{"title":"6 Localized model reduction for parameterized problems","authors":"","doi":"10.1515/9783110671490-006","DOIUrl":"https://doi.org/10.1515/9783110671490-006","url":null,"abstract":"","PeriodicalId":229517,"journal":{"name":"Snapshot-Based Methods and Algorithms","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124332191","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"7 Data-driven methods for reduced-order modeling","authors":"S. Brunton, J. Kutz","doi":"10.1515/9783110671490-007","DOIUrl":"https://doi.org/10.1515/9783110671490-007","url":null,"abstract":": Data-driven mathematical methods are increasingly important for characterizing complex systems across the physical, engineering, and biological sciences. These methods aim to discover and exploit a relatively small subset of the full high-dimensional state space where low-dimensional models can be used to describe the evolution of the system. Emerging dimensionality reduction methods, such as the dynamic mode decomposition (DMD) and its Koopman generalization, have garnered at-tention due to the fact that they can (i) discover low-rank spatio-temporal patterns of activity, (ii) embed the dynamics in the subspace in an equation-free manner (i. e., the governing equations are unknown), unlike Galerkin projection onto proper orthogonal decomposition modes, and (iii) provide approximations in terms of linear dynamical systems, which are amenable to simple analysis techniques. The selection of observables (features) for the DMD/Koopman architecture can yield accurate low-dimensionalembeddingsfornonlinearpartialdifferentialequations(PDEs)while limiting computational costs. Indeed, a good choice of observables, including time delay embeddings, can often linearize the nonlinear manifold by making the spatiotemporal dynamics weakly nonlinear. In addition to DMD/Koopman decompositions, coarse-grained models for spatio-temporal systems can also be discovered using the sparse identification of nonlinear dynamics (SINDy) algorithm which allows one to construct reduced-order models in low-dimensional embeddings. These methods can be used in a nonintrusive, equation-free manner for improved computational performance on parametric PDE systems.","PeriodicalId":229517,"journal":{"name":"Snapshot-Based Methods and Algorithms","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121600673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"2 Model order reduction by proper orthogonal decomposition","authors":"Carmen Gräßle, M. Hinze, S. Volkwein","doi":"10.1515/9783110671490-002","DOIUrl":"https://doi.org/10.1515/9783110671490-002","url":null,"abstract":": We provide an introduction to proper orthogonal decomposition (POD) model order reduction with focus on (nonlinear) parametric partial differential equations (PDEs) and (nonlinear) time-dependent PDEs, and PDE-constrained optimization with POD surrogate models as application. We cover the relation of POD and singular value decomposition, POD from the infinite-dimensional perspective, reduction of nonlinearities, certification with a priori and a posteriori error estimates, spatial and temporal adaptivity, input dependency of the POD surrogate model, POD basis update strategies in optimal control with surrogate models, and sketch related algorithmic frameworks. The perspective of the method is demonstrated with several numerical examples.","PeriodicalId":229517,"journal":{"name":"Snapshot-Based Methods and Algorithms","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128336459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"3 Proper generalized decomposition","authors":"F. Chinesta, P. Ladevèze","doi":"10.1515/9783110671490-003","DOIUrl":"https://doi.org/10.1515/9783110671490-003","url":null,"abstract":": The so-called “reduced” models have always been very popular and often essential in engineering to analyze the behavior of structures and materials, especially in dynamics. They highlight the relevant information and lead, moreover, to less expensive and more robust calculations. In addition to conventional reduction methods, a generation of reduction strategies is now being developed, such as proper generalized decomposition (PGD), which is the subject of this chapter. The primary feature of these strategies is to be very general and to offer enormous potential for solving problems beyond the reach of industrial computing codes. It is typically the case when trying to take into account the uncertainties or the variations of parameters or nonlinear problems with very large number of degrees of freedom, in the presence of several scales or interactions between several physics. These methods, along with the notions of “offline” and “online” calculations, also open the way to new approaches where simulation and analysis can be carried out almost in real-time. What distin-guishes PGD from proper orthogonal decomposition (POD) and reduced basis is the calculation procedure that does not differentiate between the different variables pa-rameters/time/space. In other