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Department of Computer Science Numerical Analysis Seminar
10:10 AM Friday 12 November in Bahen Center (BA) Room 1210
"TOWARDS A MULTI-SCALE FRAMEWORK FOR COMPUTATIONAL
FLOW CONTROL AND ESTIMATION"
Bartosz PROTAS
Department of Mathematics and Statistics
McMaster University,
Hamilton, ON, Canada
In this talk we will discuss a range of issues
related to numerical solution of optimal control and
estimation problems for systems governed by the Navier-Stokes
equation. The physical objective that a control or estimation
strategy seeks to achieve is represented by a suitably selected
cost functional which is minimized with respect to the control
variable. We will first derive an optimality system and show
how an optimal solution can be found in computations using a
gradient-based approach. The sensitivity of the cost functional
to control (i.e. the gradient) can be conveniently expressed
using an adjoint field. This method is applied to the problem
of wake control for drag reduction in the laminar regime. In the
numerical simulations, the Navier-Stokes system and the adjoint
system are both solved using a Vortex Method which is briefly
outlined and benchmarked. Application of this control strategy
to multi-scale systems, such as high Reynolds number turbulence,
requires some form of regularization. It may be introduced into
an optimization problem by modifying the form of the
evolution equation and the forms of the norms, duality pairings,
and inner products used to frame the adjoint analysis.
Typically, L_2 brackets are used in the definition of the cost
functional, the adjoint operator, and the cost functional
gradient. If instead we adopt the more general Sobolev brackets,
the various fields involved in the adjoint analysis may be made
smoother and therefore easier to resolve numerically. We will
identify several relationships which illustrate how the
different regularization options fit together to form a general
framework. Many commonly-used strategies for regularization,
including implicit Tikhonov regularization and ad hoc smoothing
of the gradient with the inverse Laplacian, are shown to fit
into the present framework as special cases. The regularization
strategies proposed are exemplified using control and estimation
problems for the Kuramoto-Sivashinsky equation and the
Navier-Stokes equation in various configurations. Computational
examples will be provided to exhibit utility of the presented
strategies. Future directions will also be discussed.
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