Reduced Order Methods for Modeling and Computational Reduction

This paper presents a model reduction and control approach for a class of active flow control problem. Computational fluid dynamics (CFD) has been used extensively to study fluid dynamic systems. It is a powerful tool to gain insight and understanding, but it is also extremely computationally intensive and thus unsuitable for control design and iteration. Various model reduction schemes have been proposed in the past to approximate the Navier-Stokes equation with a low-dimensional model. There are essentially two approaches: input/output model identification and modal decomposition based on snapshots of high-dimensional states such as proper orthogonal decomposition (POD). The former captures mostly the local behavior near a steady state and the latter is highly dependent on the snapshots of the flow state used to extract the projection. This paper presents a novel model reduction approach that attempts to combine the attractive features of the two approaches. The local behavior is first obtained through the identification of a linear (or semi-linear) time invariant model using the input/output data. To capture the large nonlinear structure, we project the difference between CFD response and the linear identified model response onto a set of POD basis. This trajectory is fit to a nonlinear dynamical model to augment the input/output linear model. The resulting linear+nonlinear model is used to design a feedback control law. The difference between the predicted trajectory and CFD simulated trajectory under this control law is further used to refine the POD basis, until a good match is obtained. The proposed method is applied to 2D compressible flow example with outlet flow rate control for a contraction section. Preliminary results for lift control of a 2D airfoil is also included.

In this thesis we analyze and develop methods based on model order reduction (MOR) for the solution of optimization problems constrained by time-dependent partial differential equations (PDEs). The methods combine a direct solution approach with model reduction via proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM). The reduced-order models (ROMs) are used to approximate the high-dimensional dynamic systems originating from a spatial discretization of a PDE. However, when used in an optimization algorithm, conventional POD/DEIM ROMs often lack the ability to give adequate approximations of the gradient. We propose methods for a suitable enhancement of the ROMs for the optimization purpose which are based on the inclusion of derivative information in the POD and DEIM subspaces. We distinguish two types of error between quantities evaluated with the high-dimensional model and its ROM approximation in dependency on the optimization variable q: The reconstruction error which is evaluated with the same q0 which is used constructing the ROM and the prediction error which assesses approximations at q with a ROM constructed at q0 different to q. The novel reconstruction results we present include estimates for solutions of the adjoint equation and the sensitivity equations as well as for the gradient of the objective function. Based on the estimates we explain how the POD and DEIM bases should be extended with either adjoint or sensitivity information. The enhanced ROMs allow control of the reconstruction error for the objective and its gradient up to machine precision. Moreover, we propose a POD prediction estimate for the objective of the optimization problem in a neighborhood of q where the ROM is constructed. In case of sensitivity-extended POD and DEIM bases we give an analogous result for solutions of the states. The derivative-extended ROMs are then used to develop adaptive algorithms for the solution of optimal control and parameter estimation problems which results in great runtime improvements for the optimization while ensuring high approximation quality of the solution of the original problem. For the parameter estimation case a novel a posteriori error estimate is proposed which assesses the quality of suboptimal solutions obtained with the ROM. A further fundamental contribution is a discussion of discretize-then-optimize (DTO) vs. optimize-then-discretize (OTD) approaches in the context of MOR for optimization. We analyze advantages and disadvantages of both approaches and discuss to which extent our methods exhibit properties of either strategy. We also give examples of representative optimization problems in which standard POD/DEIM ROMs show an inacceptable behavior and can be successfully solved by derivative-extended ROMs. We have further implemented the developed methods emphasizing an efficient realization which is important for the investigation of the MOR potential. We showcase the practical performance of the proposed algorithms and the superiority of derivative-extended over conventional ROMs on two academic and one industry-relevant application which exhibit a variety of challenges for the model reduction approach in optimization.

Model reduction methods are relevant when the computation time of a full convection–diffusion–reaction simulation based on detailed chemical reaction mechanisms is too large. In this article, we consider a model reduction approach based on optimization of trajectories and its applicability to realistic combustion models. As many model reduction methods, it identifies points on a slow invariant manifold based on time scale separation in the dynamics of the reaction system. The numerical approximation of points on the manifold is achieved by solving a semi-infinite optimization problem, where the dynamics enter the problem as constraints. The proof of existence of a solution for an arbitrarily chosen dimension of the reduced model (slow manifold) is extended to the case of realistic combustion models including thermochemistry by considering the properties of proper maps. The model reduction approach is finally applied to two models based on realistic reaction mechanisms: ozone decomposition as a small test case and syngas combustion as a test case including all features of a detailed combustion mechanism.

