Pith. sign in

REVIEW 2 major objections 4 minor 70 references

Optimality of quasi-Monte Carlo methods and suboptimality of the sparse-grid Gauss--Hermite rule in Gaussian Sobolev spaces

T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper proves sparse-grid Gauss–Hermite quadrature converges only at half the optimal rate for Gaussian Sobolev spaces, while several quasi-Monte Carlo rules, one with a cotangent change of variables, achieve the optimal rate.

desk verdict Sparse-grid Gauss–Hermite is suboptimal at N^{-α/2}; a Möbius-transformed digital net hits the optimal rate—the paper is right, with the main caveat being dependence on the authors' earlier univariate theorem. read the letter →

arxiv 2509.18712 v3 pith:AOLT2GQQ submitted 2025-09-23 math.NA cs.NA

Yoshihito Kazashi , Yuya Suzuki , Takashi Goda This is my paper
classification math.NAcs.NA MSC 65D3065D3265C05
keywords quasi-MonteCarloGauss–HermitequadraturesparsegridsGaussianSobolevspaceworst-caseerrorcotangenttransformdigitalnetsrank-1lattices
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks how fast any numerical integration rule can converge when integrating functions of finite smoothness α against the standard Gaussian measure on R^d. It proves a sharp gap between two popular methods: sparse-grid quadrature built from Gauss–Hermite rules achieves, up to logarithms, only the rate N^{-α/2} with N function evaluations, whereas several quasi-Monte Carlo rules achieve the optimal rate N^{-α} with a logarithmic factor. One construction—a higher-order digital net combined with a cotangent change of variables—attains the optimal rate including the exact logarithmic exponent (d−1)/2. The lower bound for the sparse-grid rule holds for any choice of quadrature weights, so reweighting the Gauss–Hermite nodes cannot rescue it. A sympathetic reader would take this as a definitive comparison: for Gaussian integration of finite smoothness, QMC with a suitable change of variables is the rate-optimal choice, and sparse-grid Gauss–Hermite is not.

What carries the argument

The lower bound rests on a univariate fooling function p_n: a piecewise polynomial supported on the gaps between consecutive Hermite roots, defined as ((t−ξ_j)/(ξ_{j+1}−ξ_j))^α (1−(t−ξ_j)/(ξ_{j+1}−ξ_j))^α on the j-th gap and zero elsewhere. Because it vanishes at all n Gauss–Hermite nodes, any quadrature using those points integrates it to zero, and a cited univariate estimate gives |∫ p_n ρ| / ||p_n||_{H^α_ρ} ≥ c n^{-α/2}, which is the engine of the lower bound. The upper bounds for QMC use a componentwise cotangent change of variables Ψ(t)=−cot(πt) that maps the unit cube to R^d; under this map the transformed integrand extends continuously by zero to the boundary and lies in mixed-derivat

What would settle it

Take p_n supported on gaps between Hermite roots for α=1,2,3 and compute the ratio R_n = |∫ p_n ρ|/(Σ_{r=0}^α ||D^r p_n||^2_{L2ρ})^{1/2} numerically for n=10,...,10^4; if R_n decays like n^{-α} or faster rather than n^{-α/2}, the lower-bound engine fails.

Watch

Extended reading notes

Core claim

On the Gaussian Sobolev space H^α_ρ(R^d), the worst-case error of sparse-grid Gauss–Hermite quadrature is bounded below by c N^{-α/2} (up to logarithms) for any downward-closed index set, and bounded above by C N^{-α/2} (ln N)^{(d−1)(1+α/2)} for the isotropic sparse grid, so the true rate is N^{-α/2} up to a log factor. In contrast, quasi-Monte Carlo rules with affine or cotangent changes of variables achieve worst-case error of order N^{-α} times a power of ln N; the cotangent-transformed higher-order digital net attains N^{-α}(ln N)^{(d−1)/2}, matching the known lower bound for all algorithms. The key to the suboptimality is a fooling function that vanishes at every Gauss–Hermite node, for

Load-bearing premise

The sparse-grid lower bound rests on a cited univariate inequality for the fooling function built between consecutive Hermite roots: that |∫ p_n ρ|/||p_n||_{H^α_ρ} ≥ c n^{-α/2}; the paper does not reproduce the proof, and if that inequality were wrong the suboptimality claim would collapse.

