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Numerical Analysis Seminar: Numerical Analysis of High-Dimensional Problems

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401-3650-14L / 401-3650-00S

Dozenten Prof. Christoph Schwab, Mr. Vladimir Kazeev
Vorbesprechung 15:15 20.02.2014, HG F 26.3
Kontakt Prof. Christoph Schwab, Mr. Vladimir Kazeev
Programm
22.05.2014 (Thursday) IFW C 31

13:15–14:45. Oded Stein. Hierarchical Tensor Representation (HTR)
14:45–16:15. Syang Zhou. The tensor-structured solution of linear systems and optimization: ALS and DMRG-type algorithms
See the building on a map (there are an entrance to RZ from Clausiusstrasse and a passageway from RZ to IFW), the room on a floor plan.

23.05.2014 (Friday) ML F 40

09:00–10:30. Tanja Almeroth. Alternating minimal energy methods for the tensor-structured solution of linear systems
10:30–12:00. Mathilde Gianolli. The convolution of multi-dimensional vectors in the QTT format
See the building on a map (there is an entrance from Sonneggstrasse), the room on a floor plan.

26.05.2014 (Monday) HG G 19.2

13:15–14:45. Timo Welti. The tensor-structured solution of the chemical master equation
14:45–16:15. Robert Gantner. The tensor-structured solution of multi-parametric linear systems arising from stochastic elliptic PDEs

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), Quantized 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 2014. 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 6. The preference is given to ETH students of the following programs:

  1. ETH MSc Applied Math,
  2. ETH MSc RW/CSE,
  3. ETH BSc MATH,
  4. ETH MSc Math.

The seminar will start with a presentation and discussion of possible topics at 15:15 on 20.02.2014 in HG F 26.3.

The seminar's page in the course catalogue: http://www.vvz.ethz.ch/Vorlesungsverzeichnis/lerneinheitPre.do?semkez=2014S&lang=en&ansicht=LEHRVERANSTALTUNGEN&lerneinheitId=90948.

