3
Approximation, sampling, and compression in high dimensional problems
http://sms.cam.ac.uk/collection/3007735
In a number of problems, both in theory and applications, one faces a situation when the ambient dimension is extremely high. Such problems often include approximating, sampling, or compressing functions on highdimensional domains. Classical methods fail to be effective in this case due to the effect known as `curse of dimensionality'; hence new tools and algorithms need to be devised. Compressed sensing, which has gained great popularity in this century, is one example of a circle of ideas which make highdimensional problems feasible. Methods which allow one to overcome the curse of dimensionality come from a mixture of mathematical fields: approximation, probability, functional and harmonic analysis, linear algebra, combinatorics, geometry, etc. In addition to pure mathematical interest, this field has great importance in numerous applications, in particular in data science and signal processing. Despite decades of research, many important questions in this area are still open. This workshop will bring together researchers in pure and applied mathematics, who attack highdimensional problems.
1440
2019
Fri, 21 Jun 2019 15:38:25 +0100
Tue, 18 Jun 2019 08:28:17 +0100
en
smssupport@ucs.cam.ac.uk
Approximation, sampling, and compression in high dimensional problems
http://sms.cam.ac.uk/collection/3007735
http://rss.sms.cam.ac.uk/images/cam/identifier2.png
http://video.search.yahoo.com/mrss
Approximation, sampling, and compression in high dimensional problems
In a number of problems, both in theory and applications, one faces a situation when the ambient dimension is extremely high. Such problems often include approximating, sampling, or compressing functions on highdimensional domains. Classical methods fail to be effective in this case due to the effect known as `curse of dimensionality'; hence new tools and algorithms need to be devised. Compressed sensing, which has gained great popularity in this century, is one example of a circle of ideas which make highdimensional problems feasible. Methods which allow one to overcome the curse of dimensionality come from a mixture of mathematical fields: approximation, probability, functional and harmonic analysis, linear algebra, combinatorics, geometry, etc. In addition to pure mathematical interest, this field has great importance in numerous applications, in particular in data science and signal processing. Despite decades of research, many important questions in this area are still open. This workshop will bring together researchers in pure and applied mathematics, who attack highdimensional problems.
Approximation, sampling, and compression in high dimensional problems
In a number of problems, both in theory and applications, one faces a situation when the ambient dimension is extremely high. Such problems often include approximating, sampling, or compressing functions on highdimensional domains. Classical methods fail to be effective in this case due to the effect known as `curse of dimensionality'; hence new tools and algorithms need to be devised. Compressed sensing, which has gained great popularity in this century, is one example of a circle of ideas which make highdimensional problems feasible. Methods which allow one to overcome the curse of dimensionality come from a mixture of mathematical fields: approximation, probability, functional and harmonic analysis, linear algebra, combinatorics, geometry, etc. In addition to pure mathematical interest, this field has great importance in numerous applications, in particular in data science and signal processing. Despite decades of research, many important questions in this area are still open. This workshop will bring together researchers in pure and applied mathematics, who attack highdimensional problems.
Cambridge University
K. Kher
kk693@cam.ac.uk
http://sms.cam.ac.uk/collection/3007735
Approximation, sampling, and compression in high dimensional problems
20190618T08:28:17+01:00
INIMS
002281
no

