227 research outputs found
POD for optimal control of the Cahn-Hilliard system using spatially adapted snapshots
The present work considers the optimal control of a convective Cahn-Hilliard
system, where the control enters through the velocity in the transport term. We
prove the existence of a solution to the considered optimal control problem.
For an efficient numerical solution, the expensive high-dimensional PDE systems
are replaced by reduced-order models utilizing proper orthogonal decomposition
(POD-ROM). The POD modes are computed from snapshots which are solutions of the
governing equations which are discretized utilizing adaptive finite elements.
The numerical tests show that the use of POD-ROM combined with spatially
adapted snapshots leads to large speedup factors compared with a high-fidelity
finite element optimization
Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential
This paper is concerned with a distributed optimal control problem for a
nonlocal phase field model of Cahn-Hilliard type, which is a nonlocal version
of a model for two-species phase segregation on an atomic lattice under the
presence of diffusion. The local model has been investigated in a series of
papers by P. Podio-Guidugli and the present authors; the nonlocal model studied
here consists of a highly nonlinear parabolic equation coupled to an ordinary
differential inclusion of subdifferential type. The inclusion originates from a
free energy containing the indicator function of the interval in which the
order parameter of the phase segregation attains its values. It also contains a
nonlocal term modeling long-range interactions. Due to the strong nonlinear
couplings between the state variables (which even involve products with time
derivatives), the analysis of the state system is difficult. In addition, the
presence of the differential inclusion is the reason that standard arguments of
optimal control theory cannot be applied to guarantee the existence of Lagrange
multipliers. In this paper, we employ recent results proved for smooth
logarithmic potentials and perform a so-called `deep quench' approximation to
establish existence and first-order necessary optimality conditions for the
nonsmooth case of the double obstacle potential.Comment: Key words: distributed optimal control, phase field systems, double
obstacle potentials, nonlocal operators, first-order necessary optimality
conditions. The interested reader can also see the related preprints
arXiv:1511.04361 and arXiv:1605.07801 whose results are recalled and used for
the analysis carried out in this pape
Simulation and control of a nonsmooth Cahn--Hilliard Navier--Stokes system with variable fluid densities
We are concerned with the simulation and control of a two phase flow model governed by a coupled Cahn--Hilliard Navier--Stokes system involving a nonsmooth energy potential.We establish the existence of optimal solutions and present two distinct approaches to derive suitable stationarity conditions for the bilevel problem, namely C- and strong stationarity. Moreover, we demonstrate the numerical realization of these concepts at the hands of two adaptive solution algorithms relying on a specifically developed goal-oriented error estimator.In addition, we present a model order reduction approach using proper orthogonal decomposition (POD-MOR) in order to replace high-fidelity models by low order surrogates. In particular, we combine POD with space-adapted snapshots and address the challenges which are the consideration of snapshots with different spatial resolutions and the conservation of a solenoidal property
Optimal velocity control of a convective Cahn-Hilliard system with double obstacles and dynamic boundary conditions: a `deep quench' approach
In this paper, we investigate a distributed optimal control problem for a
convective viscous Cahn-Hilliard system with dynamic boundary conditions. Such
systems govern phase separation processes between two phases taking place in an
incompressible fluid in a container and, at the same time, on the container
boundary. The cost functional is of standard tracking type, while the control
is exerted by the velocity of the fluid in the bulk. In this way, the coupling
between the state (given by the associated order parameter and chemical
potential) and control variables in the governing system of nonlinear partial
differential equations is bilinear, which presents a difficulty for the
analysis. In contrast to the previous paper arXiv:1709.02335 [math.AP] by the
same authors, the bulk and surface free energies are of double obstacle type,
which renders the state constraint nondifferentiable. It is well known that for
such cases standard constraint qualifications are not satisfied so that
standard methods do not apply to yield the existence of Lagrange multipliers.
