{"title":"9 From the POD-Galerkin method to sparse manifold models","authors":"Jean-Christophe Loiseau, S. Brunton, B. R. Noack","doi":"10.1515/9783110499001-009","DOIUrl":"https://doi.org/10.1515/9783110499001-009","url":null,"abstract":"Reduced-order models are essential for the accurate and efficient prediction, estimation, and control of complex systems. This is especially true in fluid dynamics, where the fully resolved state space may easily contain millions or billions of degrees of freedom. Because these systems typically evolve on a low-dimensional attractor, model reduction is defined by two essential steps: 1) identify a good state space for the attractor, and 2) identify the dynamics on this attractor. The leading method for model reduction in fluids is Galerkin projection of the Navier-Stokes equations onto a linear subspace of modes obtained via proper orthogonal decomposition (POD). However, there are serious challenges in this approach, including truncation errors, stability issues, difficulty handling transients, and mode deformation with changing boundaries and operating conditions. Many of these challenges result from the choice of a linear POD subspace in which to represent the dynamics. In this chapter, we describe an alternative approach, feature-based manifold modeling (FeMM), in which the low-dimensional attractor and nonlinear dynamics are characterized from typical experimental data: time-resolved sensor data and optional non-time-resolved particle image velocimetry (PIV) snapshots. FeMM consists of three steps: First, the sensor signals are lifted to a dynamic feature space. Second, we identify a sparse human-interpretable nonlinear dynamical system for the feature state based on the sparse identification of nonlinear dynamics (SINDy). Third, if PIV snapshots are available, a local linear mapping from the feature state to the velocity field is performed to reconstruct the full-state of the system. We demonstrate this approach, and compare with POD-Galerkin modeling, on the incompressible two-dimensional flow around a circular cylinder. Best practices and perspectives for future research are also included, along with open-source code for this example.","PeriodicalId":32642,"journal":{"name":"Genetics Applications","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81810200","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 Case studies of model order reduction for acoustics and vibrations","authors":"E. Deckers, W. Desmet, K. Meerbergen, F. Naets","doi":"10.1515/9783110499001-003","DOIUrl":"https://doi.org/10.1515/9783110499001-003","url":null,"abstract":"This chapter presents several case studies to illustrate specific aspects in setting up reduced-order models of acoustic and vibration models in mechanical applications. Modal truncation approaches have been a provenworkhorse for over half a century in civil andmechanical engineering, but, formany (recent) applications, these techniques are too limited. Inmechanical engineering,model users are interested in a range ofmodel applications: frequency and time domain, linear and nonlinear, single domain and multiphysics, etc. This broad range of applications makes it particularly challenging to devise appropriate reduced-order model schemes, as a scheme for one model use might be completely inadequate for other applications. Krylov methods for example have been a go-to technique inmany domains, but face particular challenges in mechanical finite element models as the system’s eigenvalues lie along the imaginary axis and the high frequencies are irrelevant for a givenmesh size from a physical perspective. In the current chapter we explore these particularities for different types of mechanical models and simulation purposes, in order to surface several good practices and points of attention when applying model order reduction on these models. We bring together two different viewpoints: the application of model order reduction from a purely mathematical point of view and the physical interpretation of models and expected properties of reduced-order models based on physical arguments from the field of mechanics. While we touch upon a range of novel model order reduction techniques, we do not discuss parametric model order reduction as it is expected that the presented guidelines can be exploited in parametric problems without additional specific concerns.","PeriodicalId":32642,"journal":{"name":"Genetics Applications","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75518862","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":"13 MOR software","authors":"B. Haasdonk","doi":"10.1515/9783110499001-013","DOIUrl":"https://doi.org/10.1515/9783110499001-013","url":null,"abstract":": This chapter is devoted to an important requirement of successful model order reduction (MOR) application, namely, the software aspect. The most common situation is the existence of a so-called full model, i. e., a high-fidelity, high-dimensional simulation model, that needs to be accelerated by MOR techniques, optimally without reimplementing the partially complex reduction techniques, as presented in the first volume of this handbook. Initially, as neither full simulation models nor MOR algorithms are to be repro-grammed, but ideally are reused from existing implementations, we concentrate on the aspect of the interplay of such packages. We will discriminate, discuss, and exem-plifydifferentlevelsofsolver“intrusiveness”thatallowcorrespondingreductiontech- niques to be applied. On the one hand, most effective MOR techniques require deep access into the full model’s simulation code. On the other hand, application-specific full model simulators may only offer very restricted access to internals, especially in case of commercial packages. This gap in requirements and practical accessibility