Optimal Control of a Phase-Field Model Using Proper Orthogonal Decomposition

Volkwein, Stefan

doi:10.1002/1521-4001(200102)81:2<83::aid-zamm83>3.0.co;2-r

Cited by 70 publications

(49 citation statements)

References 13 publications

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“…The results confirm the good approximation properties of such schemes which was already reported in [3][4][5][6]8,11,13,14,19], for example. We also compare V -and H-norm based schemes and schemes based on POD-ensembles with and without difference quotients.…”

Section: Numerical Experimentssupporting

confidence: 87%

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Galerkin proper orthogonal decomposition methods for parabolic problems

Kunisch

Volkwein

2001

Numerische Mathematik

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534

538

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Section: Numerical Experimentssupporting

confidence: 87%

“…We infer from (4) and (5) (19). Taking ψ = U k as the test function in (19b) we obtain from (3) and (5)…”

Section: Corollarymentioning

confidence: 99%

Galerkin proper orthogonal decomposition methods for parabolic problems

Kunisch

Volkwein

2001

Numerische Mathematik

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534

538

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“…This section will focus on the framework specific to this paper; for more detailed presentations about POD, see, e.g., [9,15,16,31].…”

Section: Notation and Preliminariesmentioning

confidence: 99%

An Ensemble-Proper Orthogonal Decomposition Method for the Nonstationary Navier--Stokes Equations

Gunzburger¹,

Jiang²,

Schneier³

2017

SIAM J. Numer. Anal.

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Abstract. The definition of partial differential equation (PDE) models usually involves a set of parameters whose values may vary over a wide range. The solution of even a single set of parameter values may be quite expensive. In many cases, e.g., optimization, control, uncertainty quantification, and other settings, solutions are needed for many sets of parameter values. We consider the case of the time-dependent Navier-Stokes equations for which a recently developed ensemble-based method allows for the efficient determination of the multiple solutions corresponding to many parameter sets. The method uses the average of the multiple solutions at any time step to define a linear set of equations that determines the solutions at the next time step. To significantly further reduce the costs of determining multiple solutions of the Navier-Stokes equations, we incorporate a proper orthogonal decomposition (POD) reduced-order model into the ensemble-based method. The stability and convergence results for the ensemble-based method are extended to the ensemble-POD approach. Numerical experiments are provided that illustrate the accuracy and efficiency of computations determined using the new approach.Key words. Ensemble methods, proper orthogonal decomposition, reduced-order models, Navier-Stokes equations.1. Introduction. Computing an ensemble of solutions of fluid flow equations for a set of parameters or initial/boundary conditions for, e.g., quantifying uncertainty or sensitivity analyses or to make predictions, is a common procedure in many engineering and geophysical applications. One common problem faced in these calculations is the excessive cost in terms of both storage and computing time. Thanks to recent rapid advances in parallel computing as well as intensive research in ensemble-based data assimilation, it is now possible, in certain settings, to obtain reliable ensemble predictions using only a small set of realizations. Successful methods that are currently used to generate perturbations in initial conditions include the Bred-vector method, [30], the singular vector method, [5], and the ensemble transform Kalman filter, [4]. Despite all these efforts, the current level of available computing power is still insufficient to perform high-accuracy ensemble computations for applications that deal with large spatial scales such as numerical weather prediction. In such applications, spatial resolution is often sacrificed to reduce the total computational time. For these reasons the development of efficient methods that allow for fast calculation of flow ensembles at a sufficiently fine spatial resolution is of great practical interest and significance.Only recently, a first step was taken in [22,23] where a new algorithm was proposed for computing an ensemble of solutions of the time-dependent Navier-Stokes equations (NSE) with different initial condition and/or body forces. At each time step, the new method employs the same coefficient matrix for all ensemble members. This reduces the problem of solving multi...

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“…Among the challenges one has to overcome when one wants to apply ROM techniques for optimization problems are the efficient computation of ROMs for use in optimization and the derivation of error estimates for ROMs. Some aspects of these questions have been addressed in [3,4,15,16,17,29,30,31,34,35,36,38,39,44,45,50,51,52]. In most of these applications, estimates for the error between the solution of the original optimization problem and the optimization problem governed by the reduced order model are not available; if they exist, then only for a restricted class of problems.…”

Section: Introductionmentioning

confidence: 99%

Projection Based Model Reduction for Optimal Design of the Time-dependent Stokes System

Franke

Hoppe

Linsenmann

et al. 2011

International Series of Numerical Mathematics

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Abstract. The optimal design of structures and systems described by partial differential equations (PDEs) often gives rise to large-scale optimization problems, in particular if the underlying system of PDEs represents a multi-scale, multi-physics problem. Therefore, reduced order modeling techniques such as balanced truncation model reduction, proper orthogonal decomposition, or reduced basis methods are used to significantly decrease the computational complexity while maintaining the desired accuracy of the approximation. In particular, we are interested in such shape optimization problems where the design issue is restricted to a relatively small portion of the computational domain. In this case, it appears to be natural to rely on a full order model only in that specific part of the domain and to use a reduced order model elsewhere. A convenient methodology to realize this idea consists in a suitable combination of domain decomposition techniques and balanced truncation model reduction. We will consider such an approach for shape optimization problems associated with the time-dependent Stokes system and derive explicit error bounds for the modeling error. As an application in life sciences, we will be concerned with the optimal design of capillary barriers as part of a network of microchannels and reservoirs on microfluidic biochips that are used in clinical diagnostics, pharmacology, and forensics for high-throughput screening and hybridization in genomics and protein profiling in proteomics.Mathematics Subject Classification (2000). Primary 65K10; Secondary 49M05; 49M15; 65M55; 65M60; 76Z99; 90C06.

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Optimal Control of a Phase-Field Model Using Proper Orthogonal Decomposition

Cited by 70 publications

References 13 publications

Galerkin proper orthogonal decomposition methods for parabolic problems

Galerkin proper orthogonal decomposition methods for parabolic problems

An Ensemble-Proper Orthogonal Decomposition Method for the Nonstationary Navier--Stokes Equations

Projection Based Model Reduction for Optimal Design of the Time-dependent Stokes System

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