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Traian Iliescu - One of the best experts on this subject based on the ideXlab platform.
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Are the Snapshot Difference Quotients Needed in the Proper Orthogonal Decomposition
arXiv: Numerical Analysis, 2013Co-Authors: Traian Iliescu, Zhu WangAbstract:This paper presents a theoretical and numerical investigation of the following practical question: Should the time difference quotients of the snapshots be used to generate the Proper Orthogonal Decomposition basis functions? The answer to this question is important, since some published numerical studies use the time difference quotients, whereas other numerical studies do not. The criterion used in this paper to answer this question is the rate of convergence of the error of the reduced order model with respect to the number of Proper Orthogonal Decomposition basis functions. Two cases are considered: the no_DQ case, in which the snapshot difference quotients are not used, and the DQ case, in which the snapshot difference quotients are used. The error estimates suggest that the convergence rates in the $C^0(L^2)$-norm and in the $C^0(H^1)$-norm are optimal for the DQ case, but suboptimal for the no_DQ case. The convergence rates in the $L^2(H^1)$-norm are optimal for both the DQ case and the no_DQ case. Numerical tests are conducted on the heat equation and on the Burgers equation. The numerical results support the conclusions drawn from the theoretical error estimates. Overall, the theoretical and numerical results strongly suggest that, in order to achieve optimal pointwise in time rates of convergence with respect to the number of Proper Orthogonal Decomposition basis functions, one should use the snapshot difference quotients.
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Variational Multiscale Proper Orthogonal Decomposition: Navier-Stokes Equations
arXiv: Numerical Analysis, 2012Co-Authors: Traian Iliescu, Zhu WangAbstract:We develop a variational multiscale Proper Orthogonal Decomposition reduced-order model for turbulent incompressible Navier-Stokes equations. The error analysis of the full discretization of the model 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 Proper Orthogonal Decomposition truncation error. Numerical tests for a three-dimensional turbulent flow past a cylinder at Reynolds number Re=1000 show the improved physical accuracy of the new model over the standard Galerkin and mixing-length Proper Orthogonal Decomposition reduced-order models. 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.
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Proper Orthogonal Decomposition closure models for turbulent flows: A numerical comparison
Computer Methods in Applied Mechanics and Engineering, 2012Co-Authors: Zhu Wang, Jeff Borggaard, Imran Akhtar, Traian IliescuAbstract:This paper puts forth two new closure models for the Proper Orthogonal Decomposition reduced-order modeling of structurally dominated turbulent flows: the dynamic subgrid-scale model and the variational multiscale model. These models, which are considered state-of-the-art in large eddy simulation, together with the mixing length and the Smagorinsky closure models, are tested in the numerical simulation of 3D turbulent flow past a circular cylinder at Re=1000. Five criteria are used to judge the performance of the Proper Orthogonal Decomposition reduced-order models: the kinetic energy spectrum, the mean velocity, the Reynolds stresses, the root mean square values of the velocity fluctuations, and the time evolution of the POD coefficients. All the numerical results are benchmarked against a direct numerical simulation. Based on these numerical results, we conclude that the dynamic subgrid-scale and the variational multiscale models are the most accurate.
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Proper Orthogonal Decomposition closure models for turbulent flows: A numerical comparison
Computer Methods in Applied Mechanics and Engineering, 2012Co-Authors: Zhu Wang, Jeff Borggaard, Imran Akhtar, Traian IliescuAbstract:This paper puts forth two new closure models for the Proper Orthogonal Decomposition reduced-order modeling of structurally dominated turbulent flows: the dynamic subgrid-scale model and the variational multiscale model. These models, which are considered state-of-the-art in large eddy simulation, together with the mixing length and the Smagorinsky closure models, are tested in the numerical simulation of a 3D turbulent flow around a circular cylinder at Re = 1,000. Two criteria are used in judging the performance of the Proper Orthogonal Decomposition reduced-order models: the kinetic energy spectrum and the time evolution of the POD coefficients. All the numerical results are benchmarked against a direct numerical simulation. Based on these numerical results, we conclude that the dynamic subgrid-scale and the variational multiscale models perform best.Comment: 28 pages, 6 figure
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Two-level discretizations of nonlinear closure models for Proper Orthogonal Decomposition
Journal of Computational Physics, 2011Co-Authors: Zhu Wang, Jeff Borggaard, Imran Akhtar, Traian IliescuAbstract:Proper Orthogonal Decomposition has been successfully used in the reduced-order modeling of complex systems. Its original promise of computationally efficient, yet accurate approximation of coherent structures in high Reynolds number turbulent flows, however, still remains to be fulfilled. To balance the low computational cost required by reduced-order modeling and the complexity of the targeted flows, appropriate closure modeling strategies need to be employed. Since modern closure models for turbulent flows are generally nonlinear, their efficient numerical discretization within a Proper Orthogonal Decomposition framework is challenging. This paper proposes a two-level method for an efficient and accurate numerical discretization of general nonlinear closure models for Proper Orthogonal Decomposition reduced-order models. The two-level method computes the nonlinear terms of the reduced-order model on a coarse mesh. Compared with a brute force computational approach in which the nonlinear terms are evaluated on the fine mesh at each time step, the two-level method attains the same level of accuracy while dramatically reducing the computational cost. We numerically illustrate these improvements in the two-level method by using it in three settings: the one-dimensional Burgers equation with a small diffusion parameter @n=10^-^3, the two-dimensional flow past a cylinder at Reynolds number Re=200, and the three-dimensional flow past a cylinder at Reynolds number Re=1000.
