Galerkin Reduced Order Models Research Articles

An efficient data-driven multiscale stochastic reduced order modeling framework for complex systems

Suitable reduced order models (ROMs) are computationally efficient tools in characterizing key dynamical and statistical features of nature. In this paper, a systematic multiscale stochastic ROM framework is developed for complex systems with strong chaotic or turbulent behavior. The new ROMs fundamentally differ from the traditional Galerkin ROM (G-ROM) or those deterministic closure ROMs that aim to minimize path-wise errors and apply mainly to laminar systems. Here, the new ROM focuses on recovering the large-scale dynamics to the maximum extent. At the same time, it also exploits cheap but effective conditionally linear functions as the closure terms to capture the statistical features of the medium-scale variables and their feedback to the large scales. The new ROM mimics the model structure of many complex systems, allowing it to be more physically explainable. In addition, physics constraints of energy-conserving nonlinearities are incorporated into the new ROM. One unique feature of the resulting ROM is that it facilitates an efficient and accurate scheme for nonlinear data assimilation, the solution of which is provided by closed analytic formulae that do not require ensemble methods. Such an analytical solvable data assimilation solution significantly accelerates computational efficiency. It allows the new ROM to avoid many potential numerical and sampling issues in recovering the unobserved states from partial observations. The closure term calibration is efficient and systematic via explicit mathematical formulae. The new ROM framework is applied to complex nonlinear systems in which the intrinsic turbulent behavior is either triggered by external random forcing or deterministic nonlinearity. The out-of-sample numerical results show that the new ROM significantly outperforms the G-ROM in both scenarios regarding reproducing the dynamical and statistical features and recovering unobserved states via the efficient data assimilation scheme.

Hyper-reduction for Petrov–Galerkin reduced order models

Projection-based Reduced Order Models are based on the idea of minimizing the discrete residual of a “full order model” (FOM) while at the same time constraining the unknowns to live in a space of reduced dimension. For problems with symmetric positive definite (SPD) Jacobians, this minimization can be achieved optimally by projecting the full order residual onto the approximation basis (Galerkin Projection). This approach is sub-optimal for problems with non-SPD Jacobians since it only guarantees that the projection of the residual onto the chosen basis is minimized and not the residual itself. One possible alternative in such cases is to directly minimize the 2-norm of the residual. This minimization can be achieved either by using QR factorization to solve the resulting least-squares problem or by employing the method of the normal equations (LSPG) to the same end. The first approach involves constructing and factorizing a rectangular tall and skinny matrix of size proportional to the number of unknowns of the FOM. The LSPG method avoids the use of the large matrix by directly constructing the product by its transpose. Unfortunately, constructing such a product element by element is not feasible and requires the use of a complementary mesh, which adds an extra layer of complexity to the hyper-reduction process when performing mesh sampling. The main idea of this work is to propose an alternative technique based on the idea of Petrov–Galerkin minimization. Essentially, we choose a left basis so that a least-squares minimization procedure can be carried out on a reduced problem while guaranteeing that the discrete full order residual is minimized. The resulting procedure is applicable to problems with both SPD and non-SPD Jacobians. Additionally, the resulting minimization problem can be assembled element by element, avoiding the use of the complementary mesh and simplifying implementation in the context of finite elements. The resulting technique is amenable to hyper-reduction by the use of the Empirical Cubature Method and can be readily applied in the context of nonlinear reduction procedures.

A deep learning approach to Reduced Order Modelling of parameter dependent partial differential equations

Within the framework of parameter dependent Partial Differential Equations (PDEs), we develop a constructive approach based on Deep Neural Networks for the efficient approximation of the parameter-to-solution map. The research is motivated by the limitations and drawbacks of state-of-the-art algorithms, such as the Reduced Basis method, when addressing problems that show a slow decay in the Kolmogorov n n -width. Our work is based on the use of deep autoencoders, which we employ for encoding and decoding a high fidelity approximation of the solution manifold. To provide guidelines for the design of deep autoencoders, we consider a nonlinear version of the Kolmogorov n n -width over which we base the concept of a minimal latent dimension. We show that the latter is intimately related to the topological properties of the solution manifold, and we provide theoretical results with particular emphasis on second order elliptic PDEs, characterizing the minimal dimension and the approximation errors of the proposed approach. The theory presented is further supported by numerical experiments, where we compare the proposed approach with classical Principal Orthogonal Decomposition (POD)-Galerkin reduced order models. In particular, we consider parametrized advection-diffusion PDEs, and we test the methodology in the presence of strong transport fields, singular terms and stochastic coefficients.

