Peter T. Frame, Lin Cong, Oliver T. Schmidt et al. · Computer Methods in Applied Mechanics and Engineering · 2025
Most model reduction methods reduce the state dimension and then temporally evolve a set of coefficients that encode the state in the reduced representation. In this paper, we instead employ an efficient representation of the entire trajectory of the state over some time interval of interest and then solve for the static coefficients that encode the trajectory on the interval. We use spectral proper orthogonal decomposition (SPOD) modes, which are provably optimal for representing long trajectories and substantially outperform any representation of the trajectory in a purely spatial basis (e.g., POD). We develop a method to solve for the SPOD coefficients that encode the trajectories for forced linear dynamical systems given the forcing and initial condition, thereby obtaining the accurate prediction of the dynamics afforded by the SPOD representation of the trajectory. The method, which we refer to as spectral solution operator projection (SSOP), is derived by projecting the general time-domain solution for a linear time-invariant system onto the SPOD modes. We demonstrate the new method using two examples: a linearized Ginzburg-Landau equation and an advection-diffusion problem. In both cases, the error of the proposed method is orders of magnitude lower than that of POD-Galerkin projection and balanced truncation. The method is also fast, with CPU time comparable to or lower than both benchmarks in our examples. Finally, we describe a data-free space-time method that is a derivative of the proposed method and show that it is also more accurate than balanced truncation in most cases.
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Peter T. Frame, Cong Lin, O. Schmidt et al. · arXiv (Cornell University) · 2024
Most model reduction methods reduce the state dimension and then temporally evolve a set of coefficients that encode the state in the reduced representation. In this paper, we instead employ an efficient representation of the entire trajectory of the state over some time interval of interest and then solve for the static coefficients that encode the trajectory on the interval. We use spectral proper orthogonal decomposition (SPOD) modes, which are provably optimal for representing long trajectories and substantially outperform any representation of the trajectory in a purely spatial basis (e.g., POD). We develop a method to solve for the SPOD coefficients that encode the trajectories for forced linear dynamical systems given the forcing and initial condition, thereby obtaining the accurate prediction of the dynamics afforded by the SPOD representation of the trajectory. The method, which we refer to as spectral solution operator projection (SSOP), is derived by projecting the general time-domain solution for a linear time-invariant system onto the SPOD modes. We demonstrate the new method using two examples: a linearized Ginzburg-Landau equation and an advection-diffusion problem. In both cases, the error of the proposed method is orders of magnitude lower than that of POD-Galerkin projection and balanced truncation. The method is also fast, with CPU time comparable to or lower than both benchmarks in our examples. Finally, we describe a data-free space-time method that is a derivative of the proposed method and show that it is also more accurate than balanced truncation in most cases.
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Tzuu‐Hseng S. Li, Chiu‐Pin Cheng, York‐Yih Sun · International Journal of Systems Science · 1992
Abstract The design of model-following control (MFC) foT (A, B, C, D) quadruple systems is considered. The main result is that a non-minimum-phase (NMP) plant can be controlled so that it will follow a prespecificd model in the sense of optima. The characteristics of the proposed control mechanism are: (1) the control design problem appears to be a two-point boundary-value problem (TPBVP); (2) this TPBVP can be easily solved by a modified Runge-Kutta method; (3) the existence of the control for a typical NMP plant is proved; (4) some poor properties of a NMP plant are also improved. Two examples are used to verify the control mechanism presented
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Vincent Lenaerts, Gaëtan Kerschen, Jean‐Claude Golinval · Mechanical Systems and Signal Processing · 2001
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Hai Dang Nguyen · KOPS (University of Konstanz) · 2022
