Techniques in Model Reduction for Large-Scale Systems

Paraskevopoulos, P.N.

doi:10.1016/b978-0-12-012723-8.50010-0

Cited by 7 publications

(4 citation statements)

References 78 publications

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“…In the investigation of eigenproblems of large-scale system theory, the various reduced state space methods [9] cannot ensure that the eigenvalues of the original system remain unchanged. For the invariance of eigensolutions of the original system, it is necessary to select a subspace from the w-hole state space, which corresponds to the theory of contact transformation of the Hamiltonian system theory, i.e., to select the adjoint symplectic subspaces [ 10,111.…”

Section: Introductionmentioning

confidence: 99%

“…From the analogy between structural mechanics and optimal control, the displacement corresponds to the state vector in control theory. Hence the semianalytical method, which is based on the minimum potential energy variational principle, corresponds to the reduced state space method in large-scale system theory [9]. Unfortunately, this method corresponds to the extended point transformation in the Hamiltonian system theory, but not the contact transformation [ 131.…”

Section: Introductionmentioning

confidence: 99%

“…In the large-dimensional Hamiltonian matrix case, the eigenproblem can be solved by the adjoint symplectic subspace inverse substitution method [ l l ] , such that the subspace solutions can converge to the exact solutions of the large-dimensional eigenequation.Based on Eq. (2.23), from eigenequation (3.3) one gets -(JHJ) (Jyi) =z -HT(JYi) = p,(JY,)9 (3.4) hence the adjoint whole state vector (JY,) is the eigenvector of the operator matrix HT, and the eigenvalue is -pi. So, if p is an eigenvalue, then -p is also an eigenvalue of H. Thus the eigenvalues of the operator matrix H can be naturally subdivided into two groups:The corresponding whole state eigenfunction vectors can be nominated as Yui and Ybi, respectively.…”

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confidence: 99%

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Method of separation of variables and Hamiltonian system

Zhong

1993

Numerical Methods Partial

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In the theory of mechanics and/or mathematical physics problems in a prismatic domain, the method of separation of variables usually leads to the Sturm-Liouville-type eigenproblems of self-adjoint operators, and then the eigenfunction expansion method can be used in equation solving. However, a number of important application problems cannot lead to self-adjoint operator for the transverse coordinate. From the minimum potential energy variational principle, by selection of the state and its dual variables, the generalized variational principle is deduced. Then, based on the analogy between the theory of structural mechanics and optimal control, the present article leads the problem to the Hamiltonian system. The finite-dimensional theory for the Hamiltonian system is extended to the corresponding theory of the Hamiltonian operator matrix and adjoint symplectic spaces. The adjoint symplectic orthonormality relation is proved for the whole state eigenfunction vectors, and then the expansion of an arbitrary whole state function vector by the eigenfunction vectors is established. Thus the range of classical method of separation of variables is considerably extended. The eigenproblem derived from a plate bending problem in a strip domain is used for illustration.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Method of separation of variables and Hamiltonian system

Zhong

1993

Numerical Methods Partial

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“…Simplifications are justified in stochastic models, as local heterogeneities have only limited long-range effects. Furthermore, simplifications can make the simulations more manageable and still capture the essential behaviour of the system [452][453][454].…”

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confidence: 99%

A Critical Review of the Modelling Tools for the Reactive Transport of Organic Contaminants

Samborska-Goik,

Pogrzeba

2024

Applied Sciences

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The pollution of groundwater and soil by hydrocarbons is a significant and growing global problem. Efforts to mitigate and minimise pollution risks are often based on modelling. Modelling-based solutions for prediction and control play a critical role in preserving dwindling water resources and facilitating remediation. The objectives of this article are to: (i) to provide a concise overview of the mechanisms that influence the migration of hydrocarbons in groundwater and to improve the understanding of the processes that affect contamination levels, (ii) to compile the most commonly used models to simulate the migration and fate of hydrocarbons in the subsurface; and (iii) to evaluate these solutions in terms of their functionality, limitations, and requirements. The aim of this article is to enable potential users to make an informed decision regarding the modelling approaches (deterministic, stochastic, and hybrid) and to match their expectations with the characteristics of the models. The review of 11 1D screening models, 18 deterministic models, 7 stochastic tools, and machine learning experiments aimed at modelling hydrocarbon migration in the subsurface should provide a solid basis for understanding the capabilities of each method and their potential applications.

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Model reduction of 2-D systems via orthogonal series

Paraskevopoulos

Panagopoulos

Vaitsis

et al. 1991

Multidim Syst Sign Process

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Abstract. In this article, the problem of model reduction of 2-D systems is studied via orthogonal series. The algorithm proposed reduces the problem to an overdetermined linear algebraic system of equations, which may readily be solved to yield the simplified model. When this model approximates adequately the original system, it has many important advantages, e.g., it simplifies the analysis and simulation of the original system, it reduces the computational effort in design procedures, it reduces the hardware complexity of the system, etc. Several examples are included which illustrate the efficiency of the proposed method and gives some comparison with other model reduction techniques.

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Techniques in Model Reduction for Large-Scale Systems

Cited by 7 publications

References 78 publications

Method of separation of variables and Hamiltonian system

Method of separation of variables and Hamiltonian system

A Critical Review of the Modelling Tools for the Reactive Transport of Organic Contaminants

Model reduction of 2-D systems via orthogonal series

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