Integration on manifolds by mapped low-discrepancy ...

Integration on manifolds by mapped low-discrepancy points and greedy minimal a,

Stefano De Marchi

a University

ks -energy

points

∗, Giacomo Elefantea

of Padova, Department of Mathematics, Via Trieste, 63, I-35121 Padova

Abstract Thanks to the

Poppy-seed Bagel Theorem

with minimal Riesz

s-energy,

we know that the class of points

under appropiate assumptions, are asymptot-

ically uniformly distributed with respect to the normalized Hausdor measure, and so they are an appropiate choice of points to integrate functions on manifolds via the Quasi-Monte Carlo method. A

greedy algorithm

minimal Riesz

is a method for computing an approximation of the

s-energy points that works by extracting selected points from

a larger discretization of the manifolds. However, we do not know if these extracted points remain a good choice to integrate functions. Through numerical experiments we attempt to answer to this question.

Keywords:

cubature on manifolds, Quasi-Monte Carlo method, greedy

minimal Riesz energy points, low-discrepancy points.

2010 MSC:

1. Introduction Monte Carlo (MC) and Quasi-Monte Carlo (QMC) methods are wellknown techniques in numerical analyis to integrate functions numerically. by taking the average of the function evaluated at a set of distributed uniformly with respect to a measure

∗

µ,

R

f (x)dµ(x) points on X

MC and QMC methods consist in the approximation of the integral

X

random for the Monte

Corresponding author

Email addresses: [email protected] (Stefano [email protected] (Giacomo Elefante)

Preprint submitted to Elsevier

De Marchi ),

March 4, 2016

Carlo method or deterministic for the Quasi-Monte Carlo method. The er-

[0, 1)d

ror bound in

for the Quasi-Monte Carlo method is the well known

Koksma-Hlawka inequality but there are also other similar inequalities for on other cases (see [1],[2], [3]).

1 Hd (M)

Now, let us consider the integral

d-dimensional e.g.

manifold and

[4, 11.2]).

Hd

is the

R

M f (x)dHd (x), where

d-dimensional

M

is a

Hausdor measure (see

In this case we prefer QMC method to other techinques

because it requires only a well-distributed point set on which to evaluate the function, whereas other cubature techinques may require more information, for example those based on nonnegative least squares or on approximate Fekete points (cf. [5]) require a polynomial basis on the manifold. In order to deduce convergence to the integral we may choose a deterministic point sequence that is uniformly distributed with respect to the Hausdor measure. We know from the Poppy-seed Bagel Theorem (see [6],[7]) that minimal Riesz

s-energy

points, with the right assumptions, are uniformly distributed

with respect to the Hausdor measure

Hd

so they represent an appropiate

set of points to integrate functions via the Quasi-Monte Carlo method. Furthermore, it is also possible to use a CPD (continuous and positive on the diagonal) weight function to distribute the points uniformly with respect to a given density (see [8]). We decided to compute these points approximating them via a greedy algorithm in order to generate the greedy

(w, s)-energy

ks -energy

points and the greedy

points in the weighted case.

We do not know if this greedy approximation to the minimal conguration of points is a suciently good choice to integrate functions with Quasi-Monte Carlo method. In [9] it is proved for these greedy points that the uniform distribution holds on the unit sphere

Sd

with

d≥1

so, encour-

aged by this, we decided to test numerically their cubature properties on manifolds. In the rst few sections we are going to introduce minimal Riesz energy points and greedy minimal

(w, s)-energy points in order to provide a suitable

background for the numerical experiments of Section 5.

2. Preliminaries Let

X

denote a compact Hausdor space and

measure on

µ

a regular unit Borel

X.

Denition 1. A sequence of points S = (xn )n≥1 in a compact Hausdor space X is uniformly distributed with respect to a measure µ (or µ-u.d.) if 2

for any real-valued bounded continuous function f : X → R we have that Z N 1 X f (xn ) = f (x)dµ(x). N →∞ N X lim

n=1

Furthermore we have that

Theorem 1. The sequence (xn )n∈N is µ-u.d. in X if and only if lim

N →∞

#(J; N ) = µ(J) N

holds for all µ-continuity sets J ⊆ X ; that is, Borel sets J such that µ(∂J) = where by #(J; N ) we mean the cardinality of J ∩ {χn }N n=1 .

0,

An equivalent way to describe the uniform distribution of a sequence is

in terms of For

#(J;N ) N − µ(J) .

