Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 1, pp. 341-363.

We consider the linear elliptic equation -div(au)=f on some bounded domain D, where a has the form a=exp(b) with b a random function defined as b(y)=j1yjψj where y=(yj) are i.i.d. standard scalar Gaussian variables and (ψj)j1 is a given sequence of functions in L(D). We study the summability properties of Hermite-type expansions of the solution map yu(y)V:=H01(D), that is, expansions of the form(D) , that is, expansions of the form u(y)=νFuνHν(y), where Hν(y)=j1Hνj(yj) are the tensorized Hermite polynomials indexed by the set H of finitely supported sequences of nonnegative integers. Previous results [V.H. Hoang and C. Schwab, M3AS 24 (2014) 797−826] have demonstrated that, for any 0<p1, the p summability of the sequence (jψjL)j1 implies p summability of the sequence (uνV)ν. Such results ensure convergence rates n-s with s=1p-12 of polynomial approximations obtained by best n-term truncation of Hermite series, where the error is measured in the mean-square sense, that is, in L2(N,V,γ), where γ is the infinite-dimensional Gaussian measure. In this paper we considerably improve these results by providing sufficient conditions for the p summability of (uνV)ν expressed in terms of the pointwise summability properties of the sequence (|ψj|)j1. This leads to a refined analysis which takes into account the amount of overlap between the supports of the ψj. For instance, in the case of disjoint supports, our results imply that, for all 0<p<2 the p summability of (uνV)νfollows from the weaker assumption that (ψjL)j1is q summable for q:=2p2-p>p. In the case of arbitrary supports, our results imply that the p summability of (uνV)ν follows from the p summability of (jβψjL)j1 for some >12 which ch still represents an improvement over the condition in [V.H. Hoang and C. Schwab, M3AS 24 (2014) 797−826]. We also explore intermediate cases of functions with local yet overlapping supports, such as wavelet bases. One interesting observation following from our analysis is that for certain relevant examples, the use of the Karhunen−Loève basis for the representation of might be suboptimal compared to other representations, in terms of the resulting summability properties of (uνV)ν. While we focus on the diffusion equation, our analysis applies to other type of linear PDEs with similar lognormal dependence in the coefficients.

DOI : 10.1051/m2an/2016051
Classification : 41A10, 41A58, 41A63, 65N15, 65T60
Mots-clés : Stochastic PDEs, lognormal coefficients, n-term approximation, Hermite polynomials
Bachmayr, Markus 1 ; Cohen, Albert 1 ; DeVore, Ronald 2 ; Migliorati, Giovanni 1

1 Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4 place Jussieu, 75005 Paris, France.
2 Department of Mathematics, Texas A&M University, College Station, TX 77840, USA.
@article{M2AN_2017__51_1_341_0,
     author = {Bachmayr, Markus and Cohen, Albert and DeVore, Ronald and Migliorati, Giovanni},
     title = {Sparse polynomial approximation of parametric elliptic {PDEs.} {Part} {II:} lognormal coefficients},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {341--363},
     publisher = {EDP-Sciences},
     volume = {51},
     number = {1},
     year = {2017},
     doi = {10.1051/m2an/2016051},
     mrnumber = {3601011},
     zbl = {1366.41005},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2016051/}
}
TY  - JOUR
AU  - Bachmayr, Markus
AU  - Cohen, Albert
AU  - DeVore, Ronald
AU  - Migliorati, Giovanni
TI  - Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2017
SP  - 341
EP  - 363
VL  - 51
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2016051/
DO  - 10.1051/m2an/2016051
LA  - en
ID  - M2AN_2017__51_1_341_0
ER  - 
%0 Journal Article
%A Bachmayr, Markus
%A Cohen, Albert
%A DeVore, Ronald
%A Migliorati, Giovanni
%T Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2017
%P 341-363
%V 51
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2016051/
%R 10.1051/m2an/2016051
%G en
%F M2AN_2017__51_1_341_0
Bachmayr, Markus; Cohen, Albert; DeVore, Ronald; Migliorati, Giovanni. Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 1, pp. 341-363. doi : 10.1051/m2an/2016051. http://www.numdam.org/articles/10.1051/m2an/2016051/