This thesis shows how Proper Orthogonal Decomposition (POD) techniques and the Discrete Empirical Interpolation Method (DEIM) can be applied to decrease the computational effort in solving a Semi-linear Parabolic Reaction-Diffusion System, a class of Partial Differential Equa- tions (PDE), which describes the system dynamics of a time-space-dependent SIR-compartment- model.<br />We investigate this model, which is widely-known as a mathematical tool for describing the course of an epidemic by quantifying the number of individuals of a population, which are classified in susceptible (S), infected (I) and removed (R) compartments. We introduce assumptions on the SIR-model and establish initial conditions to specify the dynamics of the model. This leads to a PDE, based on the SIR-model, which is solved using the Finite Element (FE) method and then Finite Difference (FD) schemes. We formulate a model order reduced prob- lem, obtained by the POD-method, in order to approximate the FE-solution and to decrease the computational effort. To overcome unnecessary back and forth projections between the full and reduced space in the non-linear part, we introduce the DEIM.<br /><br />For verifying theoretical results we formulate an SIR-problem on a domain with the shape of the state of Berlin. Based on that, we conduct detailed numerical investigations for the Semi-implicit Euler, implicit Euler and the Crank-Nicolson method, which belong to the class of FD-schemes. Investigations on the convergence and parameter perturbation analysis of the DEIM-solution are made for validation purpose.<br /><br />Last but not least, we conclude that the reduced method is advantageous in terms of computational effort, since the speedup of the computation time is improved significantly, while the error is in a satisfactory range.
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Nguyen Pham, Hai Dang · KOPS (University of Konstanz) · 2022
This thesis shows how Proper Orthogonal Decomposition (POD) techniques and the Discrete Empirical Interpolation Method (DEIM) can be applied to decrease the computational effort in solving a Semi-linear Parabolic Reaction-Diffusion System, a class of Partial Differential Equa- tions (PDE), which describes the system dynamics of a time-space-dependent SIR-compartment- model.<br />We investigate this model, which is widely-known as a mathematical tool for describing the course of an epidemic by quantifying the number of individuals of a population, which are classified in susceptible (S), infected (I) and removed (R) compartments. We introduce assumptions on the SIR-model and establish initial conditions to specify the dynamics of the model. This leads to a PDE, based on the SIR-model, which is solved using the Finite Element (FE) method and then Finite Difference (FD) schemes. We formulate a model order reduced prob- lem, obtained by the POD-method, in order to approximate the FE-solution and to decrease the computational effort. To overcome unnecessary back and forth projections between the full and reduced space in the non-linear part, we introduce the DEIM.<br /><br />For verifying theoretical results we formulate an SIR-problem on a domain with the shape of the state of Berlin. Based on that, we conduct detailed numerical investigations for the Semi-implicit Euler, implicit Euler and the Crank-Nicolson method, which belong to the class of FD-schemes. Investigations on the convergence and parameter perturbation analysis of the DEIM-solution are made for validation purpose.<br /><br />Last but not least, we conclude that the reduced method is advantageous in terms of computational effort, since the speedup of the computation time is improved significantly, while the error is in a satisfactory range.
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John R. Singler, B Batten · 2007
In this paper, we extend a method for reduced order model derivation for finite dimensional systems developed by Rowley to infinite dimensional systems. The method is related to standard balanced truncation, but includes aspects of the proper orthogonal decomposition in its computational approach. The method is also applicable to nonlinear systems. The method is applied to a convection diffusion equation.
Cited by 4
Allen H. Reed, Gaw Tzu Wu · Communication in Statistics- Theory and Methods · 1977
We examine the effect of applying classical linear regression when measurement error is present in x and the appropriate model is functional linear regression. The bias in the classical slop emate is characterized in terms of a single parameter, τ. An unbiased estimator, is obtained assuming a prior estimate is available of $sigma;xA confidenceinterval for x is derived which is also a confidence interval for the noncentrality parameter of th noncentral F. Then the jackknife method is used to set confidencs limits for the true slope.Results are applied to comparison of assay methods in the clinical chemistry laboratory.