[0, 1)d , where λd

is the Lebesgue measure, we introduce the following

notion of discrepancy:

Denition 2. Let P = {x0 , . . . , xN −1 } denote a nite point set in [0, 1)d and B a nonempty family of Lebesgue measurable subsets of [0, 1)d . Then #(B; N ; P ) DN (B; P ) := sup − λd (B) . N B∈B

Depending on the family B we have the following most important discrepancies. The star discrepancy DN∗ (P ) of the point set P is dened by DN∗ (P ) = DN (J ∗ ; P ), where J ∗ is the family of all subintervals of [0, 1)d of the form Q d i=1 [0, ai ). The extreme discrepancy DN (P ) of the point set P is dened by DN (P ) = d D N Qd (J ; P ), where J is the family of all subintervals of [0, 1) of the form i=1 [ai , bi ). The isotropic discrepancy JN (P ) of the point set P is dened by JN (P ) = DN (C; P ), where C is the family of all convex subsets of [0, 1)d . Star-discrepancy is important since it arises in estimating the error of the Quasi-Monte Carlo (QMC) method to integrate functions. In fact, on

[0, 1)d

we have the Koksma-Hlawka inequality (see e.g. [10])

Theorem 2. (Koksma-Hlawka inequality) If f has bounded variation V (f ) on [0, 1)d in the sense of Hardy and Krause, then for every P = 3

{x1 , . . . , xN } ⊂ [0, 1)d Z N 1 X ∗ f (x ) − f (x)dx (P ). ≤ V (f )DN n N d [0,1)

(1)

n=1

where DN∗ (P ) is the star-discrepancy of P . For the QMC method, because of this inequality it is natural to use

[0, 1)d . Low-discrepancy d order log(N ) /N . Some

low-discrepancy sequences to integrate functions in sequences are those whose star discrepancy has

examples of low-discrepancy sequences are the Halton sequence, Hammersley point sets, Sobol sequences or the Fibonacci lattice (for more information about them see e.g. [11]). If we integrate a function using QMC method in a convex subset of

[0, 1)d

we have a similar inequality due to Zaremba (see [1] ) where the isotropic discrepancy is used.

Theorem 3. Let B ⊆ [0, 1)d be a convex subset and f a function with bounded variation V (f ) on [0, 1)d in the sense of Hardy and Krause. Then, for any point set P = {x1 , . . . , xN } ⊆ [0, 1)d , we have that Z N 1 X f (xi ) − f (x)dx ≤ (V (f ) + |f (1)|)JN (P ), N B i=1

(2)

xi ∈B

where 1 = (1, . . . , 1). In the same way on manifold we have this Koksma-Hlawka like inequality (see [2] )

Theorem 4. Let M be a smooth compact d-dimensional manifold with a d normalized measure dx. Fix a family of local charts {ϕk }K k=1 , ϕk : [0, 1) → K M, and a smooth partition of unity {ψk }k=1 subordinate to these charts. Then there exists a number c > 0, which depends on the local charts but not on the function f or the measure µ, such that Z

M

f (y)dµ(y) ≤ cD(µ)||f ||W d,1 (M) ,

where D(µ) is dened as D(µ) = supU ∈A

R

4

U

dµ(y) , A

(3)

is the collection of

all intervals in M and ||f ||W n,p (M) =

X

Z

X

[0,1)

1≤k≤K |α|≤n In fact, if

dµ =

1 N

P

x∈XN

α p !1/p ∂ (ψk (ϕk (x))f (ϕk (x))) dx . α d ∂x

δx − dx

in (3), we have the analogue of the

Koksma-Hlawka inequality for manifolds.This suggest that we should minimize that discrepancy to have a smaller error. We remark that is not easy to compute an estimate of the error using this inequality and this discrepancy. In the spherical case we can use another method. Let us consider

S2 ,

it is proved (see e.g. [12]) that the worst case

error

Z 1 X 1 f (x) − f (x)dσ(x) , sup 4π S2 f ∈Ht (S2 ) N x∈X N

3 1 0, the s-energy of XN is given by Es (XN ) :=

X x6=y ∈ XN

X 1 = s |x − y|

X

y ∈ XN x ∈ XN x6=y

1 , |x − y|s

N

distinct (8)

where | · | denotes Euclidean distance in Rd . We dene also the N -point minimal s-energy over A by 0

Es (A, N ) := inf Es (XN ) XN ⊂A

8

(9)

By convention, the sum over an empty set of indices is taken to be zero and the inmum over an empty set is ∞. Notice that

¯ N) Es (A, N ) = Es (A,

and

Es (A, N ) = 0

if

A

is unbounded.

