Optimization on fractal sets

TL;DR

This paper investigates the conditions for the existence of extrema of functions on self-similar fractal sets and introduces a discrete gradient algorithm to locate these extrema.

Contribution

It provides necessary and sufficient conditions for extrema on fractals and proposes a novel discrete gradient method for their computation.

Findings

Established criteria for extrema existence on fractals

Developed a discrete gradient algorithm for extremum search

Demonstrated the algorithm's effectiveness on self-similar sets

Abstract

We outline necessary and sufficient condition to the existence of extrmas of a function on a self-similar set, and we describe discrete gradient algorithm to find the extrema.

Equations

$$K=\underset{i=1}{\bigcup ^N}{f}_i\left(K\right)$$

$$\left\{P_1,...,P_N\right\}$$

$$V_m=\underset{i=1}{\bigcup ^N}{f}_i\left(V_{m-1}\right).$$

$${\mathcal{F}}_m=\underset{i=1}{\bigcup ^N}{f}_i\left({\mathcal{F}}_{m-1}\right).$$

$$V_m\subset V_{m+1}.$$

$$\mathcal{W}=\left({\mathcal{W}}_1,\dots ,{\mathcal{W}}_m\right).$$

$$f_{\mathcal{W}}=f_{{\mathcal{W}}_1}\circ \dots \circ f_{{\mathcal{W}}_m}.$$

$$(X,Y)\in V_m^2$$

$$X\underset{m}{\sim }Y$$

$$f_{\mathcal{W}}{V}_0=\left(f_{{\mathcal{W}}_1}\circ \dots \circ f_{{\mathcal{W}}_m}\right)V_0$$

$$X\underset{m}{\sim }Y$$

$$X=f_{\mathcal{W}}\left(P_i\right)$$

$${\mathcal{F}}_m=\bigcup _{|\mathcal{W}|=m}{f}_{\mathcal{W}}\left({\mathcal{F}}_0\right)$$

$$\Sigma =\bigcup _{m\in {\mathbb{N}}^{\star }}{\Sigma }_m$$

$$\left(\mathcal{F},\Sigma ,{\left(f_i\right)}_{i=1}^N\right)$$

$$\sigma (12233\dots )=2233\dots$$

$$\pi (W)=\bigcap _{m\in {\mathbb{N}}^{\star }}{f}_{{\mathcal{W}}_1\dots {\mathcal{W}}_m}(\mathcal{F})$$

$$C_{L,\mathcal{F}}=\bigcup _{(i,j)\in {\Sigma }^2,i\ne j}\left({\mathcal{F}}_i\cap {\mathcal{F}}_j\right)$$

$$C_L={\pi }^{-1}\left(C_{L,\mathcal{F}}\right)$$

$$P=\bigcup _{m\in {\mathbb{N}}^{\star }}{\sigma }^m(C_L)$$

$$V_0=\pi \left(\bigcup _{m\in {\mathbb{N}}^{\star }}{\sigma }^m(C_L)\right)$$

$${\mathcal{N}}_0=N_0$$

$${\mathcal{N}}_m=N\times {\mathcal{N}}_{m-1}-C,\#A_m=N\times \#A_{m-1}$$

$$A_m=N^m{A}_0=N^m(N_0(N_0-1)),{\mathcal{N}}_m=N^m{N}_0-\left(\frac{1-N^m}{1-N}\right)C=\mathcal{O}(N^m).$$

$${\mathcal{F}}_{m,X}=\bigcup _{\mathcal{W}\in {\Sigma }_m,X\in f_{\mathcal{W}}(\mathcal{F})}{f}_{\mathcal{W}}(\mathcal{F})$$

$$\sum _{i=1}^N{\mu }_i=1$$

$$\mu =\sum _{i=1}^N{\mu }_i\mu \circ f_i^{-1}$$

$${\mu }_i=R_i^{D_H\left(\mathcal{F}\right)}$$

$$\sum _{i=1}^N{R}_i^{D_H\left(\mathcal{F}\right)}=1.$$

$$\mu =\sum _{i=1}^N{\mu }_i\mu \circ f_i^{-1}$$

Full text

† Université Mohammed V, Agdal, Rabat, Maroc111nizar.riane@gmail.com

‡ Sorbonne Université

CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4, place Jussieu 75005, Paris, France222Claire.David@Sorbonne-Universite.fr

Keywords: Extrema - Fractal - Laplacian - Discrete gradient - Dynamic programming.

