Lattice-based Deep Neural Networks: Regularity and Tailored Regularization.

Dirk Nuyens https://zbmath.org/authors/?q=ai:nuyens.dirk https://orcid.org/0000-0002-4555-2314 https://www.wikidata.org/entity/Q57160655

Alexander Keller 0001 Frances Y. Kuo Dirk Nuyens Ian H. Sloan Lattice-based Deep Neural Networks: Regularity and Tailored Regularization. 2026 March abs/2603.02809 CoRR https://doi.org/10.48550/arXiv.2603.02809 db/journals/corr/corr2603.html#abs-2603-02809 streams/journals/corr

Ivan G. Graham Frances Y. Kuo Dirk Nuyens Ian H. Sloan Euan A. Spence Quasi-Monte Carlo methods for uncertainty quantification of wave propagation and scattering problems modelled by the Helmholtz equation. 2025 February abs/2502.12451 CoRR https://doi.org/10.48550/arXiv.2502.12451 db/journals/corr/corr2502.html#abs-2502-12451 streams/journals/corr

Alexander Keller 0001 Frances Y. Kuo Dirk Nuyens Ian H. Sloan Regularity and Tailored Regularization of Deep Neural Networks, with application to parametric PDEs in uncertainty quantification. 2025 February abs/2502.12496 CoRR https://doi.org/10.48550/arXiv.2502.12496 db/journals/corr/corr2502.html#abs-2502-12496 streams/journals/corr

Yiqing Zhou Karsten Naert Dirk Nuyens Modeling Scrap Composition in Electric Arc and Basic Oxygen Furnaces. 2025 April abs/2504.09382 CoRR https://doi.org/10.48550/arXiv.2504.09382 db/journals/corr/corr2504.html#abs-2504-09382 streams/journals/corr

Yiqing Zhou Karsten Naert Dirk Nuyens Sensitivity Analysis of State Space Models for Scrap Composition Estimation in EAF and BOF. 2025 April abs/2504.11319 CoRR https://doi.org/10.48550/arXiv.2504.11319 db/journals/corr/corr2504.html#abs-2504-11319 streams/journals/corr

Ronald Cools Dirk Nuyens Laurence Wilkes Error bounds for function approximation using generated sets. 2025 May abs/2505.00440 CoRR https://doi.org/10.48550/arXiv.2505.00440 db/journals/corr/corr2505.html#abs-2505-00440 streams/journals/corr

Arne Bouillon Alexander Bodard Panagiotis Patrinos Dirk Nuyens Giovanni Samaey A localized consensus-based sampling algorithm. 2025 May abs/2505.24861 CoRR https://doi.org/10.48550/arXiv.2505.24861 db/journals/corr/corr2505.html#abs-2505-24861 streams/journals/corr

Alexander D. Gilbert Frances Y. Kuo Dirk Nuyens Graham Pash Ian H. Sloan Karen E. Willcox Quasi-Monte Carlo methods for uncertainty quantification of tumor growth modeled by a parametric semi-linear parabolic reaction-diffusion equation. 2025 September abs/2509.25753 CoRR https://doi.org/10.48550/arXiv.2509.25753 db/journals/corr/corr2509.html#abs-2509-25753 streams/journals/corr

Frances Y. Kuo Dirk Nuyens Laurence Wilkes Random-prime-fixed-vector randomised lattice-based algorithm for high-dimensional integration. 101785 2023 December 79 J. Complex. https://doi.org/10.1016/j.jco.2023.101785 db/journals/jc/jc79.html#KuoNW23

Frances Y. Kuo Weiwen Mo Dirk Nuyens Ian H. Sloan Abirami Srikumar Comparison of Two Search Criteria for Lattice-based Kernel Approximation. 2023 abs/2304.01685 CoRR https://doi.org/10.48550/arXiv.2304.01685 db/journals/corr/corr2304.html#abs-2304-01685

Frances Y. Kuo Dirk Nuyens Laurence Wilkes Random-prime-fixed-vector randomised lattice-based algorithm for high-dimensional integration. 2023 abs/2304.10413 CoRR https://doi.org/10.48550/arXiv.2304.10413 db/journals/corr/corr2304.html#abs-2304-10413