Preconditioned least‐squares Petrov–Galerkin reduced order models

AbstractIn this article, we introduce a methodology for improving the accuracy and efficiency of reduced order models (ROMs) constructed using the least‐squares Petrov–Galerkin (LSPG) projection method through the introduction of preconditioning. Unlike prior related work, which focuses on preconditioning the linear systems arising within the ROM numerical solution procedure to improve linear solver performance, our approach leverages a preconditioning matrix directly within the minimization problem underlying the LSPG formulation. Applying preconditioning in this way has the potential to improve ROM accuracy for several reasons. First, preconditioning the LSPG formulation changes the norm defining the residual minimization, which can improve the residual‐based stability constant bounding the ROM solution's error. The incorporation of a preconditioner into the LSPG formulation can have the additional effect of scaling the components of the residual being minimized to make them roughly of the same magnitude, which can be beneficial when applying the LSPG method to problems with disparate scales (e.g., dimensional equations, multi‐physics problems). Importantly, we demonstrate that an “ideal preconditioned” LSPG ROM (a ROM in which the preconditioner is the inverse of the Jacobian of its corresponding full order model) emulates projection of the full order model solution increment onto the reduced basis. This quantity defines a lower bound on the error of a ROM solution for a given reduced basis. By designing preconditioners that approximate the Jacobian inverse—as is common in designing preconditioners for solving linear systems—it is possible to obtain a ROM whose error approaches this lower bound. The proposed approach is evaluated on several mechanical and thermo‐mechanical problems implemented within the Albany HPC code and run in the predictive regime, with prediction across material parameter space. We demonstrate numerically that the introduction of simple Jacobi, Gauss‐Seidel, and ILU preconditioners into the proper orthogonal decomposition/LSPG formulation reduces significantly the ROM solution error, the reduced Jacobian condition number, the number of nonlinear iterations required to reach convergence, and the wall time (thereby improving efficiency). Moreover, our numerical results reveal that the introduction of preconditioning can deliver a robust and accurate solution for test cases in which the unpreconditioned LSPG method fails to converge.

Parametric reduced order models based on a Riemannian barycentric interpolation

AbstractA new strategy for constructing parametric Galerkin reduced order models is presented in this article. This strategy is achieved thanks to the Riemannian manifold, quotient of the set of full‐rank matrices by the orthogonal group. Starting from a set of training parametrized subspaces of the same dimension, namely obtained by the proper orthogonal decomposition, the projection subspace for a new untrained parameter value is sought as the Karcher barycenter of the data points. The principal advantage of this strategy is that it leads to parametric reduced order models that are naturally flexible with respect to parameter variations. This property is a result of the simple expressions of the geodesic exponential and logarithmic mappings involved in the calculations of the approximated projection subspace. Confrontation with the usual interpolation in the tangent space to the Grassmann manifold is carried out for the flow problem past a circular cylinder and the flow in a lid‐driven cavity. Comparable results in terms of accuracy are obtained, while the proposed approach is shown to be computationally cheaper by allowing real‐time update of Galerkin projections.

Calibration of projection-based reduced-order models for unsteady compressible flows

