{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,3,21]],"date-time":"2024-03-21T06:15:55Z","timestamp":1711001755527},"reference-count":32,"publisher":"University of Zielona G\u00f3ra, Poland","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2009,9,1]]},"abstract":"A Novel Interval Arithmetic Approach for Solving Differential-Algebraic Equations with ValEncIA-IVP<\/jats:title>The theoretical background and the implementation of a new interval arithmetic approach for solving sets of differential-algebraic equations (DAEs) are presented. The proposed approach computes guaranteed enclosures of all reachable states of dynamical systems described by sets of DAEs with uncertainties in both initial conditions and system parameters. The algorithm is based on ValEncIA-IVP, which has been developed recently for the computation of verified enclosures of the solution sets of initial value problems for ordinary differential equations. For the application to DAEs, ValEncIA-IVP has been extended by an interval Newton technique to solve nonlinear algebraic equations in a guaranteed way. In addition to verified simulation of initial value problems for DAE systems, the developed approach is applicable to the verified solution of the so-called inverse control problems. In this case, guaranteed enclosures for valid input signals of dynamical systems are determined such that their corresponding outputs are consistent with prescribed time-dependent functions. Simulation results demonstrating the potential of ValEncIA-IVP for solving DAEs in technical applications conclude this paper. The selected application scenarios point out relations to other existing verified simulation techniques for dynamical systems as well as directions for future research.<\/jats:p>","DOI":"10.2478\/v10006-009-0032-4","type":"journal-article","created":{"date-parts":[[2009,9,29]],"date-time":"2009-09-29T10:04:23Z","timestamp":1254218663000},"page":"381-397","source":"Crossref","is-referenced-by-count":22,"title":["A Novel Interval Arithmetic Approach for Solving Differential-Algebraic Equations with ValEncIA-IVP"],"prefix":"10.61822","volume":"19","author":[{"given":"Andreas","family":"Rauh","sequence":"first","affiliation":[]},{"given":"Michael","family":"Brill","sequence":"additional","affiliation":[]},{"given":"Clemens","family":"G\u00fcnther","sequence":"additional","affiliation":[]}],"member":"37438","reference":[{"key":"1","first-page":"1","article-title":"Validated modeling of mechanical systems with SmartMOBILE: Improvement of Performance by ValEncIA-IVP","volume":"5045","author":"E. Auer","year":"2008"},{"key":"2","unstructured":"Bendsten, C. and Stauning, O. (2007). FADBAD++, Version 2.1<\/i>, Available at: http:\/\/www.fadbad.com<\/a>"},{"key":"3","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1023\/A:1024467732637","article-title":"Verified integration of ODEs and flows using differential algebraic methods on highorder Taylor models","volume":"4","author":"M. Berz","year":"1998","journal-title":"Reliable Computing"},{"key":"4","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1145\/146847.146922","article-title":"An MEBDF code for stiff initial value problems","volume":"2","author":"J. Cash","year":"1992","journal-title":"ACM Transactions on Mathematical Software (TOMS)"},{"key":"5","author":"L. Chua","year":"1990","journal-title":"Linear and Nonlinear Circuits"},{"key":"6","first-page":"2607","article-title":"Testing algorithms for inverse simulation","author":"P. Czechowski","year":"2006"},{"key":"7","unstructured":"de Swart, J. J. B., Lioen, W. M. and van der Veen, W. A. (1998). Specification of PSIDE, Technical Report MAS-R9833<\/i>, CWI, Amsterdam, Available at: http:\/\/walter.lioen.com\/papers\/SLV98.pdf<\/a>"},{"key":"8","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1023\/A:1020573518783","article-title":"Consistency techniques for ordinary differential equations","volume":"3-4","author":"Y. Deville","year":"2002","journal-title":"Constraint"},{"key":"9","author":"P. Eijgenraam","year":"1981","journal-title":"The solution of initial value problems using interval arithmetic"},{"key":"10","unstructured":"Galassi, M. (2006). GNU Scientific Library Reference Manual. Revised Second Edition (v1.8), Available at: http:\/\/www.gnu.org\/software\/gsl\/<\/a>"},{"key":"11","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0093947","volume":"1409","author":"E. Hairer","year":"1989","journal-title":"The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods"},{"key":"12","author":"E. Hairer","year":"1991","journal-title":"Solving Ordinary Differential Equations