Model reduction techniques with a-posteriori error analysis for linear-quadratic optimal control problems

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.

Many of the tools of dynamical systems and control theory have gone largely unused for fluids, because the governing equations are so dynamically complex, both high-dimensional and nonlinear. Model reduction involves finding low-dimensional models that approximate the full high-dimensional dynamics. This paper compares three different methods of model reduction: proper orthogonal decomposition (POD), balanced truncation, and a method called balanced POD. Balanced truncation produces better reduced-order models than POD, but is not computationally tractable for very large systems. Balanced POD is a tractable method for computing approximate balanced truncations, that has computational cost similar to that of POD. The method presented here is a variation of existing methods using empirical Gramians, and the main contributions of the present paper are a version of the method of snapshots that allows one to compute balancing transformations directly, without separate reduction of the Gramians; and an output projection method, which allows tractable computation even when the number of outputs is large. The output projection method requires minimal additional computation, and has a priori error bounds that can guide the choice of rank of the projection. Connections between POD and balanced truncation are also illuminated: in particular, balanced truncation may be viewed as POD of a particular dataset, using the observability Gramian as an inner product. The three methods are illustrated on a numerical example, the linearized flow in a plane channel.

This paper presents a comparative analysis of two model reduction techniques: balanced truncation and projection onto the dominant eigenspace of the Gramian. Both methods are evaluated through the lens of Laguerre functions which is a reduction technique based on Laguerre function expansion, that approximates the low-rank factors of the Controllability and Observability Gramians to produce approximately balanced systems without solving Lyapunov equations. We demonstrate the effectiveness of each approach in reducing the complexity of linear time-invariant (LTI) systems while preserving system dynamics. Numerical experiments highlight the strengths and weaknesses of both methods, showcasing their applicability to large-scale systems. By thoroughly analyzing both methods in conjunction with Laguerre functions, we contribute to the ongoing discourse in the field and lay the groundwork for future advancements in model reduction techniques.

Inverse viscoelastic material characterization using POD reduced-order modeling in acoustic–structure interaction

Balanced truncation (BT) model order reduction (MOR) is known for its superior accuracy and computable error bounds. Balanced stochastic truncation (BST) is a particular BT procedure that provides a general, structure-independent MOR framework to preserve both passivity and stability of original models. Its application toward large scale systems, however, has been limited by the complexity of solving large size continuous time algebraic Riccati equations (CAREs). This paper introduces a novel quadratic extension of the alternating direction implicit (ADI) iteration, called QADI, that efficiently solves a CARE. A Cholesky factor variant of QADI, called CFQADI, further exploits low rank matrices and and produces solution in factor form that greatly accelerates BST. Remarkable efficiency of the proposed BST/(CF)QADI integration is demonstrated with numerical examples.

In this work linear-quadratic optimal control problems for parabolic equations with mixed control-state constraints are considered. These problems arise when a Lavrentiev regularization is utilized for state constrained linear-quadratic optimal control problems. For the numerical solution a Galerkin discretization is applied utilizing proper orthogonal decomposition (POD). Based on a perturbation method it is determined how far the suboptimal control, computed on the basis of the POD method, is from the (unknown) exact one. Numerical examples illustrate the theoretical results. In particular, the POD Galerkin scheme is applied to a problem with state constraints.

This paper presents some new findings in model order reduction of multi input-multi output (MIMO) systems of modern power grid with wind generation. A widely-used model reduction technique called Balanced Truncation (BT) is compared with a relatively new Interpolatory approach of Iterative Rational Krylov Algorithm (IRKA). It is demonstrated that both BT and IRKA can produce an acceptable accuracy during model reduction for power grids with synchronous generators. Whereas, when modern grid with wind generation is considered, BT fails to produce a stable reduced order model (ROM) while IRKA is still able to derive accurate stable ROM retaining the poorly-damped inter-area modes of the system.

A comparison of model reduction techniques from structural dynamics, numerical mathematics and systems and control

We address the problem of model order reduction (MOR) of parametrized dynamical systems. Motivated by reduced basis (RB) methods for partial differential equations, we show that some characteristic components can be transferred to model reduction of parametrized linear dynamical systems. We assume an affine parameter dependence of the system components, which allows an offline/online decomposition and is the basis for efficient reduced simulation. Additionally, error control is possible by a posteriori error estimators for the state vector and output vector, based on residual analysis and primal-dual techniques. Experiments demonstrate the applicability of the reduced parametrized systems, the reliability of the error estimators and the runtime gain by the reduction technique. The a posteriori error estimation technique can straightforwardly be applied to all traditional projection-based reduction techniques of non-parametric and parametric linear systems, such as model reduction, balanced truncation, moment matching, proper orthogonal decomposition (POD) and so on.