Editorial extensions

If this is right

  • Users of sparse-grid Gauss–Hermite quadrature for Gaussian integrals of functions with finite Sobolev smoothness cannot expect better than N^{-α/2} worst-case error, no matter how the weights are chosen.
  • The cotangent-transformed higher-order digital net achieves the sharp rate N^{-α}(ln N)^{(d−1)/2}, so equal-weight rules with positive weights suffice for optimal Gaussian integration.
  • The boundary extension result means boundary points of the unit cube—where the cotangent transform is undefined—can be assigned value zero with no loss of rate.
  • The same fooling-function argument yields analogous lower bounds for sparse-grid Gauss–Hermite interpolation in the L1ρ norm.
  • An affine-mapped lattice and affine-mapped digital net also reach the optimal polynomial rate, but with larger logarithmic exponents than the cotangent construction.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The fooling-function construction depends only on the distribution of the quadrature nodes, not on the weights, so the same approach could be used to test other node families (e.g., Leja points) for suboptimality, as the paper itself suggests.
  • The rate gap N^{-α/2} versus N^{-α} means the relative penalty grows with smoothness α; for α=1 the sparse grid has error ~N^{-1/2}, far worse than the ~N^{-1} of optimal QMC.
  • One might conjecture that the cotangent transform is the 'right' universal change of variables for Gaussian measures in this Sobolev setting; a testable extension would be to prove that only transformations whose derivatives grow like polynomials can yield the optimal log exponent.
  • The dimension-dependent constants in the error bounds may limit practical use in very high dimensions; the paper's own remark points to weighted Sobolev spaces as the standard route to dimension-independent constants.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies worst-case numerical integration over R^d with respect to the standard Gaussian measure in Gaussian Sobolev spaces H^α_ρ of dominating mixed smoothness. Its main results are: (i) a lower bound e(S_Λ) ≳ N^{-α/2} for sparse-grid Gauss–Hermite quadrature for downward-closed index sets, and N^{-α/2}(ln N)^{α(d-1)/2} for the classical isotropic Smolyak index set, matched up to a logarithmic factor by an upper bound; and (ii) upper bounds N^{-α}(ln N)^γ for four QMC constructions (affine-mapped rank-1 lattices and higher-order digital nets, and cotangent/Möbius-mapped rank-1 lattices and higher-order digital nets), with γ=(d-1)/2 for the last construction, matching the information-theoretic lower bound of Dick et al. [11]. The proofs use a univariate fooling function from the authors' earlier paper [41], a new mixed-Sobolev embedding (Prop. 2.3), Smolyak theory, and a chain-rule analysis for the cotangent transform.

Significance. If the results hold, the paper rigorously explains the observed gap between sparse-grid Gauss–Hermite and QMC methods in a clean model setting: sparse-grid Gauss–Hermite is suboptimal by a square root in N, while QMC, especially higher-order digital nets with the cotangent transform, is optimal including the logarithmic factor. The paper is carefully written and contains several self-contained ingredients, notably the Sobolev embedding in Prop. 2.3 and the chain-rule analysis of §4.2. However, the central lower bound is imported from the authors' earlier [41, Thm 3.2] without proof, and the proof of Theorem 3.3 contains a mismatch between quadrature and interpolation nodes. These issues need to be addressed before the claims are fully supported.

major comments (2)
  1. [§3.2, Theorem 3.3] The proof asserts the equality |∫ p_n ρ − Q^uni_{ℓ̄_1}(p_n)| = |∫ (p_n − I^uni_{ℓ̄_1}(p_n)) ρ| because I^uni is interpolatory. This equality requires Q^uni_{ℓ̄_1}(p_n) = ∫ I^uni_{ℓ̄_1}(p_n) ρ. But Q^uni_ℓ uses the n_ℓ roots of H_{n_ℓ}, while I^uni_ℓ is defined to use the zeros of H_{n_ℓ+1}. These node sets differ, and in general Q^uni(p_n) − ∫ I^uni(p_n)ρ = Σ_j w_j [p_n(ξ_j) − I^uni(p_n)(ξ_j)] ≠ 0. Thus the reduction in the proof does not follow. Please either change the definition of I^uni_ℓ to use the same nodes as Q^uni_ℓ, or supply a different argument. This issue does not affect Theorems 3.1–3.2, but it leaves Theorem 3.3 unsupported as written.
  2. [§3.1, Theorems 3.1 and 3.2] The multivariate lower-bound proof correctly reduces the worst-case error to the univariate ratio in Eq. (5), but the N^{-α/2} rate is then taken verbatim from [41, Theorem 3.2] without statement or proof. Since that univariate estimate is the load-bearing step for the paper's central suboptimality claim and is a self-citation, the manuscript should state the theorem and include a proof or at least a complete proof sketch, in particular the n-dependence of ||p_n||_{H^α_ρ} and the constant in the lower bound. This is not an accusation of circularity, but a request for self-contained verification of the only step that gives the polynomial rate.
minor comments (4)
  1. [Eq. (5)] The denominator on the right-hand side of Eq. (5) is missing the square root: it should be (Σ_{r=0}^α ||D^r p_n||^2_{L^2_ρ})^{1/2}, consistent with the definition of the H^α_ρ norm.
  2. [Theorems 3.2 and 3.3] The logarithmic exponent in the statements is written as (ln N)^{a(d-1)/2} with 'a' instead of α. The proofs also use 'a' in the same way; this should be corrected.
  3. [§3.1] After the telescoping identity S_Λ(h_Λ) = Q^uni_{ℓ̄_1}(p_n), it would help to state explicitly that lower-level Gauss–Hermite nodes cancel and never appear. This is what justifies the abstract's claim that the lower bound is independent of quadrature weights.
  4. [§2.2] The sentence explaining that a continuous linear functional can be viewed as an operator into H^α_ρ(R) is confusing: a functional maps into R, not into H^α_ρ(R). The intended embedding via constants should be spelled out more clearly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the sparse-grid lower bound reduces to a published univariate theorem and the QMC results are inherited from independent QMC theory.