Literatur
  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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://epubs.siam.org/doi/abs/10.1137/090757861.
  6. 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.
  7. 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.
  8. 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/.
  9. L. Grasedyck. Hierarchical Singular Value Decomposition of Tensors // SIAM Journal on Matrix Analysis and Applications. 2010. V. 31, No. 4. P. 2029–2054. http://link.aip.org/link/?SML/31/2029/1.
  10. 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.
  11. J. Ballani, L. Grasedyck. A Projection Method to Solve Linear Systems in Tensor Format// Numerical Linear Algebra with Applications, 2012. V. 20, No. 1. P. 27–43. http://dx.doi.org/10.1002/nla.1818.
  12. S. Holtz, T. Rohwedder, R. Schneider. The alternating linear scheme for tensor optimisation in the TT format // SIAM Journal on Scientific Computing, 2012. V. 34, No. 2. P. A683–A713. http://epubs.siam.org/doi/abs/10.1137/100818893.
  13. S. V. Dolgov, I. V. Oseledets. Solution of linear systems and matrix inversion in the TT-format // SIAM Journal on Scientific Computing, 2012. V. 34, No. 5. P. A2718–A2739. http://epubs.siam.org/doi/abs/10.1137/110833142.
  14. D. Kressner, C. Tobler. Preconditioned low-rank methods for high-dimensional elliptic PDE eigenvalue problems // Computational Methods in Applied Mathematics, 2011. V.11, No. 3. P.363–381. http://www.degruyter.com/view/j/cmam.2011.11.issue-3/cmam-2011-0020/cmam-2011-0020.xml.
  15. A. Uschmajew. Local convergence of the Alternating Least Squares algorithm for canonical tensor approximation // SIAM Journal on Matrix Analysis and Applications, 2012. V. 33, No. 2. P. 639–652. http://epubs.siam.org/doi/abs/10.1137/110843587.
  16. M. Espig, W. Hackbusch, S. Handschuh, R. Schneider. Optimization problems in contracted tensor networks // Computing and Visualization in Science, 2011. V. 14, No. 6. P. 271–285. http://dx.doi.org/10.1007/s00791-012-0183-y.
  17. I. V. Oseledets. Constructive representation of functions in tensor formats // Constructive Approximation, 2013. V. 37, No. 1. P. 1–18. http://link.springer.com/article/10.1007/s00365-012-9175-x.
  18. I. V. Oseledets. DMRG approach to fast linear algebra in the TT-format // Computational Methods in Applied Mathematics, 2011. V. 11, No. 3. P. 382–393. http://www.degruyter.com/view/j/cmam.2011.11.issue-3/cmam-2011-0021/cmam-2011-0021.xml.
  19. 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.degruyter.com/view/j/cmam.2011.11.issue-3/cmam-2011-0022/cmam-2011-0022.xml.
  20. I. V. Oseledets, E. E. Tyrtyshnikov. Algebraic wavelet transform via Quantics Tensor Train decomposition // SIAM Journal on Scientific Computing, 2011. V. 33, No. 3. P. 1315–1328. http://epubs.siam.org/doi/abs/10.1137/100811647.
  21. S. Dolgov, B. N. Khoromskij, D. Savostyanov. Superfast Fourier transform using QTT approximation // Journal of Fourier Analysis and Applications, 2012. V. 18, No. 5. P. 915–953. http://dx.doi.org/10.1007/s00041-012-9227-4.
  22. V. A. Kazeev, B. N. Khoromskij, E. E. Tyrtyshnikov. Multilevel Toeplitz matrices generated by tensor-structured vectors and convolution with logarithmic complexity // SIAM Journal on Scientific Computing, 2013. V. 35, No. 3. P. A1511–A1536. http://epubs.siam.org/doi/abs/10.1137/110844830.
  23. S. Holtz, T. Rohwedder, R. Schneider. On manifolds of tensors of fixed TT-rank // Numerische Mathematik, 2012. V. 120, No. 4. P. 701–731. http://www.springerlink.com/content/v24733342nhh2742.
  24. 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.degruyter.com/view/j/cmam.2010.10.issue-4/cmam-2010-0023/cmam-2010-0023.xml.
  25. 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.
  26. 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.
  27. B. N. Khoromskij, Ch. Schwab. Tensor-structured Galerkin approximation of parametric and stochastic elliptic PDEs // SIAM Journal on Scientific Computing, 2011. V. 33, No. 1. http://epubs.siam.org/sisc/resource/1/sjoce3/v33/i1/p364_s1.
  28. V. Kazeev, O. Reichmann, Ch. Schwab. hp-DG-QTT solution of high-dimensional degenerate diffusion equations // Seminar for Applied Mathematics, ETH Zürich, 2012. Technical report No. 11. http://www.sam.math.ethz.ch/reports/2012/11.
  29. V. Kazeev, O. Reichmann, Ch. Schwab. Low-rank tensor structure of linear diffusion operators in the TT and QTT formats // Linear Algebra and its Applications. V. 438, No. 11. P. 4204–4221. http://www.sciencedirect.com/science/article/pii/S002437951300058X.
