144 research outputs found
Optimal control of multiscale systems using reduced-order models
We study optimal control of diffusions with slow and fast variables and
address a question raised by practitioners: is it possible to first eliminate
the fast variables before solving the optimal control problem and then use the
optimal control computed from the reduced-order model to control the original,
high-dimensional system? The strategy "first reduce, then optimize"--rather
than "first optimize, then reduce"--is motivated by the fact that solving
optimal control problems for high-dimensional multiscale systems is numerically
challenging and often computationally prohibitive. We state sufficient and
necessary conditions, under which the "first reduce, then control" strategy can
be employed and discuss when it should be avoided. We further give numerical
examples that illustrate the "first reduce, then optmize" approach and discuss
possible pitfalls
Interaction of scales for a singularly perturbed degenerating nonlinear Robin problem
We study the asymptotic behavior of the solutions of a boundary value problem
for the Laplace equation in a perforated domain in , ,
with a (nonlinear) Robin boundary condition on the boundary of the small hole.
The problem we wish to consider degenerates under three aspects: in the limit
case the Robin boundary condition may degenerate into a Neumann boundary
condition, the Robin datum may tend to infinity, and the size of the
small hole where we consider the Robin condition collapses to . We study how
these three singularities interact and affect the asymptotic behavior as
tends to , and we represent the solution and its energy integral
in terms of real analytic maps and known functions of the singular perturbation
parameters
Eigenvalue problem in a solid with many inclusions: asymptotic analysis
We construct the asymptotic approximation to the first eigenvalue and
corresponding eigensolution of Laplace's operator inside a domain containing a
cloud of small rigid inclusions. The separation of the small inclusions is
characterised by a small parameter which is much larger compared with the
nominal size of inclusions. Remainder estimates for the approximations to the
first eigenvalue and associated eigenfield are presented. Numerical
illustrations are given to demonstrate the efficiency of the asymptotic
approach compared to conventional numerical techniques, such as the finite
element method, for three-dimensional solids containing clusters of small
inclusions.Comment: 55 pages, 5 figure
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