The reduced basis method for the electric field integral equation

Fares, M’Barek; Hesthaven, Jan S.; Maday, Yvon; Stamm, Benjamin

doi:10.1016/j.jcp.2011.03.023

Cited by 61 publications

(73 citation statements)

References 36 publications

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“…increasing M ) [6,12]; or completely recomputing the subspace using the original high-fidelity model [10,11]. Both approaches retain the "global" nature of the subspace while incurring costs that reduce the efficiency of the technology.…”

Section: Adaptive Rom Methodsmentioning

confidence: 99%

“…This data is then used to construct a lower dimensional basis using techniques such as the proper orthogonal decomposition (POD, also known as principle component analysis or the Karhunen-Loéve decomposition) [1,2,3] or the reduced basis method (RBM) [1,2,5,6]. In the POD approach, for instance, the N snapshots are reduced into a small subspace by computing the singular-value-decomposition (SVD) of…”

Section: An Overview Of Dynamic Romsmentioning

confidence: 99%

“…Of particular importance to LLNL applications is the adaptation of the reduced model to efficiently capture highly localized spatially and temporally evolving physics with many variable parameters. Current ROMs cannot efficiently represent local features in highly complex and dynamic simulations of physical phenomenon without enlarging the entire reduced model [4,5,6,2,7,8]. Some approaches provide a degree of dynamic adaptation by interrupting the ROM's execution in order to reconstruct the subspace using expensive full-fidelity simulations [9,10,11].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Reduced order modelling for dynamic simulations: LDRD feasibility study final report

Chand¹,

Henshaw²,

Vassilevska³

2012

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Section: Adaptive Rom Methodsmentioning

confidence: 99%

Section: An Overview Of Dynamic Romsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Reduced order modelling for dynamic simulations: LDRD feasibility study final report

Chand¹,

Henshaw²,

Vassilevska³

2012

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“…Both cases are clearly sub-optimal. Another hp approached is introduced for empirical interpolation methods, see [4,3] . The idea is that if the method does not converges in N bases, the parameter domain is decomposed into 2 p subdomains, and a new basis is build in these 2 p subdomains.…”

Section: Introductionmentioning

confidence: 99%

“…Like the hp algorithm for the empirical interpolation method [4,3] , our hp reduced basis method first defines a maximum number of reduced bases that is affordable during the online stage. If the offline greedy procedure for a given parameter domain converges with equal or less than the maximum bases, the offline algorithm terminates.…”

Section: Introductionmentioning

confidence: 99%

On the Use of ANOVA Expansions in Reduced Basis Methods for Parametric Partial Differential Equations

Hesthaven

Zhang

2016

J Sci Comput

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Abstract. We propose two different improvements of reduced basis (RB) methods to enable the efficient and accurate evaluation of an output functional based on the numerical solution of parametrized partial differential equations with a possibly high-dimensional parameter space. The element that combines these two techniques is that they both utilize ANOVA expansions to achieve the improvements.The first method is a three-step RB-ANOVA-RB method, aiming at using a combination of reduced basis methods and ANOVA expansions to effectively compress the parameter space without impact the accuracy of the output of interest. This is achieved by first building a low-accuracy reduced model for the full high-dimensional parametric problem. This model is used to recover an approximate ANOVA expansion for the output functional at marginal cost, allowing the estimation of the sensitivity of the output functional to parameter variation and enabling a subsequent compression of the parameter space. A new accurate reduced model can then be constructed for the compressed parametric problem at a substantially lower computational cost than for the full problem.In the second approach we explore the ANOVA expansion to drive an hp reduced basis method. This is initiated by setting up a maximum number of reduced bases that can be afforded during the online stage. If the offline greedy procedure for a given parameter domain converges with equal or less than the maximum bases, the offline algorithm stops. Otherwise, an approximate ANOVA expansion is performed for the output functional. The parameter domain is decomposed into several subdomains where the most important parameters according to the ANOVA expansion are split. The offline greedy algorithms are performed in these parameter subdomains. The algorithm is applied recursively until the offline greedy algorithms converge across all parameter subdomains.We demonstrate the accuracy, efficiency, and generality of these two approaches through a number of test cases.Keywords reduced basis method · RB · ANOVA · high-dimentional parameter space · hp method Division

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Reduced‐order modelling for linear heat conduction with parametrised moving heat sources

2016

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The purpose of Reduced‐Order Modelling (ROM) is to substantially lower the computational cost of numerical simulations. As an example proper orthogonal decomposition (POD), which is optimal in a least‐square sense, is applied to compute basis functions using existing solutions, e.g. from previously computed simulations. These reduced basis functions are subsequently used for the Galerkin projection of the governing equations. There is intense research on the application of ROM to parametrised partial differential equations (pPDE). However, in the case of transient equations research on ROM is mainly focused on models with constant parameters. In this paper an approach to cope with the linear heat conduction problem with a moving heat source is introduced. Problems of such kind are encountered in the simulation of laser or electron beam melting processes in additive manufacturing. To construct the reduced basis a nested POD method is used to mitigate the com‐putational costs of large‐scale eigenproblems. To further gain computational efficiency the discrete empirical interpolation method (DEIM) is applied to deal with non‐affine parameter dependencies. Based on various numerical examples the approximation quality of the reduced models is discussed through direct comparison with large‐scale Finite Element simulations. (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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The reduced basis method for the electric field integral equation

Cited by 61 publications

References 36 publications

Reduced order modelling for dynamic simulations: LDRD feasibility study final report

Reduced order modelling for dynamic simulations: LDRD feasibility study final report

On the Use of ANOVA Expansions in Reduced Basis Methods for Parametric Partial Differential Equations

Reduced‐order modelling for linear heat conduction with parametrised moving heat sources

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