In this paper, we overcome this difficulty by taking advantage of results
established in the quoted paper for logarithmic nonlinearities, using a
so-called `deep quench approximation'. We derive results concerning the
existence of optimal controls and the first-order necessary optimality
conditions in terms of a variational inequality and the associated adjoint
system.Comment: Key words: Cahn-Hilliard system, convection term, dynamic boundary
conditions, double obstacle potentials, optimal velocity control, optimality
conditions. arXiv admin note: text overlap with arXiv:1702.0190
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Optimal control of geometric partial differential equations
Optimal control problems for geometric (evolutionary) partial differential inclusions are considered. The focus is on problems which, in addition to the nonlinearity due to geometric evolution, contain optimization theoretic challenges because of non-smoothness. The latter might stem from energies containing non-smooth constituents such as obstacle-type potentials or terms modeling, e.g., pinning phenomena in microfluidics. Several techniques to remedy the resulting constraint degeneracy when deriving stationarity conditions are presented. A particular focus is on Yosida-type mollifications approximating the original degenerate problem by a sequence of nondegenerate nonconvex optimal control problems. This technique is also the starting point for the development of numerical solution schemes. In this context, also dual-weighted residual based error estimates are addressed to facilitate an adaptive mesh refinement. Concerning the underlying state model, sharp and diffuse interface formulations are discussed. While the former always allows for accurately tracing interfacial motion, the latter model may be dictated by the underlying physical phenomenon, where near the interface mixed phases may exist, but it may also be used as an approximate model for (sharp) interface motion. In view of the latter, (sharp interface) limits of diffuse interface models are addressed. For the sake of presentation, this exposition confines itself to phase field type diffuse interface models and, moreover, develops the optimal control of either of the two interface models along model applications. More precisely, electro-wetting on dielectric is used in the sharp interface context, and the control of multiphase fluids involving spinodal decomposition highlights the phase field technique. Mathematically, the former leads to a Hele-Shaw flow with geometric boundary conditions involving a complementarity system due to contact line pinning, and the latter gives rise to a Cahn-Hilliard Navier-Stokes model including a non-smooth obstacle type potential leading to a variational inequality constraint
Optimal control of geometric partial differential equations
Optimal control problems for geometric (evolutionary) partial differential inclusions are considered. The focus is on problems which, in addition to the nonlinearity due to geometric evolution, contain optimization theoretic challenges because of non-smoothness. The latter might stem from energies containing non-smooth constituents such as obstacle-type potentials or terms modeling, e.g., pinning phenomena in microfluidics. Several techniques to remedy the resulting constraint degeneracy when deriving stationarity conditions are presented. A particular focus is on Yosida-type mollifications approximating the original degenerate problem by a sequence of nondegenerate nonconvex optimal control problems. This technique is also the starting point for the development of numerical solution schemes. In this context, also dual-weighted residual based error estimates are addressed to facilitate an adaptive mesh refinement. Concerning the underlying state model, sharp and diffuse interface formulations are discussed. While the former always allows for accurately tracing interfacial motion, the latter model may be dictated by the underlying physical phenomenon, where near the interface mixed phases may exist, but it may also be used as an approximate model for (sharp) interface motion. In view of the latter, (sharp interface) limits of diffuse interface models are addressed. For the sake of presentation, this exposition confines itself to phase field type diffuse interface models and, moreover, develops the optimal control of either of the two interface models along model applications. More precisely, electro-wetting on dielectric is used in the sharp interface context, and the control of multiphase fluids involving spinodal decomposition highlights the phase field technique. Mathematically, the former leads to a Hele-Shaw flow with geometric boundary conditions involving a complementarity system due to contact line pinning, and the latter gives rise to a Cahn-Hilliard Navier-Stokes model including a non-smooth obstacle type potential leading to a variational inequality constraint
Optimal boundary control of a viscous Cahn-Hilliard system with dynamic boundary condition and double obstacle potentials
In this paper, we investigate optimal boundary control problems for
Cahn-Hilliard variational inequalities with a dynamic boundary condition
involving double obstacle potentials and the Laplace-Beltrami operator. The
cost functional is of standard tracking type, and box constraints for the
controls are prescribed. We prove existence of optimal controls and derive
first-order necessary conditions of optimality. The general strategy, which
follows the lines of the recent approach by Colli, Farshbaf-Shaker, Sprekels
(see the preprint arXiv:1308.5617) to the (simpler) Allen-Cahn case, is the
following: we use the results that were recently established by Colli, Gilardi,
Sprekels in the preprint arXiv:1407.3916 [math.AP] for the case of
(differentiable) logarithmic potentials and perform a so-called "deep quench
limit". Using compactness and monotonicity arguments, it is shown that this
strategy leads to the desired first-order necessary optimality conditions for
the case of (non-differentiable) double obstacle potentials.Comment: Key words: optimal control; parabolic obstacle problems; MPECs;
dynamic boundary conditions; optimality conditions. arXiv admin note:
substantial text overlap with arXiv:1308.561
Optimal control of Allen-Cahn systems
Optimization problems governed by Allen-Cahn systems including elastic
effects are formulated and first-order necessary optimality conditions are
presented. Smooth as well as obstacle potentials are considered, where the
latter leads to an MPEC. Numerically, for smooth potential the problem is
solved efficiently by the Trust-Region-Newton-Steihaug-cg method. In case of an
obstacle potential first numerical results are presented
A boundary control problem for the pure Cahn-Hilliard equation with dynamic boundary conditions
A boundary control problem for the pure Cahn-Hilliard equations with possibly
singular potentials and dynamic boundary conditions is studied and first-order
necessary conditions for optimality are proved.
Key words: Cahn-Hilliard equation, dynamic boundary conditions, phase
separation, singular potentials, optimal control, optimality conditionsComment: arXiv admin note: text overlap with arXiv:1407.391
From the viscous Cahn-Hilliard equation to a regularized forward-backward parabolic equation
A rigorous proof is given for the convergence of the solutions of a viscous
Cahn-Hilliard system to the solution of the regularized version of the
forward-backward parabolic equation, as the coefficient of the diffusive term
goes to 0. Non-homogenous Neumann boundary condition are handled for the
chemical potential and the subdifferential of a possible non-smooth double-well
functional is considered in the equation. An error estimate for the difference
of solutions is also proved in a suitable norm and with a specified rate of
convergence.Comment: Key words and phrases: Cahn-Hilliard system, forward-backward
parabolic equation, viscosity, initial-boundary value problem, asymptotic
analysis, well-posednes
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