Incremental proper orthogonal decomposition for PDE simulation data

We propose an incremental algorithm to compute the proper orthogonal decomposition (POD) of simulation data for a partial differential equation. Specifically, we modify an incremental matrix SVD algorithm of Brand to accommodate data arising from Galerkin-type simulation methods for time dependent PDEs. The algorithm is applicable to data generated by many numerical methods for PDEs, including finite element and discontinuous Galerkin methods. The algorithm initializes and efficiently updates the dominant POD eigenvalues and modes during the time stepping in a PDE solver without storing the simulation data. We prove that the algorithm without truncation updates the POD exactly. We demonstrate the effectiveness of the algorithm using finite element computations for a 1D Burgers’ equation and a 2D Navier–Stokes problem.     

本征正交分解在数据处理中的应用及展望

本征正交分解(Proper Orthogonal Decomposition,POD)是对高维复杂非线性系统进行降维处理的有效方法之一.本文对POD方法在一系列实际工程领域降维中的研究进行了综述.首先简要介绍POD方法的发展历史,简述POD方法分类,随后详细列举POD方法在粒子图像测速(Particle Image Velocimetry,PIV)技术、计算流体力学(Computational Fluid Dynamics,CFD)数据处理中的应用.对比了POD方法和动态模态分解(Dynamic Mode Decomposition,DMD)方法在实际工程应用中各自的优缺点,结果表明在流场稳定脉动时可采用DMD方法,而其他随时间变化的流场采用POD方法更合适.最后对POD方法的发展尤其是在人工智能领域的应用做出展望.     

Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition

The proper orthogonal decomposition (POD) has been widely used in fluid dynamic applications for extracting dominant flow features. The “gappy” POD is an extension to this method that allows the consideration of incomplete data sets. In this paper, the gappy POD is extended to handle unsteady flow reconstruction problems, such as those encountered when limited flow measurement data is available. In addition, a systematic approach for effective sensor placement is formulated within the gappy framework using a condition number criterion. This criterion allows for accurate flow reconstruction results and yields sensor configurations that are robust to sensor noise. Two applications are considered. The first aims to reconstruct the unsteady flow field using a small number of surface pressure measurements for a subsonic airfoil undergoing plunging motion. The second considers estimation of POD modal content of a cylinder wake flow for active control purposes. In both cases, using the dominant POD basis vectors and a small number of sensor signals, the gappy approach is found to yield accurate flow reconstruction results.     

Real-time optimization using proper orthogonal decomposition: Free surface shape prediction due to underwater bubble dynamics

A new data-driven reduced-order modeling approach for real-time optimization applications is presented. The proper orthogonal decomposition (POD) technique is used for the reduced-order model, with the basis functions determined from an ensemble of offline high-fidelity simulations. For optimization in real time, a rapid two-stage approach is taken in the online phase: the POD coefficients are first determined by solving a small optimization problem, and the desired parameters are subsequently obtained by interpolation of the POD coefficients using precomputed information from the simulation ensemble. This method is applied to optimizing parameters for underwater bubble explosions so that a desired free surface shape is generated. For time-critical applications, such as using the water barrier generated to stop sea-skimming objects, the time available for online optimization is limited to about 30 s. Results for two-dimensional simulation, using a personal computer (dual CPU running at 2.8 GHz), show that our new methodology is able to meet such a critical time requirement. For three-dimensional simulations, the time taken for computation increases, and a faster computer or parallel implementation is required.     

Optimal control of a nonlinear induction heating system using a proper orthogonal decomposition based reduced order model

This paper considers the problem of obtaining control inputs for induction heating processes involving nonlinearities and nontrivial geometries. The considered system falls within the much broader class of multiphysics or coupled, distributed parameter systems that are well known for the challenges they pose towards modeling, simulation and control. The approach is based on the versatile finite element method (FEM) widely popular in industry for modeling of such systems. The large dynamic model obtained through FEM is reduced using proper orthogonal decomposition (POD) technique, applicable to nonlinear systems, while preserving the input–output characteristics almost exactly. In the present work, using the reduced model, an optimal control input current for the induction heating system is obtained, for transferring the temperature profile over an axi-symmetric work piece from any initial temperature profile to any other desired temperature profile. The reduced model enables application and numerical implementation of established optimal control techniques based on Hamilton–Jacobi–Bellman theory for the system. Further, although the controller is developed using the reduced model, it is shown to perform perfectly well when used with the original unreduced model as well. Results of the numerical implementation are presented to evaluate the overall proposed approach.     

