Proper Orthogonal Decomposition Model Reduction Research Articles

A discontinuous Galerkin Method based on POD model reduction for Euler equation

<abstract> <p>This paper considers the work of combining the proper orthogonal decomposition (POD) reduced-order method with the discontinuous Galerkin (DG) method to solve three-dimensional time-domain Euler equations. The POD-DG formulation is established by constructing the POD base vector space, based on POD technology one can apply the Galerkin projection of the DG scheme to this dimension reduction space for calculation. Its overall goal is to overcome the disadvantages of high computational cost and memory requirement in the DG algorithm, reduce the degrees of freedom (DOFs) of the calculation model, and save the calculation time while ensuring acceptable accuracy. Numerical experiments verify these advantages of the proposed POD-DG method.</p> </abstract>

Nonlinear Model Order Reduction via Lifting Transformations and Proper Orthogonal Decomposition

This paper presents a structure-exploiting nonlinear model reduction method for systems with general nonlinearities. First, the nonlinear model is lifted to a model with more structure via variable transformations and the introduction of auxiliary variables. The lifted model is equivalent to the original model; it uses a change of variables but introduces no approximations. When discretized, the lifted model yields a polynomial system of either ordinary differential equations or differential-algebraic equations, depending on the problem and lifting transformation. Proper orthogonal decomposition (POD) is applied to the lifted models, yielding a reduced-order model for which all reduced-order operators can be precomputed. Thus, a key benefit of the approach is that there is no need for additional approximations of nonlinear terms, which is in contrast with existing nonlinear model reduction methods requiring sparse sampling or hyper-reduction. Application of the lifting and POD model reduction to the FitzHugh–Nagumo benchmark problem and to a tubular reactor model with Arrhenius reaction terms shows that the approach is competitive in terms of reduced model accuracy with state-of-the-art model reduction via POD and discrete empirical interpolation while having the added benefits of opening new pathways for rigorous analysis and input-independent model reduction via the introduction of the lifted problem structure.

Model reduction of dynamical systems by proper orthogonal decomposition: Error bounds and comparison of methods using snapshots from the solution and the time derivatives

Abstract We consider two proper orthogonal decomposition (POD) methods for dimension reduction of dynamical systems. The first method (M1) uses only time snapshots of the solution, while the second method (M2) augments the snapshot set with time-derivative snapshots. The goal of the paper is to analyze and compare the approximation errors resulting from the two methods by using error bounds. We derive several new bounds of the error from POD model reduction by each of the two methods. The new error bounds involve a multiplicative factor depending on the time steps between the snapshots. For method M1 the factor depends on the second power of the time step, while for method 2 the dependence is on the fourth power of the time step, suggesting that method M2 can be more accurate for small between-snapshot intervals. However, three other factors also affect the size of the error bounds. These include (i) the norm of the second (for M1) and fourth derivatives (M2); (ii) the first neglected singular value and (iii) the spectral properties of the projection of the system’s Jacobian in the reduced space. Because of the interplay of these factors neither method is more accurate than the other in all cases. Finally, we present numerical examples demonstrating that when the number of collected snapshots is small and the first neglected singular value has a value of zero, method M2 results in a better approximation.

Adaptive POD model reduction for solute transport in heterogeneous porous media

We study the applicability of a model order reduction technique to the solution of transport of passive scalars in homogeneous and heterogeneous porous media. Transport dynamics are modeled through the advection-dispersion equation (ADE) and we employ Proper Orthogonal Decomposition (POD) as a strategy to reduce the computational burden associated with the numerical solution of the ADE. Our application of POD relies on solving the governing ADE for selected times, termed snapshots. The latter are then employed to achieve the desired model order reduction. We introduce a new technique, termed Snapshot Splitting Technique (SST), which allows enriching the dimension of the POD subspace and damping the temporal increase of the modeling error. Coupling SST with a modeling strategy based on alternating over diverse time scales the solution of the full numerical transport model to its reduced counterpart allows extending the benefit of POD over a prolonged temporal window so that the salient features of the process can be captured at a reduced computational cost. The selection of the time scales across which the solution of the full and reduced model are alternated is linked to the Péclet number (P e), representing the interplay between advective and dispersive processes taking place in the system. Thus, the method is adaptive in space and time across the heterogenous structure of the domain through the combined use of POD and SST and by way of alternating the solution of the full and reduced models. We find that the width of the time scale within which the POD-based reduced model solution provides accurate results tends to increase with decreasing P e. This suggests that the effects of local-scale dispersive processes facilitate the POD method to capture the salient features of the system dynamics embedded in the selected snapshots. Since the dimension of the reduced model is much lower than that of the full numerical model, the methodology we propose enables one to accurately simulate transport at a markedly reduced computational cost.