AMS Classification: 37F20- 28A80-05C63-05C85.

1 Introduction

In 1636, in a correspondence with Martin Mersenne, Pierre de Fermat established a necessary condition for the existence of the minimum and the maximum of a function [dF36] :

“When a quantity, for example the ordinate of a curve, reached its maximum or its minimum, in a situation infinitely close, his increase or decrease is null.”

Since then, many results have been set, ranging from free and constraint conditions, giving birth to numerical algorithms that enable one to find the extremas of a function.

But, until now, optimization has mainly concerned regular domains, without there being really specific results for fractal sets. The following citation of G. Hardy [Har16], on a close subject, and even if some may here call it a truism, perfectly reflects the fact that it was “in consequence of the methods employed”.

Recently, the birth of analysis on fractals, especially, the work of J. Kigami [Kig89], [Kig93], [Kig98], [Kig01], [Kig03], bridged this gap, by giving rise to the building of local operators, which, for a function $u$ defined on a specific fractal set $\cal F$, at a given point $X\,\in\,{\cal F}$, are “equals to the limit, in a suitable renormalized sense, of the difference between an average value of the function in a neighborhood of $X$ and $u(X)$” [Str99]. The intrinsinc properties of those operators, analogous to differential ones, make them play the role of Laplacians. If related numerical methods have been developped, for instance, by R. Strichartz [RSS95], [Str99], [Str12b], [Str01], [Str03], [Str12a], [Str06], the field of analysis on fractals remains not completely explored, in particular as regards optimization.

In the sequel, in the spirit of Fermat paper, we try to extend results of smooth analysis on extremas in the case of fractals. The novelty of our work lays in the use of the aforementioned differential operators, specifically designed for fractal sets. To begin with, we examin existence conditions, then, we present a numerical algorithm that enable us to find local extrema of a continuous function defined on a fractal set.

2 Framework of the sudy

In the following, we place ourselves in the euclidian space of dimension $d\,\in\,\left\{1,2,3\right\}$.

Notation**.**

We will classically denote by ${\mathbb{N}}^{\star}$ the set of strictly positive integers.

2.1 Self-similar sets

Notation**.**

In the sequel, $\cal F$ denotes a fractal domain of Hausdorff dimension $D_{H}\left({\cal F}\right)$ ; $N$ is a strictly positive integer, and $\left\{f_{1},\ldots,f_{N}\right\}$ is a set of contractive maps, where, for any integer $i$ of $\left\{1,\ldots,N\right\}$, $R_{i}\,\in\,]0,1[$ is the contraction ratio of $f_{i}$, and $P_{i}\,\in\,{\mathbb{R}}^{d}$ the fixed point of $f_{i}$.

Theorem 2.1**.**

**Gluing Lemma [BD85a]

**

Given a complete metric space $(E,\delta)$, a strictly positive integer $N$, and a set $\left\{f_{i}\right\}_{1\leqslant i\leqslant N}$ of contractions on $E$ with respect to the metric $\delta$, there exists a unique non-empty compact subset $K\subset E$ such that:

[TABLE]

The set $K$ is said self-similar with respect to the family $\left\{f_{1},\ldots,f_{N}\right\}$, and called attractor of the iterated function system (IFS) $\left\{f_{1},\ldots,f_{N}\right\}$.

Definition 2.1**.**

**Boundary (or initial) graph

**

We will denote by $V_{0}$ the ordered set of the (boundary) points:

[TABLE]

The set of points $V_{0}$, where, for any $i$ of $\left\{1,...,N-1\right\}$, the point $P_{i}$ is linked to the point $P_{i+1}$, constitutes an oriented graph, that we will denote by ${{\cal F}_{0}}$.

$V_{0}$ is called the set of vertices of the graph ${{\cal F}_{0}}$.

If the point $P_{N-1}$ is linked to the point $P_{0}$, the boundary graph is closed.

Definition 2.2**.**

$m^{th}$** order graph, $m\,\in\,{\mathbb{N}}^{\star}$

**

For any strictly positive integer $m$, we set:

[TABLE]

The set of points $V_{m}$, where two consecutive points are linked, is an oriented graph, which we will denote by ${{\cal F}_{m}}$.

$V_{m}$ is called the set of vertices of the graph ${\cal F}_{m}$.