Dirk Nuyens Laurence Wilkes A randomised lattice rule algorithm with pre-determined generating vector and random number of points for Korobov spaces with 0 < α≤ 1/2. 2023 abs/2308.03138 CoRR https://doi.org/10.48550/arXiv.2308.03138 db/journals/corr/corr2308.html#abs-2308-03138

Dirk Nuyens Yuya Suzuki Scaled lattice rules for integration on ℝ^{𝕕} achieving higher-order convergence with error analysis in terms of orthogonal projections onto periodic spaces. 307-347 2022 September 92 Math. Comput. 339 https://doi.org/10.1090/mcom/3754 db/journals/moc/moc92.html#NuyensS22

Frances Y. Kuo Weiwen Mo Dirk Nuyens Constructing Embedded Lattice-based Algorithms for Multivariate Function Approximation with a Composite Number of Points. 2022 abs/2209.01002 CoRR https://doi.org/10.48550/arXiv.2209.01002 db/journals/corr/corr2209.html#abs-2209-01002

Adrian Ebert Peter Kritzer Dirk Nuyens Onyekachi Osisiogu Digit-by-digit and component-by-component constructions of lattice rules for periodic functions with unknown smoothness. 101555 2021 66 J. Complex. https://doi.org/10.1016/j.jco.2021.101555 db/journals/jc/jc66.html#EbertKNO21

Tobias Hartung Karl Jansen Frances Y. Kuo Hernan Leövey Dirk Nuyens Ian H. Sloan Lattice meets lattice: Application of lattice cubature to models in lattice gauge theory. 110527 2021 443 J. Comput. Phys. https://doi.org/10.1016/j.jcp.2021.110527 db/journals/jcphy/jcphy443.html#HartungJKLNS21

Ronald Cools Frances Y. Kuo Dirk Nuyens Ian H. Sloan Fast component-by-component construction of lattice algorithms for multivariate approximation with POD and SPOD weights. 787-812 2021 90 Math. Comput. 328 https://doi.org/10.1090/mcom/3586 db/journals/moc/moc90.html#CoolsKNS21

Frances Y. Kuo Giovanni Migliorati Fabio Nobile Dirk Nuyens Function integration, reconstruction and approximation using rank-1 lattices. 1861-1897 2021 90 Math. Comput. 330 https://doi.org/10.1090/mcom/3595 db/journals/moc/moc90.html#KuoMNN21

Pieterjan Robbe Dirk Nuyens Stefan Vandewalle Enhanced multi-index Monte Carlo by means of multiple semicoarsened multigrid for anisotropic diffusion problems. 2021 28 Numer. Linear Algebra Appl. 3 https://doi.org/10.1002/nla.2281 https://www.wikidata.org/entity/Q126460194 db/journals/nla/nla28.html#RobbeNV21

Dong T. P. Nguyen Dirk Nuyens MDFEM: Multivariate decomposition finite element method for elliptic PDEs with uniform random diffusion coefficients using higher-order QMC and FEM. 633-669 2021 148 Numerische Mathematik 3 https://doi.org/10.1007/s00211-021-01212-9 https://www.wikidata.org/entity/Q114231026 db/journals/nm/nm148.html#NguyenN21

Dirk Nuyens Yuya Suzuki Scaled lattice rules for integration on Rd achieving higher-order convergence with error analysis in terms of orthogonal projections onto periodic spaces. 2021 abs/2108.12639 CoRR https://arxiv.org/abs/2108.12639 db/journals/corr/corr2108.html#abs-2108-12639

Ronald Cools Frances Y. Kuo Dirk Nuyens Ian H. Sloan Lattice algorithms for multivariate approximation in periodic spaces with general weight parameters. 2020 75 Years of Mathematics of Computation http://www.ams.org/books/conm/754/15150 books/ams/20/BSSS2020 db/books/collections/BSSS2020.html#CoolsKNS20

Adrian Ebert Peter Kritzer Dirk Nuyens Onyekachi Osisiogu Digit-by-digit and component-by-component constructions of lattice rules for periodic functions with unknown smoothness. 2020 abs/2001.02978 CoRR http://arxiv.org/abs/2001.02978 db/journals/corr/corr2001.html#abs-2001-02978

