[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-201908-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-201908-en":131},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","improved-a-posteriori-error-bounds-for-reduced-port-hamiltonian-systems","Improved a posteriori Error Bounds for Reduced port-Hamiltonian Systems","","Projection-based model order reduction for dynamical systems introduces an unknown reduction error between a high-fidelity model and its lower-dimensional counterpart. Residual-based techniques can bound this error but often overestimate it. This work introduces two improved, structure-aware a posteriori error bounding methods for port-Hamiltonian systems: a hierarchical error bound and an auxiliary-linear-problem based bound. Both methods require an additional approximation of the system and its error system, and their relationship is analyzed theoretically and numerically, validated on a 3D port-Hamiltonian guitar model with fluid-structure interaction.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & 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the paper address in projection-based model order reduction?","Question",{"text":63,"@type":64},"Projection-based MOR introduces an unknown reduction error between the high-fidelity model and its reduced version, which needs reliable a posteriori control.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"Which improved error bounding techniques are presented?",{"text":68,"@type":64},"The paper applies and adapts a hierarchical error bound and an auxiliary linear problem (ALP) based error bound for port-Hamiltonian systems.",{"name":70,"@type":61,"acceptedAnswer":71},"How are the methods validated?",{"text":72,"@type":64},"Their effectiveness is demonstrated using a challenging three-dimensional port-Hamiltonian model of a classical guitar with fluid-structure interaction.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},201908,1788540084,{"code":4,"msg":82,"data":83},"success",[84,88,92,96,101,106,111,115,120,123,127],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & 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port-Hamiltonian Systems  \nJohannes Rettberg ∗ Robin Herkert†  \nDominik Wittwar † Patrick Buchfink† Jörg Fehr∗ Bernard Haasdonk†  \nMarch 31, 2023  \nAbstract Projection-based model order reduction of dynamical systems usually introduces an error between the high-fidelity model and its counterpart of lower dimension. This unknown error can be bounded by residual-based methods, which are typically known to be highly pessimistic in the sense of largely overestimating the true error. This work applies two improved error bounding techniques, namely (a) a hierarchical error bound and (b) an error bound based on an auxiliary linear problem, to the case of port-Hamiltonian systems. The approaches rely on a second approximation of (a) the dynamical system and (b) the error system. In this paper, these methods are for the first time adapted to port-Hamiltonian systems by exploiting their structure. The mathematical relationship between the two methods is discussed both, theoretically and numerically. The effectiveness of the described methods is demonstrated using a challenging three-dimensional port-Hamiltonian model of a classical guitar with fluid-structure interaction.  \nKeywords — structure-preserving model order reduction, port-Hamiltonian system, a posteriori error control, fluid-structure interaction  \n1 Introduction  \nThe development of modern products and the understanding of complex processes is only possible using modeling and simulation tools. In order to further optimize the products and get a detailed understanding of the processes, it is typically required to describe real examples over multiple scalesand consider multiple physical domains. In an increasingly digitalized world, complex devices can considerably benefit from a digital twin throughout their life cycle, e.g. through increased efficiencyand possible improvements through insights into data. The digital twin should be able to switch to the respective required accuracy class through a hierarchical model description. From precise descriptions for process understanding to detailed descriptions for repeated simulations to coarse models for real-time control and optimization [1] . The energy-based port-Hamiltonian (pH) framework offers ideal prerequisites for meeting the requirements of a modern simulation-based product lifecycle. The methodology allows systems to be built up modularly and to be coupled mathematically across different scales and physical domains. Through structure-preserving model reduction  \n∗ Institute of Engineering and Computational Mechanics, University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germany. (johannes.rettberg,[joerg.fehr@itm.uni-stuttgart.de](joerg.fehr@itm.uni-stuttgart.de) )  \n†Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany. (dominik.wittwar,patrick.buchfink,robin.herkert,haasdonk@mathematik . [uni-stuttgart.de](uni-stuttgart.de))  \nand flexibility in time and space discretization of the entire system or subsystems, the desire for a hierarchical model is fulfilled. Besides, under mild assumptions, these systems also satisfy helpful system-theoretical properties such as passivity and stability [1 , 2] .  \nThe usual modeling process involves the spatial discretization of the continua and the partial differential equations using, for example, finite element methods that yield a high-dimensional system of ordinary differential or differential-algebraic equations. These high-fidelity models are computationally demanding, and one way of abstracting these models in a smaller subspace is projection-based model order reduction (MOR) . In order to preserve the worthwhile properties of pH systems in the coarse models, this model reduction must be done in a structure-preserving manner [3] .  \nBy approximating the high-fidelity model in a low-dimensional subspace, MOR introduces an err","cbCaiu33kvOUUA3n","https://ap.wps.com/l/cbCaiu33kvOUUA3n","pdf",882205,18,"English","# Introduction\n## Digital twin and hierarchical modeling\n## Projection-based model order reduction and error control\n## Error bounds: standard, hierarchical, and ALP\n## Contributions and outlook","[{\"question\":\"What problem does the paper address in projection-based model order reduction?\",\"answer\":\"Projection-based MOR introduces an unknown reduction error between the high-fidelity model and its reduced version, which needs reliable a posteriori control.\"},{\"question\":\"Which improved error bounding techniques are presented?\",\"answer\":\"The paper applies and adapts a hierarchical error bound and an auxiliary linear problem (ALP) based error bound for port-Hamiltonian systems.\"},{\"question\":\"How are the methods validated?\",\"answer\":\"Their effectiveness is demonstrated using a challenging three-dimensional port-Hamiltonian model of a classical guitar with fluid-structure interaction.\"}]","Improved a posteriori Error Bounds for Reduced port-Hamiltonian Systems | PDF",45]