By extension, we will write:

[TABLE]

Property 2.2**.**

For any natural integer $m$:

[TABLE]

Property 2.3**.**

The set $\underset{m\in{\mathbb{N}}}{{\bigcup}}\,V_{m}$ is dense in ${\cal F}$.

Definition 2.3**.**

Word

Given a strictly positive integer $m$, we will call number-letter any integer ${\cal W}_{i}$ of $\left\{1,\ldots,N\right\}$, and word of length $|{\cal W}|=m$, on the graph ${\cal F}_{m}$, any set of number-letters of the form:

[TABLE]

We will write:

[TABLE]

Definition 2.4**.**

**Vertices

**Two points $X$ and $Y$ of ${{\cal F}}$ will be called vertices of the graph ${\cal F}$ if there exists a natural integer $m$ such that:

[TABLE]

Definition 2.5**.**

**Edge relation

**

Given a natural integer $m$, two points $X$ and $Y$ of ${{\cal F}_{m}}$ will be called adjacent if and only if $X$ and $Y$ are neighbors in ${{\cal F}_{m}}$. We will write:

[TABLE]

This edge relation ensures the existence of a word ${\cal W}=\left({\cal W}_{1},\ldots,{\cal W}_{m}\right)$ of length $m$, such that $X$ and $Y$ both belong to the iterate:

[TABLE]

Given two points $X$ and $Y$ of ${\cal F}$, we will say that $X$ and $Y$ are adjacent if and only if there exists a natural integer $m$ such that:

[TABLE]

Definition 2.6**.**

**Adresses

**

Given a natural integer $m$, and a vertex $X$ of ${\cal F}_{m}$, we will call address of the vertex $X$ an expression of the form

[TABLE]

where $\cal W$ is a word of length $m$, and $i$ a natural integer in $\left\{1,\ldots,N\right\}$.

Property 2.4**.**

**Subcell - Junction points

**

Given a natural integer $m$, the graph ${\cal F}_{m}$ can be written as the finite union of $N^{m}$ subgraphs:

[TABLE]

For any word $\cal W$ of length $m$, $f_{\cal W}\left({\cal F}_{0}\right)$ will be called $m^{th}$-order cell, or subcell.

Notation**.**

• i.

Given a strictly positive integer $m$, we will denote by $\Sigma_{m}$ to be the set of words ${\cal W}\,\in\,\{1,\ldots,N\}^{m}$ of length $m$.

• ii.

We then set:

[TABLE]

Notation**.**

For the sake of clarity, we will, from now on, refer to a self-similar set either by $\cal F$ or by:

[TABLE]

Notation**.**

We will denote by:

• i.

$\sigma$, the shift map from $\Sigma$ to $\Sigma$ which, for any word $\cal W$, deletes the first "letter" i.e.:

[TABLE]

• ii.

$\pi$, the surjective map from $\Sigma$ to $\cal F$ defined for every infinite "word" ${\cal W}={\cal W}_{1}{\cal W}_{2}\ldots\,\in\,\Sigma$, by:

[TABLE]

where $f_{{\cal W}_{1}\ldots{\cal W}_{m}}=f_{{\cal W}_{1}}\circ\ldots\circ f_{{\cal W}_{m}}$.

• iii.

[TABLE]

Definition 2.7**.**

We define:

• i.

The critical set:

[TABLE]

• ii.

The post-critical set:

[TABLE]

Property 2.5**.**

[TABLE]

Notation**.**

Given a natural integer $m$, we will denote by $\mathcal{N}_{m}$ the number of vertices of the graph ${\cal F}_{m}$, $\mathcal{A}_{m}$ the edge set of ${\cal F}_{m}$, and by $C$ the cardinal of the set $C_{L,{\cal F}}$.

Proposition 2.6**.**

One has:

[TABLE]

and, for any strictly positive integer $m$:

[TABLE]

Proof.

The graph ${\cal F}_{m}$ is the union of $N$ copies of the graph ${\cal F}_{m}$. Each copy has $\#A_{m-1}$ edges, and shares vertices with others copies. One can thus consider the copies as the vertices of a complete graph which has a number of edges equal to $C$, so there are $C$ vertices to discount. ∎

Remark 2.1*.*

One may check that:

[TABLE]

Definition 2.8**.**

**System of neighborhood [Kig01]

**

Let $\left({\cal F},\Sigma,\left(f_{i}\right)_{i=1}^{N}\right)$ be a self-similar structure. For any $X\,\in\,{\cal F}$, and any natural integer $m$, we set:

[TABLE]

which will be called system of neighborhood of $X$.

