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Quadrature Points via Heat Kernel Repulsion

Abstract

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.

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Notes

  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 \(\mathbb {R}^{3}\).

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Correspondence to Matthias Sachs.

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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

$$\begin{aligned} \begin{aligned} \mathrm{d}{\varvec{x}}(t)&= {\varvec{p}}(t) \mathrm{d}t,\\ \mathrm{d}{\varvec{p}}(t)&= - \nabla _{{\varvec{x}}}U({\varvec{x}}(t)) \mathrm{d}t - \gamma {\varvec{p}}(t) \mathrm{d}t + \sqrt{2 \gamma \beta ^{-1}(t)} \mathrm{d}{\varvec{W}}(t),\\ \end{aligned} \end{aligned}$$
(11)

where

  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 \(\left( U({\varvec{x}}^{(k)})\right) _{1\le k \le T}\).

figurea

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

$$\begin{aligned} g({\varvec{x}}) = {\varvec{0}} \in \mathbb {R}^{N} \end{aligned}$$
(12)

and

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

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.

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Lu, J., Sachs, M. & Steinerberger, S. Quadrature Points via Heat Kernel Repulsion. Constr Approx 51, 27–48 (2020). https://doi.org/10.1007/s00365-019-09471-4

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Keywords

  • Quadrature
  • Heat kernel
  • Numerical Integration

Mathematics Subject Classification

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