POD reduced-order model for steady laminar flow based on the body-fitted coordinate

ABSTRACTCurrent reduced-order models (ROM) by the proper orthogonal decomposition (POD) for fluid flow are on a fixed domain. We expect to combine the POD model reduction and the body-fitted coordinate methods so that the POD reduced-order model for fast calculations of fluid flow can be expanded to domains with different shapes. In this paper, the principle is studied for obtaining basis functions of body-fitted POD model for fluid flow. Three methods to obtain basis functions are discussed theoretically. It is pointed out that contravariant components must be utilized to construct the sample matrix for the fast calculation of steady laminar flow in different-shape domains. Taking this principle into account, a standard POD ROM for steady laminar flow problems is established. It is found that the model appears instable when the geometric parameters vary a lot. To overcome this problem, supplementary matrix is introduced to establish a calibrated model with much better accuracy and stability. The test cases show that the calibrated POD reduced-order model, with higher accuracy and stability than the standard one, can fulfill the fast calculations of steady laminar flow on different-shape domains.

Optimal Control Approaches for Analysis of Energy Use Minimization of Hybrid Ground-Coupled Heat Pump Systems

In this paper, we present a prediction-based dynamic programming (DP) control approach, a nonlinear model predictive control (NMPC) approach, and a linear optimal control (LOC) approach to analyze the minimization of the total energy use of a hybrid ground-coupled heat pump (hp) system (incorporating a ground-coupled hp, a gas boiler, a passive cooler, and an active chiller) under operational constraints. A large-scale emulator model (based on finite-volume method and the equivalent-diameter approach) is used for the borehole system and for the assessment of different control algorithms. A nonlinear autoregressive exogenous model is identified from the input-output data generated by the emulator model to be used in a DP-based controller. Since DP is a global optimal control method, it was used as a reference for performance assessment. Next, a state-space reduced-order control-oriented model with a larger sampling time is obtained from the emulator model using the so-called proper orthogonal decomposition model reduction technique. This model is used in an NMPC algorithm to see how much NMPC is suboptimal with respect to the DP in terms of annual energy use minimization. Finally, a series of LOCs based on constant hp coefficients of performance is tested to see how much the system performance deteriorates. The control algorithms are used for the satisfaction of heating-cooling demands of three types of buildings: 1) heating dominated; 2) cooling dominated; and 3) thermally balanced. The effects of constraining thermal buildup/depletion of ground, variable electricity prices, and marginal violation of thermal comfort on the performance of the different controllers applied are also separately analyzed.

Optimality of balanced proper orthogonal decomposition for data reconstruction II: Further approximation results

In our earlier paper Singler (2010) [7], we showed two separate data sets can be optimally approximated using balanced proper orthogonal decomposition (POD) modes derived from the data. In this work, we prove new results concerning the approximation capability of the balanced POD modes. We give exact computable expressions for the errors between the individual data sets and the low order balanced POD data reconstructions. We also consider approximating elements of the Hilbert space using various projections onto the balanced POD modes. We discuss the relevance of these results to balanced POD model reduction of nonlinear partial differential equations.

