Proper Orthogonal Decomposition in PDE-Constrained Optimization
Karl Kunisch, University of Graz, Austria.
Abstract: In this talk we focus on three issues that we addressed in our recent research on POD, which primarily motivated by problems in PDE constrained optimization. If either the size of a particular problem, or real time requirements make model reduction necessary, POD methods can certainly prove to be very usefull.
In the context of optimization with PDEs as constraints the difficulty of unmodelled dynamcis has to be addressed. We propose a method, called "optimality system POD" (OSPOD) which automatically updates the basis as the iterative optimization proceeds. - We also address the issue of error estimates of POD approximation of nonlinear dynamcial systems, and comment on recent developpemts on optimal snapshot locations.
This is jointwork with S. Volkwein.
Slides (pdf 3.0Mo)
Shear flow compositions on the Galerkin piano: A unified theory for instabilities, strange attractors, statistical mechanics, and attractor control
Bernd. R. Noack, Berlin Institute of Technology, Germany.
Abstract: A well-tuned Galerkin melody about fluid flows requires a piano with the right keys (called modes, e.g.\ POD), musical notes (physics resolution), an audience (observer), and a composer (control law) in case the music shall be changed. Here, we briefly review key enablers for these tasks. Then, a finite-time thermodynamics is proposed bridging fluid mechanics disciplines and theoretical physics treasures. This formalism opens the path to fully nonlinear attractor control, as required for turbulence manipulation in experimental demonstrators. Presented studies include shear flows with simple to complex dynamics.
Slides (pdf 3.7Mo)
Sequential Approximation of Velocity Fields Using Episodic POD
D. Rempfer & P. Mokhasi, Illinois Institute of Technology, Chicago, IL, USA
Abstract: The motivation for our work is the problem of predicting contaminant dispersion in complex turbulent flows, such as the atmospheric boundary layer in urban areas. Since real-time simulation is not a realistic alternative at these scales, our approach relies on estimating the velocity field based on a small number of sensors, coupled with methods for the short-term prediction of the velocity field. Our principal tool, Proper Orthogonal Decomposition, allows us to represent the velocity fields using an optimally small number of expansion coefficients.
This talk will focus on the progress we have made in developing a model that allows for approximation of velocity fields at past and future instances in time based on noisy velocity information available at the present time step. What we have developed is a sequential model that updates and corrects previous estimates of velocity fields when new information is provided. We will show that this model is robust to noise and outliers in the sensor data, and dynamically consistent with respect to the underlying dynamical equations.
Slides (pdf 2.8Mo)
An accurate reduced order model for unsteady flows controlled by synthetic jets
Jessie Weller, Institut de Mathématiques de Bordeaux
Abstract: We consider the problem of building an accurate low order model for relatively complicated flows, actuated by devices that affect the velocity and pressure field as a function of time. We study a 2D case of flow past a confined square cylinder. Two actuators are placed on the lower and upper part of the cylinder. Several simulations, over a fixed time interval, are performed using precomputed, and feedback, control laws. The POD technique is then applied to this database to derive a low order model. A model obtained by Galerkin projection of the Navier-Stokes equations onto the POD basis proves to be inaccurate, but we show that a variation, obtained by minimizing the residual of the model equations, is capable of reproducing the flow dynamics, at least for those cases that were used to build the model. We then test the model for a range of other control laws.
Slides (pdf 0.4Mo)
A short overview of model reduction methods and some new results
Anthasios C. Antoulas, Dept. of Electrical and Comput. Engr., Rice University, Houston, USA.
Abstract: Direct numerical simulation has become one of few available means for the systematic study of physical phenomena for which experiments are dangerous, expensive, or illegal to perform. But without the aid of systematic strategies for reducing model complexity, the burdens of complex geometries, multiphysics, and hostile operating environments coupled with an ever increasing appetite for accuracy and model fidelity would likely render simulation an ineffective tool.
In this talk, we will first provide a brief overview of the various model reduction methods. Then we will discuss some new results concerning model reduction directly from measured data, and model reduction with preservation of passivity.
Slides (pdf 7.3Mo)
POD model reduction of large scale geophysical models
I. Michael Navon, Department of Mathematics, Florida State University, USA.
Abstract: Four-dimensional variational data assimilation (4DVAR) is a powerful tool for data assimilation in meteorology and oceanography. However, a major hurdle in use of 4DVAR for realistic models is the dimension of the control space (generally equal to the size of the model state variable and typically of order 107–108) and the high computational cost in computing the cost function and its gradient requiring integration of model and its adjoint.
This led to the introduction of a reduced model approach of POD type by projecting the full model dynamics into the reduced space. Experience with a large-scale POD-based reduced model for an unstructured ocean model, the 3-D finite element Imperial College Ocean Model (ICOM) with inclusion of adaptive mesh capability is presented along with work on a ocean reduced gravity model. An adaptive POD 4-D Var is employed to update the POD bases as minimization advances and loses control. Issues of time weighting the snapshots using a dual-weighted approach to order reduction in 4D-Var Data Assimilation system tested with NASA finite volume global shallow water equations model will be discussed. Problems of suboptimal control with POD reduced model and error estimation will be addressed as well highlighting limitations of the POD approach. Finally an approach using 4-D VAR data assimilation with reduced order POD model will be outlined.
Slides (pdf 5.0Mo)
Reduced order approaches for variational data assimilation in an ocean circulation model
Eric Blayo, University of Grenoble and INRIA, France.
Abstract: A reduced order approach for 4D-Var data assimilation is presented in the context of a tropical Pacific ocean model. The control space is defined as the span of a few vectors representing a significant part of the system variability, while the model itself is not reduced. It is shown that such an approach can lead to significant improvements, both in terms of the quality of the solution and of the computational efficiency, with regard to data assimilation with a full control vector. However several limitations of the approach will also be discussed, as well as a first step towards an hybrid variational-sequential algorithm.