Beating the Curse of Dimensionality: A Theoretical Analysis of Deep Neural Networks and Parametric PDEs
ucs_sms_3007735_3009550
http://sms.cam.ac.uk/media/3009550
Beating the Curse of Dimensionality: A Theoretical Analysis of Deep Neural Networks and Parametric PDEs
Kutyniok, G
Thursday 20th June 2019  14:20 to 15:10
Fri, 21 Jun 2019 08:37:22 +0100
Isaac Newton Institute
Kutyniok, G
9e3cc6be02d1b2d406b82f6774a3d2f3
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Kutyniok, G
Thursday 20th June 2019  14:20 to 15:10
Kutyniok, G
Thursday 20th June 2019  14:20 to 15:10
Cambridge University
3334
http://sms.cam.ac.uk/media/3009550
Beating the Curse of Dimensionality: A Theoretical Analysis of Deep Neural Networks and Parametric PDEs
Kutyniok, G
Thursday 20th June 2019  14:20 to 15:10
Highdimensional parametric partial differential equations (PDEs) appear in various contexts including control and optimization problems, inverse problems, risk assessment, and uncertainty quantification. In most such scenarios the set of all admissible solutions associated with the parameter space is inherently low dimensional. This fact forms the foundation for the socalled reduced basis method. Recently, numerical experiments demonstrated the remarkable efficiency of using deep neural networks to solve parametric problems. In this talk, we will present a theoretical justification for this class of approaches. More precisely, we will derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric PDEs. In fact, without any knowledge of its concrete shape, we use the inherent lowdimensionality of the solution manifold to obtain approximation rates which are significantly superior to those provided by classical approximation results. We use this lowdimensionality to guarantee the existence of a reduced basis. Then, for a large variety of parametric PDEs, we construct neural networks that yield approximations of the parametric maps not suffering from a curse of dimensionality and essentially only depending on the size of the reduced basis. This is joint work with Philipp Petersen (Oxford), Mones Raslan, and Reinhold Schneider.
20190621T08:37:22+01:00
3334
3009550
true
16x9
false
no

On some theorems on the restriction of operator to coordinate subspace
ucs_sms_3007735_3009543
http://sms.cam.ac.uk/media/3009543
On some theorems on the restriction of operator to coordinate subspace
Kashin, B
20th June 2019  11:10 to 12:00
Fri, 21 Jun 2019 08:34:48 +0100
Isaac Newton Institute
Kashin, B
d12e2004c5e754e539739b939f19e8a0
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Kashin, B
20th June 2019  11:10 to 12:00
Kashin, B
20th June 2019  11:10 to 12:00
Cambridge University
2830
http://sms.cam.ac.uk/media/3009543
On some theorems on the restriction of operator to coordinate subspace
Kashin, B
20th June 2019  11:10 to 12:00
20190621T08:34:48+01:00
2830
3009543
true
16x9
false
no

A sequence of wellconditioned polynomials
ucs_sms_3007735_3008334
http://sms.cam.ac.uk/media/3008334
A sequence of wellconditioned polynomials
OrtegaCerdà, J
Tuesday 18th June 2019  13:30 to 14:20
Wed, 19 Jun 2019 09:00:50 +0100
Isaac Newton Institute
OrtegaCerdà, J
47bddf694332e59dea1a428a0fb3ce2d
da56e7864bc0ab97ebfcad9345352d5f
411f79c24bedb142e5814f016fcd1aee
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OrtegaCerdà, J
Tuesday 18th June 2019  13:30 to 14:20
OrtegaCerdà, J
Tuesday 18th June 2019  13:30 to 14:20
Cambridge University
2634
http://sms.cam.ac.uk/media/3008334
A sequence of wellconditioned polynomials
OrtegaCerdà, J
Tuesday 18th June 2019  13:30 to 14:20
We find an explicit sequence of polynomials of arbitrary degree with small condition number. This solves a problem posed by Michael Shub and Stephen Smale in 1993. This is joint work together with Carlos Beltran, Ujué Etayo and Jordi Marzo.
20190619T09:00:50+01:00
2634
3008334
true
16x9
false
no

Basis properties of the Haar system in various function spaces
ucs_sms_3007735_3008896
http://sms.cam.ac.uk/media/3008896
Basis properties of the Haar system in various function spaces
Seeger, A
Wednesday 19th June 2019  09:00 to 09:50
Thu, 20 Jun 2019 08:37:22 +0100
Isaac Newton Institute
Seeger, A
7c948ca26c02fb00bd2f31f8db19c6cc
c572cab331187d487ab97ce3b3a65f40
d49c72524974837439b06cfd89b25896
c0730ea4f23b09124056da39c12994cf
Seeger, A
Wednesday 19th June 2019  09:00 to 09:50
Seeger, A
Wednesday 19th June 2019  09:00 to 09:50
Cambridge University
2996
http://sms.cam.ac.uk/media/3008896
Basis properties of the Haar system in various function spaces
Seeger, A
Wednesday 19th June 2019  09:00 to 09:50
We present recent results on the Haar system in Besov and TriebelLizorkin spaces, with an emphasis on endpoint results. Joint work with Gustavo Garrigós and Tino Ullrich.
20190620T08:37:22+01:00
2996
3008896
true
16x9
false
no