A new approach to model reduction of nonlinear control systems using smooth orthogonal decomposition

A new approach to model order reduction of nonlinear control systems is aimed at developing persistent reduced order models (ROMs) that are robust to the changes in system's energy level. A multivariate analysis method called smooth orthogonal decomposition (SOD) is used to identify the dynamically relevant modal structures of the control system. The identified SOD subspaces are used to develop persistent ROMs. Performance of the resultant SOD‐based ROM is compared with proper orthogonal decomposition (POD)–based ROM by evaluating their robustness to the changes in system's energy level. Results show that SOD‐based ROMs are valid for a relatively wider range of the nonlinear control system's energy when compared with POD‐based models. In addition, the SOD‐based ROMs show considerably faster computations compared to the POD‐based ROMs of same order. For the considered dynamic system, SOD provides more effective reduction in dimension and complexity compared to POD.     

基于正交分解的递推子空间闭环辨识方法

为实现闭环系统在线辨识,提出递推正交分解闭环子空间辨识方法(RORT)。首先,根据闭环系统状态空间模型和数据间投影关系,构建确定-随机模型,并利用GIVENS变换实现投影向量的递推QR分解;然后,引入带遗忘因子的辨识算法,构建广义能观测矩阵的递推更新形式,以减少子空间辨识算法中QR分解和SVD分解的计算量;最后,针对某型号陀螺仪闭环系统进行实验。实验结果表明, RORT法的辨识拟合度高于91%,能够对陀螺仪闭环系统模型参数进行在线监测。     

Commutative encryption and watermarking based on orthogonal decomposition

Commutative Encryption and Watermarking (CEW) is a technology combined with encryption and watermarking, which is used for providing comprehensive security protection for multimedia information. This paper presents a novel CEW scheme based on orthogonal decomposition. Different from current CEW, the proposed CEW not only achieves the integration of encryption and watermarking in the final data but also has no specific restrictions in selecting encryption and watermarking algorithms. Therefore, the proposed CEW possesses higher security and applicability. Experimental results demonstrate that the proposed CEW can keep the performances of selected encryption and watermarking algorithm and show more robustness than other current CEW schemes.     

Speaker identification using discrete wavelet packet transform technique with irregular decomposition

This paper presents the study of speaker identification for security systems based on the energy of speaker utterances. The proposed system consisted of a combination of signal pre-process, feature extraction using wavelet packet transform (WPT) and speaker identification using artificial neural network. In the signal pre-process, the amplitude of utterances, for a same sentence, were normalized for preventing an error estimation caused by speakers’ change in volume. In the feature extraction, three conventional methods were considered in the experiments and compared with the irregular decomposition method in the proposed system. In order to verify the effect of the proposed system for identification, a general regressive neural network (GRNN) was used and compared in the experimental investigation. The experimental results demonstrated the effectiveness of the proposed speaker identification system and were compared with the discrete wavelet transform (DWT), conventional WPT and WPT in Mel scale.     

离散空间中正交小波分解重构算法的实现

针对目前离散空间中分解重构算法的过程较为复杂,主要研究离散空间中周期小波和非周期小波分解重构算法的实现.首先证明离散空间中的多层小波分解重构算法可以按照Mallat分解重构算法的塔式结构实现,从而将离散序列空间与函数空间中的小波理论联系起来;其次,举例说明离散空间中的分解重构算法比函数空间中的Mallat分解重构算法在滤波器的选择上更加灵活;最后,数值结果表明基于离散小波对信号进行处理在很多应用中可以取得更好的效果.     

Redundant versus orthogonal wavelet decomposition for multisensor image fusion

The wavelet decomposition has become an attractive tool for fusing multisensor images. Usually, the input images are decomposed with an orthogonal wavelet in order to extract features, which are combined through an appropriate fusion rule. The fused image is then reconstructed by applying the inverse wavelet transform. In this paper, we investigate the use of the nonorthogonal (or redundant) wavelet decomposition as an alternative approach for feature extraction. By using test and remote sensing images, various fusion rules are considered and the detailed comparison indicates the superiority of this approach compared to the standard orthogonal wavelet decomposition for image fusion.     