Slides (pdf 13.4Mo)
Model reduction for large-scale applications in probabilistic analysis and inverse problems
Karen Willcox, Department of Aeronautics and Astronautics Massachusetts Institute of Technology, USA.
Abstract: For large-scale optimal design, probabilistic analysis, and inverse problem applications, a key challenge is deriving reduced models that capture variation over a parametric input space, which, for many applications, is of high dimension. This talk presents recent developments in methodology that addresses this challenge. Our scalable algorithm for sampling high-dimensional input spaces makes tractable the task of obtaining reduced models that incorporate system parametric dependencies. The methodology is demonstrated for a range of applications, including probabilistic analysis of the effects of blade geometric variations on unsteady aerodynamics, and solution of stochastic inverse problems in contaminant transport, reacting flow, and porous media flow.
Slides (pdf 3.6Mo)
Reduced-order models for fluids, using approximate balanced truncation and dynamically scaling modes
Clarence W. Rowley, Dept. of Mechanical and Aerospace Engineering, Princeton University.
Abstract: This work addresses model-reduction techniques for fluids, using several variants of the standard Proper Orthogonal Decomposition (POD) method. In the first variant, we use POD as an approximation of balanced truncation, using adjoint simulations to quantify the dynamical importance of different flow states. We illustrate the method on a number of examples, including a 3D linearized channel flow, and the separating flow over an airfoil at large angles of attack, and we show that this method significantly outperforms standard POD, producing low-order models that have much greater accuracy for a given number of modes. In the second variant, we present reduced-order models of a temporally evolving shear layer, using POD modes that scale in space as the shear layer spreads. Using this approach, very low-order models (4 complex modes) capture much of the relevant nonlinear behavior over a wide range of shear layer thicknesses. We conclude with some open questions and challenges for POD-based reduced order models.
Slides (pdf 11.3Mo)
Error Estimates for POD based Reduced Order Model of an MHD System
S.S. Ravindran, University of Alabama, Huntsville.
Abstract: Reduced order modeling for the purpose of constructing a low dimensional model from high dimensional or infinite dimensional model has important applications in science and engineering such as fast model evaluations and optimization/control. A popular method for constructing reduced-order model is based on finding a suitable low dimensional basis by proper orthogonal decomposition (POD) and forming a model by Galerkin projection of the infinite dimensional model onto the basis.
In this talk we will discuss error estimates for Galerkin proper orthogonal decomposition method for an unsteady nonlinear coupled partial differential equations arising in magnetohydrodynamics (MHD). A specific finite element in space and finite difference in time discretization scheme will be discussed. Numerical results will be presented for a control problem in MHD.
Slides (pdf 3.0Mo)
A POD based non-linear observer for unsteady flows
Edoardo Lombardi, Institut de Mathématiques de Bordeaux
Abstract: A method is proposed to estimate the velocity field of an unsteady flow using a limited number of flow measurements. The method is based on a non-linear low-dimensional model of the flow and on expanding the velocity field in terms of empirical basis functions. The proposed technique is applied to the flow around a confined square cylinder, both in two- and three-dimensional flow regimes. In the two-dimensional case the action of feed-back synthetic jets on the cylinder is considered in the reduced order model. The estimation procedure is able to effectively reconstruct instantaneous velocity fields. In the three-dimensional case, where more complex dynamics appear in the flow, the estimation procedure gives an accurate prediction only for the POD modes that are related to the vortex shedding. For the remaining modes, the accuracy is lower. It appears that for flows characterized by complex dynamics, the major limitation of all estimation techniques based on POD is indeed the ability of the retained POD modes to adequately representing the flow field. In order to increase the accuracy of the POD modes, two filtering procedures of the high frequency dynamics of the velocity fields used to obtain the POD basis are performed. These POD bases, built from the filtered snapshots, are used within the non-linear observer method and the application of such a technique to the three-dimensional unsteady flow is presented.
Slides (pdf 1.4Mo)
Calibration of POD-based Reduced Order Models
Laurent Cordier LEA, Poitiers
Slides (pdf 1.0Mo)
Improvement of Reduced Order Modeling based on Proper Orthogonal Decomposition
Michel Bergmann, INRIA Bordeaux Sud Ouest, IMB, Bordeaux
Abstract: This study focuses on stabilizing Reduced Order Model based on Proper Orthogonal Decomposition (POD) and on improving the POD functional subspace. A modified reduced order model (ROM) that incorporates directly the pressure term is proposed. The ROM is obtained by seeking a solution that lives in the POD subspace and at the same time minimizes the Navier-Stokes residuals. Both ROM stabilization and POD subspace adaptation make use of methods based on the fine scale equation that is approximated using the residuals of the Navier-Sokes equations. Results are shown for the 2D confined cylinder wake flow.
Slides (pdf 9.5Mo)
Model reduction for fluid flows in a probabilistic framework. Application to control
Lionel Mathelin, LIMSI-CNRS, France
Abstract: In the characterization and/or control perspective, reducing the number of degrees of liberty is often necessary for application to real physical systems. One of the problems associated with this reduction lies on the consistency and fidelity of the derived reduced model when the conditions of the system change or if control is applied. A specific treatment is then required to ensure the consistency of the reduction along the whole path, say, from an uncontrolled to a target state. An approach allowing to deal with this issue and to derive a reduced basis representative of the original system on a whole range of parameters will be presented. A specific time-marching scheme will also be discussed. As far as the time-evolution of the system is concerned, it allows to directly rely on the full details physics rather than restricting to the time-evolution of a small dimensional subspace. Application will be shown on a simple 2-D drag reduction flow control problem.
Slides (pdf 16.3Mo)