Calculation of the solution of fuzzy partial differential equations is in general very difficult. We can find the exact solution only in some special cases. Fortunately, in most of engineering applications relations between the solutions and uncertain parameters are monotone (we can assume that, when the uncertainty of the parameters is sufficiently small). In this case, the exact solution can be calculated using only endpoints of given intervals. In order to improve the efficiency of calculation we can apply sensitivity analysis. In this paper, a very efficient algorithm of solution was presented. This algorithm is based on finite element method (or any other numerical method of solution PDE like for example FEM or BEM) and sensitivity analysis. Using this method we can solve engineering problems with thousands degree of freedom. Fuzzy partial differential equations can be applied for modeling of mechanical system (structures) with uncertain parameters. To construct the fuzzy membership function random sets can be applied. This theory contains fuzzy sets and probability theory as special cases. Using algorithms, which are described in this paper we can solve partial differential equations with random and fuzzy parameters.

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.

1 Methods for Model Reduction.- 2 Multiplicative Approximation.- 3 Low Order Controller Design.- 4 Model and Controller Reduction Based On Coprime Factorizations.

A model reduction approach aimed at systematically reducing computational complexity of multiple scale homogenization for nonlinear history-dependent problems is developed. The so-called N-scale model reduction (NMR) approach is based on reformulating a sequence of nonlinear unit cell problems at multiple scales in terms of inelastic deformation modes that a priori satisfy equilibrium equations at multiple scales and thus eliminating the need for costly solution of discretized nonlinear equilibrium. The N-scale model reduction approach is validated on tube crush tests.

Model reduction is a long-standing problem. As model reduction as a frequency domain optimization problem can be conjectured NP-hard, several suboptimal but efficient reduction methods were proposed. The obtained suboptimal reduced model are sometimes far away from the optimal reduced model. In this paper, we propose phase model reduction for obtaining reduced models of stable and strictly minimum phase models, possibly close to optimal model. An efficient solution is presented as a finite dimensional convex optimization problem involving linear matrix inequalities. The motivation for phase model reduction is the practical importance of a good phase approximation. Furthermore, our approach allows to constrain a good approximation on a priori specified frequency intervals, without the introduction of weighting functions. The strong interest of this approach for the model reduction with respect to existing model reduction methods is illustrated by two examples.

We describe a control-oriented model reduction process from partial differential equation (PDE) descriptions of aerodynamic flow systems, and the design of low-order feedback controllers using these procedures. An effective approach for flow model reduction for active control using balanced truncation approaches was developed. The techniques were applied to a flow control problem representative of aerospace problems: active control of flow over a modified 2D backward-facing step, which can be handled by most commercial finite element packages. Although the geometry is rather simple, the richness of the flow field is well documented. Feedback controllers were developed for disturbance rejection and tracking of the reattachment point for both high and low Reynolds numbers.

Dimensionally reduced Krylov subspace model reduction for large scale systems

Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for which the equations governing the system behavior depend on a set of parameters. Examples include parameterized partial differential equations and large-scale systems of parameterized ordinary differential equations. The goal of parametric model reduction is to generate low-cost but accurate models that characterize system response for different values of the parameters. This paper surveys state-of-the-art methods in projection-based parametric model reduction, describing the different approaches within each class of methods for handling parametric variation and providing a comparative discussion that lends insights to potential advantages and disadvantages in applying each of the methods. We highlight the important role played by parametric model reduction in design, control, optimization, and uncertainty quantification---settings that require repeated model evaluations over different parameter values.