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Dennis P. Prill, Andreas G. Class · Annals of Nuclear Energy · 2013
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DWI WAHYU NUGROHO, 1517031006 · Digilib Repository Unila (Lampung University) · 2019
The purpose of this research is to estimate nonparametric regression model using linear spline methods. The simulation data from trigonometric functions by utilize linear and nonlinear functions. The results show that the linear spline method is proper in estimating linear functions. However, it is not proper enough in predicting nonlinear function. \nKeywords: Nonparametrik Regression, Linear Spline , Generalized Cross \nValidation \n \nTujuan dari penelitian ini adalah untuk melakukan estimasi model regresi nonparametrik menggunakan metode spline linear. Data simulasi dari fungsi trigonometri dengan menggunakan fungsi linear dan nonlinear. Hasil penelitian menunjukkan bahwa metode spline linear baik dalam mengestimasi fungsi linear. Tetapi, tidak cukup baik dalam menduga fungsi nonlinear. \nKata kunci: Regresi Nonparametrik, Spline Linear, Generalized Cross Validation
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Min Chih Lin, Francisco J. Soulignac, Jayme L. Szwarcfiter · 2009
A circular-arc model M = (C, A) is a circle C together with a collection A of arcs of C. If no arc is contained in any other then M is a proper circular-arc model, and if some point of C is not covered by any arc then M is an interval model. A (proper) (interval) circulararc graph is the intersection graph of a (proper) (interval) circular-arc model. Circular-arc graphs and their subclasses have been the object of a great deal of attention in the literature. Linear time recognition algorithms have been described both for the general class and for some of its subclasses. For the isomorphism problem, there exists a polynomial time algorithm for the general case, and a linear time algorithm for interval graphs. In this work we develop a linear time algorithm for the isomorphism problem in proper circular-arc graphs, based on uniquely encoding a proper circular-arc model. Our method relies on results about uniqueness of certain PCA models, developed by Deng, Hell and Huang in [6]. The algorithm is easy to code and uses only basic tools available in almost every programming language.
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Satyabrata Pal, Samarendranath Pal · Communication in Statistics- Theory and Methods · 1988
Abstract This communication deals with the construction and optimality of non-proper (unequal block sized) variance balanced (VB) designs obtainable under linear homoscedastic normal model. Several methods of construction of non-proper VB designs have been given. Some constructed designs are universally optimal non-proper variance balanced designs. Keywords: proper and non-proper designsvariance balanced designsintra and inter group balanced block designsuniversal optimality
Cited by 10
Min Chih Lin, Ross M. McConnell, Francisco J. Soulignac et al. · 2009
A circular-arc graph is the intersection graph of a set of arcs on the circle. It is a Helly circular-arc graph if it has a Helly model, where every maximal clique is the set of arcs that traverse some clique point on the circle. A clique model is a Helly model that identifies one clique point for each maximal clique. A Helly circular-arc graph is proper if it has a Helly model where no arc is a subset of another. In this paper, we show that the clique intersection graphs of Helly circular-arc graphs are related to the proper Helly circular-arc graphs. This yields the first polynomial (linear) time recognition algorithm for the clique graphs of Helly circular-arc graphs. Next, we develop an O(n) time algorithm to obtain a clique model of Helly model, improving the previous O(n 2) bound. This gives a linear-time algorithm to find a proper Helly model for the clique graph of a Helly circular-arc graph. As an application, we find a maximum weighted clique of a Helly circular-arc graph in linear time.
