Online Reduced Basis Construction Procedure for Model Reduction of Parametrized Evolution Systems

Dihlmann, Markus; Kaulmann, Sven; Haasdonk, Bernard

doi:10.3182/20120215-3-at-3016.00020

Cited by 9 publications

(25 citation statements)

References 10 publications

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“…The dictionary-based approach is more natural than the classical hp-refinement method [14,15] and it should always provide a better approximation (see Section 4.1). The potential of approximation with dictionaries for problems with a slow decay of the Kolmogorov r-widths of the solution manifold was revealed in [13,23]. Although they improved classical approaches, the algorithms proposed in [13,23] still involve in general heavy computations in both offline and online stages, and can suffer from round-off errors.…”

Section: Contributionsmentioning

confidence: 99%

“…The potential of approximation with dictionaries for problems with a slow decay of the Kolmogorov r-widths of the solution manifold was revealed in [13,23]. Although they improved classical approaches, the algorithms proposed in [13,23] still involve in general heavy computations in both offline and online stages, and can suffer from round-off errors. If n and r are the dimensions of the full solution space and the (parameter-dependent) reduced approximation space, respectively, and K is the cardinality of the dictionary, then the offline complexity and the memory consumption associated with post-processing of the snapshots in [13,23] are at least O(n(K 2 m 2 A + m 2 b )) and O(nK + K 2 m 2 A + m 2 b ), respectively.…”

Section: Contributionsmentioning

confidence: 99%

“…Although they improved classical approaches, the algorithms proposed in [13,23] still involve in general heavy computations in both offline and online stages, and can suffer from round-off errors. If n and r are the dimensions of the full solution space and the (parameter-dependent) reduced approximation space, respectively, and K is the cardinality of the dictionary, then the offline complexity and the memory consumption associated with post-processing of the snapshots in [13,23] are at least O(n(K 2 m 2 A + m 2 b )) and O(nK + K 2 m 2 A + m 2 b ), respectively. Furthermore, the online stage in [23] requires O((r 3 + m 2 A r)K + m 2 b ) flops and O(m 2 A K 2 + m 2 b ) bytes of memory.…”

Section: Contributionsmentioning

confidence: 99%

“…It follows from Proposition 3.5 (along with Proposition 3.3) that considering (20) can provide better numerical stability than solving reduced systems of equations with standard methods. Furthermore, since affine expansions of V Θ r (µ) and b Θ (µ) have less terms than affine expansions of A r (µ) and b r (µ) in (13), their online assembling should also be much more stable.…”

Section: Sketched Minimal Residual Projectionmentioning

confidence: 99%

“…Remark 4.7. In general, our approach is more stable than the algorithms from [13,23]. These algorithms basically proceed with the solution of the reduced system of equations…”

Section: By Definition 23 and The Union Bound For The Probability Ofmentioning

confidence: 99%

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Randomized linear algebra for model reduction. Part I: Galerkin methods and error estimation

Balabanov

Nouy

2019

Adv Comput Math

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A methodology for using random sketching in the context of model order reduction for high-dimensional parameter-dependent systems of equations was introduced in [Balabanov and Nouy 2019, Part I]. Following this framework, we here construct a reduced model from a small, efficiently computable random object called a sketch of a reduced model, using minimal residual methods. We introduce a sketched version of the minimal residual based projection as well as a novel nonlinear approximation method, where for each parameter value, the solution is approximated by minimal residual projection onto a subspace spanned by several vectors picked (online) from a dictionary of candidate basis vectors. It is shown that random sketching technique can improve not only efficiency but also numerical stability. A rigorous analysis of the conditions on the random sketch required to obtain a given accuracy is presented. These conditions may be ensured a priori with high probability by considering for the sketching matrix an oblivious embedding of sufficiently large size. Furthermore, a simple and reliable procedure for a posteriori verification of the quality of the sketch is provided. This approach can be used for certification of the approximation as well as for adaptive selection of the size of the random sketching matrix. We also propose a two-step procedure for an efficient and stable estimation of an inner product between parameter-dependent vectors having affine decompositions with many terms (possibly expensive to maintain). This procedure can be used for extraction of a quantity of interest (linear or quadratic functional of the solution), estimation of the primal-dual correction, etc.

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Section: Contributionsmentioning

confidence: 99%

Section: Contributionsmentioning

confidence: 99%

Section: Contributionsmentioning

confidence: 99%

Section: Sketched Minimal Residual Projectionmentioning

confidence: 99%

“…Remark 4.7. In general, our approach is more stable than the algorithms from [13,23]. These algorithms basically proceed with the solution of the reduced system of equations…”

Section: By Definition 23 and The Union Bound For The Probability Ofmentioning

confidence: 99%

See 3 more Smart Citations

Randomized linear algebra for model reduction. Part I: Galerkin methods and error estimation

Balabanov

Nouy

2019

Adv Comput Math

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Accelerated solutions of convection‐dominated partial differential equations using implicit feature tracking and empirical quadrature

Mirhoseini,

Zahr

2023

Numerical Methods in Fluids

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SummaryThis work introduces an empirical quadrature‐based hyperreduction procedure and greedy training algorithm to effectively reduce the computational cost of solving convection‐dominated problems with limited training. The proposed approach circumvents the slowly decaying ‐width limitation of linear model reduction techniques applied to convection‐dominated problems by using a nonlinear approximation manifold systematically defined by composing a low‐dimensional affine space with bijections of the underlying domain. The reduced‐order model is defined as the solution of a residual minimization problem over the nonlinear manifold. An online‐efficient method is obtained by using empirical quadrature to approximate the optimality system such that it can be solved with mesh‐independent operations. The proposed reduced‐order model is trained using a greedy procedure to systematically sample the parameter domain. The effectiveness of the proposed approach is demonstrated on two shock‐dominated computational fluid dynamics benchmarks.

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Learning Projection-Based Reduced-Order Models

Ryckelynck,

Casenave,

Akkari

2024

SpringerBriefs in Computer Science

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In this chapter, we introduce the solution space for high-fidelity models based on partial differential equations and the finite element model. The manifold learning approach to model order reduction requires simulated data. Hence, learning projection-based reduced order models (ROM) has two steps: (i) an offline step for the computation of simulated data and for consecutive machine learning tasks, (ii) an online step where the reduced order model is used as a surrogate for the high fidelity model. The offline step generates a train set and a validation set of simulated data. The accuracy and the generalisation of the reduced order model is evaluated in the online step by using a test set of data forecast by the high-fidelity model. The test set aims also to check the computational speedups of the reduced-order model compare to the high-fidelity model.

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Online Reduced Basis Construction Procedure for Model Reduction of Parametrized Evolution Systems

Cited by 9 publications

References 10 publications

Randomized linear algebra for model reduction. Part I: Galerkin methods and error estimation

Randomized linear algebra for model reduction. Part I: Galerkin methods and error estimation

Accelerated solutions of convection‐dominated partial differential equations using implicit feature tracking and empirical quadrature

Learning Projection-Based Reduced-Order Models

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