full rationale

The central lower-bound proof (Theorem 3.1) is a genuine reduction: it constructs the univariate fooling function p_n, shows by the telescoping structure of Smolyak's formula that S_Λ(h_Λ) = Q^uni_ℓbar(p_n) = 0 because p_n vanishes at the Gauss–Hermite nodes, and derives equation (5), which expresses the multivariate worst-case ratio as the same univariate ratio studied in [41]. The statement that this ratio is at least c_α n^{-α/2} is taken from [41, Theorem 3.2], a published, parameter-free theorem about univariate Gauss–Hermite quadrature whose assumptions do not include the present multivariate sparse-grid claim. That is independent support, not a self-referential definition. The logarithmic factor in Theorem 3.2 follows from counting sparse-grid nodes via equations (6)–(7), not from the univariate theorem. The QMC optimality results similarly build on published theorems ([11], [12], [29], [55]) together with a new embedding result, Proposition 4.7, which is proved in detail using Lemmas 4.3–4.6; the cotangent-transform membership of H^α_ρ in Korobov and unanchored Sobolev spaces is established rather than assumed. The 'independent of weights' consequence follows from the telescoping identity and p_n = 0 at the nodes, so it is a proved corollary, not a renaming. No fitted parameter is relabeled as a prediction, and no uniqueness or ansatz is imported from the authors' earlier work in a load-bearing circular way. The paper does not reproduce the proof of [41, Theorem 3.2], but citing a published theorem is standard mathematical practice and does not itself constitute circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central sparse-grid suboptimality result rests on the univariate fooling-function analysis of the authors' own prior paper [41], a published, peer-reviewed result, which is independent support rather than circular reasoning. The QMC results rely on standard CBC lattice constructions and higher-order digital nets with published error bounds. The only hand-chosen parameter in the analysis is the affine-map scale b, which is tuned to balance errors, not fitted to data.

free parameters (1)
  • b (affine map scale) = sqrt(alpha/2 ln N) or 2 sqrt(alpha) ln N depending on QMC construction
    Domain truncation parameter chosen by hand to balance truncation error and QMC error; affects constants and log exponents, not the polynomial rate.
assumptions (4)
  • domain assumption Optimal lower bound for general algorithms in H^alpha_rho (Dick et al. 2018, Theorem 2.1): worst-case error >= c (ln N)^{(d-1)/2} / N^alpha.
    Serves as the benchmark; QMC optimality is measured against this rate.
  • domain assumption Univariate Gauss-Hermite error bounds from Kazashi-Suzuki-Goda [41]: lower bound (their Theorem 3.2) and upper bound (their Proposition 3.4).
    The multivariate sparse-grid lower and upper bounds reduce to these; the proofs are not reproduced in this paper.
  • domain assumption Node-count growth assumption: n_l satisfies M^{l-1} <= n_l <= M0(M^l - 1) with M > 1 (used in Theorem 3.2 and Proposition 3.5).
    Connects the level l to the total point count N in the Smolyak construction; standard in sparse-grid analysis.
  • domain assumption Published QMC error bounds: rank-1 lattice construction [12, Algorithm 3.14] gives N^{-alpha} (ln N)^{d*alpha} in Korobov spaces; higher-order digital nets [29, Theorem 1] give N^{-alpha} (ln N)^{(d-1)/2} in unanchored Sobolev spaces; scaled lattice result from [55].
    The QMC optimality results inherit these bounds after the function-space embedding is established.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Optimality of quasi-Monte Carlo methods and suboptimality of the sparse-grid Gauss--Hermite rule in Gaussian Sobolev spaces." pith.science (2026). https://pith.science/paper/AOLT2GQQ