Discrete translates in function spaces
ucs_sms_3007735_3008308
http://sms.cam.ac.uk/media/3008308
Discrete translates in function spaces
Olevskii, A
Tuesday 18th June 2019  09:50 to 10:40
Wed, 19 Jun 2019 08:59:25 +0100
Isaac Newton Institute
Olevskii, A
8111c3669929f9165c230173043164b0
91da4f1dbb8e2ee99fa00350878dee94
5a954d0043c90a99b03bc43062e5834b
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Olevskii, A
Tuesday 18th June 2019  09:50 to 10:40
Olevskii, A
Tuesday 18th June 2019  09:50 to 10:40
Cambridge University
2527
http://sms.cam.ac.uk/media/3008308
Discrete translates in function spaces
Olevskii, A
Tuesday 18th June 2019  09:50 to 10:40
Given a Banach function space on R^n, does there exist a uniformly discrete set of translates of a single function, which spans the space? I'll present a survey on the problem and discuss recent results, joint with A.Ulanovskii.
20190619T08:59:26+01:00
2527
3008308
true
16x9
false
no

Dynamical sampling and frames generated from powers of exponential operators
ucs_sms_3007735_3007755
http://sms.cam.ac.uk/media/3007755
Dynamical sampling and frames generated from powers of exponential operators
Aldroubi, A
Monday 17th June 2019  09:50 to 10:40
Tue, 18 Jun 2019 09:22:55 +0100
Isaac Newton Institute
Aldroubi, A
6a5cbd466ac98e828a2767ea0298b337
713c9a10e9e7fbb104f3381853c4493c
e637377762996e77ec050f87458cc65e
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Aldroubi, A
Monday 17th June 2019  09:50 to 10:40
Aldroubi, A
Monday 17th June 2019  09:50 to 10:40
Cambridge University
2885
http://sms.cam.ac.uk/media/3007755
Dynamical sampling and frames generated from powers of exponential operators
Aldroubi, A
Monday 17th June 2019  09:50 to 10:40
In this talk, I will give a brief review of the problem of frame generation from operator powers of exponentials acting on a set of vectors. I will discuss its relation to dynamical sampling, review some of the previous results and present several new ones.
20190618T09:22:55+01:00
2885
3007755
true
16x9
false
no

High Dimensional Approximation via Sparse Occupancy Trees
ucs_sms_3007735_3007769
http://sms.cam.ac.uk/media/3007769
High Dimensional Approximation via Sparse Occupancy Trees
Binev, P
Monday 17th June 2019  14:20 to 15:10
Tue, 18 Jun 2019 09:22:32 +0100
Isaac Newton Institute
Binev, P
e4cc530566e40226a5247a8486633e06
945aea7ab94037b335881164978e7b51
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Binev, P
Monday 17th June 2019  14:20 to 15:10
Binev, P
Monday 17th June 2019  14:20 to 15:10
Cambridge University
2547
http://sms.cam.ac.uk/media/3007769
High Dimensional Approximation via Sparse Occupancy Trees
Binev, P
Monday 17th June 2019  14:20 to 15:10
Adaptive domain decomposition is often used in finite elements methods for solving partial differential equations in low space dimensions. The adaptive decisions are usually described by a tree. Assuming that can find the (approximate) error for approximating a function on each element of the partition, we have shown that a particular coarsetofine method provides a nearbest approximation. This result can be extended to approximating point clouds any space dimension provided that we have relevant information about the errors and can organize properly the data. Of course, this is subject to the curse of dimensionality and nothing can be done in the general case. In case the intrinsic dimensionality of the data is much smaller than the space dimension, one can define algorithms that defy the curse. This is usually done by assuming that the data domain is close to a low dimensional manifold and first approximating this manifold and then the function defined by it. A few years ago, together with Philipp Lamby, Wolfgang Dahmen, and Ron DeVore, we proposed a direct method (without specifically identifying any low dimensional set) that we called "sparse occupancy trees". The method defines a piecewise constant or linear approximation on general simplicial partitions. This talk considers an extension of this method to find a similar approximation on conforming simplicial partitions following an idea from a recent result together with Francesca Fierro and Andreas Veeser about nearbest approximation on conforming triangulations.
20190618T09:22:33+01:00
2547
3007769
true
16x9
false
no