Reduced order borehole induction modelling

This article presents a new reduced order model based on proper orthogonal decomposition (POD) for solving the electromagnetic equation for borehole modelling applications. The method aims to accurately and efficiently predict the electromagnetic fields generated by an array induction tool – an instrument that transmits and receives electrical signals along different positions within a borehole. The motivation for this approach is in the generation of an efficient ‘forward model’ (which provides solutions to the electromagnetic equation) for the purpose of improving the efficiency of inversion calculations (which typically require a large number of forward solutions) that are used to determine surrounding material properties. This article develops a reduced order model for this purpose as it can be significantly more efficient to compute than standard models, for example, those based on finite elements. It is shown here how the POD basis functions are generated from the snapshot solutions of a high resolution model, and how the discretised equations can be generated efficiently. The novelty is that this is the first time such a POD model reduction approach has been developed for this application, it is also unique in its use of separate POD basis functions for the real and complex solution fields. A numerical example for predicting the electromagnetic field is used to demonstrate the accuracy of the POD method for use as a forward model. It is shown that the method retains accuracy whilst reducing the costs of the computation by several orders of magnitude in comparison to an established method.

Selective proper orthogonal decomposition model reduction for forming simulations

AbstractRobot‐based incremental sheet metal forming is a cost‐effective and flexible method for prototype and low batch size production. The simulation of such processes is very challenging and elaborate from the computational point of view. To reduce the computational effort model reduction techniques such as proper orthogonal decomposition (POD) can be applied. But the reduction of highly non‐linear models in solid mechanics for example forming simulation still leads to problems of efficiency and accuracy. Therefore, the aim of this paper is to present an alternative way to use POD for forming processes. The presented selective POD (SPOD) method is used to split the model into two domains depending on the degree of plastic strain. Only the domain with approximately linear elastic behavior will be reduced by using POD. Utilizing the SPOD method for the example of forming a horizontal flute reduces the computational time up to around 30 per cent. High accuracy with approximation errors smaller than one per mill is achieved. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Model reduction of a coupled numerical model using proper orthogonal decomposition

Summary Numerical models for variable-density flow and solute transport (VDFST) are widely used to simulate seawater intrusion and related problems. The mathematical model for VDFST is a coupled nonlinear dynamical system, so the numerical discretizations in time and space are usually required to be as fine as possible. As a result, fine-scale transient models require large computational time, which is a disadvantage for state estimation, forward prediction or model inversion. The purpose of this research is to develop mathematical and numerical methods to simulate VDFST via a model order reduction technique called Proper Orthogonal Decomposition (POD) designed for nonlinear dynamical systems. POD was applied to extract leading “model features” (basis functions) through singular value decomposition (SVD) from observational data or simulations (snapshots) of high-dimensional systems. These basis functions were then used in the Galerkin projection procedure that yielded low-dimensional (reduced-order) models. The original full numerical models were also discretized by the Galerkin Finite-Element Method (GFEM). The implementation of the POD reduced-order method was straightforward when applied to the full order model to the complex model. The developed GFEM-POD model was applied to solve two classic VDFST cases, the Henry problem and the Elder problem, in order to investigate the accuracy and efficiency of the POD model reduction method. Once the snapshots from full model results are obtained, the reduced-order model can reproduce the full model results with acceptable accuracy but with less computational cost in comparison with the full model, which is useful for model calibration and data assimilation problems. We found that the accuracy and efficiency of the POD reduced-order model is mainly determined by the optimal selection of snapshots and POD bases. Validation and verification experiments confirmed our POD model reduction procedure.

Experimental design for estimating unknown groundwater pumping using genetic algorithm and reduced order model

[1] An optimal experimental design algorithm is developed to select locations for a network of observation wells that provide maximum information about unknown groundwater pumping in a confined, anisotropic aquifer. The design uses a maximal information criterion that chooses, among competing designs, the design that maximizes the sum of squared sensitivities while conforming to specified design constraints. The formulated optimization problem is non-convex and contains integer variables necessitating a combinatorial search. Given a realistic large-scale model, the size of the combinatorial search required can make the problem difficult, if not impossible, to solve using traditional mathematical programming techniques. Genetic algorithms (GAs) can be used to perform the global search; however, because a GA requires a large number of calls to a groundwater model, the formulated optimization problem still may be infeasible to solve. As a result, proper orthogonal decomposition (POD) is applied to the groundwater model to reduce its dimensionality. Then, the information matrix in the full model space can be searched without solving the full model. Results from a small-scale test case show identical optimal solutions among the GA, integer programming, and exhaustive search methods. This demonstrates the GA's ability to determine the optimal solution. In addition, the results show that a GA with POD model reduction is several orders of magnitude faster in finding the optimal solution than a GA using the full model. The proposed experimental design algorithm is applied to a realistic, two-dimensional, large-scale groundwater problem. The GA converged to a solution for this large-scale problem.