mo-tivates the discrimination into “white-box,” “gray-box,” and “black-box” simulation scenarios. In particular, we exemplify the ideal case of MOR for white-box situations on two examples, namely, parametric linear elliptic PDE and parametric nonlinear ODE systems. Depending on those access classes, different corresponding reduction techniques can be applied. The second part of the current chapter then discusses existing MOR software. Several program packages exist which provide MOR techniques. They differ in availability, licensing, programming language, system types, physical application domains, external simulator bindings, etc. We give an overview of the most relevant of those MOR packages, such that applicants can identify potential suitable software library candidates. ODE systems, parametric problems, elliptic PDEs, parabolic PDEs, hyperbolic PDEs, first-order systems, second-order systems, differential (DAEs), and or","PeriodicalId":32642,"journal":{"name":"Genetics Applications","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87267134","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":"11 Reduced-order modeling of large-scale network systems","authors":"Xiaodong Cheng, J. Scherpen, H. Trentelman","doi":"10.1515/9783110499001-011","DOIUrl":"https://doi.org/10.1515/9783110499001-011","url":null,"abstract":"Large-scale network systems describe a wide class of complex dynamical systems composed of many interacting subsystems. A large number of subsystems and their high-dimensional dynamics often result in highly complex topology and dynamics, which pose challenges to network management and operation. This chapter provides an overview of reduced-order modeling techniques that are developed recently for simplifying complex dynamical networks. In the first part, clustering-based approaches are reviewed, which aim to reduce the network scale, i.e., find a simplified network with a fewer number of nodes. The second part presents structure-preserving methods based on generalized balanced truncation, which can reduce the dynamics of each subsystem.","PeriodicalId":32642,"journal":{"name":"Genetics Applications","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83206157","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 Model order reduction in neuroscience","authors":"B. Karasözen","doi":"10.1515/9783110499001-007","DOIUrl":"https://doi.org/10.1515/9783110499001-007","url":null,"abstract":"The human brain contains approximately 109 neurons, each with approximately 103 connections, synapses, with other neurons. Most sensory, cognitive, and motor functions of our brains depend on the interaction of a large population of neurons. In recent years, many technologies have been developed for recording large numbers of neurons either sequentially or simultaneously. Increases in computational power and algorithmic developments have enabled advanced analyses of the neuronal population parallel to the rapid growth of quantity and complexity of the recorded neuronal activity. Recent studies made use of dimensionality and model order reduction techniques to extract coherent features which are not apparent at the level of individual neurons. It has been observed that the neuronal activity evolves on low-dimensional subspaces. The aim of model reduction of large-scale neuronal networks is the accurate and fast prediction of patterns and their propagation in different areas of the brain. Spatiotemporal features of brain activity are identified on low-dimensional subspaces with methods such as dynamic mode decomposition, proper orthogonal decomposition, the discrete empirical interpolation method, and combined parameter and state reduction. In this chapter, we give an overview of the currently used dimensionality reduction and model order reduction techniques in neuroscience.","PeriodicalId":32642,"journal":{"name":"Genetics Applications","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85561698","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 Complexity reduction of electromagnetic systems","authors":"D. Ioan, G. Ciuprina, W. Schilders","doi":"10.1515/9783110499001-005","DOIUrl":"https://doi.org/10.1515/9783110499001-005","url":null,"abstract":": This chapterhas two main objectives: first, topropose a computer-aidedcon-sistent and accurate description of the behavior of electromagnetic devices at various speeds or frequencies and, second, to describe procedures to generate compact electrical circuits for them, with an approximatively equivalent behavior. The extracted models should have a finite complexity as low as possible, while yielding an acceptable accuracy, as well as preserve essential characteristics, such as passivity. A suc-cessful complexity reduction can be obtained if a priori and on-the-fly reduction strategies are applied before and during the model discretization, followed by a posteriori complexity reduction.","PeriodicalId":32642,"journal":{"name":"Genetics Applications","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88374706","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":"6 Model reduction in computational aerodynamics","authors":"","doi":"10.1515/9783110499001-006","DOIUrl":"https://doi.org/10.1515/9783110499001-006","url":null,"abstract":"","PeriodicalId":32642,"journal":{"name":"Genetics Applications","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89569443","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 in mechanical engineering","authors":"","doi":"10.1515/9783110499001-002","DOIUrl":"https://doi.org/10.1515/9783110499001-002","url":null,"abstract":"","PeriodicalId":32642,"journal":{"name":"Genetics Applications","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88158882","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}
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