Integral norm discretization and related problems
ucs_sms_3007735_3009687
http://sms.cam.ac.uk/media/3009687
Integral norm discretization and related problems
Dai, F
Friday 21st June 2019  11:10 to 12:00
Fri, 21 Jun 2019 13:17:05 +0100
Isaac Newton Institute
Dai, F
7c4b9b981069e3eb87996aaef4f01779
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1a9de4c28baf3832e7a659e9f23cbadf
39ba84344ed40779af6582e4fc2d7178
Dai, F
Friday 21st June 2019  11:10 to 12:00
Dai, F
Friday 21st June 2019  11:10 to 12:00
Cambridge University
2753
http://sms.cam.ac.uk/media/3009687
Integral norm discretization and related problems
Dai, F
Friday 21st June 2019  11:10 to 12:00
In this talk, we will discuss the problem of replacing an integral norm with respect to a given probability measure by the corresponding integral norm with respect to a discrete measure. We study the problem for elements of finite dimensional spaces in a general setting, paying a special attention to the case of the multivariate trigonometric polynomials with frequencies from a finite set with fixed cardinality. Both new results and a survey of known results will be presented. This is a joint work with A. Prymak, V.N. Temlyakov and S. Tikhonov.
20190621T13:17:05+01:00
2753
3009687
true
16x9
false
no

Linear and onebit compressive sensing with subsampled random convolutions
ucs_sms_3007735_3008341
http://sms.cam.ac.uk/media/3008341
Linear and onebit compressive sensing with subsampled random convolutions
Rauhut, H
Tuesday 18th June 2019  14:20 to 15:10
Wed, 19 Jun 2019 09:02:22 +0100
Isaac Newton Institute
Rauhut, H
044a615eefc605d1ffc59abe72a3c6f0
aa8ce70122af3396a6fc66706bfb1fdc
c712fd55ba11efb117b4a350f6037e80
0a312fcedea1dfb9e67c389c1e540f58
Rauhut, H
Tuesday 18th June 2019  14:20 to 15:10
Rauhut, H
Tuesday 18th June 2019  14:20 to 15:10
Cambridge University
3054
http://sms.cam.ac.uk/media/3008341
Linear and onebit compressive sensing with subsampled random convolutions
Rauhut, H
Tuesday 18th June 2019  14:20 to 15:10
Compressive sensing predicts that sparse vectors can recovered from incomplete linear measurements with efficient algorithms in a stable way. While many theoretical results work with Gaussian random measurement matrices, practical applications usually demand for structure. The talk covers the particular case of structured random measurements defined via convolution with a random vector and subsampling (deterministic or random as well). We will give an overview on the corresponding theory and will cover also recent results concerning recovery from onebit measurements arising in quantized compressive sensing. Based on joint works with Felix Krahmer, Shahar Mendelson, Sjoerd Dirksen and HansChristian Jung.
20190619T09:02:22+01:00
3054
3008341
true
16x9
false
no

