJB PDEs) and corresponding nonlinear control problems. Although max-plus basis expansion and max-plus finite-element methods can provide substantial computational-speed advantages, they still generally suffer from the curse-of-dimensionality. Here we consider HJB PDEs where the Hamiltonian takes the form of a (pointwise) maximum of linear/quadratic forms. The approach to solution will be rather general, but in order to ground the work, we consider only constituent Hamiltonians corresponding to long-run average-cost-per-unit-time optimal control problems for the development. We consider a previously obtained numerical method not subject to the curse-of-dimensionality. The method is based on construction of the dual-space semigroup corresponding to the HJB PDE. This dual-space semigroup is constructed from the dual-space semigroups corresponding to the constituent linear/quadratic Hamiltonians. The dual-space semigroup is particularly useful due to its form as a max-plus integral operator with kernel obtained from the originating semigroup. One considers repeated application of the dual-space semigroup to obtain the solution. Although previous work indicated that the method was not subject to the curse-of-dimensionality, it did not indicate any error bounds or convergence rate. Here we obtain specific error bounds.
Год издания: 2009
Авторы: William M. McEneaney, L. Jonathan Kluberg
Издательство: Society for Industrial and Applied Mathematics
Источник: SIAM Journal on Control and Optimization
Ключевые слова: Numerical methods for differential equations, Advanced Numerical Methods in Computational Mathematics, Fractional Differential Equations Solutions
Другие ссылки: SIAM Journal on Control and Optimization (HTML)
DSpace@MIT (Massachusetts Institute of Technology) (PDF)
DSpace@MIT (Massachusetts Institute of Technology) (HTML)