MODEL REDUCTION FOR FLUIDS, USING BALANCED PROPER ORTHOGONAL DECOMPOSITION

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 project, methods of model reduction that integrate feedback active flow control with applications to nonlinear convection and turbulent flows governed by Navier-Stokes equations are developed. A new methodology which extracts boundary conditions in reduced order proper orthogonal decomposition (POD) and finite difference models is developed. A new model reduction method based on empirical data and balanced truncation was developed and applied to nonlinear Galerkin models. Based on this method a new empirical Hankel norm model reduction algorithm is proposed. These methods are applied to a prototype nonlinear convective problem governed by the two-dimensional (2D) Burgers' equation. The reduced models are used in the design of robust boundary controllers that achieve tracking, and implemented on the full order Computational Fluid Dynamics (CFD) models.

The main focus of this paper is on an a-posteriori analysis for differentmodel-order strategies applied to optimal control problems governed bylinear parabolic partial differential equations. Based on a perturbationmethod it is deduced how far the suboptimal control, computed on the basisof the reduced-order model, is from the (unknown) exact one.For the model-order reduction, $\mathcal H_{2,\alpha}$-norm optimal modelreduction (H2), balanced truncation (BT), and proper orthogonaldecomposition (POD) are studied.The proposed approach is based on semi-discretization of the underlyingdynamics for the state and the adjoint equations as a large scale lineartime-invariant (LTI) system. This system is reduced to a lower-dimensionalone using Galerkin (POD) or Petrov-Galerkin (H2, BT) projection. The sizeof the reduced-order system is iteratively increased until the error in theoptimal control, computed with the a-posteriori error estimator, satisfiesa given accuracy. The method is illustrated with numerical tests.

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.

Balanced truncation is a standard method for model reduction, possessing many desirable properties such as preservation of model stability and a priori error bounds. Balanced truncation is conducted using controllability and observability Gramians. Generalized Gramians can be found by solving a set of linear matrix inequalities. In this paper, we show that these linear matrix inequalities can be extended so that the number of decision variables are at least doubled, leading to the concept of extended Gramians. Herein it is shown that the desirable properties of balanced truncation also hold with extended Gramians. The extended Gramians are especially useful for improving error bounds and for models possessing additional structural constraints, as is demonstrated by means of examples herein.

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.

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 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.

The multiphase fluid flow in porous media is one of the most fundamental phenomena in various physical processes, such as oil/gas flow in reservoir, subsurface contamination dispersion, chemical separation, etc. Due to its importance, the efficient and accurate solution and prediction of multiphase flow in porous media is highly required in engineering applications and mechanism studies, which has been a research hot spot with increasing interest in recent years. However, the strong nonlinearity implicated in the multiphase flow model has brought great challenges for the computation and analysis. In addition, the permeability in Darcy-type pressure equation is always represented as a high contrast coefficient, which further complicates the simulation difficulty especially for high fidelity prediction and decision-making process. In this study, we attempt to apply one popular global model reduction method, namely Proper Orthogonal Decomposition (POD), to accelerate the simulation of incompressible multiphase flow in porous media. The cornerstone of POD is processing information from a sequence of snapshots (or instantaneous solutions) to identify a set of basis functions to construct a low-dimensional space, the most energetic part of these basis functions is then selected to represent the solution of the original system. In our work, one key point is how to establish the accurate POD reduced order model (ROM) for incompressible mul-tiphase porous flow problems. According to Li et al. [1], the traditional Galerkin projection method (directly project the original governing equation into the low-dimensional space spanned by basis functions) is incapable of introducing the fluid saturation and permeability located on the cell face in POD-ROM for incompressible multiphase flow whether in homogeneous or heterogeneous porous media, thus we adopt the matrix operation method (project the discrete governing equation instead of the original governing equation into the low-dimensional space spanned by basis functions) to build the POD-ROM in this study. Based on the proposed POD-ROM framework, we use the implicit method to solve the (non-) wetting-phase pressure and saturation equations. However, in the implicit solution of the pressure and saturation equations, the standard POD-ROM alone sometimes cannot yield an expected acceleration owing to two main reasons: (1) the nonlinearity between pressure and saturation is sensitive, thus the saturation would change largely even with slight pressure change. And the small error of pressure results would lead to a large deviation of saturation, which in turn exerts negative impacts on the computation of pressure; (2) the complexity for computing a projected nonlinear term in POD-ROM still depends on the dimension of the original full-order system, it is unavoidable in conventional POD-ROM. Therefore, more POD modes are needed to be used in order to achieve the prescribed numerical accuracy, which could worsen the speed up efficiency of proposed POD-ROM. To further reduce the computational complexity of POD-ROM to speed up the implicit solution of pressure and saturation equations, another model reduction method called Discrete Empirical Interpolation Method (DEIM) [2] is applied to treat the projected nonlinear term in the proposed incompressible multiphase porous flow POD-ROM. Different from previous studies, we develop an adaptive DEIM (ADEIM) in this work to achieve a more flexible selection of the interpolation points. Combined the POD-ROM with ADEIM, a POD-ADEIM-ROM for incompressible multiphase flow in porous media is established. The computational efficiency and numerical accuracy of the proposed model is validated through several representative numerical examples. Numerical results indicate that an outstanding acceleration (2~3 orders of magnitude faster) without sacrificing numerical accuracy significantly is obtained from the proposed model when comparing to the traditional solution method that without any acceleration technique. In addition, the effects of POD mode number, the distribution of permeability field, and the mesh density on the overall performances of the proposed model are investigated in detail.

