71 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
An Iterative Model Reduction Scheme for Quadratic-Bilinear Descriptor Systems with an Application to Navier-Stokes Equations
We discuss model reduction for a particular class of quadratic-bilinear (QB)
descriptor systems. The main goal of this article is to extend the recently
studied interpolation-based optimal model reduction framework for QBODEs
[Benner et al. '16] to a class of descriptor systems in an efficient and
reliable way. Recently, it has been shown in the case of linear or bilinear
systems that a direct extension of interpolation-based model reduction
techniques to descriptor systems, without any modifications, may lead to poor
reduced-order systems. Therefore, for the analysis, we aim at transforming the
considered QB descriptor system into an equivalent QBODE system by means of
projectors for which standard model reduction techniques for QBODEs can be
employed, including aforementioned interpolation scheme. Subsequently, we
discuss related computational issues, thus resulting in a modified algorithm
that allows us to construct \emph{near}--optimal reduced-order systems without
explicitly computing the projectors used in the analysis. The efficiency of the
proposed algorithm is illustrated by means of a numerical example, obtained via
semi-discretization of the Navier-Stokes equations
Model order reduction approaches for infinite horizon optimal control problems via the HJB equation
We investigate feedback control for infinite horizon optimal control problems
for partial differential equations. The method is based on the coupling between
Hamilton-Jacobi-Bellman (HJB) equations and model reduction techniques. It is
well-known that HJB equations suffer the so called curse of dimensionality and,
therefore, a reduction of the dimension of the system is mandatory. In this
report we focus on the infinite horizon optimal control problem with quadratic
cost functionals. We compare several model reduction methods such as Proper
Orthogonal Decomposition, Balanced Truncation and a new algebraic Riccati
equation based approach. Finally, we present numerical examples and discuss
several features of the different methods analyzing advantages and
disadvantages of the reduction methods
Model Order Reduction for Rotating Electrical Machines
The simulation of electric rotating machines is both computationally
expensive and memory intensive. To overcome these costs, model order reduction
techniques can be applied. The focus of this contribution is especially on
machines that contain non-symmetric components. These are usually introduced
during the mass production process and are modeled by small perturbations in
the geometry (e.g., eccentricity) or the material parameters. While model order
reduction for symmetric machines is clear and does not need special treatment,
the non-symmetric setting adds additional challenges. An adaptive strategy
based on proper orthogonal decomposition is developed to overcome these
difficulties. Equipped with an a posteriori error estimator the obtained
solution is certified. Numerical examples are presented to demonstrate the
effectiveness of the proposed method
A POD reduced order model for resolving angular direction in neutron/photon transport problems
publisher: Elsevier articletitle: A POD reduced order model for resolving angular direction in neutron/photon transport problems journaltitle: Journal of Computational Physics articlelink: http://dx.doi.org/10.1016/j.jcp.2015.04.043 content_type: article copyright: Copyright © 2015 Elsevier Inc. All rights reserved.publisher: Elsevier articletitle: A POD reduced order model for resolving angular direction in neutron/photon transport problems journaltitle: Journal of Computational Physics articlelink: http://dx.doi.org/10.1016/j.jcp.2015.04.043 content_type: article copyright: Copyright © 2015 Elsevier Inc. All rights reserved.publisher: Elsevier articletitle: A POD reduced order model for resolving angular direction in neutron/photon transport problems journaltitle: Journal of Computational Physics articlelink: http://dx.doi.org/10.1016/j.jcp.2015.04.043 content_type: article copyright: Copyright © 2015 Elsevier Inc. All rights reserved
Model Reduction Opportunities in Detailed Simulations of Combustion Dynamics
Rocket and gas turbine combustion dynamics involves a confluence of diverse physics and interaction across a number of system components. Any comprehensive, self-consistent numerical model is burdened by a very large computational mesh, stiff unsteady processes which limit the permissible time step, and the need to perform tedious, repeated calculations for a broad parametric range. Predictive CFD models rely on very large scale simulations and advanced hardware. Reduced Basis Methods (RBM) have grown in usage during the past decade, as promising new techniques in making large simulations more accessible. These methods create models with far fewer unknown quantities than the original system, by generating “proper” fundamental solutions and their Galerkin projections, while guaranteeing accuracy and computational efficiency. RBMs seek to reproduce full CFD solutions, rather than solutions to a simplified or linearized set of equations. We present here some recent work in this area, focusing on approaches to model large scale combustor systems. The maturation of methods leading to LES-based turbulent combustion modeling is discussed, and model reduction goals and strategies are explored from the perspective of applicability in real life problems in both gas turbine, as well as rocket engines
Model Order Reduction in Fluid Dynamics: Challenges and Perspectives
This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities — which are mainly related either to nonlinear convection terms and/or some geometric variability — that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and — in the unsteady case — long-time stability of the reduced model. Moreover, we provide an extensive list of literature references
Loop Aggregation for Approximate Scientific Computing
International audienceTrading off some accuracy for better performances in scientific computing is an appealing approach to ease the exploration of various alternatives on complex simulation models. Existing approaches involve the application of either time-consuming model reduction techniques or resource-demanding statistical approaches. Such requirements prevent any opportunistic model exploration, e.g., exploring various scenarios on environmental models. This limits the ability to analyse new models for scientists, to support trade-off analysis for decision-makers and to empower the general public towards informed environmental intelligence. In this paper, we present a new approximate computing technique, aka. loop aggregation, which consists in automatically reducing the main loop of a simulation model by aggregating the corresponding spatial or temporal data. We apply this approximate scientific computing approach on a geophysical model of a hydraulic simulation with various input data. The experimentation demonstrates the ability to drastically decrease the simulation time while preserving acceptable results with a minimal set-up. We obtain a median speed-up of 95.13% and up to 99.78% across all the 23 case studies
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