Zhu Wang - One of the best experts on this subject based on the ideXlab platform.
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Are the Snapshot Difference Quotients Needed in the Proper Orthogonal Decomposition
arXiv: Numerical Analysis, 2013Co-Authors: Traian Iliescu, Zhu WangAbstract:This paper presents a theoretical and numerical investigation of the following practical question: Should the time difference quotients of the snapshots be used to generate the Proper Orthogonal Decomposition basis functions? The answer to this question is important, since some published numerical studies use the time difference quotients, whereas other numerical studies do not. The criterion used in this paper to answer this question is the rate of convergence of the error of the reduced order model with respect to the number of Proper Orthogonal Decomposition basis functions. Two cases are considered: the no_DQ case, in which the snapshot difference quotients are not used, and the DQ case, in which the snapshot difference quotients are used. The error estimates suggest that the convergence rates in the $C^0(L^2)$-norm and in the $C^0(H^1)$-norm are optimal for the DQ case, but suboptimal for the no_DQ case. The convergence rates in the $L^2(H^1)$-norm are optimal for both the DQ case and the no_DQ case. Numerical tests are conducted on the heat equation and on the Burgers equation. The numerical results support the conclusions drawn from the theoretical error estimates. Overall, the theoretical and numerical results strongly suggest that, in order to achieve optimal pointwise in time rates of convergence with respect to the number of Proper Orthogonal Decomposition basis functions, one should use the snapshot difference quotients.
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Variational Multiscale Proper Orthogonal Decomposition: Navier-Stokes Equations
arXiv: Numerical Analysis, 2012Co-Authors: Traian Iliescu, Zhu WangAbstract:We develop a variational multiscale Proper Orthogonal Decomposition reduced-order model for turbulent incompressible Navier-Stokes equations. The error analysis of the full discretization of the model 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 Proper Orthogonal Decomposition truncation error. Numerical tests for a three-dimensional turbulent flow past a cylinder at Reynolds number Re=1000 show the improved physical accuracy of the new model over the standard Galerkin and mixing-length Proper Orthogonal Decomposition reduced-order models. 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.
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Proper Orthogonal Decomposition closure models for turbulent flows: A numerical comparison
Computer Methods in Applied Mechanics and Engineering, 2012Co-Authors: Zhu Wang, Jeff Borggaard, Imran Akhtar, Traian IliescuAbstract:This paper puts forth two new closure models for the Proper Orthogonal Decomposition reduced-order modeling of structurally dominated turbulent flows: the dynamic subgrid-scale model and the variational multiscale model. These models, which are considered state-of-the-art in large eddy simulation, together with the mixing length and the Smagorinsky closure models, are tested in the numerical simulation of 3D turbulent flow past a circular cylinder at Re=1000. Five criteria are used to judge the performance of the Proper Orthogonal Decomposition reduced-order models: the kinetic energy spectrum, the mean velocity, the Reynolds stresses, the root mean square values of the velocity fluctuations, and the time evolution of the POD coefficients. All the numerical results are benchmarked against a direct numerical simulation. Based on these numerical results, we conclude that the dynamic subgrid-scale and the variational multiscale models are the most accurate.