Hence, without loss of generality, we could restrict ourselves to the case where

A

is compact.

Let us dene now a CPD function.

Denition 4. Let A ⊂ Rd0 be an innite compact set whose d-dimensional Hausdor measure Hd (A) is nite. A symmetric function w : A × A → [0, +∞] is called a CPD-weight function (CPD stands for continuous and positive on the diagonal) on A × A if (i) w is continuous as function on A × A at Hd -almost every point of the diagonal D(A) = {(x, x) : x ∈ A}, (ii) there is some neighborhood G of D(A) such that inf G w > 0, (iii) w is bounded on any closed subset B ⊂ A × A such that B ∩ D(A) = ∅. By means of the previous denition we can dene the weighted Riesz

s-energy.

Denition 5. Let s > 0. Given a collection of N points XN A, the weighted Riesz s-energy of XN is dened by X

Esw (XN ) :=

1≤i6=j≤N

= {x1 , . . . , xN } ⊂

w(xi , xj ) , |xi − xj |s

while the N -point weighted Riesz s-energy of A is Esw (A, N ) = inf{Esw (XN ) : XN ⊂ A, #XN = N }.

And the weighted Hausdor measure Hds,w on Borel sets B ⊂ A is dened by Hds,w (B)

Z

(w(x, x))−d/s dHd (x).

= B

Recall moreover that is a set

A

A (see e.g. [13]), i.e. φ(B) = A and

onto s.t.

is said to be

d-rectiable

if and only if

Rd d there exist a constant L and a compact set B ⊂ R

there exist a Lipschitzian function

φ

mapping some bounded subset of

|φ(x) − φ(y)| ≤ L|x − y| ∀ x, y ∈ B.

9

The connection between the Riesz energy and a sequence uniformly distributed with respect to the Hausdor measure is given by the Poppy-seed Bagel Theorem (cf. [6],[8]). 0 Theorem 5. [Weighted Poppy-seed Bagel Theorem] Let A ⊂ Rd be a 0 a compact subset of a d-dimensional C 1 -manifold in Rd , d < d0 , and assume that w : A × A → [0, +∞] is a CDP-weight function on A × A. Then

d Edw (A, N ) Vol(B ) = d,w , 2 N →∞ N log N Hd (A)

lim

(10)

Furthermore, if Hd (A) > 0 and XN∗ is a sequence of congurations on A satisfying (10), with Edw (A, N ) replaced by Edw (XN∗ ), then d,w 1 X ∗ Hd (·)|A δx (·) −→ d,w N Hd (A) x∈X ∗

as N → ∞.

(11)

N

Assume now that A ⊂ Rd is a closed d-rectiable set. Then for s > d, 0

Cs,d Esw (A, N ) , = s,w 1+s/d N →∞ N (Hd (A))s/d lim

(12)

where Cs,d is a nite positive number independent of A and d0 . Moreover, if Hd (A) > 0, any sequence XN∗ of congurations on A satisfying (12), with ∗ ) instead of E w (A, N ), satises (11). Esw (XN s 3.1. Greedy minimal Riesz-energy points Using a greedy algorithm we could have an approximation of the minimal Riesz s-energy points.

Denition 6. Let k : X × X → R ∪ {∞} be a simmetric kernel on a locally compact Hausdor space X , and let A ⊂ X be a compact set. A sequence (an )∞ n=1 ⊂ A is called a greedy minimal k-energy sequence on A if it is generated in the following way: (i)

a1

is selected arbitrarily on A.

(ii) Assuming that a1 , . . . , an are already selected, then such way that it satises n X i=1

k(an+1 , ai ) = inf

x∈A

n X i=1 10

k(x, ai ),

an+1

is chosen in

for every n ≥ 1.

For the M. Riesz kernel in

[0, +∞)

0

X = Rd

which depends on a parameter

ks (x, y) := Ks (|x − y|), where

|·|

in

0

x, y ∈ Rd ,

is the Euclidean norm and

Ks (t) := For

s

we set

k = ks

t−s if s > 0 − log(t) if s = 0,

greedy minimal ks -energy points. k(x, y) = w(x, y)Ks (|x − y|) we will build the greedy

we obtain the

Instead, if we use

minimal (w, s)-energy points.