@misc{pith2026250918712,
  author       = {Pith},
  title        = {Pith review of: Optimality of quasi-Monte Carlo methods and suboptimality of the sparse-grid Gauss--Hermite rule in Gaussian Sobolev spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AOLT2GQQ}},
  note         = {Machine review of arXiv:2509.18712}
}
abstract

Optimality of several quasi-Monte Carlo methods and suboptimality of the sparse-grid quadrature based on the univariate Gauss--Hermite rule is proved in the Sobolev spaces of mixed dominating smoothness of order $\alpha$, where the optimality is in the sense of worst-case convergence rate. For sparse-grid Gauss--Hermite quadrature, lower and upper bounds are established, with rates coinciding up to a logarithmic factor. The dominant rate is found to be only $N^{-\alpha/2}$ with $N$ function evaluations, although the optimal rate is known to be $N^{-\alpha}(\ln N)^{(d-1)/2}$. The lower bound is obtained by exploiting the structure of the Gauss--Hermite nodes and is independent of the quadrature weights; consequently, no modification of the weights can improve the rate $N^{-\alpha/2}$. In contrast, several quasi-Monte Carlo methods with a change of variables are shown to achieve the optimal rate, some up to, and one including, the logarithmic factor.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

70 extracted references · 25 canonical work pages

  1. [41]

    Suboptimality of Gauss–Hermite quadrature and op- timality of the trapezoidal rule for functions with finite smoothness

    Y. Kazashi, Y. Suzuki, and T. Goda. “Suboptimality of Gauss–Hermite quadrature and op- timality of the trapezoidal rule for functions with finite smoothness”. SIAM J. Numer. Anal. 61 (2023), pp. 1426–1448. doi: 10.1137/22M1480276

    doi:10.1137/22m1480276 2023

  2. [55]

    Scaled lattice rules for integration on Rd achieving higher-order convergence with error analysis in terms of orthogonal projections onto periodic spaces

    D. Nuyens and Y. Suzuki. “Scaled lattice rules for integration on Rd achieving higher-order convergence with error analysis in terms of orthogonal projections onto periodic spaces”.Math. Comp. 92 (2022), pp. 307–347. doi: 10.1090/mcom/3754

    doi:10.1090/mcom/3754 2022

  3. [11]

    On the Optimal Order of Integra- tion in Hermite Spaces with Finite Smoothness

    J. Dick, C. Irrgeher, G. Leobacher, and F. Pillichshammer. “On the Optimal Order of Integra- tion in Hermite Spaces with Finite Smoothness”. SIAM J. Numer. Anal. 56 (2018), pp. 684–

    2018

  4. [1]

    Quasi-monte carlo feature maps for shift-invariant kernels

    H. Avron, V. Sindhwani, J. Yang, and M. W. Mahoney. “Quasi-monte carlo feature maps for shift-invariant kernels”. J. Mach. Learn. Res. 17 (2016), pp. 1–38

    2016

  5. [2]

    On the optimality of linear methods for operator approximation in convex classes of functions

    N. S. Bakhvalov. “On the optimality of linear methods for operator approximation in convex classes of functions”. USSR Computational Mathematics and Mathematical Physics 11 (1971), pp. 244–249. doi: 10.1016/0041-5553(71)90017-6

    work page doi:10.1016/0041-5553(71)90017-6 1971

  6. [3]

    T. D. Barfoot. State Estimation for Robotics: Second Edition . 2nd ed. Cambridge University Press, 2024. doi: 10.1017/9781009299909

    work page doi:10.1017/9781009299909 2024

  7. [4]

    The Gaussian wave packet transform via quadrature rules

    P. Bergold and C. Lasser. “The Gaussian wave packet transform via quadrature rules”. IMA J. Numer. Anal. 44 (2024), pp. 1785–1820. doi: 10.1093/imanum/drad049

    doi:10.1093/imanum/drad049 2024

  8. [5]

    H. Brezis. Functional analysis, sobolev spaces and partial differential equations . Springer New York, 2010

    2010

Show all 70 references
  1. [6]

    Sparse grids

    H.-J. Bungartz and M. Griebel. “Sparse grids”. Acta Numer. 13 (2004), pp. 147–269. doi: 10.1017/S0962492904000182

    2004 doi

  2. [7]

    Sparse quadrature for high-dimensional integration with Gaussian measure

    P. Chen. “Sparse quadrature for high-dimensional integration with Gaussian measure”. ESAIM: M2AN 52 (2018), pp. 631–657. doi: 10.1051/m2an/2018012

    2018

  3. [8]

    High-Dimensional Adaptive Sparse Polynomial Inter- polation and Applications to Parametric PDEs

    A. Chkifa, A. Cohen, and C. Schwab. “High-Dimensional Adaptive Sparse Polynomial Inter- polation and Applications to Parametric PDEs”. Found Comput Math 14 (2014), pp. 601–633. doi: 10.1007/s10208-013-9154-z . 22

