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Local/global model order reduction strategy for the simulation of
, 2011
"... quasibrittle fracture ..."
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A Survey of Model Reduction Methods for Parametric Systems
, 2013
"... Numerical simulation of largescale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent largescale nature of the models leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational bu ..."
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Cited by 12 (4 self)
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Numerical simulation of largescale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent largescale nature of the models leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original largescale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey stateoftheart in parametric model reduction methods. Parametric model reduction targets the broad class of problems for which the equations governing the system behavior depend on a set of parameters. Examples include parameterized partial differential equations and largescale systems of parameterized ordinary differential
Application of Least Square MPE technique in the reduced proper modeling of electrical circuits
 In Proceedings of the International Symposium on MTNS, Kyoto
"... Verhoeven Abstract Reduced order models are usually derived by performing the Galerkin projection procedure, where the original equations are projected onto the space spanned by a set of approximating basis functions. For Differential Algebraic Equations this projection scheme may yield an unsolvab ..."
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Verhoeven Abstract Reduced order models are usually derived by performing the Galerkin projection procedure, where the original equations are projected onto the space spanned by a set of approximating basis functions. For Differential Algebraic Equations this projection scheme may yield an unsolvable reduced order model. This means that a model of an electrical circuit can become illposed if it is reduced by the Galerkin technique. As a remedy to the problem, in this paper the reformulation of the reduced order model problem in the least squares sense is suggested. The space where the original is projected is different to the space used in the Galerkin procedure. It is shown that the resulting reduced order model will be guaranteed to be wellposed when the problem of nding a reduced order model is cast into a least
LOCALIZED DISCRETE EMPIRICAL INTERPOLATION METHOD
"... Abstract. This paper presents a new approach to construct more efficient reducedorder models for nonlinear partial differential equations with proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM). Whereas DEIM projects the nonlinear term onto one global subsp ..."
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Abstract. This paper presents a new approach to construct more efficient reducedorder models for nonlinear partial differential equations with proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM). Whereas DEIM projects the nonlinear term onto one global subspace, our localized discrete empirical interpolation method (LDEIM) computes several local subspaces, each tailored to a particular region of characteristic system behavior. Then, depending on the current state of the system, LDEIM selects an appropriate local subspace for the approximation of the nonlinear term. In this way, the dimensions of the local DEIM subspaces, and thus the computational costs, remain low even though the system might exhibit a wide range of behaviors as it passes through different regimes. LDEIM uses machine learning methods in the offline computational phase to discover these regions via clustering. Local DEIM approximations are then computed for each cluster. In the online computational phase, machinelearningbased classification procedures select one of these local subspaces adaptively as the computation proceeds. The classification can be achieved using either the system parameters or a lowdimensional representation of the current state of the system obtained via feature extraction. The LDEIM approach is demonstrated for a reacting flow example of an H2Air flame. In this example, where the system state has a strong nonlinear dependence on the parameters, the LDEIM provides speedups of two orders of magnitude over standard DEIM.
Comparison of some reduced representation approximations, submitted
, 2013
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Statistical extraction of process zones and representative subspaces in fracture of random composites. Accepted for publication
 in International Journal for Multiscale Computational Engineering, arXiv:1203.2487v2, 2012. hal00780840, version 3  2 Sep 2013
"... fracture of random composites ..."
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Comparison of POD reduced order strategies for the nonlinear 2D Shallow Water Equations
, 2014
"... ..."
PRESERVING LAGRANGIAN STRUCTURE IN NONLINEAR MODEL REDUCTION WITH APPLICATION TO STRUCTURAL DYNAMICS
"... Abstract. This work proposes a modelreduction methodology that preserves Lagrangian structure (equivalently Hamiltonian structure) and achieves computational efficiency in the presence of highorder nonlinearities and arbitrary parameter dependence. As such, the resulting reducedorder model retai ..."
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Abstract. This work proposes a modelreduction methodology that preserves Lagrangian structure (equivalently Hamiltonian structure) and achieves computational efficiency in the presence of highorder nonlinearities and arbitrary parameter dependence. As such, the resulting reducedorder model retains key properties such as energy conservation and symplectic timeevolution maps. We focus on parameterized simple mechanical systems subjected to Rayleigh damping and external forces, and consider an application to nonlinear structural dynamics. To preserve structure, the method first approximates the system’s ‘Lagrangian ingredients’—the Riemannian metric, the potentialenergy function, the dissipation function, and the external force—and subsequently derives reducedorder equations of motion by applying the (forced) Euler–Lagrange equation with these quantities. From the algebraic perspective, key contributions include two efficient techniques for approximating parameterized reduced matrices while preserving symmetry and positive definiteness: matrix gappy POD and reducedbasis sparsification (RBS). Results for a parameterized trussstructure problem demonstrate the importance of preserving Lagrangian structure and illustrate the proposed method’s merits: it reduces computation time while maintaining high accuracy and stability, in contrast to existing nonlinear modelreduction techniques that do not preserve structure.