An analysis of calibration for reduced-order models (ROMs) is presented in this work. The Galerkin and least-squares Petrov-Galerkin (LSPG) methods are tested on compressible flows involving a disparity of temporal scales. A novel calibration strategy is proposed for the LSPG method and two test cases are analyzed. The first consists of a subsonic airfoil flow where boundary layer instabilities are responsible for trailing-edge noise generation and the second comprises a supersonic airfoil flow with a transient period where a detached shock wave propagates upstream at the same time that shock-vortex interaction occurs at the trailing edge. Results show that calibration produces stable and long-time accurate Galerkin and LSPG ROMs for both cases investigated. The impact of hyper-reduction is tested on LSPG models via an accelerated greedy missing point estimation (MPE) algorithm. For the first case investigated, LSPG solutions obtained with hyper-reduction show good comparison with those obtained by the full order model. However, for the supersonic case the transient features of the flow need to be properly captured by the sampled points of the accelerated greedy MPE method. Otherwise, the dynamics of the moving shock wave are not fully recovered. The impact of different time-marching schemes is also assessed and, differently than reported in literature, Galerkin models are shown to be more accurate than those computed by LSPG when the non-conservative form of the Navier-Stokes equations is solved. For the supersonic case, the Galerkin and LSPG models (without hyper-reduction) capture the overall dynamics of the detached and oblique shock waves along the airfoil. However, when shock-vortex interaction occurs at the trailing-edge, the Galerkin ROM is able to capture the high-frequency fluctuations from vortex shedding while the LSPG presents a more dissipative solution, not being able to recover the flow dynamics.

Improving the predictable accuracy of fluid Galerkin reduced-order models using two POD bases

A fundamental limitation of fluid flow reduced-order models (ROMs) which utilize the proper orthogonal decomposition is that there is little capability to determine one’s confidence in the fidelity of the ROM a priori. One reason why fluid ROMs are plagued by this issue is that nonlinear fluid flows are fundamentally multi-scale, often chaotic dynamical systems and a single linear spatial basis, however carefully selected, is incapable of ensuring that these characteristics are captured. In this paper, the velocity and the velocity gradient were decomposed using differently optimized linear bases. This enabled an optimization for several dynamically significant flow characteristics within the modal bases. This was accomplished while still ensuring the resulting model is accurate and without iterative methods for constructing the modal bases.

Networked-oscillator-based modeling and control of unsteady wake flows.

A networked-oscillator-based analysis is performed to examine and control the transfer of kinetic energy for periodic bluff body flows. The dynamics of energy fluctuations in the flow field are described by a set of oscillators defined by conjugate pairs of spatial proper orthogonal decomposition (POD) modes. To extract the network of interactions among oscillators, impulse responses of the oscillators to amplitude and phase perturbations are tracked. Tracking small energy inputs and using linear regression, a networked-oscillator model is constructed that reveals energy exchange among the modes. The model captures the nonlinear interactions among the modal oscillators through a linear approximation. A large collection of system responses is aggregated to capture the general network structure of oscillator interactions. The present networked-oscillator model describes the modal perturbation dynamics more accurately than the empirical Galerkin reduced-order model. The linear network model for nonlinear dynamics is subsequently utilized to design a model-based feedback controller. The controller suppresses the modal amplitudes that result in wake unsteadiness leading to drag reduction. The strength of the proposed approach is demonstrated for a canonical example of two-dimensional unsteady flow over a circular cylinder. The present formulation enables the characterization of modal interactions to control fundamental energy transfers in unsteady bluff body flows.

A Galerkin-free model reduction approach for the Navier–Stokes equations

Galerkin projection of the Navier-Stokes equations on Proper Orthogonal Decomposition (POD) basis is predominantly used for model reduction in fluid dynamics. The robustness for changing operating conditions, numerical stability in long-term transient behavior and the pressure-term consideration are generally the main concerns of the Galerkin Reduced-Order Models (ROM). In this article, we present a novel procedure to construct an off-reference solution state by using an interpolated POD reduced basis. A linear interpolation of the POD reduced basis is performed by using two reference solution states. The POD basis functions are optimal in capturing the averaged flow energy. The energy dominant POD modes and corresponding base flow are interpolated according to the change in operating parameter. The solution state is readily built without performing the Galerkin projection of the Navier-Stokes equations on the reduced POD space modes as well as the following time-integration of the resulted Ordinary Differential Equations (ODE) to obtain the POD time coefficients. The proposed interpolation based approach is thus immune from the numerical issues associated with a standard POD-Galerkin ROM. In addition, a posteriori error estimate and a stability analysis of the obtained ROM solution are formulated. A detailed case study of the flow past a cylinder at low Reynolds numbers is considered for the demonstration of proposed method. The ROM results show good agreement with the high fidelity numerical flow simulation.