    2014 doi

  4. [9]

    Multilevel and quasi-Monte Carlo methods for uncertainty quantification in particle travel times through random heterogeneous porous media

    D. Crevill´ en-Garc ´ ıa and H. Power. “Multilevel and quasi-Monte Carlo methods for uncertainty quantification in particle travel times through random heterogeneous porous media”. R. Soc. Open Sci. 4 (2017), p. 170203. doi: 10.1098/rsos.170203

    2017 doi

  5. [10]

    Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order

    J. Dick. “Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order”. SIAM J. Numer. Anal. 46 (2008), pp. 1519–1553. doi: 10.1137/060666639

    2008 doi

  6. [12]

    J. Dick, P. Kritzer, and F. Pillichshammer. Lattice Rules: Numerical Integration, Approxi- mation, and Discrepancy . Vol. 58. Springer Series in Computational Mathematics. Springer International Publishing, 2022. doi: 10.1007/978-3-031-09951-9

    2022 doi

  7. [13]

    High-dimensional integration: The quasi-Monte carlo way

    J. Dick, F. Y. Kuo, and I. H. Sloan. “High-dimensional integration: The quasi-Monte carlo way”. Acta Numer. 22 (2013), pp. 133–288. doi: 10.1017/S0962492913000044

    2013 doi

  8. [14]

    Dick and F

    J. Dick and F. Pillichshammer. Digital Nets and Sequences: Discrepancy Theory and Quasi– Monte Carlo Integration. 1st ed. Cambridge University Press, 2010.doi: 10.1017/CBO9780511761188

    2010 doi

  9. [15]

    Sparse-grid polynomial interpolation approximation and integration for parametric and stochastic elliptic PDEs with lognormal inputs

    D. D˜ ung. “Sparse-grid polynomial interpolation approximation and integration for parametric and stochastic elliptic PDEs with lognormal inputs”. ESAIM: M2AN 55 (2021), pp. 1163–

    2021

  10. [16]

    Optimal numerical integration and approximation of functions on Rd equipped with Gaussian measure

    D. D˜ ung and V. Kien Nguyen. “Optimal numerical integration and approximation of functions on Rd equipped with Gaussian measure”. IMA J. Numer. Anal. 44 (2024), pp. 1242–1267. doi: 10.1093/imanum/drad051

    2024 doi

  11. [17]

    D˜ ung, V

    D. D˜ ung, V. K. Nguyen, C. Schwab, and J. Zech. Analyticity and Sparsity in Uncertainty Quantification for PDEs with Gaussian Random Field Inputs . Vol. 2334. Lecture Notes in Mathematics. Springer International Publishing, 2023. doi: 10.1007/978-3-031-38384-7

    2023 doi

  12. [18]

    An abstract approach to Marcinkiewicz-Zygmund inequalities for approximation and quadrature in modulation spaces

    M. Ehler and K. Gr¨ ochenig. “An abstract approach to Marcinkiewicz-Zygmund inequalities for approximation and quadrature in modulation spaces”. Math. Comp. (2023). doi: 10.1090/ mcom/3930

    2023

  13. [19]

    Multidimensional projection filters via automatic dif- ferentiation and sparse-grid integration

    M. F. Emzir, Z. Zhao, and S. S¨ arkk¨ a. “Multidimensional projection filters via automatic dif- ferentiation and sparse-grid integration”. Signal Processing 204 (2023), p. 108832. doi: 10. 1016/j.sigpro.2022.108832

    2023

  14. [20]

    O. G. Ernst, H. Gottschalk, T. Kowalewitz, and P. Kr¨ uger. Learning to Integrate. 2025. doi: 10.48550/arXiv.2506.11801. arXiv: 2506.11801 [math]. Pre-published

    2025 doi arXiv

  15. [21]

    Convergence of Sparse Collocation for Functions of Countably Many Gaussian Random Variables (with Application to Elliptic PDEs)

    O. G. Ernst, B. Sprungk, and L. Tamellini. “Convergence of Sparse Collocation for Functions of Countably Many Gaussian Random Variables (with Application to Elliptic PDEs)”. SIAM J. Numer. Anal. 56 (2018), pp. 877–905. doi: 10.1137/17M1123079

    2018 doi

  16. [22]

    On Expansions and Nodes for Sparse Grid Col- location of Lognormal Elliptic PDEs

    O. G. Ernst, B. Sprungk, and L. Tamellini. “On Expansions and Nodes for Sparse Grid Col- location of Lognormal Elliptic PDEs”. Sparse Grids and Applications - Munich 2018 . Ed. by H.-J. Bungartz, J. Garcke, and D. Pfl¨ uger. Vol. 144. Springer International Publishing, 2021, p...

