Energy preserving model order reduction of the nonlinear Schrödinger equation

An energy preserving reduced order model is developed for two dimensional nonlinear Schrödinger equation (NLSE) with plane wave solutions and with an external potential. The NLSE is discretized in space by the symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting system of Hamiltonian ordinary differential equations are integrated in time by the energy preserving average vector field (AVF) method. The mass and energy preserving reduced order model (ROM) is constructed by proper orthogonal decomposition (POD) Galerkin projection. The nonlinearities are computed for the ROM efficiently by discrete empirical interpolation method (DEIM) and dynamic mode decomposition (DMD). Preservation of the semi-discrete energy and mass are shown for the full order model (FOM) and for the ROM which ensures the long term stability of the solutions. Numerical simulations illustrate the preservation of the energy and mass in the reduced order model for the two dimensional NLSE with and without the external potential. The POD-DMD makes a remarkable improvement in computational speed-up over the POD-DEIM. Both methods approximate accurately the FOM, whereas POD-DEIM is more accurate than the POD-DMD.     

Parallel discontinuous Galerkin unstructured mesh solvers for the calculation of three-dimensional wave propagation problems

We discuss the parallel performances of discontinuous Galerkin solvers designed on unstructured tetrahedral meshes for the calculation of three-dimensional heterogeneous electromagnetic and aeroacoustic wave propagation problems. An explicit leap-frog time-scheme along with centered numerical fluxes are used in the proposed discontinuous Galerkin time-domain (DGTD) methods. The schemes introduced are genuinely non-dissipative, in order to achieve a discrete equivalent of the energy conservation. Parallelization of these schemes is based on a standard strategy that combines mesh partitioning and a message passing programming model. The resulting parallel solvers are applied and evaluated on several large-scale, homogeneous and heterogeneous, wave propagation problems.     

Condition number of the stiffness matrix arising in POD Galerkin schemes for dynamical systems

Proper orthogonal decomposition (POD) provides a method for deriving low order models of non‐linear dynamical systems, where a so‐called POD basis is computed by a singular value decomposition and these basis functions are used in a Galerkin ansatz for the non‐linear dynamics. In this paper we prove estimates for the condition number of the stiffness matrices arising in the POD Galerkin discretization. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear dynamic analysis of a complex dual rotor-bearing system based on a novel model reduction method

In this paper, a dynamic model of a complex dual rotor-bearing system of an aero-engine is established based on the finite element method with three types of beam elements (rigid disc, cylindrical beam element and conical beam element), as well as taking into account the nonlinearities of all of the supporting rolling element bearings. To rapidly and accurately analyze dynamic behaviors of the complex dual rotor-bearing system, a two-level model order reduction (MOR) method is proposed by combining component mode synthesis (CMS) method and proper orthogonal decomposition (POD) technique. The first-level reduced-order model (ROM) of the dual rotors is obtained by CMS method with a high precision for the original system. Then, the POD method is applied to second-level model order reduction to further decrease the degrees of freedom (DOFs) of first-level ROM. Second-level ROM with mode expansion and direct second-level ROM are obtained, and the nonlinear displacement responses of the two ROMs are compared with the first-level ROM. The numerical results demonstrate that the proposed method has a higher computational efficiency and accuracy in terms of mode expansion than the direct model reduction by using POD method. In addition, the nonlinear vibration responses of the dual rotor-bearing system are studied by this second-level ROM in the case of different clearances of the inter-shaft bearing. The results indicate that the dynamic characteristics of the dual rotor-bearing system are very complicated for a large clearance.     