Tobias Hartung Karl Jansen Frances Y. Kuo Hernan Leövey Dirk Nuyens Ian H. Sloan Lattice meets lattice: Application of lattice cubature to models in lattice gauge theory. 2020 abs/2011.05451 CoRR https://arxiv.org/abs/2011.05451 db/journals/corr/corr2011.html#abs-2011-05451

Peter Kritzer Frances Y. Kuo Dirk Nuyens Mario Ullrich Lattice rules with random n achieve nearly the optimal O(n-α-1∕2) error independently of the dimension. 96-113 2019 240 J. Approx. Theory https://doi.org/10.1016/j.jat.2018.09.011 db/journals/jat/jat240.html#KritzerKNU19

Tianfeng Hou Dirk Nuyens Staf Roels Hans Janssen Quasi-Monte Carlo based uncertainty analysis: Sampling efficiency and error estimation in engineering applications. 2019 191 Reliab. Eng. Syst. Saf. https://doi.org/10.1016/j.ress.2019.106549 https://www.wikidata.org/entity/Q127598133 db/journals/ress/ress191.html#HouNRJ19

Pieterjan Robbe Dirk Nuyens Stefan Vandewalle Recycling Samples in the Multigrid Multilevel (Quasi-)Monte Carlo Method. S37-S60 2019 41 SIAM J. Sci. Comput. 5 https://doi.org/10.1137/18M1194031 https://www.wikidata.org/entity/Q126856814 db/journals/siamsc/siamsc41.html#RobbeNV19

Yuya Suzuki Gowri Suryanarayana Dirk Nuyens Strang Splitting in Combination with Rank-1 and Rank-r Lattices for the Time-Dependent Schrödinger Equation. B1254-B1283 2019 41 SIAM J. Sci. Comput. 6 https://doi.org/10.1137/18M1207879 db/journals/siamsc/siamsc41.html#SuzukiSN19

Pieterjan Robbe Dirk Nuyens Stefan Vandewalle Enhanced Multi-Index Monte Carlo by means of Multiple Semi-Coarsened Multigrid for Anisotropic Diffusion Problems. 2019 abs/1907.12334 CoRR http://arxiv.org/abs/1907.12334 db/journals/corr/corr1907.html#abs-1907-12334

Frances Y. Kuo Giovanni Migliorati Fabio Nobile Dirk Nuyens Function integration, reconstruction and approximation using rank-1 lattices. 2019 abs/1908.01178 CoRR http://arxiv.org/abs/1908.01178 db/journals/corr/corr1908.html#abs-1908-01178

Ronald Cools Frances Y. Kuo Dirk Nuyens Ian H. Sloan Lattice algorithms for multivariate approximation in periodic spaces with general weight parameters. 2019 abs/1910.06604 CoRR http://arxiv.org/abs/1910.06604 db/journals/corr/corr1910.html#abs-1910-06604

Ronald Cools Frances Y. Kuo Dirk Nuyens Ian H. Sloan Fast component-by-component construction of lattice algorithms for multivariate approximation with POD and SPOD weights. 2019 abs/1910.06606 CoRR http://arxiv.org/abs/1910.06606 db/journals/corr/corr1910.html#abs-1910-06606

Ivan G. Graham Frances Y. Kuo Dirk Nuyens Robert Scheichl Ian H. Sloan Circulant embedding with QMC: analysis for elliptic PDE with lognormal coefficients. 479-511 2018 140 Numerische Mathematik 2 https://doi.org/10.1007/s00211-018-0968-0 https://www.wikidata.org/entity/Q91472396 db/journals/nm/nm140.html#GrahamKNSS18

Ivan G. Graham Frances Y. Kuo Dirk Nuyens Robert Scheichl Ian H. Sloan Analysis of Circulant Embedding Methods for Sampling Stationary Random Fields. 1871-1895 2018 56 SIAM J. Numer. Anal. 3 https://doi.org/10.1137/17M1149730 https://www.wikidata.org/entity/Q129646751 db/journals/siamnum/siamnum56.html#GrahamKNSS18