    2018 doi

  17. [23]

    Numerical Simulation for Asset-Liability Management in Life Insurance

    T. Gerstner, M. Griebel, M. Holtz, R. Goschnick, and M. Haep. “Numerical Simulation for Asset-Liability Management in Life Insurance”. Mathematics – Key Technology for the Future. Ed. by H.-J. Krebs and W. J¨ ager. Springer Berlin Heidelberg, 2008, pp. 319–341. doi: 10. 1007/9...

    2008

  18. [24]

    Glasserman

    P. Glasserman. Monte Carlo Methods in Financial Engineering . Red. by B. Rozovskii and M. Yor. Vol. 53. Stochastic Modelling and Applied Probability. Springer New York, 2003. doi: 10.1007/978-0-387-21617-1 . 23

    2003 doi

  19. [25]

    Countable tensor products of Hermite spaces and spaces of Gaussian kernels

    M. Gnewuch, M. Hefter, A. Hinrichs, and K. Ritter. “Countable tensor products of Hermite spaces and spaces of Gaussian kernels”. Journal of Complexity . Approximation and Geometry in High Dimensions 71 (2022), p. 101654. doi: 10.1016/j.jco.2022.101654

    2022

  20. [26]

    Infinite-dimensional integration and L2-approximation on Hermite spaces

    M. Gnewuch, A. Hinrichs, K. Ritter, and R. R¨ ußmann. “Infinite-dimensional integration and L2-approximation on Hermite spaces”. Journal of Approximation Theory 300 (2024), p. 106027. doi: 10.1016/j.jat.2024.106027

    2024

  21. [27]

    How Sharp Are Error Bounds? –Lower Bounds on Quadrature Worst-Case Errors for Analytic Functions–

    T. Goda, Y. Kazashi, and K. Tanaka. “How Sharp Are Error Bounds? –Lower Bounds on Quadrature Worst-Case Errors for Analytic Functions–”. SIAM J. Numer. Anal. 62 (2024), pp. 2370–2392. doi: 10.1137/24M1634163

    2024 doi

  22. [28]

    Recent advances in higher order quasi-Monte Carlo methods

    T. Goda and K. Suzuki. “Recent advances in higher order quasi-Monte Carlo methods”. Dis- crepancy theory. Vol. 26. Radon Ser. Comput. Appl. Math. De Gruyter, Berlin, 2020, pp. 69–

    2020

  23. [29]

    Optimal Order Quadrature Error Bounds for Infinite- Dimensional Higher-Order Digital Sequences

    T. Goda, K. Suzuki, and T. Yoshiki. “Optimal Order Quadrature Error Bounds for Infinite- Dimensional Higher-Order Digital Sequences”. Found. Comput. Math. 18 (2018), pp. 433–458. doi: 10.1007/s10208-017-9345-0

    2018 doi

  24. [30]

    Quasi- Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients

    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), pp. 329–368. doi: 10.1007/s00211-014-0689-y

    2015 doi

  25. [31]

    On the quasi-Monte Carlo method with Halton points for elliptic PDEs with log-normal diffusion

    H. Harbrecht, M. Peters, and M. Siebenmorgen. “On the quasi-Monte Carlo method with Halton points for elliptic PDEs with log-normal diffusion”. Math. Comp. 86 (2016), pp. 771–

    2016

  26. [32]

    QMC integration for lognormal-parametric, elliptic PDEs: local supports and product weights

    L. Herrmann and C. Schwab. “QMC integration for lognormal-parametric, elliptic PDEs: local supports and product weights”. Numer. Math. 141 (2019), pp. 63–102. doi: 10.1007/s00211- 018-0991-1

    2019 doi

  27. [33]

    Quadrature error bounds with applications to lattice rules

    F. J. Hickernell. “Quadrature error bounds with applications to lattice rules”. SIAM J. Numer. Anal. 33 (1996), pp. 1995–2016. doi: 10.1137/S0036142994261439

    1996 doi

  28. [34]

    Derivative polynomials for tangent and secant

    M. E. Hoffman. “Derivative polynomials for tangent and secant”. Amer. Math. Monthly 102 (1995), pp. 23–30. doi: 10.2307/2974853

    1995 doi

  29. [35]

    Integration in Hermite spaces of analytic functions

    C. Irrgeher, P. Kritzer, G. Leobacher, and F. Pillichshammer. “Integration in Hermite spaces of analytic functions”. Journal of Complexity 31 (2015), pp. 380–404. doi: 10.1016/j.jco. 2014.08.004

    2015 doi

  30. [36]

    Gaussian filters for nonlinear filtering problems

    K. Ito and K. Xiong. “Gaussian filters for nonlinear filtering problems”. IEEE Trans. Autom. Control 45 (2000), pp. 910–927. doi: 10.1109/9.855552