Variational multiscale proper orthogonal decomposition: Navier‐stokes equations

We develop a variational multiscale proper orthogonal decomposition (POD) reduced‐order model (ROM) for turbulent incompressible Navier‐Stokes equations. Under two assumptions on the underlying finite element approximation and the generation of the POD basis, the error analysis of the full discretization of the ROM is presented. All error contributions are considered: the spatial discretization error (due to the finite element discretization), the temporal discretization error (due to the backward Euler method), and the POD truncation error. Numerical tests for a three‐dimensional turbulent flow past a cylinder at Reynolds number show the improved physical accuracy of the new model over the standard Galerkin and mixing‐length POD ROMs. The high computational efficiency of the new model is also showcased. Finally, the theoretical error estimates are confirmed by numerical simulations of a two‐dimensional Navier‐Stokes problem. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 641–663, 2014     

Error estimates for space–time discontinuous Galerkin formulation based on proper orthogonal decomposition

In this study, proper orthogonal decomposition (POD) method is applied to diffusion–convection–reaction equation, which is discretized using space–time discontinuous Galerkin (dG) method. We provide estimates for POD truncation error in dG-energy norm, dG-elliptic projection, and space–time projection. Using these new estimates, we analyze the error between the dG and the POD solution, and the error between the exact and the POD solution. Numerical results, which are consistent with theoretical convergence rates, are presented.     

油藏油水两相流低阶模型算法

目前,油藏数值模拟主要采用的方法如有限元方法、有限容积法等在油藏数值计算时均需要较长的计算时间,很大程度上限制了油藏注采的实时预测与快速动态模拟.该文以一种高效的数据处理方法(最佳正交分解(POD)方法)为基础,对油藏油、水两相流抽取特征函数,并对油藏两相流模型进行Galerkin投影得到新的低阶计算模型.数值计算表明,POD方法所得到的特征向量能量具有最优的特征,能以较少的特征向量捕捉到数学模型中较大的“能量”,因此能最大限度地描述油藏的特征(压力、饱和度),对油藏偏微分方程模型起到较好的降阶作用.结论表明,低阶模型的计算结果与隐压显饱(IMPES)所得计算结果吻合较好,且能节省更多的计算时间,因此能较好地在油藏注采数值模拟中进行历史拟合与仿真计算.     

Finite difference scheme based on proper orthogonal decomposition for the nonstationary Navier-Stokes equations

The proper orthogonal decomposition(POD)and the singular value decomposition(SVD) are used to study the finite difference scheme(FDS)for the nonstationary Navier-Stokes equations. Ensembles of data are compiled from the transient solutions computed from the discrete equation system derived from the FDS for the nonstationary Navier-Stokes equations.The optimal orthogonal bases are reconstructed by the elements of the ensemble with POD and SVD.Combining the above procedures with a Galerkin projection approach yields a new optimizing FDS model with lower dimensions and a high accuracy for the nonstationary Navier-Stokes equations.The errors between POD approximate solutions and FDS solutions are analyzed.It is shown by considering the results obtained for numerical simulations of cavity flows that the error between POD approximate solution and FDS solution is consistent with theoretical results.Moreover,it is also shown that this validates the feasibility and efficiency of POD method.     

Nonintrusive reduced‐order modeling of parametrized time‐dependent partial differential equations

We propose a nonintrusive reduced‐order modeling method based on the notion of space‐time‐parameter proper orthogonal decomposition (POD) for approximating the solution of nonlinear parametrized time‐dependent partial differential equations. A two‐level POD method is introduced for constructing spatial and temporal basis functions with special properties such that the reduced‐order model satisfies the boundary and initial conditions by construction. A radial basis function approximation method is used to estimate the undetermined coefficients in the reduced‐order model without resorting to Galerkin projection. This nonintrusive approach enables the application of our approach to general problems with complicated nonlinearity terms. Numerical studies are presented for the parametrized Burgers' equation and a parametrized convection‐reaction‐diffusion problem. We demonstrate that our approach leads to reduced‐order models that accurately capture the behavior of the field variables as a function of the spatial coordinates, the parameter vector and time. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Applying proper orthogonal decomposition to the computation of particle dispersion in a two-dimensional ventilated cavity

The Lagrangian technique is used to compute particle dispersion in a two-dimensional ventilated cavity. The instantaneous air velocity field at the particle’s location is obtained by proper orthogonal decomposition (POD). The low-dimensional dynamic model is obtained by performing a Galerkin projection of the Navier–Stokes equations onto each POD eigenfunction and it is coupled with the particle’s equation of motion. A substantial decrease in computing time (when comparing with LES computations) is noted. Two different cases of particles’ injection are modelled: the particle source is located in the inlet in the first simulation, and close to the floor in the second simulation.     

