Research Interests

• Function spaces with dominating mixed smoothness

• Point-wise multiplication, approximation theory

• Second-order elliptic PDEs

• Deep neural networks 

Preprints

1. V. K. Nguyen, Hilbert numbers of Sobolev spaces with mixed smoothness in the sup-norm, 2025

Dissertation and books

1. D. Dũng, V. K. Nguyen, Ch. Schwab, J. Zech, Analyticity and sparsity in uncertainty quantification for PDEs with Gaussian random field inputs, Lecture Notes in Mathematics 2334, 2023, https://link.springer.com/book/9783031383854

2. Van Kien Nguyen, Function spaces of dominating mixed smoothness, Weyl and Bernstein numbers, PhD thesis, Friedrich Schiller University Jena, 2017, https://www.db-thueringen.de/receive/dbt_mods_00032611

 Journal Papers

1. V. K. Nguyen, Optimality conditions for change of variable operators in Sobolev spaces with mixed derivatives, to appear in Analysis Mathematica.

2. V. K. Nguyen, Pseudo s-Numbers of Embeddings of Gaussian Weighted Sobolev Spaces, Journal of Approximation Theory 309 (2025), paper no. 106159, https://doi.org/10.1016/j.jat.2025.106159 

3. D. Dũng, V. K. Nguyen, Optimal numerical integration and approximation of functions on Rd equipped with Gaussian measure, IMA Journal of Numerical Analysis 44 (2024), 1242–1267,  https://doi.org/10.1093/imanum/drad051

4. D. Dũng, V. K. Nguyen, D. T. Pham,  Deep ReLU neural network approximation in  Bochner spaces and applications to  parametric PDEs, Journal of Complexity  79 (2023), paper no. 101779, https://doi.org/10.1016/j.jco.2023.101779

5. V. K. Nguyen, Analyticity of Parametric Elliptic Eigenvalue Problems and Applications to Quasi-Monte Carlo Methods, Complex Variables and Elliptic Equations, 2023, https://doi.org/10.1080/17476933.2023.2205136

6. V. K. Nguyen, V. D. Nguyen, Best n-term approximation of diagonal operators and application to function spaces with mixed smoothness, Analysis Mathematica 48 (2022), 1127--1152, https://doi.org/10.1007/s10476-022-0169-z

7. V. D. Nguyen, V. K. Nguyen, W. Sickel, s-Numbers of embeddings of weighted Wiener algebras,  Journal of Approximation Theory 279 (2022), paper no. 105745, https://doi.org/10.1016/j.jat.2022.105745

8. D. Dũng, V. K. Nguyen, Deep ReLU neural networks in high-dimensional approximation, Neural Networks 142 (2021), 619-635, https://doi.org/10.1016/j.neunet.2021.07.027

9. D. Dũng, V. K. Nguyen, M. X. Thao, Computation complexity of deep ReLU neural networks in high-dimensional approximation, Journal of Computer Science and Cybernetics 37 (2021), 291-320, https://doi.org/10.15625/1813-9663/37/3/15902 

10. M. Bachmayr, I. G. Graham, V. K. Nguyen, R. Scheichl, Unified Analysis of Periodization-based Sampling Methods for Mat\'ern Covariances, SIAM Journal on Numerical Analysis, 58 (2020), 2953–2980, https://doi.org/10.1137/19M1269877 

11. V. K. Nguyen, Greedy Algorithms and Kolmogorov Widths in Banach Spaces,  Journal of Approximation Theory 251 (2020), 105344, https://doi.org/10.1016/j.jat.2019.105344 

12. M. Bachmayr, V. K. Nguyen, Identifiability of Diffusion Coefficients for Source Terms of Non-Uniform Sign, Inverse Problems and Imaging, 13 (2019), 1007--1021,  10.3934/ipi.2019045 

13. G. Byrenheid, R. J. Kunsch, V. K. Nguyen, Monte Carlo Methods for Uniform Approximation on Periodic Sobolev Spaces with Mixed Smoothness, Journal of Complexity, 46 (2018), 90--102, https://doi.org/10.1016/j.jco.2017.12.002

14. V.K. Nguyen, W. Sickel, On a Problem of Jaak Peetre Concerning Pointwise Multipliers of Besov Spaces, Studia Math., 243 (2018), 207--231, 10.4064/sm170522-16-8

15. V.K. Nguyen, W. Sickel, Pointwise Multipliers for Besov Spaces of Dominating Mixed Smoothness - II, Science China Mathematics,  60 (2017), 2241--2262, https://doi.org/10.1007/s11425-017-9140-9

16. V.K. Nguyen, W. Sickel, Isotropic and dominating mixed Lizorkin - Triebel spaces - a comparison, Analysis Mathematica, 43 (2017), 303--337, https://doi.org/10.1007/s10476-017-0309-z

17. V.K. Nguyen, W. Sickel, Pointwise multipliers for Sobolev and Besov spaces of dominating mixed smoothness,  Journal of Mathematical Analysis and Applications 452 (2017), 62--90, https://doi.org/10.1016/j.jmaa.2017.02.046

18. V.K. Nguyen, Gelfand numbers of embeddings of mixed Besov spaces, Journal of Complexity 41 (2017), 35--57, https://doi.org/10.1016/j.jco.2016.11.004 

19. V.K. Nguyen, W. Sickel, Isotropic and dominating mixed Besov spaces - a comparison, Functional Analysis, Harmonic Analysis and Image Processing: A collection of papers in honor of Bj\"orn Jawerth, Contemporary Mathematics, AMS, 693 (2017), 363--389, https://doi.org/10.1090/conm/693

20. V.K. Nguyen, M. Ullrich, T. Ullrich, Change of variable in spaces of mixed smoothness and numerical integration of multivariate functions on the unit cube, Constructive Approximation 46 (2017), 69--108, https://doi.org/10.1007/s00365-017-9371-9

21. V.K. Nguyen, Weyl and Bernstein numbers of embeddings of Sobolev spaces with dominating mixed smoothness, Journal of Complexity 36 (2016), 46--73, https://doi.org/10.1016/j.jco.2016.04.004 

22. V.K. Nguyen, W. Sickel, Weyl numbers of embeddings of tensor product Besov spaces,  Journal of Approximation Theory 200 (2015), 170--220, https://doi.org/10.1016/j.jat.2015.07.003

23. V.K. Nguyen, Bernstein numbers of embeddings of isotropic and dominating mixed Besov spaces, Mathematische Nachrichten 288 (2015), 1694--1717,  https://doi.org/10.1002/mana.201500018