II\u2014Stiff and Differential-Algebraic Problems"},{"key":"13","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-009-5819-7","author":"J. Hammersley","year":"1964","journal-title":"Monte-Carlo Methods"},{"key":"14","unstructured":"Hoefkens, J. (2001). Rigorous Numerical Analysis with High-Order Taylor Models<\/i>, Ph.D. thesis, Michigan State University, East Lansing, MI, Available at: http:\/\/www.bt.pa.msu.edu\/cgi-bin\/display.pl?name=hoefkensphd<\/a>"},{"key":"15","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/S0168-9274(98)00039-7","article-title":"Solving ordinary differential equations by generalized Adams methods: Properties and implementation techniques","volume":"2","author":"F. Iavernaro","year":"1998","journal-title":"Applied Numerical Mathematics"},{"key":"16","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4471-0249-6","author":"L. Jaulin","year":"2001","journal-title":"Applied Interval Analysis"},{"key":"17","unstructured":"Keil, C. (2007). Profil\/BIAS, Version 2.0.4<\/i>, Available at: www.ti3.tu-harburg.de\/keil\/profil\/<\/a>"},{"key":"18","first-page":"189","article-title":"Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken","volume":"3","author":"R. Krawczyk","year":"1969","journal-title":"Computing"},{"key":"19","unstructured":"Kunkel, P., Mehrmann, V., Rath, W. and Weickert, J. (1997). GELDA: A Software Package for the Solution of General Linear Differential Algebraic Equations<\/i>, pp. 115-138, Available at: http:\/\/www.math.tu-berlin.de\/numerik\/mt\/NumMat\/Software\/GELDA\/<\/a>"},{"key":"20","article-title":"Deterministic global optimization for dynamic systems using interval analysis","author":"Y. Lin","year":"2007"},{"key":"21","author":"R. Moore","year":"1966","journal-title":"Interval Arithmetic"},{"key":"22","article-title":"Interval tools for ODEs and DAEs","author":"N. Nedialkov","year":"2007"},{"key":"23","unstructured":"Nedialkov, N. S. and Pryce, J. D. (2008). DAETS\u2014Differential- Algebraic Equations by Taylor Series<\/i>, Available at: http:\/\/www.cas.mcmaster.ca\/~nedialk\/daets\/<\/a>"},{"key":"24","first-page":"65","article-title":"A description of DASSL: A differential\/algebraic systems solver","volume":"1","author":"L. Petzold","year":"1982","journal-title":"IMACS Transactions on Scientific Computation"},{"key":"25","unstructured":"Rauh, A. (2008). Theorie und Anwendung von Intervall-methoden f\u00fcr Analyse und Entwurf robuster und optimaler Regelungen dynamischer Systeme<\/i>, Fortschritt-Berichte VDI, Reihe 8, Nr. 1148, Ph.D. thesis, University of Ulm, Ulm, (in German)."},{"key":"26","article-title":"Verified simulation of ODEs and DAEs in ValEncIA-IVP","author":"A. Rauh","year":"2008"},{"key":"27","article-title":"Detection and reduction of overestimation in guaranteed simulations of hamiltonian systems","author":"A. Rauh","year":"2008"},{"key":"28","article-title":"ValEncIA-IVP: A comparison with other initial value problem solvers","author":"A. Rauh","year":"2007a"},{"key":"29","first-page":"1023001","article-title":"Extensions of ValEncIA-IVP for reduction of overestimation, for simulation of differential algebraic systems, and for dynamical optimization","volume":"7","author":"A. Rauh","year":"2007b"},{"key":"30","first-page":"397","article-title":"Interval methods for optimal control","author":"A. Rauh","year":"2009"},{"key":"31","first-page":"175","article-title":"Towards the development of an interval arithmetic environment for validated computer-aided design and verification of systems in control engineering","volume":"5492","author":"A. Rauh","year":"2009"},{"key":"32","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1007\/3-540-45802-6_26","volume":"281","author":"K. R\u00f6benack","year":"2002","journal-title":"Nonlinear and Adaptive Control"}],"container-title":["International Journal of Applied Mathematics and Computer Science"],"original-title":[],"link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/amcs\/19\/3\/article-p381.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/view\/j\/amcs.2009.19.issue-3\/v10006-009-0032-4\/v10006-009-0032-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,29]],"date-time":"2024-02-29T10:27:40Z","timestamp":1709202460000},"score":1,"resource":{"primary":{"URL":"https:\/\/content.sciendo.com\/doi\/10.2478\/v10006-009-0032-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,9,1]]},"references-count":32,"journal-issue":{"issue":"3"},"URL":"https:\/\/doi.org\/10.2478\/v10006-009-0032-4","relation":{},"ISSN":["1641-876X"],"issn-type":[{"value":"1641-876X","type":"print"}],"subject":[],"published":{"date-parts":[[2009,9,1]]}}}