Learning patient-specific parameters for a diffuse interface glioblastoma model from neuroimaging data

Parameters in mathematical models for glioblastoma multiforme (GBM) tumour growth are highly patient specific. Here, we aim to estimate parameters in a Cahn–Hilliard type diffuse interface model in an optimised way using model order reduction (MOR) based on proper orthogonal decomposition (POD). Based on snapshots derived from finite element simulations for the full-order model (FOM), we use POD for dimension reduction and solve the parameter estimation for the reduced-order model (ROM). Neuroimaging data are used to define the highly inhomogeneous diffusion tensors as well as to define a target functional in a patient-specific manner. The ROM heavily relies on the discrete empirical interpolation method, which has to be appropriately adapted in order to deal with the highly nonlinear and degenerate parabolic partial differential equations. A feature of the approach is that we iterate between full order solvers with new parameters to compute a POD basis function and sensitivity-based parameter estimation for the ROM problems. The algorithm is applied using neuroimaging data for two clinical test cases, and we can demonstrate that the reduced-order approach drastically decreases the computational effort.     

二维非稳态对流扩散边界控制问题的简化算法

针对二维非稳态对流扩散边界控制问题计算量大的问题,提出了基于降阶模型的最优实时控制方法.利用POD(the Proper Orthogonal Decomposition)和奇异值分解以及Galerkin投影方法得到了具有高精度离散形式的状态空间降阶模型.在所得的降阶状态空间模型中,利用离散时间线性二次调节器方法设计出了最优控制器.对流-扩散过程的控制模拟结果说明了所提方法的有效性和准确性.     

Multiscale failure Modeling of composites using generalized finite element method

In this work multiscale failure modeling of composites is made using generalized finite element method (GFEM). In this method the global approximation are constructed by combining the local basis with partition of unity functions. The enrichment functions for the GFEM approximation are computed using a proper orthogonal decomposition (POD) technique. The approximation is then used in a two scale Galerkin scheme for failure modeling of composites. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

POD a-posteriori error analysis for optimal control problems with mixed control-state constraints

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.     

Sliding mode control of the forced generalized Burgers equation

^** Email: smaoui{at}mcs.sci.kuniv.edu.kw This paper deals with the sliding mode control (SMC) of theforced generalized Burgers equation via the Karhunen-Loève(K-L) Galerkin method. The decomposition procedure of the K-Lmethod is presented to illustrate the use of this method inanalysing the numerical simulations data which represent thesolutions of the forced generalized Burgers equation for viscosityranging from 1 to 100. The K-L Galerkin projection is used asa model reduction technique for non-linear systems to derivea system of ordinary differential equations (ODEs) that mimicsthe dynamics of the forced generalized Burgers equation. Thedata coefficients derived from the ODE system are then usedto approximate the solutions of the forced Burgers equation.Finally, static and dynamic SMC schemes with the objective ofenhancing the stability of the forced generalized Burgers equationare proposed. Simulations of the controlled system are givento illustrate the developed theory.     

