Ngọc Trình Trần, T. N. Tran, H. Matthies, G. Stavroulakis, Manfred Staat
{"title":"SHAKEDOWN ANALYSIS OF PLATE BENDING UNDER STOCHASTIC UNCERTAINTY BY CHANCE CONSTRAINED PROGRAMMING","authors":"Ngọc Trình Trần, T. N. Tran, H. Matthies, G. Stavroulakis, Manfred Staat","doi":"10.7712/100016.2012.11106","DOIUrl":"https://doi.org/10.7712/100016.2012.11106","url":null,"abstract":"1 Aachen University of Applied Sciences, Jülich Campus, Institute for Bioengineering, Biomechanics Lab., Heinrich-Mußmann-Str. 1, 52428 Jülich, Germany, trinhdhkt@gmail.com, m.staat@fhaachen.de, http://www.fh-aachen.de/forschung/institut-fuer-bioengineering/ 2 University of Duisburg-Essen, Chair of Mechanics and Robotics, Lotharstr. 1, 47057 Duisburg, Germany, thanh.tran@uni-due.de 3 TU Braunschweig, Institute of Scientific Computing, Hans-Sommer-Str. 65, 38092 Braunschweig, Germany, wire@tu-bs.de 4 Technical University of Crete, 73100 Chania, Greece, gestavr@dpem.tuc.gr","PeriodicalId":223490,"journal":{"name":"Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121330149","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":"BLACK-BOX SOLVER FOR ONE-DIMENSIONAL MULTISCALE MODELLING USING THE QTT FORMAT","authors":"I. Oseledets, M. Rakhuba, A. Chertkov","doi":"10.7712/100016.2387.10906","DOIUrl":"https://doi.org/10.7712/100016.2387.10906","url":null,"abstract":"In the present work we propose an efficient black-box solver for one-dimensional multiple scaled diffusion equation. For this problem it has been recently shown [1] that the solution can be represented in a certain low parametric representation, namely the quantized tensor train (QTT) format [2]. The key idea of the QTT format is to make the real space data multidimensional by introducing virtual dimensionalities. The next step is to apply the tensor train (TT) representation [3] to multidimensional data, which leads us to the logarithmic complexity. Hence very fine grids that describe the finest scale can be used. Since the solution of second order multi-scale problems can be represented in the QTT format, simple and efficient solvers can be developed using the existing software for the approximate solution of linear systems in the TT-format. However, if equations are discretized using standard finite element/difference methods, it is not possible to get to very fine meshes, say with 2 grid points due to the condition number. On the other hand, the theory guarantees the existence of a good QTT-FEM approximant of the continuous problem. Thus, another discretization should be used to compute it numerically. Our idea is to rewrite the initial formulation in a certain form without derivatives. After that we get an explicit formula, which consists of the inversion of a diagonal matrix and the multiplication by a dense matrix. The latter can be multiplied with logarithmic complexity in the QTT format due to a special structure. The numerical experiment show that this formula gives accurate results and can be used for 2 grid points with no problems with conditioning, while total computational time is around several seconds. Ivan V. Oseledets, Maxim V. Rakhuba, Andrei V. Chertkov","PeriodicalId":223490,"journal":{"name":"Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)","volume":"82 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130421246","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}