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Proper Orthogonal Decomposition closure models for turbulent flows: A numerical comparison
Computer Methods in Applied Mechanics and Engineering, 2012Co-Authors: Zhu Wang, Jeff Borggaard, Imran Akhtar, Traian IliescuAbstract:This paper puts forth two new closure models for the Proper Orthogonal Decomposition reduced-order modeling of structurally dominated turbulent flows: the dynamic subgrid-scale model and the variational multiscale model. These models, which are considered state-of-the-art in large eddy simulation, together with the mixing length and the Smagorinsky closure models, are tested in the numerical simulation of a 3D turbulent flow around a circular cylinder at Re = 1,000. Two criteria are used in judging the performance of the Proper Orthogonal Decomposition reduced-order models: the kinetic energy spectrum and the time evolution of the POD coefficients. All the numerical results are benchmarked against a direct numerical simulation. Based on these numerical results, we conclude that the dynamic subgrid-scale and the variational multiscale models perform best.Comment: 28 pages, 6 figure
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Two-level discretizations of nonlinear closure models for Proper Orthogonal Decomposition
Journal of Computational Physics, 2011Co-Authors: Zhu Wang, Jeff Borggaard, Imran Akhtar, Traian IliescuAbstract:Proper Orthogonal Decomposition has been successfully used in the reduced-order modeling of complex systems. Its original promise of computationally efficient, yet accurate approximation of coherent structures in high Reynolds number turbulent flows, however, still remains to be fulfilled. To balance the low computational cost required by reduced-order modeling and the complexity of the targeted flows, appropriate closure modeling strategies need to be employed. Since modern closure models for turbulent flows are generally nonlinear, their efficient numerical discretization within a Proper Orthogonal Decomposition framework is challenging. This paper proposes a two-level method for an efficient and accurate numerical discretization of general nonlinear closure models for Proper Orthogonal Decomposition reduced-order models. The two-level method computes the nonlinear terms of the reduced-order model on a coarse mesh. Compared with a brute force computational approach in which the nonlinear terms are evaluated on the fine mesh at each time step, the two-level method attains the same level of accuracy while dramatically reducing the computational cost. We numerically illustrate these improvements in the two-level method by using it in three settings: the one-dimensional Burgers equation with a small diffusion parameter @n=10^-^3, the two-dimensional flow past a cylinder at Reynolds number Re=200, and the three-dimensional flow past a cylinder at Reynolds number Re=1000.
Imran Akhtar - One of the best experts on this subject based on the ideXlab platform.
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Proper Orthogonal Decomposition closure models for turbulent flows: A numerical comparison
Computer Methods in Applied Mechanics and Engineering, 2012Co-Authors: Zhu Wang, Jeff Borggaard, Imran Akhtar, Traian IliescuAbstract:This paper puts forth two new closure models for the Proper Orthogonal Decomposition reduced-order modeling of structurally dominated turbulent flows: the dynamic subgrid-scale model and the variational multiscale model. These models, which are considered state-of-the-art in large eddy simulation, together with the mixing length and the Smagorinsky closure models, are tested in the numerical simulation of 3D turbulent flow past a circular cylinder at Re=1000. Five criteria are used to judge the performance of the Proper Orthogonal Decomposition reduced-order models: the kinetic energy spectrum, the mean velocity, the Reynolds stresses, the root mean square values of the velocity fluctuations, and the time evolution of the POD coefficients. All the numerical results are benchmarked against a direct numerical simulation. Based on these numerical results, we conclude that the dynamic subgrid-scale and the variational multiscale models are the most accurate.
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Proper Orthogonal Decomposition closure models for turbulent flows: A numerical comparison
Computer Methods in Applied Mechanics and Engineering, 2012Co-Authors: Zhu Wang, Jeff Borggaard, Imran Akhtar, Traian IliescuAbstract:This paper puts forth two new closure models for the Proper Orthogonal Decomposition reduced-order modeling of structurally dominated turbulent flows: the dynamic subgrid-scale model and the variational multiscale model. These models, which are considered state-of-the-art in large eddy simulation, together with the mixing length and the Smagorinsky closure models, are tested in the numerical simulation of a 3D turbulent flow around a circular cylinder at Re = 1,000. Two criteria are used in judging the performance of the Proper Orthogonal Decomposition reduced-order models: the kinetic energy spectrum and the time evolution of the POD coefficients. All the numerical results are benchmarked against a direct numerical simulation. Based on these numerical results, we conclude that the dynamic subgrid-scale and the variational multiscale models perform best.Comment: 28 pages, 6 figure
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Two-level discretizations of nonlinear closure models for Proper Orthogonal Decomposition
Journal of Computational Physics, 2011Co-Authors: Zhu Wang, Jeff Borggaard, Imran Akhtar, Traian IliescuAbstract:Proper Orthogonal Decomposition has been successfully used in the reduced-order modeling of complex systems. Its original promise of computationally efficient, yet accurate approximation of coherent structures in high Reynolds number turbulent flows, however, still remains to be fulfilled. To balance the low computational cost required by reduced-order modeling and the complexity of the targeted flows, appropriate closure modeling strategies need to be employed. Since modern closure models for turbulent flows are generally nonlinear, their efficient numerical discretization within a Proper Orthogonal Decomposition framework is challenging. This paper proposes a two-level method for an efficient and accurate numerical discretization of general nonlinear closure models for Proper Orthogonal Decomposition reduced-order models. The two-level method computes the nonlinear terms of the reduced-order model on a coarse mesh. Compared with a brute force computational approach in which the nonlinear terms are evaluated on the fine mesh at each time step, the two-level method attains the same level of accuracy while dramatically reducing the computational cost. We numerically illustrate these improvements in the two-level method by using it in three settings: the one-dimensional Burgers equation with a small diffusion parameter @n=10^-^3, the two-dimensional flow past a cylinder at Reynolds number Re=200, and the three-dimensional flow past a cylinder at Reynolds number Re=1000.