Markovtype inequalities and extreme zeros of orthogonal polynomials
ucs_sms_3007735_3009694
http://sms.cam.ac.uk/media/3009694
Markovtype inequalities and extreme zeros of orthogonal polynomials
Nikolov, G
Friday 21st June 2019  14:20 to 15:10
Fri, 21 Jun 2019 15:38:24 +0100
Isaac Newton Institute
Nikolov, G
cba2f86be81839819bad5f569bb09070
9aec5559d640b67fd4b3dfaae9a53950
47af5eaae0127763479e2a58a6b06bd1
9ac2222fe794295c45c3d482fee0b3e9
Nikolov, G
Friday 21st June 2019  14:20 to 15:10
Nikolov, G
Friday 21st June 2019  14:20 to 15:10
Cambridge University
3018
http://sms.cam.ac.uk/media/3009694
Markovtype inequalities and extreme zeros of orthogonal polynomials
Nikolov, G
Friday 21st June 2019  14:20 to 15:10
The talk is centered around the problem of finding (obtaining tight twosided bounds for) the sharp constants in certain MarkovBernstein type inequalities in weighted L2 norms. It turns out that, under certain assumptions, this problem is equivalent to the estimation of the extreme zeros of orthogonal polynomials with respect to a measure supported on R+. It will be shown how classical tools like the EulerRayleigh method and Gershgorin circle theorem produce surprisingly good bounds for the extreme zeros of the Jacobi, Gegenbauer and Laguerre polynomials. The sharp constants in the L2 Markov inequalities with the Laguerre and Gegenbauer weight functions and in a discrete ℓ2 MarkovBernstein inequality are investigated using the same tool.
20190621T15:38:25+01:00
3018
3009694
true
16x9
false
no

QuasiMonte Carlo integration in uncertainty quantification of elliptic PDEs with logGaussian coefficients
ucs_sms_3007735_3008348
http://sms.cam.ac.uk/media/3008348
QuasiMonte Carlo integration in uncertainty quantification of elliptic PDEs with logGaussian coefficients
Herrmann, L
Tuesday 18th June 2019  15:40 to 16:30
Wed, 19 Jun 2019 09:02:57 +0100
Isaac Newton Institute
Herrmann, L
98e5796b2a64b2543996f9aff695b987
112cbe669df4cfb46ffeb2f81cfa7328
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Herrmann, L
Tuesday 18th June 2019  15:40 to 16:30
Herrmann, L
Tuesday 18th June 2019  15:40 to 16:30
Cambridge University
3208
http://sms.cam.ac.uk/media/3008348
QuasiMonte Carlo integration in uncertainty quantification of elliptic PDEs with logGaussian coefficients
Herrmann, L
Tuesday 18th June 2019  15:40 to 16:30
QuasiMonte Carlo (QMC) rules are suitable to overcome the curse of dimension in the numerical integration of highdimensional integrands. Also the convergence rate of essentially first order is superior to Monte Carlo sampling. We study a class of integrands that arise as solutions of elliptic PDEs with logGaussian coefficients. In particular, we focus on the overall computational cost of the algorithm. We prove that certain multilevel QMC rules have a consistent accuracy and computational cost that is essentially of optimal order in terms of the degrees of freedom of the spatial Finite Element discretization for a range of infinitedimensional priors. This is joint work with Christoph Schwab. References: [L. Herrmann, Ch. Schwab: QMC integration for lognormalparametric, elliptic PDEs: local supports and product weights, Numer. Math. 141(1) pp. 63102, 2019], [L. Herrmann, Ch. Schwab: Multilevel quasiMonte Carlo integration with product weights for elliptic PDEs with lognormal coefficients, to appear in ESAIM:M2AN], [L. Herrmann: Strong convergence analysis of iterative solvers for random operator equations, SAM report, 201735, in review]
20190619T09:02:57+01:00
3208
3008348
true
16x9
false
no

Representer theorems and convex optimization
ucs_sms_3007735_3007776
http://sms.cam.ac.uk/media/3007776
Representer theorems and convex optimization
Boyer, C
Monday 17th June 2019  15:40 to 16:30
Tue, 18 Jun 2019 09:23:19 +0100
Isaac Newton Institute
Boyer, C
fd0a6ea7be2761fbc834da45930a80fb
09c522a8edd9bc2102c651ad2e49c467
3b444786d4ba33b1aaff86675770dc67
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Boyer, C
Monday 17th June 2019  15:40 to 16:30
Boyer, C
Monday 17th June 2019  15:40 to 16:30
Cambridge University
2648
http://sms.cam.ac.uk/media/3007776
Representer theorems and convex optimization
Boyer, C
Monday 17th June 2019  15:40 to 16:30
We establish a general principle which states that regularizing an inverse problem with a convex function yields solutions which are convex combinations of a small number of atoms. These atoms are identified with the extreme points and elements of the extreme rays of the regularizer level sets. As a side result, we characterize the minimizers of the total gradient variation. As an ongoing work, we will also study the geometry of the total gradient variation ball. This is a joint work with Antonin Chambolle, Yohann De Castro, Vincent Duval, Frédéric de Gournay, and Pierre Weiss.
20190618T09:23:19+01:00
2648
3007776
true
16x9
false
no

