Empirical Eigenfunctions Research Articles

Nonlinear model reduction for dynamic analysis of cell population models

Transient cell population balance models consist of nonlinear partial differential-integro equations. An accurate discretized approximation typically requires a large number of nonlinear ordinary differential equations that are not well suited for dynamic analysis and model based controller design. In this paper, proper orthogonal decomposition (also known as the method of empirical orthogonal eigenfunctions and Karhunen Loéve expansion) is used to construct nonlinear reduced-order models from spatiotemporal data sets obtained via simulations of an accurate discretized yeast cell population model. The short-term and long-term behavior of the reduced-order models are evaluated by comparison to the full-order model. Dynamic simulation and bifurcation analysis results demonstrate that reduced-order models with a comparatively small number of differential equations yield accurate predictions over a wide range of operating conditions.

Dynamic optimization of dissipative PDE systems using nonlinear order reduction

This article presents computationally efficient methods for the solution of dynamic constraint optimization problems arising in the context of spatially distributed processes governed by highly dissipative nonlinear partial differential equations (PDEs). The methods are based on spatial discretization using the method of weighted residuals with spatially global basis functions (i.e., functions that cover the entire domain of definition of the process and satisfy the boundary conditions). More specifically, we perform spatial discretization of the optimization problems using the method of weighted residuals with analytical or empirical (obtained via Karhunen–Loève expansion) eigenfunctions as basis functions, and combination of the method of weighted residuals with approximate inertial manifolds. The proposed methods account for the fact that the dominant dynamics of highly dissipative PDE systems are low dimensional in nature and lead to approximate optimization problems that are of significantly lower order compared to the ones obtained from spatial discretization using finite-difference and finite-element techniques, and thus, they can be solved with significantly smaller computational demand. The resulting dynamic nonlinear programs include equality constraints that constitute a low-order system of coupled ordinary differential equations and algebraic equations, which can then be solved with combination of standard temporal discretization and nonlinear programming techniques. We employ backward finite differences (implicit Euler) to perform temporal discretization and solve the nonlinear programs resulting from temporal and spatial discretization using reduced gradient techniques (MINOS). We use two representative examples of dissipative PDEs, a diffusion-reaction process with constant and spatially varying coefficients, and the Kuramoto–Sivashinsky equation, a model that describes incipient instabilities in a variety of physical and chemical systems, to demonstrate the implementation and evaluate the effectiveness of the proposed optimization algorithms.

DPIV/T-driven convective heat transfer simulation

We present a new approach to simulating unsteady heat transfer, with only very few degrees of freedom, by employing directly eigenmodes extracted from DPIV/DPIT experimental data. In particular, we formulate Galerkin low-dimensional systems of the coupled Navier–Stokes and energy equations using hierarchical empirical eigenfunctions extracted from an ensemble of velocity and temperature snapshots. We demonstrate that even severely truncated Galerkin representations (two velocity modes and four temperature modes) produce simulations capable of capturing the dynamics of the flow and heat transfer. This finding is documented by applying proper orthogonal decomposition to water flow past a heated circular cylinder at Reynolds number 610.

Low-order dynamical models of thermal convection in high-aspect ratio enclosures

Low-dimensional dynamical models for transitional buoyancy-driven flow in differentially heated enclosures are presented. The full governing partial differential equations with the associated boundary conditions are solved by a spectral element method. Proper orthogonal decomposition is applied to the oscillatory solutions obtained from the full model in order to construct empirical eigenfunctions. Using the most energetic empirical eigenfunctions for the velocity and temperature fields as basis functions and applying Galerkin's method, low-order models consisting of few non-linear ordinary differential equations are obtained. For all cases, close to the “design” conditions (Pr0,Gr0), the low-order model (LOM) predictions are in excellent agreement with the predictions of the full model. In particular, the critical Grashof number at the onset of the first temporal flow instability (Hopf bifurcation) as well as the frequency and amplitude of oscillations at slightly supercritical conditions are in excellent agreement with the predictions of the full model. Far from design conditions, the LOMs capture some important characteristic properties of the full model solutions. For example, the low-order model derived for a cavity of A = 20 and Gr0 = 3.2 × 104, Pr0 = 0.71, captures the multiplicity of solutions for large values of Grashof number, while it predicts a unique steady solution at small values of Grashof number. In addition, the model predicts that a stationary instability precedes the onset of oscillatory convection. On the other hand, low-order models derived for low-aspect ratio cavities predict that the solution is unique and stable for sufficiently small values of Grashof number and that the primary instability leads to oscillatory time-dependent flow in agreement with experimental and numerical studies based on the full model.

