ETH :: D-MATH :: Numerical Analysis Seminar: Adaptive Low-Rank Discretizations of High-Dimensional PDEs

15:00 -- 16:30. Simon Laumer. Polynomial Approximation in Hierarchical Tucker Format by Vector-Tensorization
16:30 -- 18:00. Jacob Steiner. Alternating Least Squares methods for the tensor-structured optimization in high dimensions

25th May, Friday. Room G 26.3

13:00 -- 14:30. Andrea Riva. Explicit representation of tensors in the TT and QTT formats
14:30 -- 16:00. Severin Thaler. Quantized-TT-Cayley transform for computing the dynamics and the spectrum of high-dimensional Hamiltonians
16:00 -- 17:30. Thomas Michaels. Tensor Train decomposition in mathematics and physics

Beginnt am

27.02.2012

Vorbesprechung

30.01.2012, 15:00-16:00, HG G57.1

Kontakt

Prof. Christoph Schwab, Mr. Vladimir Kazeev

Voraussetzungen

The prerequisites are:

for students taking the seminar for ETH BSc MATH: completed BSc examinations in Numerische Mathematik I+II;

for students taking the seminar for ETH MSc Math, Applied Math, RW/CSE: completed exam in courses Numerical solution of elliptic and parabolic PDEs, OR NumPDEs for RW/CSE, Numerical solution of stochastic PDEs.

Beschreibung

In recent years, new fundamental developments have taken place in the field of numerical multilinear algebra. Driven mainly by the need for efficient numerical processing of data in high-dimensional parameter spaces, adaptive rank reduction and low-rank techniques have been developed to the point where they can be used also for numerical solution of stochastic Partial Differential Equations (PDEs) and PDEs on high dimensional state- and parameter spaces. Such problems arise in a host of applications (e. g. quantum chemisty, electron structure computations, computational biology and computational finance).

The seminar will be concerned with newly introduced low-rank tensor representations, such as the Tensor Train (TT), Quantics Tensor Train (QTT) and Hierarchical Tensor (HT) formats. During the semester each participant is supposed to prepare a two-hours lecture, which is to be given in May 2012. It should be based on at least two of the recent research papers, a preliminary list of which is given below. Depending on the student's preferences and study program, the focus may be made on the fundamentals, implementation aspects or application of these formats. The participants are encouraged to use the adaptive low-rank tensor packages, such as TT Toolbox and Hierarchical Tucker Toolbox, which have recently become available as MATLAB implementations.

The number of participants of the seminar is limited to 5. The preference is given to ETH students of the following programs:

ETH MSc Applied Math,
ETH MSc RW/CSE,
ETH BSc MATH,
ETH MSc Math.

The first meeting for interested students will take place at 15:00 on the 30th January 2012 at HG G 57.1.

AS OF THE 31^ST JANUARY 2012, THE SEMINAR IS FULL. THE REGISTRATION IS CLOSED.

The seminar will start with a preliminary discussion in the second week of FS 2012, the tentative date is Tuesday, 27th February 2012. To attend the preliminary discussion it is necessary to sign up for the seminar.

The seminar's page in the course catalogue: http://vvz.ethz.ch/Vorlesungsverzeichnis/lerneinheitPre.do?lerneinheitId=77779&semkez=2012S&lang=de.