I. Babuška, F. Nobile and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45 (2007) 1005–1034. | DOI | MR | Zbl

M. Bachmayr, A. Cohen and G. Migliorati, Sparse polynomial approximation of parametric elliptic PDEs. Part I: Affine coefficients, ESAIM: M2AN 51 (2017) 321–339. | DOI | Numdam | MR | Zbl

J. Beck, F. Nobile, L. Tamellini and R. Tempone, On the optimal polynomial approximation of stochastic PDEs by Galerkin and collocation methods. Math. Models Methods Appl. Sci. 22 (2012) 1–33. | DOI | MR | Zbl

J. Beck, F. Nobile, L. Tamellini and R. Tempone, Convergence of quasi-optimal stochastic Galerkin methods for a class of PDEs with random coefficients. Comput. Math. Appl. 67 (2014) 732–751. | DOI | MR | Zbl

J. Charrier, Strong and weak error estimates for elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal. 50 (2012) 216–246. | DOI | MR | Zbl

A. Chkifa, A. Cohen and C. Schwab, Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs. J. Math. Pures Appl. 103 (2015) 400–428. | DOI | MR | Zbl

A. Cohen, Numerical analysis of wavelet methods, Studies in Mathematics and its Applications. Elsevier, Amsterdam (2003). | MR | Zbl

A. Cohen and R. Devore, Approximation of high-dimensional parametric PDEs. Acta Numer. 24 (2015) 1–159. | DOI | MR | Zbl

A. Cohen, R. Devore and C. Schwab, Analytic regularity and polynomial approximation of parametric and stochastic PDEs. Anal. Appl. 9 (2011) 11–47. | DOI | MR | Zbl

M. Dashti and A.M. Stuart, The Bayesian Approach to Inverse Problems. Handbook of Uncertainty Quantification, edited by R. Ghanem, D. Higdon and H. Owhadi. Springer (2015).

R. Devore, Nonlinear Approximation, Acta Numer. 7 (1998) 51–150. | DOI | MR | Zbl

O. Ernst and B. Sprungk, Stochastic Collocation for Elliptic PDEs with Random Data: The Lognormal Case, in Sparse Grids and Applications – Munich (2012), edited by J. Garcke and D. Pflüger. Vol. 97 of Lect. Notes Comput. Sci. Eng. Springer International Publishing Switzerland (2014). | MR

J. Galvis and M. Sarkis, Approximating infinity-dimensional stochastic Darcy’s equations without uniform ellipticity. SIAM J. Numer. Anal. 47 (2009) 3624–3651. | DOI | MR | Zbl

R. Ghanem and P. Spanos, Spectral techniques for stochastic finite elements. Arch. Comput. Methods Engrg. 4 (1997) 63–100. | DOI | MR

R.G. Ghanem and P.D. Spanos, Stochastic Finite Elements: A Spectral Approach, 2nd edition, Dover (2007). | MR | Zbl

C. Gittelson, Stochastic Galerkin discretization of the log-normal isotropic diffusion problem. Math. Models Methods Appl. Sci. 20 (2010) 237–263. | DOI | MR | Zbl

I.G. Graham, F.Y. Kuo, J.A. Nichols, R. Scheichl, Ch. Schwab and I. H. Sloan, Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients. Numer. Math. 131 (2015) 329–368. | DOI | MR | Zbl

M. Hairer, An Introduction to Stochastic PDEs. Lecture notes. Available at http://www.hairer.org (2009).

V.H. Hoang and C. Schwab, N-term Galerkin Wiener chaos approximation rates for elliptic PDEs with lognormal Gaussian random inputs. M3AS 24 (2014) 797–826. | MR | Zbl

O. Knio and O. Le Maitre, Spectral Methods for Uncertainty Quantication: With Applications to Computational Fluid Dynamics. Springer (2010). | MR | Zbl

