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Adaptive CVT-Based Reduced-Order Modeling of Burgers Equation
Guang Ri Piao,Qiang Du,Hyung Chun Lee 한국산업응용수학회(구 한국산업정보응용수학회) 2009 한국산업정보응용수학회 Vol.13 No.2
In this article, we consider a weighted CVT-based reduced-order modelling for Burgers equation. Brief review of the CVT (centroidal Voronoi tessellation) approaches to reduced-order bases are provided. In CVT-reduced order modelling, we start with a snapshot set just as is done in a POD (Proper Orthogonal Decomposition)-based setting. So far, the CVT was researched with uniform density (p(y)=1) to determine the basis elements for the approximating subspaces. Here, we shall investigate the technique of CVT with nonuniform density as a procedure to determine the basis elements for the approximating subspaces. Some numerical experiments including comparison of two CVT (CVT-uniform and CVT-nonuniform)-based algorithm with numerical results obtained from FEM(finite element method) and POD-based algorithm are reported.
Piao, Guang-Ri,Cui, Mei-Hong The Youngnam Mathematical Society 2017 East Asian mathematical journal Vol.33 No.4
This paper studies the relationship between the college entrance examination scores and college mathematics achievements of the freshman, who are the students of college of technology and management in Yanbian University, by employing statistical analysis. First of all, we start from the correlation analysis of the majors and regions of students. Secondly, with gender and region as the classification, we divide the sample into two parts and test the mean significance, and we find that not only different gender but also different region have significant differences to college mathematics achievements. Thirdly, we use the college entrance examination scores as independent variable, and college mathematics achievement as the dependent variable to establish regression model. The model provides a realistic basis for the education department of our school, and provides reference value for the other major research.
Distributed Feedback Control of the Benjamin-Bona-Mahony-Burgers Equation by a Reduced-Order Model
Piao, Guang-Ri,Lee, Hyung-Chun Global Science Press 2015 East Asian Journal on Applied Mathematics Vol.5 No.1
<B>Abstract</B><P>A reduced-order model for distributed feedback control of the Benjamin-Bona-Mahony-Burgers (BBMB) equation is discussed. To retain more information in our model, we first calculate the functional gain in the full-order case, and then invoke the proper orthogonal decomposition (POD) method to design a low-order controller and thereby reduce the order of the model. Numerical experiments demonstrate that a solution of the reduced-order model performs well in comparison with a solution for the full-order description.</P>
GUANG-RI PIAO,HYUNG-CHEN LEE,JUNE-YUB LEE 한국산업응용수학회 2009 Journal of the Korean Society for Industrial and A Vol.13 No.4
In this article, we study a reduced-order modelling for distributed feedback control problem of the Burgers equations. Brief review of the centroidal Voronoi tessellation (CVT) are provided. A weighted (nonuniform density) CVT is introduced and low-order approximate solution and compensator-based control design of Burgers equation is discussed. Through weighted CVT (or CVT-nonuniform) method, obtained low-order basis is applied to low-order functional gains to design a low-order controller, and by using the low-order basis order of control modelling was reduced. Numerical experiments show that a solution of reduced-order controlled Burgers equation performs well in comparison with a solution of full order controlled Burgers equation.
INTERNAL FEEDBACK CONTROL OF THE BENJAMIN-BONA-MAHONY-BURGERS EQUATION
GUANG-RI PIAO,HYUNG-CHEN LEE 한국산업응용수학회 2014 Journal of the Korean Society for Industrial and A Vol.18 No.3
A numerical scheme is proposed to control the BBMB (Benjamin-Bona-Mahony-Burgers) equation, and the scheme consists of three steps. Firstly, BBMB equation is converted to a finite set of nonlinear ordinary differential equations by the quadratic B-spline finite element method in spatial. Secondly, the controller is designed based on the linear quadratic regulator (LQR) theory; Finally, the system of the closed loop compensator obtained on the basis of the previous two steps is solved by the backward Euler method. The controlled numerical solutions are obtained for various values of parameters and different initial conditions. Numerical simulations show that the scheme is efficient and feasible.
ADAPTIVE CVT-BASED REDUCED-ORDER MODELING OF BURGERS EQUATION
GUANG-RI PIAO,QIANG DU,HYUNG-CHUN LEE 한국산업응용수학회 2009 Journal of the Korean Society for Industrial and A Vol.13 No.2
In this article, we consider a weighted CVT-based reduced-order modelling for Burgers equation. Brief review of the CVT (centroidal Voronoi tessellation) approaches to reduced-order bases are provided. In CVT-reduced order modelling, we start with a snapshot set just as is done in a POD (Proper Orthogonal Decomposition)-based setting. So far, the CVT was researched with uniform density (p(y) = 1) to determine the basis elements for the approximating subspaces. Here, we shall investigate the technique of CVT with nonuniform density as a procedure to determine the basis elements for the approximating subspaces. Some numerical experiments including comparison of two CVT (CVT-uniform and CVT-nonuniform)-based algorithm with numerical results obtained from FEM(finite element method) and POD-based algorithm are reported.
GUANG-RI PIAO,Mei-Hong Cui 영남수학회 2017 East Asian mathematical journal Vol.33 No.4
This paper studies the relationship between the college en- trance examination scores and college mathematics achievements of the freshman, who are the students of college of technology and management in Yanbian University, by employing statistical analysis. First of all, we start from the correlation analysis of the majors and regions of students. Secondly, with gender and region as the classification, we divide the sample into two parts and test the mean significance, and we find that not only dif- ferent gender but also different region have significant differences to college mathematics achievements. Thirdly, we use the college entrance examina- tion scores as independent variable, and college mathematics achievement as the dependent variable to establish regression model. The model pro- vides a realistic basis for the education department of our school, and provides reference value for the other major research.
A FINITE ELEMENT SOLUTION FOR THE CONSERVATION FORM OF BBM-BURGERS' EQUATION
Yang Ning,Mingzhe Sun,GUANG-RI PIAO 영남수학회 2017 East Asian mathematical journal Vol.33 No.5
With the accuracy of the nonlinearity guaranteed, plenty of time and large memory space are needed when we solve the finite ele- ment numerical solution of nonlinear partial differential equations. In this paper, we use the Group Element Method (GEM) to deal with the non- linearity of the BBM-Burgers Equation with Conservation form and per- form a numerical analysis for two particular initial-boundary value (the Dirichlet boundary conditions and Neumann-Dirichlet boundary condi- tions) problems with the Finite Element Method (FEM). Some numerical experiments are performed to analyze the error between the exact solution and the FEM solution in MATLAB.
QUADRATIC B-SPLINE FINITE ELEMENT METHOD FOR THE BENJAMIN-BONA-MAHONY-BURGERS EQUATION
Yong-Xue Yin,GUANG-RI PIAO 영남수학회 2013 East Asian mathematical journal Vol.29 No.5
A quadratic B-spline finite element method for the spatial variable combined with a Newton method for the time variable is proposed to approximate a solution of Benjamin-Bona-Mahony-Burgers (BBMB) equation. Two examples were considered to show the efficiency of the proposed scheme. The numerical solutions obtained for various viscosity were compared with the exact solutions. The numerical results show that the scheme is efficient and feasible.