Jeff Borggaard - One of the best experts on this subject based on the ideXlab platform.
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Proper Orthogonal Decomposition closure models for turbulent flows: A numerical comparison
Computer Methods in Applied Mechanics and Engineering, 2012Co-Authors: Zhu Wang, Jeff Borggaard, Imran Akhtar, Traian IliescuAbstract:This paper puts forth two new closure models for the Proper Orthogonal Decomposition reduced-order modeling of structurally dominated turbulent flows: the dynamic subgrid-scale model and the variational multiscale model. These models, which are considered state-of-the-art in large eddy simulation, together with the mixing length and the Smagorinsky closure models, are tested in the numerical simulation of 3D turbulent flow past a circular cylinder at Re=1000. Five criteria are used to judge the performance of the Proper Orthogonal Decomposition reduced-order models: the kinetic energy spectrum, the mean velocity, the Reynolds stresses, the root mean square values of the velocity fluctuations, and the time evolution of the POD coefficients. All the numerical results are benchmarked against a direct numerical simulation. Based on these numerical results, we conclude that the dynamic subgrid-scale and the variational multiscale models are the most accurate.
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Proper Orthogonal Decomposition closure models for turbulent flows: A numerical comparison
Computer Methods in Applied Mechanics and Engineering, 2012Co-Authors: Zhu Wang, Jeff Borggaard, Imran Akhtar, Traian IliescuAbstract:This paper puts forth two new closure models for the Proper Orthogonal Decomposition reduced-order modeling of structurally dominated turbulent flows: the dynamic subgrid-scale model and the variational multiscale model. These models, which are considered state-of-the-art in large eddy simulation, together with the mixing length and the Smagorinsky closure models, are tested in the numerical simulation of a 3D turbulent flow around a circular cylinder at Re = 1,000. Two criteria are used in judging the performance of the Proper Orthogonal Decomposition reduced-order models: the kinetic energy spectrum and the time evolution of the POD coefficients. All the numerical results are benchmarked against a direct numerical simulation. Based on these numerical results, we conclude that the dynamic subgrid-scale and the variational multiscale models perform best.Comment: 28 pages, 6 figure
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Two-level discretizations of nonlinear closure models for Proper Orthogonal Decomposition
Journal of Computational Physics, 2011Co-Authors: Zhu Wang, Jeff Borggaard, Imran Akhtar, Traian IliescuAbstract:Proper Orthogonal Decomposition has been successfully used in the reduced-order modeling of complex systems. Its original promise of computationally efficient, yet accurate approximation of coherent structures in high Reynolds number turbulent flows, however, still remains to be fulfilled. To balance the low computational cost required by reduced-order modeling and the complexity of the targeted flows, appropriate closure modeling strategies need to be employed. Since modern closure models for turbulent flows are generally nonlinear, their efficient numerical discretization within a Proper Orthogonal Decomposition framework is challenging. This paper proposes a two-level method for an efficient and accurate numerical discretization of general nonlinear closure models for Proper Orthogonal Decomposition reduced-order models. The two-level method computes the nonlinear terms of the reduced-order model on a coarse mesh. Compared with a brute force computational approach in which the nonlinear terms are evaluated on the fine mesh at each time step, the two-level method attains the same level of accuracy while dramatically reducing the computational cost. We numerically illustrate these improvements in the two-level method by using it in three settings: the one-dimensional Burgers equation with a small diffusion parameter @n=10^-^3, the two-dimensional flow past a cylinder at Reynolds number Re=200, and the three-dimensional flow past a cylinder at Reynolds number Re=1000.