As it is proved in [9], for greedy minimal the following result on the unit sphere

Sd

(w, d)-energy

points we have

that leads us to suspect that they

could be a good choice for integration on manifolds.

Theorem 6. Assume that w : Sd × Sd → [0, +∞) is a continuous function w } be an arbitrary greedy such that w(x, x) > 0 for all x ∈ Sd . Let {XN,d d (w, d)-energy sequence on S , d ≥ 1. Then lim

w ) Edw (XN,d

N →∞

N 2 log N

=

and furthermore 1 N

X

δx (·) →

w x∈XN,d

Vol(B d ) Hdd,w (Sd ) Hdd,w (·)|Sd Hdd,w (Sd )

,

.

In particular, any greedy kd -energy sequence (XN,d ) on Sd is asymptotically s-energy minimizing on Sd for s = d. Unfortunately this result, in general, is not extendable to

s > d.

d0 with H (A) > 0 (where On the other hand, for any compact A ⊂ R δ w w is arbitrary), the following order of growth for Es (XN,s ) holds (cf. [9]).

δ

Theorem 7. Let X Hδ∞ (A) := inf{ (diam Gi )δ : A ⊂ ∪i Gi },

0 < δ ≤ d0 .

i

Assume that A ⊂ Rd is a compact set such that Hδ (A) > 0. Let w be a bounded lower semicontinuous CDP-weight function on A × A. Consider an 0

11

w ) ⊂ A, for s ≥ δ . Then for arbitrary greedy (w, s)-energy sequence (XN,s

N ≥2 w Esw (XN,s )

≤

Ms,δ,A ||w||Hδ∞ (A)−s/δ N 1+s/δ , Mδ,A ||w||Hδ∞ (A)−1 N 2 log N,

if s > δ if s = δ,

(13)

where Ms,δ,A , Mδ,A > 0 and are indipendent of w and N , and ||w|| = sup{w(x, y) : x, y ∈ A}. The previous theorem leads us to the following corollary

Corollary 1. Let A ⊂ Rd0 be a d-rectiable set. Suppose s > d and w is a bounded lower semicontinuous CDP-weight function on A × A. Consider an w ) ⊂ A. Then (X w ) is dense in arbitrary greedy (w, s)-energy sequence (XN,s N,s A. If s = d and A is assumed to be a compact subset of a d-dimensional C 1 manifold, the same conclusion holds for any greedy (w, d)-energy sequence. Taking w = 1 the result is applicable to greedy ks -energy sequences. So we do not know exactly if greedy

(w, s)-energy

sequences are good to

integrate functions on a manifold with respect to the normalized measure

Hdw

as it could be for the minimal energy points or whether we should prefer

them to a low discrepancy sequence, such as Halton points, mapped on the manifold with preserving measure maps. In the next section we present numerical tests showing that these points are not the best ones for integrating a function when a preserving measure maps is available, whereas have a similar behavior of low-discrepancy points if instead a generic map is used.

4. Numerical experiments The test functions we consider on the cone, cylinder, sphere and torus are

x y z p (1 + z)(1 − z) cos + + , 2 3 5 1 cos(30xyz) if z < 2 f2 (x, y, z) := (x2 + y 2 + z 2 )3/2 if z ≥ 12 ,

f1 (x, y, z) :=

f3 (x, y, z) := e f4 (x, y, z) :=

− sin(2x2 +3y 2 +5z 2 )

e−

√

x2 +y 2 +z 2

1 + x2

,

cos(1 + x2 ) sin(1 − y 2 )e|z| .

12

(14)

To compute the integrals

1 Hd (M)

Z fi (x)dHd (x), i = 1, . . . , 4 M

we use a QMC method with (a) low discrepancy points mapped on the manifolds, (b) greedy minimal

ks -energy

About (b), to compute

N

points.

greedy minimal

a uniform mesh on a rectangle of

R2

ks -energy points we start from N 2 /2 points and we map

consisting of

them to the manifold - if available by using the corresponding preserving measure map - and then we extract

N

greedy minimal

ks -energy points from

this mapped mesh. We set

s=2

because of the manifolds' dimension (a surface in

R3 )

and

we use the preserving measure maps in 2.1 for the cone, cylinder and sphere. Whereas for the torus we used this map which does not preserve the Lebesgue measure

 x = (2 + cos(u)) cos(v) [0, 2π] × [0, 2π] 3 (u, v) → y = (2 + cos(u)) sin(v)  z = sin(u)

(15)

Table 1: Map for the torus (d.)