F.Y. Kuo, R. Scheichl, Ch. Schwab, I.H. Sloan and E. Ullmann, Multilevel Quasi-Monte Carlo Methods for Lognormal Diffusion Problems, , to appear in Math. of Comp. (2015). | arXiv | MR

A. Mugler and H.-J. Starkloff, On the convergence of the stochastic Galerkin methods for random elliptic partial differential equations. ESAIM: M2AN 47 (2013) 1237–1263. | DOI | Numdam | MR | Zbl

D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press (2010). | MR | Zbl

  • Van Kien Nguyen Analyticity of parametric elliptic eigenvalue problems and applications to quasi-Monte Carlo methods, Complex Variables and Elliptic Equations, Volume 69 (2024) no. 1, pp. 1-21 | DOI:10.1080/17476933.2023.2205136 | Zbl:1532.35147
  • van Harten, Wouter Gerrit; Scarabosio, Laura Exploiting locality in sparse polynomial approximation of parametric elliptic PDEs and application to parameterized domains, European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis, Volume 58 (2024) no. 5, pp. 1581-1613 | DOI:10.1051/m2an/2024050 | Zbl:7957220
  • Griebel, Michael; Seidler, Uta A DIMENSION-ADAPTIVE COMBINATION TECHNIQUE FOR UNCERTAINTY QUANTIFICATION, International Journal for Uncertainty Quantification, Volume 14 (2024) no. 2, p. 21 | DOI:10.1615/int.j.uncertaintyquantification.2023046861
  • Shivanand, Sharana Kumar; Rosić, Bojana; Matthies, Hermann G. Stochastic modelling of symmetric positive definite material tensors, Journal of Computational Physics, Volume 505 (2024), p. 30 (Id/No 112883) | DOI:10.1016/j.jcp.2024.112883 | Zbl:7842876
  • Elliott, Charles M.; Garcke, Harald; Niethammer, Barbara; Simonett, Gieri Interfaces, free boundaries and geometric partial differential equations. Abstracts from the workshop held February 11–16, 2024, Oberwolfach Rep. 21, No. 1, 389-482, 2024 | DOI:10.4171/owr/2024/8 | Zbl:1546.00064
  • Bachmayr, Markus; Djurdjevac, Ana Multilevel representations of isotropic Gaussian random fields on the sphere, IMA Journal of Numerical Analysis, Volume 43 (2023) no. 4, p. 1970 | DOI:10.1093/imanum/drac034
  • Dinh Dũng; Van Kien Nguyen; Duong Thanh Pham Deep ReLU neural network approximation in Bochner spaces and applications to parametric PDEs, Journal of Complexity, Volume 79 (2023), p. 32 (Id/No 101779) | DOI:10.1016/j.jco.2023.101779 | Zbl:1545.65035
  • Cohen, Albert; Migliorati, Giovanni Near-optimal approximation methods for elliptic PDEs with lognormal coefficients, Mathematics of Computation, Volume 92 (2023) no. 342, pp. 1665-1691 | DOI:10.1090/mcom/3825 | Zbl:1512.65013
  • Eigel, Martin; Farchmin, Nando; Heidenreich, Sebastian; Trunschke, Philipp Adaptive nonintrusive reconstruction of solutions to high-dimensional parametric PDEs, SIAM Journal on Scientific Computing, Volume 45 (2023) no. 2, p. a457-a479 | DOI:10.1137/21m1461988 | Zbl:1530.65014
  • Schwab, Christoph; Zech, Jakob Deep learning in high dimension: neural network expression rates for analytic functions in L2(Rd,γd), SIAM/ASA Journal on Uncertainty Quantification, Volume 11 (2023), pp. 199-234 | DOI:10.1137/21m1462738 | Zbl:1524.41084
  • Dũng, Dinh Collocation approximation by deep neural ReLU networks for parametric and stochastic PDEs with lognormal inputs, Sbornik: Mathematics, Volume 214 (2023) no. 4, pp. 479-515 | DOI:10.4213/sm9791e | Zbl:1535.65013
