Wavelet tensor train (WTT) is a promising tool. However, the idea came virtually unnoticed in the tensor community, and I am a little bit lazy in push the stuff forward (especially taking into the account other interesting things to do). However, the potential of WTT as a data-sparse format in data mining and compression and in the solution of PDEs is huge. With Vladimir Kazeev we just finished a paper “The tensor structure of a class of adaptive algebraicwavelet transforms” which reveals very interesting algebraic properties of the WTT transforms
- WTT transform matrix with filter ranks has rank r^2 + 1
- WTT transform with filter rank r applied to a TT-tensor of rank R gives a TT-tensor with rank r + R (which is very unusual if you think about it)
The tensor notation may not be very user-friendly; but the specific results are quite promising, to my opinion.
Recently I have joined Skolkovo Institute of Science and Technology. I do remain my position in INM RAS as a part time, and the new Scientific Computing group in Sktech will be developed within a close collaboration with the Institute of Numerical Mathematics. It is going to be an exciting journey. The draft of my Sktech homepage. The webpage for the Scientific Computing group will be available soon
By the way, there are OPEN POSITIONS (Postdocs and PhD, Phd are especially welcome). Feel free to contact me at i.oseledets (dog) skolkovotech.ru for any questions and details. The conditions are really competitive, and it is going to be fun in Moscow with a lot of opportunities.
The solution of the Poisson equation in low-rank formats is a classical topic (see the works by Khoromskij, Grasedyck, Hackbusch, Beylkin, Rokhlin). However, the methods do not always outperform standard ones for small dimensions (especially for d = 2 and d = 3) and medium ranks due to high complexity in the rank. For the simplest Poisson equation it is possible to decrease the complexity by using a textbook FFT method, combined with fast low-rank approximation in the frequency space based on the cross approximation, see Fast low-rank solution of the Poisson equation with application to the Stokes problem (joint with E. A. Muravleva). The robustness of the method is confirmed for the low-rank solver of the Stokes problem, which requires the solution of equations with the residual vectors in the Krylov method
In our recent paper
Computation of extreme eigenvalues in higher dimensions using block tensor train format (S. V. Dolgov, B. N. Khoromskij, I. V. Oseledets, D. V. Savostyanov) we have presented a new method for the computation of extereme eigenvalues using Block Rayleigh Quotient with smart “index shift” which allows to approximately reduce local problems to standard block eigenvalue problems. The method is linear in the dimension and quadratic in the mode size. In numerical experiments we confirm the effectiveness of the solver for the Henon-Heiles potential in many dimensions and for the Heisenberg Model in quantum spin systems. For the latter example, we have compared eigb to the ALPS solver and to the ITensor solver and showed that it is competitive.
The solver is implemented in Fortran (file tt_eigb.f90) and has interface in Python ( , module tt.eigb). The MATLAB interface is underway.
2013 is now in its full rights, so it is time to go on with research.
With Christian Lubich we have finished a paper on a very efficient time-stepping scheme for the dynamical low-rank approximation — so-called KLS-scheme, which is remarkably simple but efficient to compute the dynamics on low-rank manifolds. It presents a full analysis for the two-dimensional case, with multidimensional case (TT-format) in progress.
One more old paper on explicit representations of simple functions in tensor formats is published in the Constructive Approximation!
Our paper on the new time-stepping scheme based on the QTT-format has been published in SISC!
Four papers are in progress : on the block eigenvalue solver in the TT-format, on the dynamical low-rank approximation, on the tensor structure of the wavelet tensor train matrix and on the fast solution of the Stokes problem in tensor format. Hope to finish them soon.
Also, I put some time to get an implementation of the TT-Toolbox in Python. The preliminary version (ttpy 0.1) is available on the github. Please take a look on it, if you are interested.
We have recently put the publications of our research group at the Institute of Numerical Mathematics RAS on the web. The list is not yet full, but is close. Check it!
This paper with Dmitry Savostyanov published in the end of 2011 in the Proceedings of 7th International Workshop on Multidimensional Systems (nDS), doi: 10.1109/nDS.2011.6076873 is about fast adaptive methods for the approximation of high-dimensional arrays by cross-type methods (such methods are quite popular for matrices).
The method of TT-ranks adaptation is based on the DMRG-scheme, which is a “universal tool” for TT-methods. A prototype implementation (quite messy, but working) is available in the Github repository of the TT-Toolbox
. To make it work, you should install the TT-Toolbox