LOBPCG swMATH ID: 9638 Software Authors: Kressner, Daniel; Tobler, Christine Description: Preconditioned low-rank methods for high-dimensional elliptic PDE eigenvalue problems. We consider elliptic PDE eigenvalue problems on a tensorized domain, discretized such that the resulting matrix eigenvalue problem Ax=λx exhibits Kronecker product structure. In particular, we are concerned with the case of high dimensions, where standard approaches to the solution of matrix eigenvalue problems fail due to the exponentially growing degrees of freedom. Recent work shows that this curse of dimensionality can in many cases be addressed by approximating the desired solution vector x in a low-rank tensor format. In this paper, we use the hierarchical Tucker decomposition to develop a low-rank variant of LOBPCG, a classical preconditioned eigenvalue solver. We also show how the ALS and MALS (DMRG) methods known from computational quantum physics can be adapted to the hierarchical Tucker decomposition. Finally, a combination of ALS and MALS with LOBPCG and with our low-rank variant is proposed. A number of numerical experiments indicate that such combinations represent the methods of choice. Homepage: http://www.degruyter.com/view/j/cmam.2011.11.issue-3/cmam-2011-0020/cmam-2011-0020.xml Keywords: high-dimensional PDE eigenvalue problems; low-rank tensor methods; LOBPCG; ALS; DMRG Related Software: htucker; TT Toolbox; GitHub; lobpcg.m; MCTDH; ALEA; JDQZ; cross2D; Algorithm 862; RTRMC; ALPS; JDQR; N-way Toolbox; JDCG; CheMPS2; TeNPy; TensorNetwork; ITensor; AlexNet; SqueezeDet Cited in: 34 Documents all top 5 Cited by 49 Authors 5 Bachmayr, Markus 5 Kressner, Daniel 4 Kazeev, Vladimir A. 4 Uschmajew, André 3 Dahmen, Wolfgang A. 3 Schwab, Christoph 3 Tobler, Christine 3 Vandereycken, Bart 2 Hackbusch, Wolfgang 2 Khoromskiĭ, Boris Nikolaevich 2 Oseledets, Ivan Valer’evich 2 Rakhuba, Maxim V. 2 Rohwedder, Thorsten 2 Schneider, Reinhold 1 Beckermann, Bernhard 1 Benner, Peter 1 Breiten, Tobias 1 Cichocki, Andrzej 1 Deng, Lei 1 Dolgov, Sergey V. 1 Elman, Howard C. 1 Etter, Simon 1 Götte, Michael 1 Grasedyck, Lars 1 Huang, Ruihao 1 Klinvex, Alicia 1 Knyazev, Andrew V. 1 Lee, Namgil 1 Li, Guoqi 1 Litvinenko, Alexander 1 Lubich, Christian 1 Mach, Thomas 1 Matthies, Hermann Georg 1 Mu, Lin 1 Pfeffer, Max 1 Reichmann, Oleg 1 Saied, Faisal 1 Sameh, Ahmed H. 1 Savostyanov, Dmitry V. 1 Shi, Tianyi 1 Steinlechner, Michael 1 Su, Tengfei 1 Townsend, Alex 1 Wang, Dingheng 1 Wen, Zaiwen 1 Wu, Bijiao 1 Zhang, Junyu 1 Zhang, Yin 1 Zhao, Guangshe all top 5 Cited in 18 Serials 5 SIAM Journal on Matrix Analysis and Applications 5 SIAM Journal on Scientific Computing 4 Linear Algebra and its Applications 3 SIAM Journal on Numerical Analysis 2 Numerische Mathematik 2 Foundations of Computational Mathematics 1 Computers & Mathematics with Applications 1 Journal of Computational Physics 1 Applied Mathematics and Computation 1 Calcolo 1 Journal of Scientific Computing 1 Neural Networks 1 Computing and Visualization in Science 1 Computational Methods in Applied Mathematics 1 Acta Numerica 1 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 1 GAMM-Mitteilungen 1 SIAM/ASA Journal on Uncertainty Quantification all top 5 Cited in 14 Fields 31 Numerical analysis (65-XX) 18 Linear and multilinear algebra; matrix theory (15-XX) 4 Partial differential equations (35-XX) 4 Approximations and expansions (41-XX) 2 Probability theory and stochastic processes (60-XX) 2 Operations research, mathematical programming (90-XX) 1 Operator theory (47-XX) 1 Differential geometry (53-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Computer science (68-XX) 1 Fluid mechanics (76-XX) 1 Quantum theory (81-XX) 1 Statistical mechanics, structure of matter (82-XX) 1 Biology and other natural sciences (92-XX) Citations by Year