High-speed digital imaging of the medial surface of the vocal folds.

High-speed digital imaging of the medial surface of the vocal folds was performed in excised canine larynx experiments. Building on the excised larynx investigations of Baer [Ph.D. dissertation, MIT, Boston, MA (1975)] and hemilarynx investigations of Jiang and Titze [Laryngoscope 103, 872-882 (1993)], nine vocal fold fleshpoints were tracked simultaneously along the medial surface of one coronal plane of the left vocal fold using a Kodak EktaPro 4540 high-speed digital imaging system. By imaging from two distinct views, 3D reconstructions of fleshpoint trajectories were performed with a sampling frequency of 4.5 kHz and a spatial resolution of approximately 0.08 mm. Quantitative results were derived from a typical example of periodic chestlike vibrations. Furthermore, these data were decomposed into empirical eigenfunctions, the building blocks of vocal fold vibration, illuminating basic mechanisms of self-sustained oscillation. Previously, such mechanisms have only been explored theoretically using computer models of vocal fold vibration [Berry et al., J. Acoust. Soc. Am. 95, 3595-3604 (1994)]. Similar to the theoretical studies, two eigenfunctions captured 98% of the variance of the data. Because this investigation utilized high-speed technology, the methodology may also be used to examine complex, aperiodic vibrations. Thus, this technique allows mechanisms of regular and irregular vocal fold vibration to be explored using direct observations of vibrating tissues in the laboratory.

Numerical solution of optimal magnetic suppression of natural convection in magneto-hydrodynamic flows by empirical reduction of modes

The control of natural convection in magneto-hydrodynamic (MHD) flow is investigated by means of the Karhunen–Loève Galerkin procedure [Int J Numer Meth Engng 1998;41:1133–51]. The Karhunen–Loève Galerkin procedure, which is a type of Galerkin methods that employs the empirical eigenfunctions of the Karhunen–Loève decomposition as basis functions, can reduce non-linear partial differential equations to sets of minimal number of ordinary differential equations by limiting the solution space to the smallest linear subspace that is sufficient to describe the observed phenomena. In the present investigation, it is demonstrated that the Karhunen–Loève Galerkin procedure is well suited for the problems of control or optimization, where one has to solve the governing equations repeatedly but one can also estimate the approximate solution space from the range of control variable. The performance of the Karhunen–Loève Galerkin procedure for solving the optimal control problem of natural convection is assessed in comparison with the traditional technique employing the Boussinesq equation, and is found to be very accurate as well as efficient.

Computation of empirical eigenfunctions and order reduction for nonlinear parabolic PDE systems with time-dependent spatial domains

Computation of empirical eigenfunctions and order reduction for nonlinear parabolic PDE systems with time-dependent spatial domains

Analysis of asymmetric vocal fold vibrations in a finite-element model

Asymmetric vibrations of a finite-element model of the vocal folds were analyzed using the method of empirical eigenfunctions. In a previous study of symmetric vibrations from the model, periodic vibrations yielded two dominant eigenfunctions, which were related to the theoretical normal modes of the model. In a more recent study, the method of empirical eigenfunctions was applied to high-speed endoscopic imaging of vocal fold vibration on human subjects, suggesting several mechanisms of irregular vibration. However, in the in vivo study, the investigation of empirical eigenfunctions was limited to the superior view provided by endoscopy. The results of the present computational study were compared with the in vivo study, while providing additional information regarding vocal fold vibration along the medial surface of the folds. Moreover, the computational study allowed the asymmetric vibrations to be studied systematically as a function of tissue stiffness. [Work supported by Grant No. R29 DC03072 from the National Institute on Deafness and Other Communication Disorders.]