    2000 doi

  31. [37]

    Quasi-Monte Carlo methods for lattice systems: A first look

    K. Jansen, H. Leovey, A. Ammon, A. Griewank, and M. M¨ uller-Preussker. “Quasi-Monte Carlo methods for lattice systems: A first look”. Computer Physics Communications 185 (2014), pp. 948–959. doi: 10.1016/j.cpc.2013.10.011

    2014 doi

  32. [38]

    Sparse Gauss-Hermite Quadrature Filter with Application to Spacecraft Attitude Estimation

    B. Jia, M. Xin, and Y. Cheng. “Sparse Gauss-Hermite Quadrature Filter with Application to Spacecraft Attitude Estimation”. Journal of Guidance, Control, and Dynamics 34 (2011), pp. 367–379. doi: 10.2514/1.52016

    2011 doi

  33. [39]

    Fast approximation by pe- riodic kernel-based lattice-point interpolation with application in uncertainty quantification

    V. Kaarnioja, Y. Kazashi, F. Y. Kuo, F. Nobile, and I. H. Sloan. “Fast approximation by pe- riodic kernel-based lattice-point interpolation with application in uncertainty quantification”. Numer. Math. 150 (2022), pp. 33–77. doi: 10.1007/s00211-021-01242-3

    2022 doi

  34. [40]

    Quasi–Monte Carlo integration with product weights for elliptic PDEs with log- normal coefficients

    Y. Kazashi. “Quasi–Monte Carlo integration with product weights for elliptic PDEs with log- normal coefficients”. IMA J. Numer. Anal. 39 (2019), pp. 1563–1593. doi: 10.1093/imanum/ dry028. 24

    2019 doi

  35. [42]

    Quasi-Monte Carlo sampling method for simulation-based dynamic probabilistic risk assessment of nuclear power plants

    K. Kubo, S. Jang, T. Takashi, and A. Yamaguchi. “Quasi-Monte Carlo sampling method for simulation-based dynamic probabilistic risk assessment of nuclear power plants”. Journal of Nuclear Science and Technology 59 (2022), pp. 357–367. doi: 10 . 1080 / 00223131 . 2021 . 1971119

    2022

  36. [43]

    Quasi-Monte Carlo methods with applications in finance

    P. L’Ecuyer. “Quasi-Monte Carlo methods with applications in finance”. Finance Stoch. 13 (2009), pp. 307–349. doi: 10.1007/s00780-009-0095-y

    2009 doi

  37. [44]

    G. Leoni. A first course in Sobolev spaces . 2nd ed. Vol. 181. Graduate studies in mathematics. American Mathematical Society, Providence, RI, 2017. xxii+734

    2017

  38. [45]

    C. Lubich. From quantum to classical molecular dynamics: Reduced models and numerical analysis. European Mathematical Society Publishing House, 2008. doi: 10.4171/067

    2008 doi

  39. [46]

    PDE-Constrained Optimal Control Problems with Uncertain Pa- rameters using SAGA

    M. Martin and F. Nobile. “PDE-Constrained Optimal Control Problems with Uncertain Pa- rameters using SAGA”. SIAM/ASA J. Uncertainty Quantification 9 (2021), pp. 979–1012. doi: 10.1137/18M1224076

    2021 doi

  40. [47]

    Enhancing Accuracy of Deep Learning Algorithms by Training with Low-Discrepancy Sequences

    S. Mishra and T. K. Rusch. “Enhancing Accuracy of Deep Learning Algorithms by Training with Low-Discrepancy Sequences”. SIAM J. Numer. Anal. 59 (2021), pp. 1811–1834. doi: 10.1137/20M1344883

    2021 doi

  41. [48]

    Adaptive Leja Sparse Grid Constructions for Stochastic Collocation and High-Dimensional Approximation

    A. Narayan and J. D. Jakeman. “Adaptive Leja Sparse Grid Constructions for Stochastic Collocation and High-Dimensional Approximation”. SIAM J. Sci. Comput. 36 (2014), A2952– A2983. doi: 10.1137/140966368

    2014 doi

  42. [49]

    Fast CBC construction of randomly shifted lattice rules achieving convergence for unbounded integrands over in weighted spaces with POD weights

    J. A. Nichols and F. Y. Kuo. “Fast CBC construction of randomly shifted lattice rules achieving convergence for unbounded integrands over in weighted spaces with POD weights”.J. Complex. 30 (2014), pp. 444–468. doi: 10.1016/j.jco.2014.02.004

    2014 doi

  43. [50]

    Low-discrepancy and low-dispersion sequences

    H. Niederreiter. “Low-discrepancy and low-dispersion sequences”. J. Number Theory 30 (1988), pp. 51–70. doi: 10.1016/0022-314X(88)90025-X