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.

Dynamic models have proven to be helpful for determining the residual water content in combustible biomass. However, these models often require partial differential equations, which render simulations impracticable when several thousand particles need to be considered, such as in the drying of wood chips. Reduced-order models help to overcome this problem. We compare proper orthogonal decomposition (POD) based to balanced truncation based reduced-order models. Both reduced models are lean enough for an application to systems with many particles, but the model based on balanced truncation shows more accurate results.

Proper orthogonal decomposition (POD) finds an orthonormal basis yielding an optimal reconstruction of a given dataset. We consider an optimal data reconstruction problem for two general datasets related to balanced POD, which is an algorithm for balanced truncation model reduction for linear systems. We consider balanced POD outside of the linear systems framework, and prove that it solves the optimal data reconstruction problem. The theoretical result is illustrated with an example.

POD reduced-order unstructured mesh modeling applied to 2D and 3D fluid flow

This paper considers an implicitly restarted Krylov subspace method that approximates a stable, linear transfer function $f(s)$ of order n by one of order m, where $n\gg m$. It is well known that oblique projections onto a Krylov subspace may generate unstable partial realizations. To remedy this situation, the oblique projectors obtained via classical Krylov subspace methods are supplemented with further projectors which enable the formation of stable partial realizations directly from $f(s)$. A key feature of this process is that it may be incorporated into an implicit restart scheme. A second difficulty arises from the fact that Krylov subspace methods often generate partial realizations that contain nonessential modes. To this end, balanced truncation may be employed to discard the unwanted part of the reduced-order model. This paper proposes oblique projection methods for large-scale model reduction that simultaneously compute stable reduced-order models while discarding all nonessential modes. It is shown that both of these tasks may be effected by a single oblique projection process. Furthermore, the process is shown to naturally fit into an implicit restart framework. The theoretical properties of these methods are thoroughly investigated, and exact low-dimensional expressions for the $\L^{\infty}$-norm of the residual errors are derived. Finally, the behavior of the algorithm is illustrated on two large-scale examples.

A multiplicity of simulations is required to optimize systems with thermal transient processes in the presence of uncertain parameters. That is why model order reduction is applied to minimize the numerical effort. The consideration of heat radiation and convection with parameter-dependent heat transfer coefficients results in a nonlinear system with many inputs as these loads are distributed over the whole surface limiting the attainable reduced dimension. Therefore, a new input reduction method is presented approximating the input matrix based on load vector snapshots using singular value decomposition. Afterward, standard reduction methods like the Krylov subspace method or balanced truncation can be applied. Compared to proper orthogonal decomposition, the number of training simulations decreases significantly and the reduced-order model provides a high accuracy within a broad parameter range. In a second step, the discrete empirical interpolation method is used to limit the evaluation of the nonlinearity to a few degrees of freedom and proper orthogonal decomposition allows the fast adaptation of the emissivity. As a result, the reduced system becomes independent of the original dimensions and the computation time is reduced drastically. This approach enables an optimal method combination depending on the number of simulations performed with the reduced model.