  • Li, Sijing; Zhang, Cheng; Zhang, Zhiwen; Zhao, Hongkai A data-driven and model-based accelerated Hamiltonian Monte Carlo method for Bayesian elliptic inverse problems, Statistics and Computing, Volume 33 (2023) no. 4, p. 16 (Id/No 90) | DOI:10.1007/s11222-023-10262-y | Zbl:1517.62030
  • Chen, Peng; Ghattas, Omar Sparse polynomial approximations for affine parametric saddle point problems, Vietnam Journal of Mathematics, Volume 51 (2023) no. 1, pp. 151-175 | DOI:10.1007/s10013-022-00584-1 | Zbl:1528.41011
  • Dinh, Dung Коллокационная аппроксимация глубокими ReLU-нейронными сетями решений параметрических и стохастических уравнений с частными производными c логнормальными входами, Математический сборник, Volume 214 (2023) no. 4, p. 38 | DOI:10.4213/sm9791
  • Beck, Joakim; Liu, Yang; von Schwerin, Erik; Tempone, Raúl Goal-oriented adaptive finite element multilevel Monte Carlo with convergence rates, Computer Methods in Applied Mechanics and Engineering, Volume 402 (2022), p. 39 (Id/No 115582) | DOI:10.1016/j.cma.2022.115582 | Zbl:1507.65008
  • Eigel, Martin; Ernst, Oliver G.; Sprungk, Björn; Tamellini, Lorenzo On the convergence of adaptive stochastic collocation for elliptic partial differential equations with affine diffusion, SIAM Journal on Numerical Analysis, Volume 60 (2022) no. 2, pp. 659-687 | DOI:10.1137/20m1364722 | Zbl:1508.65163
  • Kämmerer, Lutz; Potts, Daniel; Taubert, Fabian The uniform sparse FFT with application to PDEs with random coefficients, Sampling Theory, Signal Processing, and Data Analysis, Volume 20 (2022) no. 2, p. 39 (Id/No 19) | DOI:10.1007/s43670-022-00037-3 | Zbl:1515.65293
  • Bonnaire, P.; Pettersson, P.; Silva, C. F. Intrusive generalized polynomial chaos with asynchronous time integration for the solution of the unsteady Navier-Stokes equations, Computers and Fluids, Volume 223 (2021), p. 15 (Id/No 104952) | DOI:10.1016/j.compfluid.2021.104952 | Zbl:1521.76632
  • Dũng, Dinh Sparse-grid polynomial interpolation approximation and integration for parametric and stochastic elliptic PDEs with lognormal inputs, European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis, Volume 55 (2021) no. 3, pp. 1163-1198 | DOI:10.1051/m2an/2021017 | Zbl:7405595
  • Nguyen, Dong T. P.; Nuyens, Dirk MDFEM: multivariate decomposition finite element method for elliptic PDEs with lognormal diffusion coefficients using higher-order QMC and FEM, European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis, Volume 55 (2021) no. 4, pp. 1461-1505 | DOI:10.1051/m2an/2021029 | Zbl:1492.65327
  • Thesing, L.; Hansen, A. C. Non-uniform recovery guarantees for binary measurements and infinite-dimensional compressed sensing, The Journal of Fourier Analysis and Applications, Volume 27 (2021) no. 2, p. 45 (Id/No 14) | DOI:10.1007/s00041-021-09813-6 | Zbl:1460.42044
  • Nguyen, Dong T. P.; Nuyens, Dirk MDFEM: multivariate decomposition finite element method for elliptic PDEs with uniform random diffusion coefficients using higher-order QMC and FEM, Numerische Mathematik, Volume 148 (2021) no. 3, pp. 633-669 | DOI:10.1007/s00211-021-01212-9 | Zbl:1495.65219
  • Ernst, Oliver G.; Sprungk, Björn; Tamellini, Lorenzo On expansions and nodes for sparse grid collocation of lognormal elliptic PDEs, Sparse grids and applications – Munich 2018. Selected papers based on the presentations at the fifth workshop, SGA2018, Munich, Germany, July 23–27, 2018, Cham: Springer, 2021, pp. 1-31 | DOI:10.1007/978-3-030-81362-8_1 | Zbl:1498.65210
