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Reduced Order Optimal Control of the Convective FitzHugh-Nagumo Equation
In this paper, we compare three model order reduction methods: the proper
orthogonal decomposition (POD), discrete empirical interpolation method (DEIM)
and dynamic mode decomposition (DMD) for the optimal control of the convective
FitzHugh-Nagumo (FHN) equations. The convective FHN equations consists of the
semi-linear activator and the linear inhibitor equations, modeling blood
coagulation in moving excitable media. The semilinear activator equation leads
to a non-convex optimal control problem (OCP). The most commonly used method in
reduced optimal control is POD. We use DEIM and DMD to approximate efficiently
the nonlinear terms in reduced order models. We compare the accuracy and
computational times of three reduced-order optimal control solutions with the
full order discontinuous Galerkin finite element solution of the convection
dominated FHN equations with terminal controls. Numerical results show that POD
is the most accurate whereas POD-DMD is the fastest
Optimal Control of Convective FitzHugh-Nagumo Equation
We investigate smooth and sparse optimal control problems for convective
FitzHugh-Nagumo equation with travelling wave solutions in moving excitable
media. The cost function includes distributed space-time and terminal
observations or targets. The state and adjoint equations are discretized in
space by symmetric interior point Galerkin (SIPG) method and by backward Euler
method in time. Several numerical results are presented for the control of the
travelling waves. We also show numerically the validity of the second order
optimality conditions for the local solutions of the sparse optimal control
problem for vanishing Tikhonov regularization parameter. Further, we estimate
the distance between the discrete control and associated local optima
numerically by the help of the perturbation method and the smallest eigenvalue
of the reduced Hessian
On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond
An improved understanding of the divergence-free constraint for the
incompressible Navier--Stokes equations leads to the observation that a
semi-norm and corresponding equivalence classes of forces are fundamental for
their nonlinear dynamics. The recent concept of {\em pressure-robustness}
allows to distinguish between space discretisations that discretise these
equivalence classes appropriately or not. This contribution compares the
accuracy of pressure-robust and non-pressure-robust space discretisations for
transient high Reynolds number flows, starting from the observation that in
generalised Beltrami flows the nonlinear convection term is balanced by a
strong pressure gradient. Then, pressure-robust methods are shown to outperform
comparable non-pressure-robust space discretisations. Indeed, pressure-robust
methods of formal order are comparably accurate than non-pressure-robust
methods of formal order on coarse meshes. Investigating the material
derivative of incompressible Euler flows, it is conjectured that strong
pressure gradients are typical for non-trivial high Reynolds number flows.
Connections to vortex-dominated flows are established. Thus,
pressure-robustness appears to be a prerequisite for accurate incompressible
flow solvers at high Reynolds numbers. The arguments are supported by numerical
analysis and numerical experiments.Comment: 43 pages, 18 figures, 2 table
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