Quadrature Points via Heat Kernel Repulsion


We discuss the classical problem of how to pick N weighted points on a d-dimensional manifold so as to obtain a reasonable quadrature rule

$$\begin{aligned} \frac{1}{|M|}\int _{M}{f(x) \mathrm{d}x} \simeq \sum _{n=1}^{N}{a_i f(x_i)}. \end{aligned}$$

This problem, naturally, has a long history; the purpose of our paper is to propose selecting points and weights so as to minimize the energy functional

$$\begin{aligned} \sum _{i,j =1}^{N}{ a_i a_j \exp \left( -\frac{d(x_i,x_j)^2}{4t}\right) } \rightarrow \min , \quad \text{ where }~t \sim N^{-2/d}, \end{aligned}$$

d(xy) is the geodesic distance, and d is the dimension of the manifold. This yields point sets that are theoretically guaranteed, via spectral theoretic properties of the Laplacian \(-\Delta \), to have good properties. One nice aspect is that the energy functional is universal and independent of the underlying manifold; we show several numerical examples.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 99

This is the net price. Taxes to be calculated in checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10


  1. 1.

    Geometrically, this projection can be interpreted as the intersection of the line connecting the point \((x_{1},x_{2},x_{3})\) and the point \((0,0,-1)\) with the hyperplane spanned by the first two canonical basis vectors in \(\mathbb {R}^{3}\).


  1. 1.

    Abrikosov, A.: On the magnetic properties of superconductors of the second type. Sov. Phys. JETP 5, 1174–1182 (1957)

  2. 2.

    Ahrens, C., Beylkin, G.: Rotationally invariant quadratures for the sphere. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465(2110), 3103–3125 (2009)

  3. 3.

    Blanc, X., Lewin, M.: The crystallization conjecture: a review. EMS Surv. Math. Sci. 2, 255–306 (2015)

  4. 4.

    Bilyk, D., Dai, F.: Geodesic distance Riesz energy on the sphere. arXiv:1612.08442

  5. 5.

    Bilyk, D., Dai, F., Steinerberger, S.: General and Refined Montgomery Lemmata. arXiv:1801.07701

  6. 6.

    Bondarenko, A., Radchenko, D., Viazovska, M.: Optimal asymptotic bounds for spherical designs. Ann. Math. 178(2), 443–452 (2013)

  7. 7.

    Bondarenko, A., Radchenko, D., Viazovska, M.: Well-separated spherical designs. Constr. Approx. 41(1), 93–112 (2015)

  8. 8.

    Brauchart, J., Grabner, P.: Distributing many points on spheres: minimal energy and designs. J. Complex. 31(3), 293–326 (2015)

  9. 9.

    Chatterjee, S.: Rigidity of the three-dimensional hierarchical Coulomb gas. arXiv:1708.01965

  10. 10.

    Chen, W., Mackey, L., Gorham, J., Briol, F.-X., Oates, C.J.: Stein points. arXiv:1803.10161

  11. 11.

    Coifman, R., Lafon, S., Lee, A., Maggioni, M., Nadler, B., Warner, F., Zucker, S.: Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps. PNAS 102, 7426–7431 (2005)

  12. 12.

    Coifman, R., Lafon, S., Lee, A., Maggioni, M., Nadler, B., Warner, F., Zucker, S.: Geometric diffusions as a tool for harmonic analysis and structure definition of data: multiscale methods. PNAS 102, 7432–7437 (2005)

  13. 13.

    Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups, Grundl. Math. Wissen. 290. Springer, New York (1999)

  14. 14.

    Dahlberg, B.: On the distribution of Fekete points. Duke Math. J. 45, 537–542 (1978)

  15. 15.

    Davis, P., Rabinowitz, P.: Methods of Numerical Integration. Computer Science and Applied Mathematics, 2nd edn. Academic Press Inc, Orlando (1984)

  16. 16.

    Delsarte, P., Goethals, J.M., Seidel, J.J.: Spherical codes and designs. Geom. Dedic. 6(3), 363–388 (1977)

  17. 17.

    Dick, J., Pillichshammer, F.: Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge (2010)

  18. 18.

    Drmota, M., Tichy, R.: Sequences, Discrepancies and Applications. Lecture Notes in Mathematics, 1651. Springer, Berlin (1997)

  19. 19.

    Ehler, M., Graef, M., Oates, C.J.: Optimal Monte Carlo integration on closed manifolds. arXiv:1707.04723

  20. 20.

    Grabner, P., Tichy, R.R.: Spherical designs, discrepancy and numerical integration. Math. Comput. 60, 327–336 (1993)

  21. 21.