Optimization of ventilation system operation in office environment using POD model reduction and genetic algorithm

The purpose of this paper is to develop a numerical optimization methodology of ventilation system operation that help improve indoor environment quality and energy costs for space heating and cooling simultaneously. By introducing proper orthogonal decomposition (POD) based model reduction technique into computational fluid dynamics (CFD) simulation, the spatial distributions of key environmental parameters are firstly considered in optimization procedure, which enable to help improve the environment comfort of occupied zones in a office room. Four types of low-dimension parameter spaces are acquired by POD method including indoor temperature, airflow, CO2 concentration and predicted mean vote (PMV) distributions. The precisions of the reconstructed spaces are compared with those of CFD-based original ones. Because of the extremely lower dimension, resulted parameter subspaces are suitable for optimization process in this study. In the proposed genetic algorithm (GA), the objective function is constructed in a way attempting to aggregate and weight indices into one indicator, such as index of PMV, space temperature gradients (STG), indoor air quality (IAQ), energy cost of heating and cooling, etc. A simulation based case study is conducted, which indicates that the presented optimization approach is able to improve indoor comfort and energy costs of ventilation system in a balanced way.

Sequential linear quadratic control of bilinear parabolic PDEs based on POD model reduction

We present a framework to solve a finite-time optimal control problem for parabolic partial differential equations (PDEs) with diffusivity-interior actuators, which is motivated by the control of the current density profile in tokamak plasmas. The proposed approach is based on reduced order modeling (ROM) and successive optimal control computation. First we either simulate the parabolic PDE system or carry out experiments to generate data ensembles, from which we then extract the most energetic modes to obtain a reduced order model based on the proper orthogonal decomposition (POD) method and Galerkin projection. The obtained reduced order model corresponds to a bilinear control system. Based on quasi-linearization of the optimality conditions derived from Pontryagin’s maximum principle, and stated as a two boundary value problem, we propose an iterative scheme for suboptimal closed-loop control. We take advantage of linear synthesis methods in each iteration step to construct a sequence of controllers. The convergence of the controller sequence is proved in appropriate functional spaces. When compared with previous iterative schemes for optimal control of bilinear systems, the proposed scheme avoids repeated numerical computation of the Riccati equation and therefore reduces significantly the number of ODEs that must be solved at each iteration step. A numerical simulation study shows the effectiveness of this approach.

A note on POD model reduction methods for DAEs

We review some known results for proper orthogonal decomposition (POD) model order reduction applied to ordinary differential equations (ODEs). Then, these results are generalized for several types of differential-algebraic equations (DAEs). We provide algorithms for the model reduction and error bounds for the reduced-order models. Some limits of the approach are pointed out and alternative methods for reduced-order subspace approximation are presented. The POD approach is tested and evaluated for a medium-sized DAE example from multi-body dynamics.

Analysis of Nonlinear Aeroelastic Panel Response Using Proper Orthogonal Decomposition

The nonlinear panel flutter problem solved by Dowell in 1966 is used to investigate the new application of the proper orthogonal decomposition model reduction technique to aeroelastic analysis. Emphasis is placed on the nonlinear structural dynamic equations with nonconservative forcing modeled assuming a supersonic, inviscid flow. Here the aeroelastic coupled equation is presented in discrete form using a finite difference approach, and subsequently in state space form, to be integrated as a set of first order differential equations. In this paper, a POD approach is developed for generalized second-order differential equations; however, the application of POD to the governing equations in state space form is also discussed. This study compares the results and effectiveness of the model reduction technique for integration of the full set of degrees of freedom. The solution is compared to Dowell’s classic results which forms the base reference for the model reduction study. The reduced order model is then created from the full simulation model. Accuracy of the solution, reduced computational time, limits of stability, and the strengths and weaknesses of the model reduction are investigated.