Alexander D. Gilbert Frances Y. Kuo Dirk Nuyens Grzegorz W. Wasilkowski Efficient Implementations of the Multivariate Decomposition Method for Approximating Infinite-Variate Integrals. A3240-A3266 2018 40 SIAM J. Sci. Comput. 5 https://doi.org/10.1137/17M1161890 https://www.wikidata.org/entity/Q129142592 db/journals/siamsc/siamsc40.html#GilbertKNW18

Dong T. P. Nguyen Dirk Nuyens Multivariate integration over Rs with exponential rate of convergence. 327-342 2017 315 J. Comput. Appl. Math. https://doi.org/10.1016/j.cam.2016.11.016 https://www.wikidata.org/entity/Q115581080 db/journals/jcam/jcam315.html#NguyenN17

Frances Y. Kuo Dirk Nuyens Leszek Plaskota Ian H. Sloan Grzegorz W. Wasilkowski Infinite-dimensional integration and the multivariate decomposition method. 217-234 2017 326 J. Comput. Appl. Math. https://doi.org/10.1016/j.cam.2017.05.031 db/journals/jcam/jcam326.html#KuoNPSW17

Pieterjan Robbe Dirk Nuyens Stefan Vandewalle A Multi-Index Quasi-Monte Carlo Algorithm for Lognormal Diffusion Problems. 2017 39 SIAM J. Sci. Comput. 5 https://doi.org/10.1137/16M1082561 db/journals/siamsc/siamsc39.html#RobbeNV17

Alexander D. Gilbert Frances Y. Kuo Dirk Nuyens Grzegorz W. Wasilkowski Efficient implementations of the Multivariate Decomposition Method for approximating infinite-variate integrals. 2017 abs/1712.06782 CoRR http://arxiv.org/abs/1712.06782 db/journals/corr/corr1712.html#abs-1712-06782

Dirk Nuyens Gowri Suryanarayana Markus Weimar Rank-1 lattice rules for multivariate integration in spaces of permutation-invariant functions - Error bounds and tractability. 55-84 2016 42 Adv. Comput. Math. 1 https://doi.org/10.1007/s10444-015-9411-6 db/journals/adcm/adcm42.html#NuyensSW16

Frances Y. Kuo Dirk Nuyens Application of Quasi-Monte Carlo Methods to Elliptic PDEs with Random Diffusion Coefficients: A Survey of Analysis and Implementation. 1631-1696 2016 16 Found. Comput. Math. 6 https://doi.org/10.1007/s10208-016-9329-5 db/journals/focm/focm16.html#KuoN16

Ronald Cools Frances Y. Kuo Dirk Nuyens Gowri Suryanarayana Tent-transformed lattice rules for integration and approximation of multivariate non-periodic functions. 166-181 2016 36 J. Complex. https://doi.org/10.1016/j.jco.2016.05.004 https://www.wikidata.org/entity/Q57778794 db/journals/jc/jc36.html#CoolsKNS16

Ronald Cools Dirk Nuyens Monte Carlo and Quasi-Monte Carlo Methods, MCQMC 2014, Leuven, Belgium, April 2014 MCQMC Springer 2016 Springer Proceedings in Mathematics and Statistics 163 978-3-319-33505-6 978-3-319-33507-0 https://doi.org/10.1007/978-3-319-33507-0 db/conf/mcqmc/mcqmc2014.html

Josef Dick Dirk Nuyens Friedrich Pillichshammer Lattice rules for nonperiodic smooth integrands. 259-291 2014 126 Numerische Mathematik 2 https://doi.org/10.1007/s00211-013-0566-0 db/journals/nm/nm126.html#DickNP14

Josef Dick Frances Y. Kuo Quoc Thong Le Gia Dirk Nuyens Christoph Schwab Higher Order QMC Petrov-Galerkin Discretization for Affine Parametric Operator Equations with Random Field Inputs. 2676-2702 2014 52 SIAM J. Numer. Anal. 6 https://doi.org/10.1137/130943984 db/journals/siamnum/siamnum52.html#DickKGNS14

Dirk Nuyens The construction of good lattice rules and polynomial lattice rules. 223-256 2014 Uniform Distribution and Quasi-Monte Carlo Methods https://doi.org/10.1515/9783110317930.223 books/daglib/0036089 db/books/collections/KNPW2014.html#Nuyens14