The Energy Operator and New Scaling Relations for the Incompressible Navier--Stokes Equations

The incompressible Navier--Stokes equations have been widely studied due to their vast ability to model hydrodynamic phenomena. The nonlinear character of these equations produces a hierarchical system of dynamical scales that offer challenging barriers to computation simulation and scale elimination modeling. Here, the incompressible Navier--Stokes equations are decomposed and studied under the context of global eigensystems that have a direct relation to the dynamical character of the scales-of-motion. In particular, the self-adjoint linear Reynolds--Orr operator is shown to correspond to the energy operator of this nonlinear system of equations and has the distinguishing property of having an eigensystem that orders the time derivative of the integral kinetic energy for the nonlinear Navier--Stokes operator. The eigensystem of the Reynolds--Orr operator is shown to yield a dynamical range of scales that can be characterized by an energy-containing range, an inertial range, and a dissipation range. It is shown that a canonical decomposition of the incompressible Navier--Stokes equations has only a finite number of scales that are responsible for the unsteady character of the nonlinear system. Furthermore, a time-dependent growth of the kinetic energy is characterized without the need for a classical linearized instability. Discussions are presented and estimates are made on the various underlying scaling characteristics, and a contrast is made to empirical eigenfunctions found by ordering the integral kinetic energy.

Reduction of modes for the solution of inverse natural convection problems

The inverse natural convection problem of determining heat flux at the bottom wall of a two-dimensional cavity from temperature measurement in the domain is investigated by means of the Karhunen–Loève Galerkin procedure. The Karhunen–Loève Galerkin procedure, which is a type of Galerkin method that employs the empirical eigenfunctions of the Karhunen–Loève decomposition as basis functions, can reduce nonlinear partial differential equations to sets of minimal number of ordinary differential equations by limiting the solution space to the smallest linear subspace that is sufficient to describe the observed phenomena. Previously, it had been demonstrated that the problems of optimal control of Burgers equation [H.M. Park, M.W. Lee, Y.D. Jang, Comput. Methods Appl. Mech. Engrg. 166 (1998) 289–308] and the Navier–Stokes equation [H.M. Park, M.W. Lee, Comput. Methods Appl. Mech. Engrg., 1999 (in press)] can be solved very efficiently through the reduction of modes based on the Karhunen–Loève Galerkin procedure. In the present investigation, this technique is applied to the solution of inverse natural convection problem of estimating unknown wall heat flux. The performance of the present technique of inverse analysis using the Karhunen–Loève Galerkin procedure is assessed in comparison with a traditional technique employing the Boussinesq equation, and is found to be very accurate as well as efficient.

The Karhunen–Loève Galerkin method for the inverse natural convection problems

The Karhunen–Loève Galerkin method, which is a type of Galerkin method that employs the empirical eigenfunctions of the Karhunen–Loève decomposition as basis functions, is shown to solve inverse natural convection problems efficiently. The specific problem investigated is the inverse natural convection problem of determining the time-varying strength of a heat source from temperature measurement in the domain. The Karhunen–Loève Galerkin procedure can reduce the Boussinesq equation to a set of minimal number of ordinary differential equations by limiting the solution space to the smallest linear subspace that is sufficient to describe the observed phenomena. The performance of the present technique of inverse analysis using the Karhunen–Loève Galerkin procedure is assessed in comparison with the traditional technique of employing the Boussinesq equation, and is found to be very accurate as well as efficient.

Optimization of transport-reaction processes using nonlinear model reduction

This article focuses on optimization problems arising in the context of transport-reaction processes which are governed by nonlinear elliptic partial differential equations and proposes a computationally efficient method for their solution. The central idea of the method is to discretize the infinite-dimensional optimization problem by utilizing the method of weighted residuals with empirical eigenfunctions obtained by applying Karhunen–Loéve expansion to an appropriately constructed ensemble of solutions of the PDE equality constraints for different values of the design variables. This model reduction procedure leads to low-dimensional nonlinear programs that represent accurate approximations of the original infinite-dimensional nonlinear program, and whose solution can be obtained with standard optimization algorithms. The key issues of construction of the ensemble used for the computation of the empirical eigenfunctions and validity of the optimal solutions computed from the finite-dimensional programs are addressed. The proposed method is applied to two representative transport-reaction processes and is shown to be more efficient compared to conventional optimization approaches based on spatial discretization with the finite-difference method.