Literatur

I. V. Oseledets, E. E. Tyrtyshnikov. Breaking the curse of dimensionality, or how to use SVD in many dimensions // SIAM Journal on Scientific Computing. 2009, October. V. 31, No. 5. P. 3744–3759. DOI: 10.1137/090748330. http://epubs.siam.org/sisc/resource/1/sjoce3/v31/i5/p3744_s1.
I. V. Oseledets. Tensor Train decomposition // SIAM Journal on Scientific Computing. 2011. V. 33, No. 5. P. 2295–2317. DOI: 10.1137/090752286. http://dx.doi.org/10.1137/090752286.
I. V. Oseledets, E. E. Tyrtyshnikov. TT-cross approximation for multidimensional arrays // Linear Algebra and its Applications. 2010, January. V. 432, No. 1. P. 70–88. DOI: 10.1016/j.laa.2009.07.024. http://www.sciencedirect.com/science/article/pii/S0024379509003747.
D. Savostyanov, I. Oseledets. Fast adaptive interpolation of multi-dimensional arrays in Tensor Train format // Multidimensional (nD) Systems (nDs), 2011 7th International Workshop on. 2011, sept. — P. 1–8. http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6076873.
I. V. Oseledets. Approximation of 2d × 2d matrices using tensor decomposition // SIAM Journal on Matrix Analysis and Applications. 2010. V. 31, No. 4. P. 2130–2145. DOI: 10.1137/090757861. http://link.aip.org/link/?SML/31/2130/1.
B. N. Khoromskij. Tensors-structured numerical methods in scientific computing: Survey on recent advances // Chemometrics and Intelligent Laboratory Systems. 2011. DOI: 10.1016/j.chemolab.2011.09.001. http://www.sciencedirect.com/science/article/pii/S0169743911001808.
B. Khoromskij. O (d log N )-Quantics Approximation of N-d Tensors in High-Dimensional Numerical Modeling // Constructive Approximation. 2011. P. 1-24. DOI: 10.1007/s00365-011-9131-1. http://dx.doi.org/10.1007/s00365-011-9131-1.
W. Hackbusch, S. Kühn. A New Scheme for the Tensor Representation // Journal of Fourier Analysis and Applications. 2009. V. 15. P. 706–722. DOI: 10.1007/s00041-009-9094-9, 10.1007/s00041-009-9094-9. http://www.springerlink.com/content/t3747nk47m368g44/.
L. Grasedyck. Hierarchical Singular Value Decomposition of Tensors // Preprint Series DFG-SPP 1324. 2009, July. No. 20. P. 1–25. http://www.dfg-spp1324.de/download/preprints/preprint020.pdf.
L. Grasedyck. Polynomial Approximation in Hierarchical Tucker Format by Vector-Tensorization // Preprint 308: Institut für Geometrie und Praktische Mathematik, RWTH Aachen, 2010, April. http://www.igpm.rwth-aachen.de/Download/reports/pdf/IGPM308_k.pdf.
J. Ballani, L. Grasedyck. A Projection Method to Solve Linear Systems in Tensor Format// Preprint 22: Max-Planck-Institut für Mathematik in den Naturwissenschaften, 2010. http://www.mis.mpg.de/publications/preprints/2010/prepr2010-22.html.
S. Holtz, T. Rohwedder, R. Schneider. The alternating linear scheme for tensor optimisation in the TT format // Preprint 71: DFG Research Center MATHEON, 2010, December. http://www.dfg-spp1324.de/download/preprints/preprint071.pdf.
S. V. Dolgov, I. V. Oseledets. Solution of linear systems and matrix inversion in the TT-format // Preprint 19: Max-Planck-Institut für Mathematik in den Naturwissenschaften, 2011. http://www.mis.mpg.de/publications/preprints/2011/prepr2011-19.html.
D. Kressner, C. Tobler. Preconditioned low-rank methods for high-dimensional elliptic PDE eigenvalue problems: Research Report 48: Seminar for Applied Mathematics, ETHZ, 2011. http://www.sam.math.ethz.ch/reports/2011/48.
A. Uschmajew. Local convergence of the Alternating Least Squares algorithm for canonical tensor approximation // Preprint 103: DFG-Schwerpunktprogramm 1324, 2011, August. http://www.dfg-spp1324.de/download/preprints/preprint103.pdf.
M. Espig, W. Hackbusch, S. Handschuh, R. Schneider. Optimization problems in contracted tensor networks // Preprint 66: Max-Planck-Institut für Mathematik in den Naturwissenschaften, 2011. http://www.mis.mpg.de/publications/preprints/2011/prepr2011-66.html.
I. V. Oseledets. Constructive representation of functions in tensor formats // Preprint 4: Institute of Numerical Mathematics of RAS, 2010, August. http://pub.inm.ras.ru/pub/inmras2010-04.pdf.
I. V. Oseledets. DMRG approach to fast linear algebra in the TT-format // Preprint: Hausdorff Research Institute for Mathematics, 2011, July. http://131.220.77.52/files/preprints/AnalysisandNumerics/mvk.pdf.
I. Oseledets, E. Tyrtyshnikov, N. Zamarashkin. Tensor-Train ranks for matrices and their inverses // Computational Methods in Applied Mathematics. 2011. V. 11, No. 3. P. 394–403. http://www.cmam.info/index.php?do=issues/art&vol=11&num=3&art=325.
I. V. Oseledets, E. E. Tyrtyshnikov. Algebraic wavelet transform via Quantics Tensor Train decomposition: Preprint 3: Institute of Numerical Mathematics of RAS, 2010, August. http://pub.inm.ras.ru/pub/inmras2010-03.pdf.
S. Dolgov, B. N. Khoromskij, D. Savostyanov. Multidimensional Fourier transform in logarithmic complexity using QTT approximation // Preprint 18: Max-Planck-Institut für Mathematik in den Naturwissenschaften, 2011. http://www.mis.mpg.de/publications/preprints/2011/prepr2011-18.html.
V. A. Kazeev, B. N. Khoromskij, E. E. Tyrtyshnikov. Multilevel Toeplitz matrices generated by tensor-structured vectors and convolution with logarithmic complexity (submitted to SISC) // Preprint 36: Max-Planck-Institut für Mathematik in den Naturwissenschaften, 2011. http://www.mis.mpg.de/publications/preprints/2011/prepr2011-36.html.
S. Holtz, T. Rohwedder, R. Schneider. On manifolds of tensors of fixed TT-rank: Preprint 61: DFG Research Center MATHEON, 2010, September. http://www.matheon.de/research/show_preprint.asp?action=details&serial=749.
B. N. Khoromskij, I. V. Oseledets. Quantics-TT collocation approximation of parameter-dependent and stochastic elliptic PDEs // Computational Methods In Applied Mathematics. 2010. V. 10, No. 4. P. 345–365. http://www.cmam.info/index.php?do=issues/art&vol=10&num=4&art=260.
B. N. Khoromskij, I. V. Oseledets. QTT+DMRG approach to high-dimensional quantum molecular dynamics: Preprint 69: Max-Planck-Institut für Mathematik in den Naturwissenschaften, 2010. http://www.mis.mpg.de/publications/preprints/2010/prepr2010-69.html.
B. N. Khoromskij, I. V. Oseledets. Quantics-TT Approximation of Elliptic Solution Operators in Higher Dimensions // Russian Journal of Numerical Analysis and Mathematical Modelling. 2011, June. V. 26, No. 3. P. 303–322. DOI: 10.1515/RJNAMM.2011.017. http://www.degruyter.com/view/j/rnam.2011.26.issue-3/rjnamm.2011.017/rjnamm.2011.017.xml.
B. N. Khoromskij, C. Schwab. Tensor-structured Galerkin approximation of parametric and stochastic elliptic PDEs // Preprint 9: Max-Planck-Institut für Mathematik in den Naturwissenschaften, 2010. http://www.mis.mpg.de/publications/preprints/2010/prepr2010-9.html.

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