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Artificial viscosity Proper Orthogonal Decomposition
Mathematical and Computer Modelling, 2010Co-Authors: Jeff Borggaard, Traian Iliescu, Zhu WangAbstract:We introduce improved reduced-order models for turbulent flows. These models are inspired from successful methodologies used in large eddy simulation, such as artificial viscosity, applied to standard models created by Proper Orthogonal Decomposition of flows coupled with Galerkin projection. As a first step in the analysis and testing of our new methodology, we use the Burgers equation with a small diffusion parameter. We present a thorough numerical analysis for the time discretization of the new models. We then test these models in two problems displaying shock-like phenomena. Of course, since the Burgers equation does not model turbulence, we next need to test our new models in realistic turbulent flow settings. This is the subject of a forthcoming report.
Suman Chakravorty - One of the best experts on this subject based on the ideXlab platform.
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A randomized balanced Proper Orthogonal Decomposition technique
Journal of Computational and Applied Mathematics, 2020Co-Authors: Suman ChakravortyAbstract:Abstract In this paper, we consider the model reduction problem of large-scale linear systems, such as systems obtained through the discretization of partial differential equations. We propose a randomized Proper Orthogonal Decomposition (RPOD ∗ ) technique to obtain the reduced order model by perturbing the primal and adjoint system using Gaussian white noise. We show that computations required by RPOD ∗ algorithm is orders of magnitude lower while its performance is much better than other state of the art algorithms. We also relate the RPOD ∗ algorithm to Krylov subspace methods and show that it constitutes a randomized approach to computational linear algebra problems that utilize Krylov subspace methods.
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ACC - A computationally optimal randomized Proper Orthogonal Decomposition technique
2016 American Control Conference (ACC), 2016Co-Authors: Suman ChakravortyAbstract:In this paper, we consider the model reduction problem of large-scale systems, such as systems obtained through the discretization of partial differential equations. We propose a computationally optimal randomized Proper Orthogonal Decomposition (RPOD*) technique to obtain the reduced order model by perturbing the primal and adjoint system using Gaussian white noise. We show that the computations required by the RPOD* algorithm is orders of magnitude cheaper when compared to the balanced Proper Orthogonal Decomposition (BPOD) algorithm while the performance of the RPOD* algorithm is better than BPOD. It is optimal in the sense that a minimal number of snapshots is needed. We also relate the RPOD* algorithm to random projection algorithms. One numerical example is given to illustrate the procedure.
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A Computationally Optimal Randomized Proper Orthogonal Decomposition Technique
arXiv: Dynamical Systems, 2015Co-Authors: Suman ChakravortyAbstract:In this paper, we consider the model reduction problem of large-scale systems, such as systems obtained through the discretization of partial differential equations. We propose a computationally optimal randomized Proper Orthogonal Decomposition (RPOD*) technique to obtain the reduced order model by perturbing the primal and adjoint system using Gaussian white noise. We show that the computations required by the RPOD* algorithm is orders of magnitude cheaper when compared to the balanced Proper Orthogonal Decomposition (BPOD) algorithm and BPOD output projection algorithm while the performance of the RPOD* algorithm is much better than BPOD output projection algorithm. It is optimal in the sense that a minimal number of snapshots is needed. We also relate the RPOD* algorithm to random projection algorithms. The method is tested on two advection-diffusion equations.
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ACC - A randomized Proper Orthogonal Decomposition technique
2015 American Control Conference (ACC), 2015Co-Authors: Suman ChakravortyAbstract:In this paper, we consider the problem of model reduction of large scale systems, such as those obtained through the discretization of PDEs. We propose a randomized Proper Orthogonal Decomposition (RPOD) technique to obtain the reduced order models by randomly choosing a subset of the inputs/outputs of the system to construct a suitable small sized Hankel matrix from the full Hankel matrix. It is shown that the RPOD technique is computationally orders of magnitude cheaper when compared to techniques such as the Eigensystem Realization Algorithm (ERA)/Balanced Proper Orthogonal Decomposition (BPOD) while obtaining the same information in terms of the number and accuracy of the dominant modes. The method is tested on a linearized channel flow problem.
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A Randomized Proper Orthogonal Decomposition Technique
arXiv: Dynamical Systems, 2013Co-Authors: Suman ChakravortyAbstract:In this paper, we consider the problem of model reduction of large scale systems, such as those obtained through the discretization of PDEs. We propose a randomized Proper Orthogonal Decomposition (RPOD) technique to obtain the reduced order models by randomly choosing a subset of the inputs/outputs of the system to construct a suitable small sized Hankel matrix from the full Hankel matrix. It is shown that the RPOD technique is computationally orders of magnitude cheaper when compared to techniques such as the Eigensystem Realization algorithm (ERA)/Balanced POD (BPOD) while obtaining the same information in terms of the number and accuracy of the dominant modes. The method is tested on several different advection-diffusion equations.