Modeling structural variability in reduced order models of machine tool assembly groups via parametric MOR

We present a parametric model order reduction (PMOR) method applied to a parameter depending generalized state-space system, which describes the evolution of the temperature field on a vertical stand of a machine tool assembly group induced by a moving tool slide. The position of this slide parametrizes the input matrix of the associated system. The main idea is to compute projection matrices V_j, W_j in certain parameter sample points μ_j and concatenate them to the projection bases V, W, respectively, as described in [1]. Instead of using the iterative rational Krylov algorithm (IRKA) to produce the projection matrices in each parameter sample point as suggested there, here we use the well known method of balanced truncation (BT). The numerical results show that for the same reduced order r obtained from V, W ∈ ℝ^n×r, BT produces a parametric reduced order model (ROM) of similar accuracy as IRKA in less time. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Development of POD reduced-order model and its closure scheme for 2D Rayleigh–Bénard convection

Reduced-order model (ROM) based on proper orthogonal decomposition (POD) is a fast computational fluid dynamics (CFD) method and has been widely applied to pure flow or heat conduction problems in the past. In this paper, the typical 2D Rayleigh–Bénard convection (RBC) in a square cavity was set as a research target. Firstly, the POD-ROM of 2D RBC problem was constructed at Ra = 10⁷, Pr = 0.71. Combining with direct numerical simulation (DNS) databases, a closure model (CM) was then proposed to correct the evolution process of POD-ROM. Based on the proposed CM, we realized the prediction of flow evolution for a new flow case under the parameters different from that used to get its POD eigenmodes. It showed that the proposed POD-ROM with CM could be able to predict the dynamics of new flow cases. Moreover, the corresponding method proposed in the present study can be also easily extended to other types of flow-heat coupling problems, such as natural heat convection, etc.     

Variational time discretizations of higher order and higher regularity

We consider a family of variational time discretizations that are generalizations of discontinuous Galerkin (dG) and continuous Galerkin–Petrov (cGP) methods. The family is characterized by two parameters. One describes the polynomial ansatz order while the other one is associated with the global smoothness that is ensured by higher order collocation conditions at both ends of the subintervals. Applied to Dahlquist’s stability problem, the presented methods provide the same stability properties as dG or cGP methods. Provided that suitable quadrature rules of Hermite type are used to evaluate the integrals in the variational conditions, the variational time discretization methods are connected to special collocation methods. For this case, we present error estimates, numerical experiments, and a computationally cheap postprocessing that allows to increase both the accuracy and the global smoothness by one order.

     

Time-discrete higher order ALE formulations: a priori error analysis

We derive optimal a priori error estimates for discontinuous Galerkin (dG) time discrete schemes of any order applied to an advection–diffusion model defined on moving domains and written in the Arbitrary Lagrangian Eulerian (ALE) framework. Our estimates hold without any restrictions on the time steps for dG with exact integration or Reynolds’ quadrature. They involve a mild restriction on the time steps for the practical Runge–Kutta–Radau methods of any order. The key ingredients are the stability results shown earlier in Bonito et al. (Time-discrete higher order ALE formulations: stability, 2013) along with a novel ALE projection. Numerical experiments illustrate and complement our theoretical results.     

The galerkin method and feedback control of linear distributed parameter systems

In the development of feedback control theory for distributed parameter systems (DPS), i.e., systems described by partial differential equations, it is important to maintain the finite dimensionality of the controller even though the DPS is infinite dimensional. Since this dimension is directly related to the available on-line computer capacity, it must be finite (and not very large) in order to make the controller implementable from an engineering standpoint. In previous work, it has been our intention to investigate what can be accomplished by finite-dimensional control of infinite-dimensional systems; in particular, we have concentrated on controller design and closed-loop stability. The starting point for all of this is some means for producing a finite-dimensional approximation—a reduced-order model—of the actual DPS. When the “modes” of the DPS are known, the natural candidate for model reduction is projection onto the modal subspace spanned by a finite number of critical modes. Unfortunately, in real engineering systems, these modes are never known exactly and some other reasonable approximation must be used. In this paper, the model reduction is based on the well-known Galerkin procedure. We generate the Galerkin reduced-order model and develop a finite-dimensional controller from it; then we analyze the stability of this controller in closed loop with the actual DPS. Our results indicate conditions under which model reduction based on consistent Galerkin approximations will lead to stable finite-dimensional control.