To compare the results obtained with the greedy minimal

k2 -energy

points, we compute the integrals by QMC method using Halton points and Fibonacci lattice and we map them on the manifolds.

Furthermore, be-

cause of the peculiarity of Fibonacci points we generate for all the sequences a number of points that follows the Fibonacci sequence starting from twelfth Fibonacci number, up to

2584,

144,

eighteenth Fibonacci number and in

the following tables we present the results using

144, 610

and

2584

points.

The exact value of the integral is calculated with a change of variable and is computed in

R2

by the built-in Matlab function

dblquad

with a tollerance

−11 . We refer to this value to compute the relative errors. of order 10 All the tests are performed on a laptop with Intel Core i3-3120M @2.50 GHz, 4 Gb of RAM, Windows 10 and MATLAB 8.5.0.197613.

Here we

present only some experiments on the cone, the cylinder and the torus. It is possible to nd other experiments, among the one here presented, in the Master's thesis of the second author [14].

13

We have implemented a Matlab package GMKs (

points )

Greedy Minimal ks

that allows to reproduce all the experiments presented in the pa-

per. Interested people can download the package at

www.math.unipd.it/∼demarchi/software/GMKs. f1 f2 f3 f4

Cone

Cylinder

Torus

Sphere

6.378e-01

7.125e-01

3.435e-01

7.295e-01

0.130e+01

5.784e-01

0.470e+01

2.809e-01

0.116e+01

0.132e+01

0.131e+01

0.1340e+01

1.458e-01

-1.160e-01

-1.269e-02

8.950e-02

Table 2: Exact values of the integrals

(a) Halton points

(b) Greedy minimal

(c) Fibonacci points Figure 1: 610 points on the cone

14

k2 -energy

points

N

Halton

Fibonacci

144

1.215e-02

5.352e-03

2.097e-01

610

4.939e-03

1.270e-03

1.470e-01

2584

1.241e-03

3.029e-04

9.817e-02

Table 3: Relative errors for

k2 -energy

f1

Halton

Fibonacci

9.101e-03

6.250e-03

2.366e-01

610

5.277e-03

1.173e-03

1.764e-01

2584

6.766e-04

3.678e-04

1.212e-01

f2

GM

k2

on the cone with Fibonacci, Halton and Greedy Minimal

points

N

Halton

Fibonacci

144

1.059e-02

1.416e-03

7.048e-02

610

3.763e-04

3.389e-04

7.172e-02

2584

1.289e-04

8.026e-05

3.790e-02


f3

GM

k2


points

N

Halton

Fibonacci

144

2.767e-02

1.384e-02

2.333e-01

610

9.623e-03

3.230e-03

1.782e-01

2584

3.068e-03

7.594e-04

1.313e-01


k2 -energy


144


k2 -energy

k2

points

N

k2 -energy

GM

f4

GM

k2


points

N

Halton

Fibonacci

144

1.092e-03

5.264e-04

2.240e-01

GM

610

3.131e-04

6.400e-05

1.603e-01

2584

1.029e-04

6.929e-06

9.533e-02

k2

Table 7: Relative errors for f1 on the cylinder with Fibonacci, Halton and Greedy Minimal k2 -energy points

15

(a) Halton points

(b) Greedy minimal

k2 -energy

points

(c) Fibonacci points Figure 2: 610 points on the cylinder

N

Halton

Fibonacci

144

6.513e-02

8.764e-03

3.830e-01

GM

610

2.900e-02

2.964e-03

2.267e-01

2584

1.436e-03

6.975e-04

1.502e-01

k2


N

Halton

Fibonacci

144

1.792e-02

2.930e-04

1.150e-01

GM

610

2.359e-03

1.125e-05

8.751e-02

2584

3.287e-04

6.267e-07

4.895e-02

k2


16

N

Halton

Fibonacci

144

6.611e-03

1.506e-04

3.879e-02

610

1.457e-02

8.382e-06

2.524e-02

2584

4.730e-04

4.671e-07

2.339e-02

Table 10: Relative errors for mal

k2 -energy

f4

GM

k2

on the cylinder with Fibonacci, Halton and Greedy Mini-

points

(a) Halton points

(b) Greedy minimal

k2 -energy

points

(c) Fibonacci points Figure 3: 610 points on the torus

N

Halton

Fibonacci

144

2.833e-03

6.448e-04

1.595e-03

GM

610

1.001e-03

7.411e-05

1.202e-03

2584

1.017e-04

8.504e-06

1.324e-03

k2

Table 11: Relative errors for f1 on the sphere with Fibonacci, Halton and Greedy Minimal k2 -energy points