Modern power systems have experienced a significant increase in its complexity, and extremely large-scale system models have to be addressed in the study of stability analysis and control. Model reduction is a technique for developing an approximate system model with lower dimensions that shares similar properties to the original system. This work proposes a computationally efficient approach for linear system model reduction in large-scale power systems that is based on the balanced truncation method using an extended Krylov subspace technique. Key algorithm improvements, including sparsity handling of the linearized power system models and efficient computational techniques for solving dual Lyapunov equations, are discussed in detail. In addition, application of the proposed model reduction is extended to unstable systems, and a power system stabilizer (PSS) parameter optimization method based on reduced system models is used to validate the effectiveness of the proposed method. Case studies show that the resulting reduced models preserve the unique characteristics of the original high-dimensional models, including time and frequency domain responses as well as eigenvalues. Numerical results based on test systems with up to 12 $\thinspace$ 685 buses demonstrate the computational efficiency and validity of the proposed approach.

ABSTRACT(Chemical) kinetic models in terms of ordinary differential equations correspond to finite-dimensional dissipative dynamical systems involving a multiple time scale structure. Most dimension reduction approaches aimed at a slow mode description of the full system compute approximations of low-dimensional attracting slow invariant manifolds and parameterize these manifolds in terms of a subset of chosen chemical species, the so called reaction progress variables. The invariance property suggests a slow invariant manifold to be constructed as (a bundle of) solution trajectories of suitable ordinary differential equation initial or boundary value problems. The focus of this work is on a discussion and exploitation for deeper insight of unifying geometric and analytical issues of various approaches to trajectory-based numerical approximation techniques of slow invariant manifolds that are in practical use for model reduction in chemical kinetics. Two basic concepts are pointed out reducing various model reduction approaches to a common denominator. In particular, we discuss our recent trajectory optimization approach in the light of these two concepts. We newly relate both of them within our variational boundary value viewpoint, propose a Hamiltonian formulation and conjecture its relation to conservation laws, (partial) integrability and symmetry issues as novel viewpoints on dimension reduction approaches that might open up new deep perspectives and approaches to the problem from dynamical systems theory.

The motivation for using polynomic combinations of system states and inputs to model nonlinear dynamical system is founded upon the classical theories of analysis and function representation. A feature of such representations is the need to make available all possible monomials in these variables, up to the degree specified, so as to provide for the description of widely varying functions within a broad class. For a particular application, however, certain monomials may be quite superfluous. This paper examines the possibility of removing monomials from the model in accordance with the level of sensitivity displayed by the residuals to their absence. Critical in these studies is the effect of system input excitation, and the effect of discarding monomial terms, upon the model parameter set. Therefore, model reduction is approached iteratively, with inputs re-designed at each iteration to ensure sufficient excitation of remaining monomials for parameter approximation. Examples are reported to illustrate the performance of such model reduction approaches.

Model reduction is a common task within the simulation, control, and optimization of complex dynamical systems. For instance, in control and optimization problems for instationary partial differential equations (PDEs), the associated large-scale linear systems resulting from a (semi-) discretization have to be solved repeatedly. Model reduction is a more and more accepted technique to attack these problems in reasonable time. Here, we investigate model reduction for parabolic PDEs based on balanced truncation (BT). The system theoretic properties of BT and the related method of singular perturbation approximation (SPA) provide desirable features for the reduced-order system. The major computational task in BT is the solution of large-scale Lyapunov or Stein equations, depending on whether the semidiscretized or fully discretized system is considered. Therefore, the method is often considered to be of limited use for really large-scale applications, i.e., problems that require $\gg1$ GByte of storage, so that memory access becomes an issue. Here, we develop effective implementations of BT and SPA by exploiting the structure of the underlying PDE problem. This is achieved by a data-sparse approximation of the (semi-)discretized differential operator using the hierarchical matrix format. Furthermore, we review how to integrate the corresponding formatted arithmetic in the sign function method for computing approximate solution factors of the Lyapunov equations and suggest an iterative scheme for solving Stein equations employing formatted arithmetic. These approaches are well suited for a class of practically relevant problems and allow the application of BT and SPA to systems coming from two-dimensional (2D) and three-dimensional (3D) finite element method (FEM) and boundary element method (BEM) discretizations. The resulting methods are most beneficial in the context of BEM discretizations for 3D problems where large-scale dense matrices define the (semi-)discretized control system. We also analyze the additional approximation error in the reduced-order model introduced by the hierarchical matrix approximation. The efficiency of the methods is demonstrated for linear, scalar convection-diffusion-reaction equations. The numerical results confirm that the model reduction/BT error and the hierarchical matrix approximation error can be balanced by choosing the tolerances and parameters involved appropriately.