terms, we can say that we minimize or make stationary a residual defined over the parameters-time-space domain. PGD with time/space separation and the classical greedy computation technique were introduced in the 1980s as part of the LATIN solver [66, 67] for solving nonlinear time-dependent problems with the terminology “time/space radial approximation.” The corpus of literature de-votedtothismethod is vast[68, 77]but remainedinthe form oftime/spaceseparations for many years. A more general separated representation was more recently employed in [5, 6] for approximating the solution of multidimensional partial differential equa-tions.In[93],suchseparatedrepresentationsarealsoconsideredforsolvingstochastic equations. PGD is the common name coined in 2010 by the authors of this chapter for these techniques because it can be viewed as an extension of the classical POD. To-day,manyworksuseanddevelopthePGDinextremelyvariedfields.Inthischapterwe In this chapter we use different notations to be consistent with the referred publica-tions. In any case, notation will be appropriately defined before being used to avoid any possible confusion.","PeriodicalId":229517,"journal":{"name":"Snapshot-Based Methods and Algorithms","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128482073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"5 Computational bottlenecks for PROMs: precomputation and hyperreduction","authors":"","doi":"10.1515/9783110671490-005","DOIUrl":"https://doi.org/10.1515/9783110671490-005","url":null,"abstract":"","PeriodicalId":229517,"journal":{"name":"Snapshot-Based Methods and Algorithms","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115450353","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Preface to the second volume of Model Order Reduction","authors":"","doi":"10.1515/9783110671490-201","DOIUrl":"https://doi.org/10.1515/9783110671490-201","url":null,"abstract":"This second volume of the Model Order Reduction handbook project mostly focuses on snapshot-based methods for parameterized partial differential equations. This approach has seen tremendous development in the past two decades, especially in the broad domain of computational mechanics. However, the main ideas were already known long before; see, e. g., the seminal work by J. L. Lumley, “The structure of inhomogeneous turbulent flows,” in Atmospheric Turbulence and Radio Wave Propagation, 1967, for proper orthogonal decomposition (POD), and the one by A. K. Noor and J. N. Peters, Reduced basis technique for nonlinear analysis of structures, AIAA Journal, Vol. 4, 1980, for the reduced basis method. The most popular mathematical strategy behind snapshot-based methods relies on Galerkin projection on finite-dimensional subspaces generated by snapshot solutions corresponding to a special choice of parameters. Because of that, it is often termed as a projection-based intrusive approach. A suitable offline-online splitting of the computational steps, as well as the use of hyperreduction techniques to be used for the nonlinear (or nonaffine) terms and nonlinear residuals, is key to efficiency. The first chapter, by G. Rozza et al., introduces all the preliminary notions and basic ideas to start delving into the topic of snapshot-based model order reduction. All the notions will be recast into a deeper perspective in the following chapters. The second chapter, byGrässle et al., provides an introduction to PODwith a focus on (nonlinear) parametric partial differential equations (PDEs) and (nonlinear) timedependent PDEs, and PDE-constrained optimization with POD surrogate models as application. Several numerical examples are provided to support the theoretical findings. A second scenario in the methodological development is provided in the third chapter, by Chinesta and Ladevèze, on proper generalized decomposition, a research line significantly grown in the last couple of decades also thanks to real-world applications. Basic concepts used here rely on the separation of variables (time, space, design parameters) and tensorization. The fourth chapter, by Maday and Patera, focuses on the reduced basis method, including a posteriori error estimation, as well as a primal-dual approach. Several combinations of these approaches have been proposed in the last few years to face problems of increasing complexity. When facing nonaffine and nonlinear problems, the development of efficient reduction strategies is of paramount importance. These strategies can require either global or local (pointwise) subspace constructions. This issue is thoroughly covered in the fifth chapter, by Farhat et al., where several front-end computational problems in the field of nonlinear structural dynamics, scattering elastoacousticwave propagation problems, and a parametric PDE-ODE wildfire model problem are presented.","PeriodicalId":229517,"journal":{"name":"Snapshot-Based Methods and Algorithms","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131013808","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
求助内容：
应助结果提醒方式：