    1988 doi

  44. [51]

    Low-discrepancy sequences and global function fields with many rational places

    H. Niederreiter and C. Xing. “Low-discrepancy sequences and global function fields with many rational places”. Finite Fields Appl. 2 (1996), pp. 241–273. doi: 10.1006/ffta.1996.0016

    1996

  45. [52]

    Niederreiter

    H. Niederreiter. Random Number Generation and Quasi-Monte Carlo Methods . Society for Industrial and Applied Mathematics, 1992. doi: 10.1137/1.9781611970081

    1992 doi

  46. [53]

    Novak and H

    E. Novak and H. Wo´ zniakowski. Tractability of Multivariate Problems. Volume I: Linear In- formation. European Mathematical Society Publishing House, 2008. doi: 10.4171/026

    2008 doi

  47. [54]

    Novak and H

    E. Novak and H. Wo´ zniakowski. Tractability of Multivariate Problems. Volume II: Standard Information for Functionals . 1st ed. Vol. 12. EMS Tracts in Mathematics. EMS Press, 2010. doi: 10.4171/084

    2010 doi

  48. [56]

    Multiple sparse-grid Gauss– Hermite filtering

    R. Radhakrishnan, A. K. Singh, S. Bhaumik, and N. K. Tomar. “Multiple sparse-grid Gauss– Hermite filtering”. Applied Mathematical Modelling 40 (2016), pp. 4441–4450. doi: 10.1016/ j.apm.2015.11.035

    2016

  49. [57]

    Reed and B

    M. Reed and B. Simon. Methods of modern mathematical physics. I . 2nd ed. Academic Press,

  50. [58]

    A Multi-Index Quasi–Monte Carlo Algorithm for Lognormal Diffusion Problems

    P. Robbe, D. Nuyens, and S. Vandewalle. “A Multi-Index Quasi–Monte Carlo Algorithm for Lognormal Diffusion Problems”. SIAM J. Sci. Comput. 39 (2017), S851–S872. doi: 10.1137/ 16M1082561

    2017

  51. [59]

    When are quasi-monte carlo algorithms efficient for high dimensional integrals?

    I. H. Sloan and H. Wo´ zniakowski. “When are quasi-monte carlo algorithms efficient for high dimensional integrals?” J. Complex. 14 (1998), pp. 1–33. doi: 10.1006/jcom.1997.0463

    1998

  52. [60]

    I. H. Sloan and S. Joe. Lattice methods for multiple integration . Oxford science publications. The Clarendon Press, Oxford University Press, New York, 1994. xii+239

    1994

  53. [61]

    Quadrature and interpolation formulas for tensor products of certain classes of functions

    S. A. Smolyak. “Quadrature and interpolation formulas for tensor products of certain classes of functions”. Dokl. Akad. Nauk SSSR 148 (1963), pp. 1042–1045

    1963

  54. [62]

    Distribution of points in a cube and approximate evaluation of integrals

    I. M. Sobol’. “Distribution of points in a cube and approximate evaluation of integrals”. ˇZ. Vyˇ cisl. Mat i Mat. Fiz.7 (1967), pp. 784–802

    1967

  55. [63]

    Optimality of the double exponential formula – functional analysis approach –

    M. Sugihara. “Optimality of the double exponential formula – functional analysis approach –”. Numer. Math. 75 (1997), pp. 379–395. doi: 10.1007/s002110050244

    1997 doi

  56. [64]

    M¨ obius-transformed trapezoidal rule

    Y. Suzuki, N. Hyv¨ onen, and T. Karvonen. “M¨ obius-transformed trapezoidal rule”. Math. Comp. (2025). doi: 10.1090/mcom/4084

    2025 doi

  57. [65]

    Explicit Cost Bounds of Algorithms for Multivari- ate Tensor Product Problems

    G. W. Wasilkowski and H. Wozniakowski. “Explicit Cost Bounds of Algorithms for Multivari- ate Tensor Product Problems”. Journal of Complexity 11 (1995), pp. 1–56. doi: 10.1006/ jcom.1995.1001

    1995

  58. [66]

    On sparse grid interpolation for American option pricing with multiple underlying assets

    J. Yang and G. Li. “On sparse grid interpolation for American option pricing with multiple underlying assets”. Journal of Computational and Applied Mathematics 464 (2025), p. 116544. doi: 10.1016/j.cam.2025.116544. 26

    2025

  59. [102]

    doi: 10.1515/9783110652581-004

    doi

  60. [707]

    doi: 10.1137/16M1087461

    doi

  61. [797]

    doi: 10.1090/mcom/3107

    doi

  62. [1198]

    doi: 10.1051/m2an/2021017

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.