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Weighted Tensor Product Algorithms for Linear Multivariate Problems

G.W Wasilkowski, H Woźniakowski
1999 Journal of Complexity  
We study the "-approximation of linear multivariate problems de ned over weighted tensor product Hilbert spaces of functions f of d variables.  ...  Relatively less is known for weighted tensor product problems, see, e.g., 10, 17].  ...  Weighted Tensor Product Algorithms In this section, we de ne a class of weighted tensor product algorithms (or WTP algorithms, for short) for approximation of the weighted multivariate solution operator  ... 
doi:10.1006/jcom.1999.0512 fatcat:6mr6awdpnfbt5b5m6egbug5244

Tractability of approximating multivariate linear functionals

Erich Novak, Henryk Woźniakowski
2010 Journal of Fixed Point Theory and Applications  
We review selected tractability results for approximating linear tensor product functionals defined over reproducing kernel Hilbert spaces.  ...  This review is based on Volume II of our book [5] Tractability of Multivariate Problems.  ...  Proof: The weighted tensor product (WTP) algorithm was introduced for product weights in [9] as a generalization of the Smolyak/sparse grid algorithm, see [8] .  ... 
doi:10.1007/s11784-010-0018-8 fatcat:xpuw3roy4zdyfkiqo3yswiqvjy

Quasi-polynomial tractability

Michael Gnewuch, Henryk Woźniakowski
2011 Journal of Complexity  
Unfortunately, many multivariate problems are not polynomially tractable. This holds for all non-trivial unweighted linear tensor product problems.  ...  It seems natural to ask what is the "smallest" non-exponential tractability of unweighted linear tensor product problems; that is, when the cost of a multivariate problem can be bounded by a multiple of  ...  Acknowledgments We thank Anargyros Papageorgiou for valuable comments on our paper. The first author acknowledges support from the German Science Foundation DFG under Grant GN 91/4-1.  ... 
doi:10.1016/j.jco.2010.07.001 fatcat:c2hnvc6dzbdtrdac7lwdlfzoyi

Finite-order weights imply tractability of linear multivariate problems

G.W. Wasilkowski, H. Woźniakowski
2004 Journal of Approximation Theory  
We prove that finite-order weights imply strong tractability or tractability of linear multivariate problems, depending on a certain condition on the reproducing kernel of the space.  ...  We study the minimal number n(ε, d) of information evaluations needed to compute a worst case ε-approximation of a linear multivariate problem.  ...  Acknowledgements We are grateful for comments from Erich Novak and Arthur G. Werschulz.  ... 
doi:10.1016/j.jat.2004.06.011 fatcat:xvngizohyjdfpgcbyhrkv5ebti

Page 5746 of Mathematical Reviews Vol. , Issue 2000h [page]

2000 Mathematical Reviews  
W. (1-K Y-C; Lexington, KY); Wozniakowski, H. (1-CLMB-C; New York, NY) Weighted tensor product algorithms for linear multivariate problems.  ...  The authors study the solution (in the sense of finding the worst- case €-approximation) of general multivariate linear problems de- fined in weighted tensor product Hilbert spaces of functions of d variables  ... 

Explicit error bounds for randomized Smolyak algorithms and an application to infinite-dimensional integration [article]

Michael Gnewuch, Marcin Wnuk
2019 arXiv   pre-print
Smolyak's method, also known as hyperbolic cross approximation or sparse grid method, is a powerful tool to tackle multivariate tensor product problems solely with the help of efficient algorithms for  ...  Randomized Smolyak algorithms can be used as building blocks for efficient methods such as multilevel algorithms, multivariate decomposition methods or dimension-wise quadrature methods to tackle successfully  ...  Acknowledgment The authors thank Stefan Heinrich for pointing out the reference [34] .  ... 
arXiv:1903.02276v1 fatcat:w3hgnnpeuvaadctqjtzosfc25a

Modeling Bidirectional Texture Functions with Multivariate Spherical Radial Basis Functions

Yu-Ting Tsai, Kuei-Li Fang, Wen-Chieh Lin, Zen-Chung Shih
2011 IEEE Transactions on Pattern Analysis and Machine Intelligence  
Finally, a hierarchical fitting algorithm for bidirectional texture functions is developed to exploit spatial coherence and reduce computational cost.  ...  First, since the surface appearance of a real-world object is frequently a mixed effect of different physical factors, the proposed sum-of-products model based on multivariate SRBFs especially provides  ...  ACKNOWLEDGMENTS The authors would like to thank the anonymous reviewers for profound comments and suggestions, Dr. Xin Tong for providing BTF data, and Jia-Yin Ji for preparing the model Cloth.  ... 
doi:10.1109/tpami.2010.211 pmid:21135438 fatcat:gy27xmnzbbgyld6alt4g4ifnee

Weighted tensor decomposition for approximate decoupling of multivariate polynomials [article]