  • Bochmann, Maximilian; Kämmerer, Lutz; Potts, Daniel A sparse FFT approach for ODE with random coefficients, Advances in Computational Mathematics, Volume 46 (2020) no. 5, p. 21 (Id/No 65) | DOI:10.1007/s10444-020-09807-w | Zbl:1455.65241
  • Hansen, A. C.; Thesing, L. On the stable sampling rate for binary measurements and wavelet reconstruction, Applied and Computational Harmonic Analysis, Volume 48 (2020) no. 2, pp. 630-654 | DOI:10.1016/j.acha.2018.08.004 | Zbl:1454.94032
  • Tsilifis, Panagiotis; Papaioannou, Iason; Straub, Daniel; Nobile, Fabio Sparse polynomial chaos expansions using variational relevance vector machines, Journal of Computational Physics, Volume 416 (2020), p. 19 (Id/No 109498) | DOI:10.1016/j.jcp.2020.109498 | Zbl:1437.62114
  • Li, Sijing; Zhang, Zhiwen; Zhao, Hongkai A data-driven approach for multiscale elliptic PDEs with random coefficients based on intrinsic dimension reduction, Multiscale Modeling Simulation, Volume 18 (2020) no. 3, pp. 1242-1271 | DOI:10.1137/19m1277485 | Zbl:1459.35194
  • Bachmayr, Markus; Graham, Ivan G.; Nguyen, Van Kien; Scheichl, Robert Unified analysis of periodization-based sampling methods for Matérn covariances, SIAM Journal on Numerical Analysis, Volume 58 (2020) no. 5, pp. 2953-2980 | DOI:10.1137/19m1269877 | Zbl:1471.60047
  • Feischl, Michael; Peterseim, Daniel Sparse compression of expected solution operators, SIAM Journal on Numerical Analysis, Volume 58 (2020) no. 6, pp. 3144-3164 | DOI:10.1137/20m132571x | Zbl:1475.65215
  • Calderbank, Robert; Hansen, Anders; Roman, Bogdan; Thesing, Laura On reconstructing functions from binary measurements, Compressed sensing and its applications. Selected papers of the third international MATHEON conference, TU Berlin, Berlin, Germany, December 4–8, 2017, Cham: Birkhäuser, 2019, pp. 97-128 | DOI:10.1007/978-3-319-73074-5_3 | Zbl:1450.94025
  • Herrmann, Lukas; Schwab, Christoph Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients, European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis, Volume 53 (2019) no. 5, pp. 1507-1552 | DOI:10.1051/m2an/2019016 | Zbl:7135561
  • Dexter, Nick; Tran, Hoang; Webster, Clayton A mixed 1 regularization approach for sparse simultaneous approximation of parameterized PDEs, European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis, Volume 53 (2019) no. 6, pp. 2025-2045 | DOI:10.1051/m2an/2019048 | Zbl:7167647
  • Kazashi, Yoshihito Quasi–Monte Carlo integration with product weights for elliptic PDEs with log-normal coefficients, IMA Journal of Numerical Analysis, Volume 39 (2019) no. 3, p. 1563 | DOI:10.1093/imanum/dry028
  • Herrmann, Lukas; Schwab, Christoph QMC integration for lognormal-parametric, elliptic PDEs: local supports and product weights, Numerische Mathematik, Volume 141 (2019) no. 1, pp. 63-102 | DOI:10.1007/s00211-018-0991-1 | Zbl:7006664
  • Cohen, Albert; Migliorati, Giovanni Multivariate approximation in downward closed polynomial spaces, Contemporary computational mathematics – a celebration of the 80th birthday of Ian Sloan. In 2 volumes, Cham: Springer, 2018, pp. 233-282 | DOI:10.1007/978-3-319-72456-0_12 | Zbl:1405.41021