Nico Achtsis Ronald Cools Dirk Nuyens Conditional Sampling for Barrier Option Pricing under the LT Method. 327-352 2013 4 SIAM J. Financial Math. 1 https://doi.org/10.1137/110855909 https://www.wikidata.org/entity/Q57778802 db/journals/siamfm/siamfm4.html#AchtsisCN13

Fred J. Hickernell Peter Kritzer Frances Y. Kuo Dirk Nuyens Weighted compound integration rules with higher order convergence for all N. 161-183 2012 59 Numer. Algorithms 2 https://doi.org/10.1007/s11075-011-9482-5 db/journals/na/na59.html#HickernellKKN12

Jan Baldeaux Josef Dick Gunther Leobacher Dirk Nuyens Friedrich Pillichshammer Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules. 403-431 2012 59 Numer. Algorithms 3 https://doi.org/10.1007/s11075-011-9497-y db/journals/na/na59.html#BaldeauxDLNP12

Ivan G. Graham Frances Y. Kuo Dirk Nuyens Robert Scheichl Ian H. Sloan Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications. 3668-3694 2011 230 J. Comput. Phys. 10 https://doi.org/10.1016/j.jcp.2011.01.023 db/journals/jcphy/jcphy230.html#GrahamKNSS11

Ronald Cools Frances Y. Kuo Dirk Nuyens Constructing lattice rules based on weighted degree of exactness and worst case error. 63-89 2010 87 Computing 1-2 https://doi.org/10.1007/s00607-009-0076-1 https://www.wikidata.org/entity/Q57778819 db/journals/computing/computing87.html#CoolsKN10

Dirk Nuyens Ronald Cools Fast component-by-component construction of rank-1 lattice rules with a non-prime number of points. 4-28 2006 22 J. Complex. 1 https://doi.org/10.1016/j.jco.2005.07.002 https://www.wikidata.org/entity/Q57778915 db/journals/jc/jc22.html#NuyensC06

Dirk Nuyens Ronald Cools Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces. 903-920 2006 75 Math. Comput. 254 https://doi.org/10.1090/S0025-5718-06-01785-6 https://www.wikidata.org/entity/Q57778912 db/journals/moc/moc75.html#NuyensC06

Ronald Cools Frances Y. Kuo Dirk Nuyens Constructing Embedded Lattice Rules for Multivariate Integration. 2162-2188 2006 28 SIAM J. Sci. Comput. 6 https://doi.org/10.1137/06065074X https://www.wikidata.org/entity/Q57778908 db/journals/siamsc/siamsc28.html#CoolsKN06

Dirk Nuyens Ronald Cools Fast Component-By-Component Construction of Rank-1 Lattice Rules for (Non-)Primes (Part II). 2004 conf/dagstuhl/2004P4401 Algorithms and Complexity for Continuous Problems http://drops.dagstuhl.de/opus/volltexte/2005/148 db/conf/dagstuhl/P4401.html#NuyensC04 Nico Achtsis Jan Baldeaux Alexander Bodard Arne Bouillon Ronald Cools Josef Dick Adrian Ebert Quoc Thong Le Gia Alexander D. Gilbert Ivan G. Graham Tobias Hartung Fred J. Hickernell Tianfeng Hou Karl Jansen Hans Janssen Alexander Keller 0001 Peter Kritzer Frances Y. Kuo Gunther Leobacher Hernan Leövey Giovanni Migliorati Weiwen Mo Karsten Naert Dong T. P. Nguyen Fabio Nobile Onyekachi Osisiogu Graham Pash Panagiotis Patrinos Friedrich Pillichshammer Leszek Plaskota Pieterjan Robbe Staf Roels Giovanni Samaey Robert Scheichl Christoph Schwab Ian Hugh SloanIan H. Sloan Euan A. Spence Abirami Srikumar Gowri Suryanarayana Yuya Suzuki Mario Ullrich Stefan Vandewalle Grzegorz W. Wasilkowski Markus Weimar Laurence Wilkes Karen WillcoxKaren E. Willcox Yiqing Zhou