    Halton, J.H.: On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer. Math. 2, 84–90 (1960)

  22. 22.

    Hannay, J.H., Nye, J.F.: Fibonacci numerical integration on a sphere. J. Phys. A Math. Gen. 37, 11591 (2004)

  23. 23.

    Hlawka, E.: For the approximate calculation of multiple integrals. Monatsh. Math. 66, 140–151 (1962)

  24. 24.

    Hsu, E.: Stochastic Analysis on Manifolds. Graduate Studies in Mathematics, 38. American Mathematical Society, Providence, RI (2002)

  25. 25.

    Jones, P.W., Maggioni, M., Schul, R.: Manifold parametrizations by eigenfunctions of the Laplacian and heat kernels. Proc. Natl. Acad. Sci. USA 105(6), 1803–1808 (2008)

  26. 26.

    Jones, P.W., Maggioni, M., Schul, R.: Universal local parametrizations via heat kernels and eigenfunctions of the Laplacian. Ann. Acad. Sci. Fenn. Math. 35(1), 131–174 (2010)

  27. 27.

    Korobov, N.M.: Number Theoretic Methods in Approximate Analysis. Fizmatgiz, Moscow (1963)

  28. 28.

    Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Pure and Applied Mathematics. Wiley-Interscience, New York-London-Sydney (1974)

  29. 29.

    Lebedev, V.I.: Quadratures on a sphere. Zh. Vȳchisl. Mat. Mat. Fiz. 16(2), 2930–29306 (1976)

  30. 30.

    Lebedev, V.I., Laikov, D.N.: A quadrature formula for the sphere of the 131st algebraic order of accuracy. Doklady Math. 59(3), 477–481 (1999)

  31. 31.

    Leimkuhler, B., Matthews, C.: Robust and efficient configurational molecular sampling via Langevin dynamics. J. Chem. Phys. 138(17), 174102 (2013)

  32. 32.

    Leimkuhler, B., Matthews, C., Stoltz, G.: The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics. IMA J. Numer. Anal. 36(1), 13–79 (2015)

  33. 33.

    Leimkuhler, B., Matthews, C.: Efficient molecular dynamics using geodesic integration and solvent-solute splitting. Proc. R. Soc. A 472(2189), 20160138 (2016)

  34. 34.

    Linderman, G., Steinerberger, S.: Numerical Integration on Graphs: Where to Sample and How to Weigh. arXiv:1803.06989

  35. 35.

    Mak, S., Roshan Joseph, V.: Support points. arXiv:1609.01811

  36. 36.

    McLaren, A.D.: Optimal numerical integration on a sphere. Math. Comput. 17(84), 361–383 (1963)

  37. 37.

    Montgomery, H.: Irregularities of distribution by means of power sums, Congress of Number Theory (Zarautz 1984). Universidad del Pais Vascom Bilbao 1989, 11–27 (1989)

  38. 38.

    Montgomery, H.: Ten Lectures at the Interface of Harmonic Analysis and Number Theory. American Mathematical Society (1994)

  39. 39.

    Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods, vol. 63. SIAM, Philadelphia (1992)

  40. 40.

    Novak, E., Wozniakowski, H.: Tractability of multivariate problems. Vol. 1: Linear information. EMS Tracts in Mathematics, 6. European Mathematical Society (EMS), Zürich (2008)

  41. 41.

    Novak, E., Wozniakowski, H.: Tractability of Multivariate Problems. Volume II: Standard Information for Functionals. EMS Tracts in Mathematics, 12. European Mathematical Society (EMS), Zürich, (2010)

  42. 42.

    Novak, E., Wozniakowski, H.: Tractability of multivariate problems. Volume III: Standard information for operators. EMS Tracts in Mathematics, 18. European Mathematical Society (EMS), Zürich (2012)

  43. 43.

    Pavliotis, G.A.: Stochastic Processes and Applications: Diffusion Processes, the Fokker–Planck and Langevin Equations, vol. 60. Springer, Berlin (2014)

  44. 44.

    Owen, A.B.: Quasi-Monte Carlo sampling. Monte Carlo Ray Tracing. Siggraph 1, 69–88 (2003)

  45. 45.

    Schwartz, R.E.: The five-electron case of Thomson’s problem. Exp. Math. 22(2), 157–186 (2013)

  46. 46.

    Serfaty, S.: Microscopic description of log and Coulomb gases. arXiv:1709.04089

  47. 47.

    Seymour, P., Zaslavsky, T.: Averaging sets: a generalization of mean values and spherical designs. Adv. Math. 52, 213–240 (1984)

  48. 48.

