PEBL-ROM: Projection-error based local reduced-order models

Amsallem, David; Haasdonk, Bernard

doi:10.1186/s40323-016-0059-7

Cited by 36 publications

(25 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The hp-refinement method may present serious drawbacks: it can be highly sensitive to the parametrization, it can require a large number of subdomains in P (especially for high-dimensional parameter domains) and it can require computing too many solution samples. These drawbacks can be partially reduced by various modifications of the hp-refinement method [2,26], but not circumvented.…”

Section: Dictionary-based Minimal Residual Methodsmentioning

confidence: 99%

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

Balabanov

Nouy

2019

Adv Comput Math

View full text Add to dashboard Cite

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.

show abstract

Section: Dictionary-based Minimal Residual Methodsmentioning

confidence: 99%

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

Balabanov

Nouy

2019

Adv Comput Math

View full text Add to dashboard Cite

show abstract

“…In this paper we compare several strategies to perform snapshot clustering, employing in any case a Galerkin projection to construct a ROM for each cluster. Neighboring snapshots can be either added or not to each cluster to obtain overlapping clusters; in our examples we do not consider any overlap among clusters, although in principle this can also be done [45,46]. The approach described above was firstly proposed in [45] to address the construction of local ROMs in the state space -although without constructing a ROM to be systematically queried over the parameter space -and further extended in [47,48,46].…”

Section: Local Reduced Basis Methodsmentioning

confidence: 99%

Numerical approximation of parametrized problems in cardiac electrophysiology by a local reduced basis method

Pagani

Manzoni

Quarteroni

2018

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

The efficient solution of coupled PDEs/ODEs problems arising in cardiac electrophysiology is of key importance whenever interested to study the electrical behavior of the tissue for several instances of relevant physical and/or geometrical parameters. This poses significant challenges to reduced order modeling (ROM) techniques-such as the reduced basis method-traditionally employed when dealing with the repeated solution of parameter dependent differential equations. Indeed, the nonlinear nature of the problem, the presence of moving fronts in the solution, and the high sensitivity of this latter to parameter variations, make the application of standard ROM techniques very problematic. In this paper we propose a local ROM built through a k-means clustering in the state space of the snapshots for both the solution and the nonlinear term. Several comparisons among alternative local ROMs on a benchmark test case show the effectivity of the proposed approach. Finally, the application to a parametrized problem set on an idealized leftventricle geometry shows the capability of the proposed ROM to face complex problems.

show abstract

“…In [16] a strategy inspired by the mesh h−refinement is proposed. In [6,5] a construction of linear subspaces is propposed, based on a k-means clustering that groups together similar snapshots of the solution. A similar approach is presented in [52] to overcome some shortcomings of the DEIM approach.…”

Section: State Of the Artmentioning

confidence: 99%

Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces

et al. 2020

View full text Add to dashboard Cite

We consider the problem of model reduction of parametrized PDEs where the goal is to approximate any function belonging to the set of solutions at a reduced computational cost. For this, the bottom line of most strategies has so far been based on the approximation of the solution set by linear spaces on Hilbert or Banach spaces. This approach can be expected to be successful only when the Kolmogorov width of the set decays fast. While this is the case on certain parabolic or elliptic problems, most transport-dominated problems are expected to present a slow decaying width and require to study nonlinear approximation methods. In this work, we propose to address the reduction problem from the perspective of general metric spaces with a suitably defined notion of distance. We develop and compare two different approaches, one based on barycenters and another one using tangent spaces when the metric space has an additional Riemannian structure. As a consequence of working in metric spaces, both approaches are automatically nonlinear. We give theoretical and numerical evidence of their efficiency to reduce complexity for one-dimensional conservative PDEs where the underlying metric space can be chosen to be the L 2 -Wasserstein space. arXiv:1909.06626v1 [math.NA]

show abstract

PEBL-ROM: Projection-error based local reduced-order models

Cited by 36 publications

References 25 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

Numerical approximation of parametrized problems in cardiac electrophysiology by a local reduced basis method

Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces

Contact Info

Product

Resources

About