Boundary control of the Navier–Stokes equation by empirical reduction of modes

The Karhunen–Loève Galerkin procedure is a type of Galerkin methods that employs the empirical eigenfunctions of the Karhunen–Loève decomposition as basis functions. This technique can reduce nonlinear partial differential equations to sets of minimal number of ordinary differential equations by limiting the solution space to the smallest linear subspace that is sufficient to describe the observed phenomena. Previously [1], it has been shown that one dimensional Burgers equation is reduced to a low dimensional model by this method, which is employed to solve boundary optimal control problems very efficiently. The present paper demonstrates that the Karhunen–Loève Galerkin procedure can be extended to solve problems of the boundary optimal control of multidimensional Navier–Stokes equations. Since the reduction of modes in the multidimensional case is much larger than that in the one dimensional case, the present technique is found to be more powerful when applied to the control problems of the Navier–Stokes equation than those of the Burgers equation.

Control of Navier–Stokes equations by means of mode reduction

AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In a previous work (Park HM, Lee MW. An efficient method of solving the Navier–Stokes equation for the flow control. International Journal of Numerical Methods in Engineering 1998; 41: 1131–1151), the authors proposed an efficient method of solving the Navier–Stokes equations by reducing their number of modes. Employing the empirical eigenfunctions of the Karhunen–Loève decomposition as basis functions of a Galerkin procedure, one can a priori limit the function space considered to the smallest linear sub-space that is sufficient to describe the observed phenomena, and consequently, reduce the Navier–Stokes equations defined on a complicated geometry to a set of ordinary differential equations with a minimum degree of freedom. In the present work, we apply this technique, termed the Karhunen–Loève Galerkin procedure, to a pointwise control problem of Navier–Stokes equations. The Karhunen–Loève Galerkin procedure is found to be much more efficient than the traditional method, such as finite difference method in obtaining optimal control profiles when the minimization of the objective function has been done by using a conjugate gradient method. Citing Literature Volume33, Issue430 June 2000Pages 535-557 RelatedInformation

Control of Burgers equation by means of mode reduction

We consider a new method of solving control problems for the viscous Burgers equation in one space dimension. It is based on the Karhunen–Loève decomposition which is a technique of obtaining empirical eigenfunctions from the experimental or numerical data of a system. Employing these empirical eigenfunctions as basis functions of a Galerkin procedure, one can limit the solution space to the smallest linear subspace that is sufficient to describe the observed phenomena, and consequently reduce the Burgers equation to a set of minimal number of ordinary differential equations. The resulting low dimensional model of Burgers equation is shown to simulate the original system almost exactly. The present algorithm is well suited for the problems of control or optimization, where one has to solve the governing equation repeatedly but one can also estimate the approximate solution space based on the range of control variables. The present method of solving control problem of Burgers equation employing the low dimensional model obtained by the Karhunen–Loève Galerkin procedure yields accurate results in a very efficient way, when the minimization of the objective function is performed using a conjugate gradient method.

Solution of an inverse heat transfer problem by means of empirical reduction of modes

An efficent method of solving the inverse heat transfer problem of estimating the unknown function of wall heat flux for laminar flow inside a duct is proposed in the present paper. It is based on the Karhunen—Loeve Galerkin procedure which employs the empirical eigenfunctions of the Karhunen—Loeve decomposition as basis functions of a Galerkin procedure. With the empirical eigenfunctions, one can a priori limit the function space considered to the smallest linear subspace that is sufficient to describe the observed phenomena, and thus convert the governing equations to a model with a minimum degree of freedom, resulting in a drastic reduction of computation time without loss of accuracy. The performance of the present technique of inverse analysis using the Karhunen—Loeve Galerkin procedure is evaluated by several numerical experiments, and is found to be very accurate as well as efficient.