    Sobol, I.M.: Distribution of points in a cube and approximate evaluation of integrals. Zh. Vycisl. Mat. i Mat. Fiz. 7, 784–802 (1967)

  49. 49.

    Sobolev, S.: Cubature formulas on the sphere which are invariant under transformations of finite rotation groups. Dokl. Akad. Nauk SSSR 146, 310–313 (1962)

  50. 50.

    Steinerberger, S.: Exponential Sums and Riesz energies. J. Number Theory 182, 37–56 (2018)

  51. 51.

    Steinerberger, S.: Spectral Limitations of Quadrature Rules and Generalized Spherical Designs. arXiv:1708.08736

  52. 52.

    Steinerberger, S.: Generalized Designs on Graphs: Sampling, Spectra, Symmetries. arXiv:1803.02235

  53. 53.

    Thomson, J.J.: On the structure of the atom: an investigation of the stability and periods of oscillation of a number of corpuscles arranged at equal intervals around the circumference of a circle; with application of the results to the theory of atomic structure. Philos. Mag. 7, 237–265 (1904)

  54. 54.

    Varadhan, S.R.S.: On the behavior of the fundamental solution of the heat equation with variable coefficients. Commun. Pure Appl. Math. 20, 431–455 (1967)

  55. 55.

    Womersley, R.:

  56. 56.

    Yudin, V.A.: Lower bounds for spherical designs. Izv. Ross. Akad. Nauk Ser. Mat. 61 (1997), no. 3, 213–223; translation in Izv. Math. 61 (1997), no. 3, 673–683

  57. 57.

    Zaremba, S.K.: Good lattice points, discrepancy, and numerical integration. Ann. Mat. Pura Appl. 73(1), 293–317 (1966)

  58. 58.

    Anderson, J.W.: Hyperbolic Geometry. Springer, berlin (2006)

Download references

Author information

Correspondence to Matthias Sachs.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was partially supported by the National Science Foundation under Grant DMS-1638521 to the Statistical and Applied Mathematical Sciences Institute. The research of J.L. was also supported in part by the National Science Foundation under award DMS-1454939.

Communicated by Edward B. Saff.

Appendix A. Simulated Annealing Scheme for \(M=\mathbb {T}^{d}\)

Appendix A. Simulated Annealing Scheme for \(M=\mathbb {T}^{d}\)

In order to find good approximations of global minimizers of the energy functions \( E_\mathrm{Gaussian}\) and \(E_{\mathrm{Riesz},s}\), we use an annealing scheme based on the stochastic dynamics described by an Itô-diffusion of the form

$$\begin{aligned} \begin{aligned} \mathrm{d}{\varvec{x}}(t)&= {\varvec{p}}(t) \mathrm{d}t,\\ \mathrm{d}{\varvec{p}}(t)&= - \nabla _{{\varvec{x}}}U({\varvec{x}}(t)) \mathrm{d}t - \gamma {\varvec{p}}(t) \mathrm{d}t + \sqrt{2 \gamma \beta ^{-1}(t)} \mathrm{d}{\varvec{W}}(t),\\ \end{aligned} \end{aligned}$$


  1. (1)

    \({\varvec{W}}= (W_{1}, \ldots , W_{Nd})\), with \(W_{i}, 1\le i \le Nd\) being independent Wiener processes,

  2. (2)

    \({\varvec{x}}= ({\varvec{x}}_{1},\ldots ,{\varvec{x}}_{N})\) so that \({\varvec{x}}_{i}(t) \in \mathbb {T}^{d},1\le i \le N\),

  3. (3)

    \(\gamma >0\), and \(\beta \in \mathcal {C}([0,\infty ), \mathbb {R}_{+} )\),

with \(U \in \left\{ E_\mathrm{Gaussian}, E_{\mathrm{Riesz},s} \right\} \). The stochastic differential equation (SDE) (11) is known in the statistical physics literature as the underdamped Langevin equation. The underdamped Langevin equation can be viewed as a stochastic perturbed version of Hamilton’s equation associated with the Hamiltonian \( H({\varvec{x}},{\varvec{p}})= U({\varvec{x}}) +\frac{1}{2}\Vert {{\varvec{p}}}\Vert _{2}^{2}\). The remaining terms in (11) model the exchange of energy with a heat bath; see, e.g., [43] for more details. The parameter \(\gamma >0\) determines the strength of the coupling and as such can be interpreted as a friction coefficient. \(\beta (t)>0\) can be interpreted as the inverse temperature of the heat bath, and the function \(\beta : [0,\infty ) \rightarrow [0,\infty )\) is commonly referred to as a cooling schedule.