Gabriel Hollander, Philippe Dreesen, Mariya Ishteva, Johan Schoukens
2016 arXiv   pre-print
For this, techniques based on tensor methods are known, but these have only been studied in the exact case.  ...  Finally, we apply the proposed weighted decoupling algorithm in the domain of system identification, and observe smaller model errors.  ...  , we summarize the proposed weighted cpd algorithm and incorporate it into the larger decoupling problem of the noisy multivariate polynomial function f .  ... 
arXiv:1601.07800v1 fatcat:cyelouf5vfebdnuqicy7hdav4e

Tractability of multivariate problems for standard and linear information in the worst case setting: Part I

Erich Novak, Henryk Woźniakowski
2016 Journal of Approximation Theory  
Furthermore, some of these linear functionals have the same norm as the linear operators. We then apply this error bound for linear (unweighted) tensor products.  ...  We present a lower error bound for approximating linear multivariate operators defined over Hilbert spaces in terms of the error bounds for appropriately constructed linear functionals as long as algorithms  ...  We thank Mario Ullrich who computed some numbers for us. We also thank Greg Wasilkowski, Markus Weimar and two referees for valuable comments.  ... 
doi:10.1016/j.jat.2016.02.017 fatcat:umtkcunfirddzbt5q2k655fxam

Nonlinear system identification with regularized Tensor Network B-splines

Ridvan Karagoz, Kim Batselier
2020 Automatica  
Algorithms for optimization in the tensor network format make it possible to fit multivariate B-spline surfaces onto high-dimensional data by directly finding a lowrank tensor network approximation of  ...  large weight tensor.  ...  In the multivariate case, the differences in adjacent weights in the weight tensor W have to be penalized along each dimension individually.  ... 
doi:10.1016/j.automatica.2020.109300 fatcat:kblqj7uyyrbmpcgjne53zpgjoa

Tractability of Multivariate Problems for Standard and Linear Information in the Worst Case Setting: Part I [article]

Erich Novak, Henryk Wozniakowski
2015 arXiv   pre-print
Furthermore, some of these linear functionals have the same norm as the linear operators. We then apply this error bound for linear (unweighted) tensor products.  ...  We present a lower error bound for approximating linear multivariate operators defined over Hilbert spaces in terms of the error bounds for appropriately constructed linear functionals as long as algorithms  ...  We thank Mario Ullrich who computed some numbers for us. We also thank Greg Wasilkowski, Markus Weimar and two referees for valuable comments.  ... 
arXiv:1511.05803v1 fatcat:6bled2gg4fec3lkiu7n65f2vmm

Lower bounds for the complexity of linear functionals in the randomized setting

Erich Novak, Henryk Woźniakowski
2011 Journal of Complexity  
Hinrichs [3] recently studied multivariate integration defined over reproducing kernel Hilbert spaces in the randomized setting and for the normalized error criterion.  ...  More specifically, let n ran (ε, INT d ) be the minimal number of randomized function samples that is needed to compute an ε-approximation for the d-variate case of multivariate integration.  ...  Acknowledgments We are grateful for valuable comments of A. Hinrichs, A. Papageorgiou, J. F. Traub and two anonymous referees.  ... 
doi:10.1016/j.jco.2010.08.002 fatcat:uaoinntnvjchro2r6wmnb2vq6a

Decoupling multivariate functions using a non-parametric Filtered CPD approach

Jan Decuyper, Koen Tiels, Siep Weiland, Johan Schoukens
2021 IFAC-PapersOnLine  
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User  ...  Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of  ...  Given the diagonal form, the collection of Jacobians may be written as a sum of r outer products (or rank-one tensors).  ... 
doi:10.1016/j.ifacol.2021.08.401 fatcat:ew3l3ctu2ze7pgrmmt2qi2i2se

Open problems for tractability of multivariate integration

Henryk Woźniakowski
2003 Journal of Complexity  
The second is a weighted tensor product Sobolev space for which necessary and sufficient conditions for tractability of multivariate integration are known.  ...  We end this note by presenting an open problem for the tractability of multivariate integration in the randomized setting. r  ...  Worst case: weighted spaces We now discuss multivariate integration for a weighted tensor product Sobolev space F d : This is a Hilbert space of functions defined on ½0; 1 d with the reproducing kernel  ... 
doi:10.1016/s0885-064x(03)00016-5 fatcat:cjdz3qhwtfbpjnldign63dbtmi

Nonlinear system identification with regularized Tensor Network B-splines [article]

Ridvan Karagoz, Kim Batselier
2020 arXiv   pre-print
large weight tensor.  ...  Tensor network theory is used to alleviate the curse of dimensionality of multivariate B-splines by representing the high-dimensional weight tensor as a low-rank approximation.  ...  In the multivariate case, the differences in adjacent weights in the weight tensor W have to be penalized along each dimension individually.  ... 
arXiv:2003.07594v1 fatcat:5eajakzpqvbt3ewwiv7uja2qia
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