Totally positive functions in sampling theory and timefrequency analysis
ucs_sms_3007735_3009680
http://sms.cam.ac.uk/media/3009680
Totally positive functions in sampling theory and timefrequency analysis
Groechenig, K
Friday 21st June 2019  09:50 to 10:40
Fri, 21 Jun 2019 13:15:59 +0100
Isaac Newton Institute
Groechenig, K
8336c90d00795fceb784a67ebe8c9e37
07163c8ea4080dd6c377c17d3a92f6d3
f0f7c5ef2e6428a399769345a8308a37
bc1fced198eb98639c6c9801a94d827d
Groechenig, K
Friday 21st June 2019  09:50 to 10:40
Groechenig, K
Friday 21st June 2019  09:50 to 10:40
Cambridge University
2985
http://sms.cam.ac.uk/media/3009680
Totally positive functions in sampling theory and timefrequency analysis
Groechenig, K
Friday 21st June 2019  09:50 to 10:40
Totally positive functions play an important role in approximation theory and statistics. In this talk I will present recent new applications of totally positive functions (TPFs) in sampling theory and timefrequency analysis. (i) We study the sampling problem for shiftinvariant spaces generated by a TPF. These spaces arise the span of the integer shifts of a TPF and are often used as a substitute for bandlimited functions. We give a complete characterization of sampling sets for a shiftinvariant space with a TPF generator of Gaussian type in the style of Beurling. (ii) A related problem is the question of Gabor frames, i.e., the spanning properties of timefrequency shifts of a given function. It is conjectured that the lattice shifts of a TPF generate a frame, if and only if the density of the lattice exceeds 1. At this time this conjecture has been proved for two important subclasses of TPFs. For rational lattices it is true for arbitrary TPFs. So far, TPFs seem to be the only window functions for which the fine structure of the associated Gabor frames is tractable. (iii) Yet another question in timefrequency analysis is the existence of zeros of the Wigner distribution (or the radar ambiguity function). So far all examples of zerofree ambiguity functions are related to TPFs, e.g., the ambiguity function of the Gaussian is zero free.
20190621T13:15:59+01:00
2985
3009680
true
16x9
false
no

Transportation cost spaces on finite metric spaces
ucs_sms_3007735_3007762
http://sms.cam.ac.uk/media/3007762
Transportation cost spaces on finite metric spaces
Kutzarova, D
Monday 17th June 2019  11:10 to 12:00
Tue, 18 Jun 2019 09:23:33 +0100
Isaac Newton Institute
Kutzarova, D
89ea24aa98c4929bcfc67a7b49f9fc5d
782d62fb872530dcdafe60936b710498
d0f09ce7de5b49adc6f3a2a7852d8d58
2b23fbf8df471939177fa46c271eead8
Kutzarova, D
Monday 17th June 2019  11:10 to 12:00
Kutzarova, D
Monday 17th June 2019  11:10 to 12:00
Cambridge University
2962
http://sms.cam.ac.uk/media/3007762
Transportation cost spaces on finite metric spaces
Kutzarova, D
Monday 17th June 2019  11:10 to 12:00
Transportation cost spaces are studied by several groups of researchers, for different reasons and under different names. The term Lipschitzfree spaces is commonly used in Banach space theory. We prove that the transportation cost space on any finite metric space contains a large wellcomplemented subspace which is close to ℓn1. We show that transportation cost spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to ℓn1 of the corresponding dimensions. These classes contain wellknown families of diamond graphs and Laakso graphs. In the particular case of diamond graphs we prove that their cycle space is spanned by even levels of Haar functions. It is curious that the subspaces generated by all the even/odd levels of the Haar functions also appear in the study of quasigreedy basic sequences in L1[0,1]. This research is joint with Stephen Dilworth and Mikhail Ostrovskii.
20190618T09:23:33+01:00
2962
3007762
true
16x9
false
no
3007735