Construction of energy-stable projection-based reduced order models

An approach for building energy-stable Galerkin reduced order models (ROMs) for linear hyperbolic or incompletely parabolic systems of partial differential equations (PDEs) using continuous projection is developed. This method is an extension of earlier work by the authors specific to the equations of linearized compressible inviscid flow. The key idea is to apply to the PDEs a transformation induced by the Lyapunov function for the system, and to build the ROM in the transformed variables. For linear problems, the desired transformation is induced by a special inner product, termed the “symmetry inner product”, which is derived herein for several systems of physical interest. Connections are established between the proposed approach and other stability-preserving model reduction methods, giving the paper a review flavor. More specifically, it is shown that a discrete counterpart of this inner product is a weighted L2 inner product obtained by solving a Lyapunov equation, first proposed by Rowley et al. and termed herein the “Lyapunov inner product”. Comparisons between the symmetry inner product and the Lyapunov inner product are made, and the performance of ROMs constructed using these inner products is evaluated on several benchmark test cases.

Practical implementation aspects of Galerkin reduced order models based on proper orthogonal decomposition for computational fluid dynamics

The significant computational cost of computational fluid dynamics (CFD) is still the major drawback to its use for multi-query analysis or optimization cycles. This paper is intended to present the formulation of a Reduced Order Model (ROM) for CFD through the traditional Galerkin projection of differential equations on the modes obtained by proper orthogonal decomposition (POD). The ROM framework was built considering the projection of the continuous equations, which allows its utilization based solely on the Full Order Model (FOM) results. This strategy enables the independence of the CFD solver. This preliminary research illustrates the application of this technique in three examples of increasing complexity. Firstly, the linear heat equation in a one-dimensional domain, then the one-dimensional Burgers’ equation. Finally, the Navier–Stokes equations in a two-dimensional domain for a backward-facing step flow is analyzed considering different Reynolds numbers. The implementation details are discussed and, using ANSYS CFX® as FOM reference, the results for the incompressible Navier–Stokes are investigated in terms of the accuracy level achieved, depending on the parameters chosen for the snapshots generation and the execution time reduction obtained.

Error estimates for Galerkin reduced-order models of the semi-discrete wave equation

Galerkin reduced-order models for the semi-discrete wave equation, that preserve the second-order structure, are studied. Error bounds for the full state variables are derived in the continuous setting (when the whole trajectory is known) and in the discrete setting when the Newmark average-acceleration scheme is used on the second-order semi-discrete equation. When the approximating subspace is constructed using the proper orthogonal decomposition, the error estimates are proportional to the sums of the neglected singular values. Numerical experiments illustrate the theoretical results.

Stable Galerkin reduced order models for linearized compressible flow

The Galerkin projection procedure for construction of reduced order models of compressible flow is examined as an alternative discretization of the governing differential equations. The numerical stability of Galerkin models is shown to depend on the choice of inner product for the projection. For the linearized Euler equations, a symmetry transformation leads to a stable formulation for the inner product. Boundary conditions for compressible flow that preserve stability of the reduced order model are constructed. Preservation of stability for the discrete implementation of the Galerkin projection is made possible using a piecewise-smooth finite element basis. Stability of the reduced order model using this approach is demonstrated on several model problems, where a suitable approximation basis is generated using proper orthogonal decomposition of a transient computational fluid dynamics simulation.

Non-linear modal properties of non-shallow cables

A non-linear mechanical model of non-shallow linearly elastic suspended cables is employed to investigate the non-linear modal characteristics of the free planar motions. An asymptotic analysis of the equations of motion is carried out directly on the partial-differential equations overcoming the drawbacks of a discretization process. The direct asymptotic treatment delivers the approximation of the individual non-linear normal modes. General properties about the non-linearity of the in-plane modes of different type—geometric, elasto-static and elasto-dynamic—are unfolded. The spatial corrections to the considered linear mode shape caused by the quadratic geometric forces are investigated for modes belonging to the three mentioned classes. Moreover, the convergence of Galerkin reduced-order models is discussed and the influence of passive modes is highlighted.