We discretize (11) using the well-studied “BAOAB”-splitting scheme (see Algorithm 1), which as a symmetric stochastic splitting scheme is of weak second order accuracy in the discretization/stepsize parameter \({\Delta t}\); see [31, 32] for details. We parametrize Algorithm 1 with a cooling schedule \(\beta ^{-1}(t) = \frac{C}{1+\log (t)}\), and we initialize \(\varvec{p}^{(k)}\) and \(\varvec{x}^{(k)}\) with \({\varvec{0}}\in \mathbb {R}^{Nd}\) and the Halton point set of appropriate size and dimension, respectively. The other parameters in Algorithm 1 and the value of the constant \(C>0\) in the cooling schedule are tuned for each optimization problem. We ensure that the system has settled in a local minimum at the end of the annealing procedure by monitoring the potential energy trajectory \(\left( U({\varvec{x}}^{(k)})\right) _{1\le k \le T}\).


A.1. Simulated Annealing Scheme for Smooth Hypersurfaces

For optimization on the sphere, the “dented” sphere, and the hyperboloid, we use a constrained version of the Langevin diffusion process (11); i.e., we consider (11) subject to

$$\begin{aligned} g({\varvec{x}}) = {\varvec{0}} \in \mathbb {R}^{N} \end{aligned}$$


$$\begin{aligned} \nabla _{{\varvec{x}}_{i}} g({\varvec{x}}) \cdot {\varvec{p}}_{i} = 0, ~1\le i \le N, \end{aligned}$$

where \(g = (g_{1},\ldots ,g_{N})\) is chosen such that the constraint (12) ensures that \({\varvec{x}}_{i}, ~ 1 \le i \le N\) are elements of the hypersurface, e.g.,

$$\begin{aligned} g_{i}({\varvec{x}}) = \Vert {{\varvec{x}}_{i}}\Vert _{2}^{2} - 1 = 0, ~~1 \le i \le N, \end{aligned}$$

in the case of \(M=\mathbb {S}^{d-1}\). We use the geodesic Langevin Integrator “g-BAOAB” (see [33] for details) in order to numerically integrate the constrained dynamics. The resulting annealing scheme resembles Algorithm 1. We use a cooling scheme of the same type as in the unconstrained case. In all examples, we set the initial velocity of each particle to zero, i.e., \({\varvec{p}}^{(0)}_{i} = {\varvec{0}}, ~~1 \le i \le N\). For \(M=\mathbb {S}^{2}\) we initialize \({\varvec{x}}^{(k)}\) by spherical Fibonacci point sets. In the case of the “dented” sphere example, we initialize \({\varvec{x}}^{(k)}\) by mapping the spherical Fibonacci point set (for \(N=89\)) onto the “dented” sphere using the projection map

$$\begin{aligned} (x_{1},x_{2},x_{3}) \mapsto \left( x_{1}, \mathrm{sign}(x_{2}) \sqrt{ (\alpha + x_{1}^2) x_{2}^2}, x_{3}\right) . \end{aligned}$$

In the example of the Poincaré disk model, we initialize the particles by first generating uniformly distributed points on the disk \(\{ (x_{1},x_{2}) : x_{1}^{2}+x_{2}^{2} \le 4/5 \}\), and then project these points onto the upper sheet of the hyperboloid model using the appropriately defined inverse of the projection (9). During simulation time the additional constraint (10) for the compactified hyperboloid is ensured to be (approximately) satisfied by adding the additional energy term \(\tilde{U}({\varvec{x}}) = \sum _{i=1}^{N} \tilde{U}_{i}({\varvec{x}}_{i})\) to the energy functional \(E_\mathrm{Gaussian}\), where

$$\begin{aligned} \tilde{U}_{i}( ({\varvec{x}}_{i,1},{\varvec{x}}_{i,2},{\varvec{x}}_{i,3})) = {\left\{ \begin{array}{ll} 0, &{}\quad \text {if } \vert {{\varvec{x}}_{i,3}}\vert \le c,\\ \kappa ( {\varvec{x}}_{i,3} - c)^{\alpha } &{}\quad \text {otherwise}, \end{array}\right. } \end{aligned}$$

with \(c = \frac{ 1+ r^{2}}{1 - r^{2}}, r =4/5\), and sufficiently large \(\alpha>1, \kappa >0\). As for the torus examples, the values of the remaining parameters in the annealing scheme are chosen problem dependently.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lu, J., Sachs, M. & Steinerberger, S. Quadrature Points via Heat Kernel Repulsion. Constr Approx 51, 27–48 (2020).

Download citation


  • Quadrature
  • Heat kernel
  • Numerical Integration

Mathematics Subject Classification

  • 41A55
  • 65D32 (primary)
  • 35K08 (secondary)