17

N

Halton

Fibonacci

144

1.476e-01

1.089e-02

8.406e-02

GM

610

4.415e-02

8.045e-05

4.872e-03

2584

6.847e-04

3.115e-06

5.177e-03

k2


N

Halton

Fibonacci

144

3.282e-03

1.025e-04

2.834e-03

GM

610

1.755e-03

5.727e-06

2.251e-03

2584

7.294e-05

3.190e-07

1.325e-03

k2


N

Halton

Fibonacci

144

1.549e-03

1.425e-03

1.912e-03

GM

610

6.631e-03

7.970e-05

3.702e-03

2584

2.725e-04

4.442e-06

4.756e-03

k2


18

(a) Halton points

(b) Greedy minimal

k2 -energy

points

(c) Fibonacci points Figure 4: 610 points on the torus

N

Halton

Fibonacci

144

2.152e-01

1.777e-01

6.894e-02

610

1.888e-01

1.780e-01

5.367e-02

2584

1.788e-01

1.780e-01

4.014e-02


k2 -energy

f1

k2

on the torus with Fibonacci, Halton and Greedy Minimal

points

N

Halton

Fibonacci

144

1.218e-01

1.690e-01

3.081e-02

610

1.453e-01

1.410e-01

4.728e-02

2584

1.414e-01

1.411e-01

2.297e-02


k2 -energy

GM

f2

GM

k2


points

19

N

Halton

Fibonacci

144

3.033e-02

3.426e-02

4.949e-03

610

2.716e-03

8.821e-03

1.349e-02

2584

8.763e-03

6.453e-03

1.673e-03


k2 -energy

k2


f3

points

N

Halton

Fibonacci

144

6.015e-01

5.339e-01

2.435e-01

610

5.109e-01

5.237e-01

1.874e-01

2584

5.252e-01

5.238e-01

1.319e-01


k2 -energy

GM

f4

GM

k2


points

N

Cone

Cylinder

Torus

144

0.217

0.218

0.248

0.208

610

20.067

21.046

19.340

19.284

1519.112

1513.211

1571.449

1511.768

2584

Sphere

Table 19: Time in seconds to compute the greedy minimal

k2 -energy

points

4.1. Greedy minimal ks -energy points: tuning the parameter s In the previous section we set can consider a tuning of

s.

The script

demo2

s

s=2

because of the dimension, but we

and see how the errors change as a function of

of the Matlab package GMKs, previously mentioned,

allows us to make the experiments that we are going to present. In the following gures we see the behavior of the relative errors of the integrals for

s ∈ [0, 10],

with step

0.05,

points.

20

using 200 greedy minimal

ks -energy

(a) Cone

(b) Cylinder

(c) Sphere

(d) Torus

Figure 5: Relative errors using 200 points for the function

(a) Cone

(b) Cylinder

(c) Sphere

(d) Torus


21

f1

f2

(a) Cone

(b) Cylinder

(c) Sphere

(d) Torus


(a) Cone

(b) Cylinder

(c) Sphere

(d) Torus


22

f3

f4

5. Conclusion In this paper we tested the integration on manifolds by mapped lowdiscrepancy points and greedy minimal

ks -energy

points.

We used four test functions, smooth, discontinuous and with singularities, integrated by the Quasi-Monte Carlo method using the previous mentioned point sets. We show how is important to use a map that preserve the measure as we can see in the case of the Torus. From the Tables of the relative errors we can realize that even if we add more points the errors in general do not change much for the greedy minimal energy points. On the other hand, from our experiments the time to extract the greedy minimal

ks -energy

points grows so it is more advisable to use less points. A

good compromise, error vs computational time, is to use Finally, by tuning the parameter

s

610

points.

we have seen that, using the same

number of points, it is better to avoid values of the parameter below the manifold dimension, in our case

s = 2.

Whereas, for

s

bigger than the

dimension, the relative errors are almost of the same order, so as foreseen from theory the optimal

s

is around the dimension. An exception is the

case of the sphere.

Acknowledgements.

This work has been supported by the University

of Padova ex 60% and the GNCS-INdAM funds Visiting Professors 2015 funds. The authors wish to thank Prof. Edward B. Sa of Vanderbilt Univerisity and Prof. Roberto Monti of University of Padova for the valuable discussions.

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