Low-Dimensional Approximation for Control of Spatio-Temporal Processes Using Principal Interaction Patterns

2012 ◽  
Vol 591-593 ◽  
pp. 1217-1220
Author(s):  
Xiang Ping Cao ◽  
Zhao Yang Li ◽  
Mei Xing Liu
Keyword(s):  
Galerkin Approximation ◽  
Basis Functions ◽  
Interaction Patterns ◽  
Infinite Dimensional ◽  
Dynamical Behaviors ◽  
Dimensional Approximation ◽  
Spatio Temporal ◽  
Low Dimensional Approximations ◽  
Low Dimensional ◽  
Advanced Model

Although the first-principal models of the spatio-temporal processes can accurately predict nonlinear and distributed dynamical behaviors, their infinite-dimensional nature does not allow their directly use. In this note, low-dimensional approximations for control of spatio-temporal processes using principal interaction patterns are constructed. Advanced model reduction approach based on spatial basis function expansion together with Galerkin method is used to obtain the low-dimensional approximation. Spatial structure called principal interaction patterns are extracted from the system according to a variational principle and used as basis functions in a Galerkin approximation. The simulations of the burgers equations has illustrated that low-dimensional approximation based on principal interaction patterns for spatio-temporal processes has smaller errors than more conventional approaches using Fourier modes or Empirical Eigenfunctions as basis functions.

On Low-Dimensional Galerkin Modeling of Confined Twin-Jet Flow and Heat Transfer

2001 ◽  
Author(s):  
H. Gunes ◽  
K. Gocmen ◽  
L. Kavurmacioglu
Keyword(s):  
Heat Transfer ◽  
Jet Flow ◽  
Basis Functions ◽  
Galerkin Projection ◽  
Temperature Fields ◽  
Transitional Flow ◽  
Flow And Heat Transfer ◽  
Galerkin Models ◽  
Velocity And Temperature Fields ◽  
Low Dimensional

Abstract The two-dimensional incompressible non-isothermal confined twin-jet flow has been numerically studied in the transitional flow regime by a finite volume technique. Results have been obtained for the velocity and temperature distributions close to the onset of temporal oscillations. Next, the proper orthogonal decomposition (POD) is applied to the instantaneous flow and temperature data to obtain POD-based basis functions for both velocity and temperature fields. These basis functions are capable to identify the coherent structures in the velocity and temperature fields. The low-dimensional Galerkin models of the full Navier-Stokes and energy equations are constructed by the Galerkin projection onto basis functions. Since the low-dimensional Galerkin models are much easier to analyze than the full governing equations, basic insights into important mechanisms of dynamically complex flow and heat transfer (e.g. flow instabilities) can be easily studied by these models. The numerical implications, the validity of the models and their performance characteristics are discussed.

scholarly journals Exponential Family Graph Embeddings

2020 ◽  
Vol 34 (04) ◽  
pp. 3357-3364
Author(s):  
Abdulkadir Celikkanat ◽  
Fragkiskos D. Malliaros
Keyword(s):  
Random Walk ◽  
Exponential Family ◽  
Representation Learning ◽  
Learning Problems ◽  
Interaction Patterns ◽  
Network Representation ◽  
Learning Tasks ◽  
Learning Techniques ◽  
Real World Datasets ◽  
Low Dimensional

Representing networks in a low dimensional latent space is a crucial task with many interesting applications in graph learning problems, such as link prediction and node classification. A widely applied network representation learning paradigm is based on the combination of random walks for sampling context nodes and the traditional Skip-Gram model to capture center-context node relationships. In this paper, we emphasize on exponential family distributions to capture rich interaction patterns between nodes in random walk sequences. We introduce the generic exponential family graph embedding model, that generalizes random walk-based network representation learning techniques to exponential family conditional distributions. We study three particular instances of this model, analyzing their properties and showing their relationship to existing unsupervised learning models. Our experimental evaluation on real-world datasets demonstrates that the proposed techniques outperform well-known baseline methods in two downstream machine learning tasks.

scholarly journals Applied Koopman Theory for Partial Differential Equations and Data-Driven Modeling of Spatio-Temporal Systems

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-16 ◽  
Author(s):  
J. Nathan Kutz ◽  
J. L. Proctor ◽  
S. L. Brunton
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Data Driven ◽  
Nonlinear Pdes ◽  
Dynamic Mode ◽  
Partial Differential ◽  
Koopman Operator ◽  
Dimensional Approximation ◽  
Spatio Temporal ◽  
The Impact

We consider the application of Koopman theory to nonlinear partial differential equations and data-driven spatio-temporal systems. We demonstrate that the observables chosen for constructing the Koopman operator are critical for enabling an accurate approximation to the nonlinear dynamics. If such observables can be found, then the dynamic mode decomposition (DMD) algorithm can be enacted to compute a finite-dimensional approximation of the Koopman operator, including its eigenfunctions, eigenvalues, and Koopman modes. We demonstrate simple rules of thumb for selecting a parsimonious set of observables that can greatly improve the approximation of the Koopman operator. Further, we show that the clear goal in selecting observables is to place the DMD eigenvalues on the imaginary axis, thus giving an objective function for observable selection. Judiciously chosen observables lead to physically interpretable spatio-temporal features of the complex system under consideration and provide a connection to manifold learning methods. Our method provides a valuable intermediate, yet interpretable, approximation to the Koopman operator that lies between the DMD method and the computationally intensive extended DMD (EDMD). We demonstrate the impact of observable selection, including kernel methods, and construction of the Koopman operator on several canonical nonlinear PDEs: Burgers’ equation, the nonlinear Schrödinger equation, the cubic-quintic Ginzburg-Landau equation, and a reaction-diffusion system. These examples serve to highlight the most pressing and critical challenge of Koopman theory: a principled way to select appropriate observables.

Online Long-Term Object Tracking Based on Compressed Haar-Like Features via Sparse Representation

2014 ◽  
Vol 989-994 ◽  
pp. 1610-1614
Author(s):  
Ming Zhao ◽  
Lu Ping Wang ◽  
Lu Ping Zhang
Keyword(s):  
Sparse Representation ◽  
Target Location ◽  
Model Updating ◽  
Dimensional Space ◽  
Reconstruction Error ◽  
Illumination Changes ◽  
Spatio Temporal ◽  
Low Dimensional ◽  
Scale Variation

Online long-term tracking is a challenging problem as data streams change over time. In this paper, sparse representation has been applied to visual tracking by finding the most correct sample with minimal reconstruction error using compressed Haar-like features. However, most sparse representation tracking algorithm introduce l1 regularization into the PCA reconstruction using samples directly, which leads to complexity computation and can not adapt to occlusion, rotation and change in size. Our model updating not only uses the samples from the training set, but also generates the warped versions (include scale variation, rotation, occlusion and illumination changes) for the previous tracking result. Also, we do not use the samples in models for sparse representation directly, but the Haar-like features instead which are compressed in a very low-dimensional space. In addition, we use a robust and fast algorithm which exploits the spatio-temporal context for predicting the target location in the next frame. This step will lead to the reduction of the searching range by the detector. We demonstrate the proposed method is able to track objects well under pose and scale variation, rotation, occlusion and illumination with great real-time performance on challenging image sequences.

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