In the present work, multiple non existing model order reduction (MOR) techniques for unstable second‐order form systems (SOSs) are proposed. For unstable SOSs, continuous‐time algebraic Lyapunov equations get unsolvable that halt the reduction process. To avoid this problem, unstable SOS is first decomposed into stable and unstable portions and balanced truncation is applied to the stable part. The obtained reduced order model (ROM) for the stable portion is augmented with the unstable portion to obtain the overall reduced system. It is observed that the second‐order structure in ROM for the first technique gets lost as well as augmented unstable dynamics degrade the ROM performance. To cater to these constraints, two structure‐preserving second‐order balanced truncation techniques for unstable SOSs are proposed in second part. System is first stabilized using the Bernoulli feedback stabilization procedure and then gramians are computed for the stabilized system. Further, gramians are partitioned into position and velocity portions to achieve structure preservation in ROM and balanced truncation is applied. As second technique retains second‐order structure as well as involves stabilized system dynamics, this technique far closely approximates original system behavior. Proposed techniques are tested on multiple systems and results certify the correct development of proposed techniques that can be utilized for MOR applications of unstable SOSs. © 2021 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

Despite great advances in reservoir simulation capabilities with the introduction of high-performance computing (HPC) platforms and enhanced solvers, high fidelity grid-based simulation still remains a challenging task. This task is especially demanding for fine-resolved geological reservoirs with multiphase and multicomponents and in production optimization and uncertainty quantification frameworks where several calls of the large scale simulation model need to be performed. Model order reduction techniques have been applied in porous media flow modeling in order to obtain fast simulation surrogates to alleviate the high computational cost associated with these simulations. The difficulty of applying model reduction techniques to porous media flow arises from the nonlinearity of the system. In order to overcome the issue of nonlinearity, we introduce the bilinear form of the dynamical system which in many cases produces a satisfying approximation of the system. The bilinear approximation is a simple form of the parent system and it is linear in the input and linear in the state but it not linear in both jointly. The bilinear form is computed using truncated multidimensional Taylor's expansion of the system terms. Examples are presented to illustrate this recent approach for the case of single phase flow modeling, and comparisons are made with the case of linearized models and the full nonlinear models. In addition, we present methods to reduce the complexity of the bilinear system of equations and results are presented that compare linear model reduction (balanced truncation) and nonlinear model reductions, such as POD techniques.

The objective of this study is to develop model order reduction capabilities for high-fidelity off-road mobility simulations. The model reduction technique using the proper orthogonal decomposition (POD) is implemented at the level of the numerical solver in order to decrease the number of equations that need to be solved at each iteration of the solution procedure. The POD is, however, limited in that the modes are dependent on snapshot data collected during the running of a full order model (FOM), limiting the modes to being accurate only for the specific scenario from which they were collected. Due to this limitation, a method of mode adaptation through interpolation on a tangent space of the Grassmann manifold is investigated to allow modes to be predicted for cases in which a full order model has not been run. Modes produced for known values of a simulation parameter are used to predict the modes for a value of the simulation parameter for which POD modes have not been directly produced. For a single tire soil bin mobility model, the POD modes are found to be effective at retaining accuracy with minimal errors while also reducing computational time.

Studies a class of balanced truncation algorithms applicable to relative/multiplicative model reduction. These algorithms seek to balance the controllability gramian of a given transfer function and the observability gramian of its right inverse. For this reason, the algorithms are referred to as inverse-weighted balanced truncation (IWBT) algorithms. It is shown that by using IWBT algorithms one can derive relative and multiplicative L/sub /spl infin// error bounds that are known to hold for other reduction algorithms. It is also shown that the balanced stochastic truncation (BST) method is actually one special, but an optimal version of the IWBT algorithms. As such, the authors' result also serves to establish the fact that the available error bounds pertaining to BST algorithms actually hold for IWBT algorithms.

In this paper, we focus on a new class of optimal control problems governed by a simple integral cost functional and isoperimetric-type constraints (constant level sets of some simple integral functionals). By using the notions of a variational differential system and adjoint equation, necessary optimality conditions are established for a feasible solution in the considered optimization problem. More precisely, under simplified hypotheses and using a modified Legendrian duality, we establish a maximum principle for the considered optimization problem.

Previous algorithms to obtain reduced-order models by balanced truncation in a single step either require a very specific way to solve a pair of Lyapunov equations or are suitable only for scalar or symmetric MIMO systems. In this paper, model reduction is revisited and an algorithm to obtain a reduced order model in one step only is proposed. As in the previous algorithms, the key point is to construct two rectangular matrices whose smaller dimensions are equal to the number of Hankel singular values to be kept in the lower model. Unlike the one-step algorithms available in the literature, the algorithm proposed here does not make any restriction to the way the Lyapunov equations necessary to obtain the controllability and observability gramians are solved. Furthermore, since the algorithm only relies on singular value decomposition, it is expected to be robust.