Finite-dimensional approximation and control of non-linear parabolic PDE systems

This article proposes a rigorous and practical methodology for the derivation of accurate finite-dimensional approximations and the synthesis of non-linear output feedback controllers for non-linear parabolic PDE systems for which the manipulated inputs, the controlled and measured outputs are distributed in space. The method consists of three steps: first, the Karhunen-Loeve expansion is used to derive empirical eigenfunctions of the non-linear parabolic PDE system, then the empirical eigenfunctions are used as basis functions within a Galerkin's and approximate inertial manifold model reduction framework to derive low-order ODE systems that accurately describe the dominant dynamics of the PDE system, and finally, these ODE systems are used for the synthesis of non-linear output feedback controllers that guarantee stability and enforce output tracking in the closed-loop system. The proposed method is used to perform model reduction and synthesize a non-linear dynamic output feedback controller for a rapid thermal chemical vapour deposition process. The controller uses measurements of wafer temperature at five locations to manipulate the power of the top lamps in order to achieve spatially uniform temperature, and thus, uniform deposition of the thin film on the wafer over the entire process cycle. The performance of the non-linear controller is successfully tested through simulations and is shown to be superior to the one of a linear controller.

Control of Navier-Stokes equations by means of mode reduction

In a previous work (Park HM, Lee MW. An efficient method of solving the Navier–Stokes equation for the flow control. International Journal of Numerical Methods in Engineering 1998; 41: 1131–1151), the authors proposed an efficient method of solving the Navier–Stokes equations by reducing their number of modes. Employing the empirical eigenfunctions of the Karhunen–Loève decomposition as basis functions of a Galerkin procedure, one can a priori limit the function space considered to the smallest linear sub-space that is sufficient to describe the observed phenomena, and consequently, reduce the Navier–Stokes equations defined on a complicated geometry to a set of ordinary differential equations with a minimum degree of freedom. In the present work, we apply this technique, termed the Karhunen–Loève Galerkin procedure, to a pointwise control problem of Navier–Stokes equations. The Karhunen–Loève Galerkin procedure is found to be much more efficient than the traditional method, such as finite difference method in obtaining optimal control profiles when the minimization of the objective function has been done by using a conjugate gradient method.

Proper orthogonal decomposition of direct numerical simulation data: Data reduction and observer construction

In this paper, direct numerical simulation (DNS) data of an opposed-jet hydrogen/air diffusion flame are, in a postprocessing step, analyzed using the proper orthogonal decomposition (POD) technique. The aim of this work is twofold. The first goal is to compute a small number of space-dependent empirical eigenfunctions, so that a low-dimensional representation of the data generated by the large model of the discretized partial differential equations can be obtained using a weighted sum of these few eigenfunctions (POD modes). It is found that only six modes are needed for an accurate representation of the data in an extended range of inflow velocities. This large data reduction takes into account not only chemical kinetics but also transport phenomena in a full two-dimensional context and constitutes the first step toward the construction of low-dimensional dynamic models for the opposed-jet system. It is also found that the PODs have very good interpolatory properties. The second goal is to use part of the available data (i.e., partial measurements), together with the computed modes, to estimate, or, in the terminology of process control, to observe, the “unmeasured” quantities. It is found that only a small number of measurements are needed to obtain accurate estimates of the rest of the data.

Output Feedback Control of Parabolic PDE Systems with Nonlinear Spatial Differential Operators

This article focuses on nonlinear static (direct) output feedback control of parabolic partial differential equations (PDE) systems with nonlinear spatial differential operators with application to a rapid thermal chemical vapor deposition (RTCVD) process. Initially, a detailed mathematical model is presented for the RTCVD process, which consists of a nonlinear parabolic PDE that describes the spatiotemporal evolution of the wafer temperature, coupled with a set of nonlinear ordinary differential equations (ODEs) that describes the time evolution of the chamber temperature and the concentrations of the various species. Then, a systematic methodology is presented for the synthesis of nonlinear static output feedback controllers for parabolic PDE systems with nonlinear spatial differential operators. Initially, the Karhunen−Loéve expansion is used to derive empirical eigenfunctions of the nonlinear parabolic PDE system, then the empirical eigenfunctions are used as basis functions within a Galerkin model reduction framework to derive low-order ODE systems that accurately describe the dominant dynamics of the PDE system, and finally, these ODE systems are used for the synthesis of nonlinear static output feedback controllers that guarantee stability and enforce output tracking in the closed-loop system. The proposed control method is employed to synthesize a nonlinear easy-to-implement controller for the RTCVD process that uses measurements of wafer temperature at five locations to manipulate the power of the top lamps in order to achieve uniform temperature and, thus, uniform deposition of a thin film on the wafer over the entire process cycle. The performance of the developed nonlinear output feedback controller is successfully tested through simulations and is shown to be superior to the one of a linear control scheme.