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I have writ­ten a very short and incom­plete intro­duc­tory page into skele­ton decom­po­si­tion and using the maxvol algo­rithm. I will update it with ref­er­ences and stuff.

http://oseledets.github.io/news/cross-approximation-intro/

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Read here about our new paper:

http://oseledets.github.io/news/convolution-via-cross-approximation/

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We have recently a launched a web­site of Sci­en­tific Group at Skoltech. I will now post updates mostly there, so if you are inter­ested, please fol­low :)

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Wavelets and tensors

August 23rd, 2013
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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

  1. WTT trans­form matrix with fil­ter ranks has rank r^2 + 1
  2. 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.
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Skoltech

August 23rd, 2013
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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.

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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

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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.

 

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Happy new year!

January 9th, 2013
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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.

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One more paper

December 22nd, 2012
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One more old paper on explicit rep­re­sen­ta­tions of sim­ple func­tions in ten­sor for­mats is pub­lished in the Con­struc­tive Approximation!

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Our paper on the new time-stepping scheme based on the QTT-format has been pub­lished in SISC!

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