  30. I. P. Gavrilyuk, B. N. Khoromskij. Quantized-TT-Cayley transform to compute dynamics and spectrum of high-dimensional Hamiltonians // Computational Methods in Applied Mathematics, 2011. V. 11, No. 3. P. 273–290. http://www.degruyter.com/view/j/cmam.2011.11.issue-3/cmam-2011-0015/cmam-2011-0015.xml.
  31. Ch. Lubich, I. Oseledets. A projector-splitting integrator for dynamical low-rank approximation. arXiv preprint 1301.1058, 2013. http://arxiv.org/abs/1301.1058.
  32. A. Uschmajew, B. Vandereycken. The geometry of algorithms using hierarchical tensors // Linear Algebra and its Applications. 2013. V. 439, No. 1. P. 133–166. DOI: http://dx.doi.org/10.1016/j.laa.2013.03.016. http://www.sciencedirect.com/science/article/pii/S0024379513002115.
  33. S. Dolgov, B. Khoromskij, I. Oseledets. Fast Solution of Parabolic Problems in the Tensor Train/Quantized Tensor Train Format with
    Initial Application to the Fokker–Planck Equation // SIAM Journal on Scientific Computing, 2012. V. 34, No. 6. P. A3016–A3038. DOI: 10.1137/120864210. http://epubs.siam.org/doi/abs/10.1137/120864210.
  34. D. Savostyanov. QTT-rank-one vectors with QTT-rank-one and full-rank Fourier images // Linear Algebra and its Applications, 2012. V. 436, No. 9. P. 3215–3224. DOI: 10.1016/j.laa.2011.11.008. http://www.sciencedirect.com/science/article/pii/S002437951100749X.
  35. S. V. Dolgov, B. N. Khoromskij. Tensor-product approach to global time-space-parametric discretization of chemical master equation // Preprint 68: Max-Planck-Institut für Mathematik in den Naturwissenschaften, 2012, November. http://www.mis.mpg.de/publications/preprints/2012/prepr2012-68.html.
  36. V. Kazeev, M. Khammash, M. Nip, C. Schwab. Direct solution of the Chemical Master Equation using Quantized Tensor Trains: Research Report 4: Seminar for Applied Mathematics, ETH Zürich, 2013. http://www.sam.math.ethz.ch/reports/2013/04.
  37. V. Kazeev, C. Schwab. Tensor approximation of stationary distributions of chemical reaction networks // Seminar for Applied Mathematics, ETH Zürich, 2013. Research Report 18. http://www.sam.math.ethz.ch/reports/2013/18.
  38. V. Kazeev, I. Oseledets. The tensor structure of a class of adaptive algebraic wavelet transforms // Seminar for Applied Mathematics, ETH Zürich, 2013. Research Report 28. http://www.sam.math.ethz.ch/reports/2013/28.
  39. D. V. Savostyanov. Quasioptimality of maximum-volume cross interpolation of tensors // arXiv preprint 1305.1818, 2013. http://arxiv.org/abs/1305.1818.
  40. S. V. Dolgov, D. V. Savostyanov. Alternating minimal energy methods for linear systems in higher dimensions. Part I: SPD systems // arXiv preprint 1301.6068, 2013. http://arxiv.org/abs/1301.6068.
  41. S. V. Dolgov, D. V. Savostyanov. Alternating minimal energy methods for linear systems in higher dimensions. Part II: Faster algorithm and application to nonsymmetric systems // arXiv preprint 1304.1222, 2013. http://arxiv.org/abs/1304.1222.
  42. D. Kressner, M. Steinlechner, B. Vandereycken. Low-rank tensor completion by Riemannian optimization // MATHICSE EPFL, 2013. Technical report 20. http://mathicse.epfl.ch/files/content/sites/mathicse/files/Mathicse reports 2013/20.2013_DK-MS-BV.pdf.
  43. J. Ballani, L. Grasedyck. Tree Adaptive Approximation in the Hierarchical Tensor Format // Institut für Geometrie und Praktische Mathematik, RWTH Aachen, 2013. Preprint 367. http://www.igpm.rwth-aachen.de/Download/reports/pdf/IGPM367_k.pdf.
  44. L. Grasedyck, M. Kluge, S. Krämer. Alternating Directions Fitting (ADF) of Hierarchical Low Rank Tensors // Preprint Series DFG-SPP 1324, 2013. Report 149. http://www.dfg-spp1324.de/download/preprints/preprint149.pdf.
  45. P.-A. Absil, I. Oseledets. Low-rank retractions: a survey and new results // Université catholique de Louvain, 2013. Technical report 2013.04. http://sites.uclouvain.be/absil/2013.04.
  46. R. Schneider, A. Uschmajew. Approximation rates for the hierarchical tensor format in periodic Sobolev spaces // MATHICSE EPFL, 2013. Technical report 6. http://mathicse.epfl.ch/ files/content/sites/mathicse/files/Mathicse reports 2013/06.2013_RS-AU-New.pdf.
  47. L. Grasedyck, D. Kressner, C. Tobler. A literature survey of low rank tensor approximation techniques // arXiv preprint 1302.7121, 2013, February. http://arxiv.org/abs/1302.7121.
 

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