  • Chen, Peng Sparse quadrature for high-dimensional integration with Gaussian measure, European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis, Volume 52 (2018) no. 2, pp. 631-657 | DOI:10.1051/m2an/2018012 | Zbl:6966736
  • Bachmayr, Markus; Cohen, Albert; Migliorati, Giovanni Representations of Gaussian random fields and approximation of elliptic PDEs with lognormal coefficients, The Journal of Fourier Analysis and Applications, Volume 24 (2018) no. 3, pp. 621-649 | DOI:10.1007/s00041-017-9539-5 | Zbl:1428.60054
  • Herrmann, Lukas; Schwab, Christoph QMC Algorithms with Product Weights for Lognormal-Parametric, Elliptic PDEs, Monte Carlo and Quasi-Monte Carlo Methods, Volume 241 (2018), p. 313 | DOI:10.1007/978-3-319-91436-7_17
  • Müller, Christopher; Ullmann, Sebastian; Lang, Jens A Bramble-Pasciak Conjugate Gradient Method for Discrete Stokes Problems with Lognormal Random Viscosity, Recent Advances in Computational Engineering, Volume 124 (2018), p. 63 | DOI:10.1007/978-3-319-93891-2_5
  • Ernst, Oliver G.; Sprungk, Björn; Tamellini, Lorenzo Convergence of sparse collocation for functions of countably many Gaussian random variables (with application to elliptic PDEs), SIAM Journal on Numerical Analysis, Volume 56 (2018) no. 2, pp. 877-905 | DOI:10.1137/17m1123079 | Zbl:6864017
  • Kamilis, Dimitris; Polydorides, Nick Uncertainty quantification for low-frequency, time-harmonic Maxwell equations with stochastic conductivity models, SIAM/ASA Journal on Uncertainty Quantification, Volume 6 (2018), pp. 1295-1334 | DOI:10.1137/17m1156010 | Zbl:1405.35264
  • Thesing, Laura; Hansen, Anders Linear Reconstructions and the Analysis of the Stable Sampling Rate, Sampling Theory in Signal and Image Processing, Volume 17 (2018) no. 1, p. 103 | DOI:10.1007/bf03549616
  • Chen, Peng; Villa, Umberto; Ghattas, Omar Hessian-based adaptive sparse quadrature for infinite-dimensional Bayesian inverse problems, Computer Methods in Applied Mechanics and Engineering, Volume 327 (2017), pp. 147-172 | DOI:10.1016/j.cma.2017.08.016
  • Bachmayr, Markus; Cohen, Albert; Migliorati, Giovanni Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients, ESAIM: Mathematical Modelling and Numerical Analysis, Volume 51 (2017) no. 1, p. 321 | DOI:10.1051/m2an/2016045
  • Griebel, Michael; Oswald, Peter Stable splittings of Hilbert spaces of functions of infinitely many variables, Journal of Complexity, Volume 41 (2017), pp. 126-151 | DOI:10.1016/j.jco.2017.01.003 | Zbl:1378.46019
  • Bachmayr, Markus; Cohen, Albert; Dũng, Dinh; Schwab, Christoph Fully discrete approximation of parametric and stochastic elliptic PDEs, SIAM Journal on Numerical Analysis, Volume 55 (2017) no. 5, pp. 2151-2186 | DOI:10.1137/17m111626x | Zbl:1377.65005
  • Dũng, Dinh; Temlyakov, Vladimir N.; Ullrich, Tino Hyperbolic Cross Approximation, arXiv (2016) | DOI:10.48550/arxiv.1601.03978 | arXiv:1601.03978
  • Griebel, Michael; Oswald, Peter Hilbert function space splittings on domains with infinitely many variables, arXiv (2016) | DOI:10.48550/arxiv.1607.05978 | arXiv:1607.05978
  • Bachmayr, Markus; Cohen, Albert; Migliorati, Giovanni Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients, arXiv (2015) | DOI:10.48550/arxiv.1509.07045 | arXiv:1509.07045

Cité par 49 documents. Sources : Crossref, NASA ADS, zbMATH