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Min Chih Lin A, Ross M. Mcconnell B, Francisco J. Soulignac A et al. · 2008
A circular-arc graph is the intersection graph of a set of arcs on the circle. It is a Helly circular-arc graph if it has a Helly model, where every maximal clique is the set of arcs that traverse some clique point on the circle. A clique model is a Helly model that identifies one clique point for each maximal clique. A Helly circular-arc graph is proper if it has a Helly model where no arc is a subset of another. In this paper, we show that the clique intersection graphs of Helly circular-arc graphs are related to the proper Helly circular-arc graphs. This yields the first polynomial (linear) time recognition algorithm for the clique graphs of Helly circular-arc graphs. Next, we develop an O(n) time algorithm to obtain a clique model of Helly model, improving the previous O(n 2) bound. This gives a linear-time algorithm to find a proper Helly model for the clique graph of a Helly circular-arc graph. As an application, we find a maximum weighted clique of a Helly circular-arc graph in linear time.
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Gaëtan Kerschen, Feeny, B. F., Golinval, Jean-Claude · Open Repository and Bibliography (University of Liège) · 2002
The present study focuses on the model reduction of non-linear systems. The proper orthogonal decomposition is exploited to compute eigenmodes from time series of displacement. These eigenmodes, called the proper orthogonal modes, are optimal with respect to energy content and are used to build a low-dimensional model of the non-linear system. For this purpose, the proper orthogonal modes obtained from a chaotic orbit are considered. Indeed, such an orbit is assumed to cover the phase space more uniformly. It is shown that the modes for this particular behaviour are more representative of the system dynamics than any other set of modes extracted from a non-chaotic response. This is applied to a buckled beam with two permanent magnets and the reduced-order model is validated using both qualitative and quantitative comparisons.
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Zhang Wg · PubMed · 1983
In this paper, a new method is suggested for calculating the absorption amount of drug. The method is based on the general linear compartment model for pharmacokinetics, hence applicable to any proper linear compartment model for pharmacokinetics. For the linear one compartment model, it is in agreement with the Wagner-Nelson method which is often used nowadays. For the linear two compartment model, it has advantages over the Loo-Riegelman method, which is also in common use in, for example, simplicity in mathematical derivation, less computation, etc. And it can be easily extended to some particular nonlinear compartment model.
Cited by 1
Nélio Machado · 2006
Polynomial Regression Models, even se can be consideied as a paiticulai case of the general linear iegression model, they present proper chaiacteiistics and specific pioblems.
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H. M. Rajashekhar Swamy, N. Krishnamoorthy, D.L. Prabakhara et al. · International Journal of Structural Engineering · 2013
The objective of the present study is to estimate the realistic displacements and force quantities in the structural members, accounting for their three dimensional behaviour and non-linear constitutive relations of soil. This demands usage of proper constitutive model for soil and its numerical model, which is validated experimentally. A program has been developed in the present work to compare soil-structure interaction analyses of a three dimensional frame resting on mat foundation for linear and non-linear constitutive relations of soil. For linear analysis initial tangent modulus is used. For non-linear analysis hypoelasticity model is adopted. This hypoelastic model, used, is validated for a plate load test. Results of non-linear analysis are compared with non-interactive analysis and linear analysis to establish its worthiness.
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Wang Guo-qiang · 2012
The purpose of this study is to select the most proper model for estimating the agricultural land productivity in Henan province.Methods of data integration,statistics test,and extension trend analysis were used.The results indicate that the relationship between the accessible grain productivity per unit samples of Henan province and their agricultural land use index is not a linear one.The statistics test and the extension trend of different models(e.g.,second-order polynomial model,exponential model,logarithmic model,power model,and linear model) indicate that the second-order polynomial model is the most proper one for Henan province.We also found that the estimation result of the model was reliable regarding model test and the spatial distribution in reality.The paper concludes that the optimal model for productivity estimation must be based on the spatial distribution of the survey data,the practical condition of the research area,and the characteristics of the model.
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Tamara Bechtold, Michael Striebel, Kasra Mohaghegh et al. · TU/e Research Portal (Eindhoven University of Technology) · 2008
In this paper we demonstrate model order reduction of a nonlinear academic model of an inverter chain. Two reduction methods, which are suitable for nonlinear differential algebraic equation systems are combined, the trajectory piecewise linear approach and the proper orthogonal decomposition.
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