结合仿射几何和正交分解的类中心分类法研究

类中心分类法是非常有效的分类算法。但由于向量空间产生空间扭曲,导致类中心分类法处理某些界限不明显的类别精确度不高。对此引入仿射几何和力的正交分解的方法,提出基于仿射几何和力的正交分解模型的类中心分类法的改进算法。在降低了计算维度的同时解决了由于向量空间特征项维被认为是正交带来的空间扭曲引起的误差。     

An approach to closed-loop subspace identification by orthogonal decomposition

A method of identifying closed-loop systems is developed by using the orthogonal decomposition (ORT) method. The idea is to project the input and output data onto the space of exogenous inputs by using the LQ decomposition to obtain their deterministic components. The ORT-based method is then applied to deterministic components like the direct approach in order to derive state-space models of the plant. We also show that the present method is a subspace version of the two-stage method for transfer function estimation from closed loop data. Some simulation results are included to show the applicability of the present method.     

Logic based Benders' decomposition for orthogonal stock cutting problems

We consider the problem of packing a set of rectangular items into a strip of fixed width, without overlapping, using minimum height. Items must be packed with their edges parallel to those of the strip, but rotation by 90° is allowed. The problem is usually solved through branch-and-bound algorithms. We propose an alternative method, based on Benders' decomposition. The master problem is solved through a new ILP model based on the arc flow formulation, while constraint programming is used to solve the slave problem. The resulting method is hybridized with a state-of-the-art branch-and-bound algorithm. Computational experiments on classical benchmarks from the literature show the effectiveness of the proposed approach. We additionally show that the algorithm can be successfully used to solve relevant related problems, like rectangle packing and pallet loading.     

Feature extraction and volume decomposition for orthogonal layered manufacturing

Robust and efficient process planning techniques play an important role in CAD/CAM integration. These techniques need to be developed for each type of manufacturing processes owing to the unique characteristics of each of these processes. In this paper, we describe feature extraction techniques that can be applied to layered manufacturing (LM). The aim is to improve the LM process efficiency by considering the specific feature information of the model, which is normally neglected by previous researches. A feature-based LM system has been developed using these techniques. Based on the proposed orthogonal LM system, features extracted from the geometric analysis are defined in the LM domain, and the algorithm for process planning and volume decomposition based on the specific LM features is proposed and implemented.     

Spectral efficiency enhancement using an antenna‐based orthogonal frequency division multiaccess technique

An approach to enhance antenna spectral efficiency is proposed based on combining spacetime electromagnetic (EM) models of Tx/Rx antennas with orthogonal frequency division multiaccess (OFDM), leading to the EM‐OFDM, a technology capable of removing intersymbol interference (ISI) in high‐data rate communication links caused by the EM‐induced distortion antenna effects. The proposed approach differs from traditional OFDM in wireless communication in several aspects. First, the technique suggests a new decoupling approach by treating each given antenna transreceive device pair as a “stand‐alone channel” with its own distortion mechanisms considered separately from the propagation channel. Moreover, the deterministic distortion caused by the nonflat pure antenna EM filtration effects is exploited to carefully design a specialized OFDM transmission techniques based on the antenna parameters, not the multipath fading channels often invoked in conventional uses of OFDM methods. (The EM‐OFDM, however, can be combined with a traditional OFDM later if fading channels are present.) In this manner, a more efficient implementation of the wireless link equalization strategy may be enacted since the EM antenna origin of ISI is very different from the traditional propagation channel one. As a proof of concept, the proposed EM‐OFDM method is implemented for a single‐input‐single‐output link comprised of half‐wavelength linear wire antennas. A careful use of finite difference time‐domain to provide EM data allowed the construction of 64 decoupled “pure antenna OFDM subchannels.” Simulation results suggests that the antenna‐based OFDM system is capable of completely neutralizing all ISI effects caused by the limited antenna matching bandwidth of the transreceive wires, therefore, supporting considerably higher data rates with low symbol error rate (SER). A concrete evaluation of the SER using quadrature phase‐shift keying (QPSK) digital carrier modulation resulted in an increase of the effective antenna digital communication spectral efficiency by ratios up to 300%. Moreover, the EM‐OFDM error rate was found to be close to the ideal QPSK level or the maximum possible theoretical limit. Thus, combining detailed EM knowledge with standard signal processing methods can lead to considerable improvement in system design without modifying the antenna physical layout. The proposed approach is expected to play a role in the forthcoming 5G/6G and millimetre wave technology systems currently under development where there is a trend toward integration of EM and digital signal processing at the physical layer level.