Definition 2.9**.**

**Self-similar measure on $\cal F$ [Str06]

**

A measure $\mu$ with full support on ${\mathbb{R}}^{d}$ will be called self-similar measure on ${\cal F}=\underset{i=1}{\overset{N}{\bigcup}}\,f_{i}\left({\cal F}\right)$ if, given a family of strictly positive weights $\left(\mu_{i}\right)_{1\leqslant i\leqslant N}$ such that:

[TABLE]

one has:

[TABLE]

Property 2.7**.**

**Building of a self-similar measure on $\cal F$

**

We set, for any integer $i$ belonging to $\left\{1,\ldots,N\right\}$:

[TABLE]

One has then:

[TABLE]

which enables us to define a self-similar measure $\mu$ on $\cal F$ through:

[TABLE]

One may note that the measure $\mu$ corresponds to the normalized $D_{H}\left({\cal F}\right)$-dimensional Hausdorff measure ($\mathcal{H}^{D_{H}\left({\cal F}\right)}$) (we refer to **[Fal85]**):

[TABLE]

for any subset $E\subset{\mathbb{R}}^{d}$.

2.2 Laplacians, on self-similar sets

Definition 2.10**.**

**Energy, on the graph ${\cal F}_{m}$, $m\,\in\,{\mathbb{N}}$, of a pair of functions

**

Given a natural integer $m$, and two real valued functions $u$ and $v$, defined on the set $V_{m}$ of the vertices of ${\cal F}_{m}$, we introduce the energy, on the graph ${\cal F}_{m}$, of the pair of functions $(u,v)$, as:

[TABLE]

For the sake of simplicity, we will write it under the form:

[TABLE]

Definition 2.11**.**

**Dirichlet form on a measured space

**

(we refer to the paper [BD85b], or the book [FOT94])

Given a measured space $(E,\mu)$, a Dirichlet form on $E$ is a bilinear symmetric form, that we will denote by ${\cal E}$, defined on a vectorial subspace $D$ dense in $L^{2}(E,\mu)$, such that:

For any real-valued function $u$ defined on $D$ : ${\cal E}(u,u)\geqslant 0$. 2. 2.

$D$, equipped with the inner product which, to any pair $(u,v)$ of $D\times D$, associates:

[TABLE]

is a Hilbert space. 3. 3.

For any real-valued function $u$ defined on $D$, if:

[TABLE]

then : ${\cal{E}}(u_{\star},u_{\star})\leqslant{\cal{E}}(u,u)$ (Markov property, or lack of memory property).

Definition 2.12**.**

Dirichlet form, on a finite set (see [Kig03])

Let $V$ denote a finite set $V$, equipped with the usual inner product which, to any pair $(u,v)$ of functions defined on $V$, associates:

[TABLE]

A Dirichlet form on $V$ is a symmetric bilinear form ${\cal E}$, such that:

For any real valued function $u$ defined on $V$: ${\cal E}(u,u)\geqslant 0$. 2. 2.

${\cal{E}}(u,u)=0$ if and only if $u$ is constant on $V$. 3. 3.

For any real-valued function $u$ defined on $V$, if: $u_{\star}=\min\,(\max(u,0),1)$, i.e. :

[TABLE]

then: ${\cal{E}}(u_{\star},u_{\star})\leqslant{\cal{E}}(u,u)$ (Markov property).

Property 2.8**.**

Given a natural integer $m$, and a real-valued function $u$, defined on the set $V_{m}$ of vertices of ${\cal F}_{m}$, the map, which, to any pair of real-valued, continuous functions $(u,v)$ defined on $V_{m}$, associates:

[TABLE]

is a Dirichlet form on ${\cal F}_{m}$.

Moreover:

[TABLE]

Proposition 2.9**.**

For any strictly positive integer $m$, if $u$ is a real-valued function defined on $V_{m-1}$, its harmonic extension, denoted by $\tilde{u}$, is obtained as the extension of $u$ to $V_{m}$ which minimizes the energy:

[TABLE]

Remark 2.2*.*

**Concretely:

**

The link between ${\cal{E}}_{{\cal F}_{m}}$ and ${\cal{E}}_{{\cal F}_{m-1}}$ is obtained through the introduction of two strictly positive constants $r_{m}$ and $r_{m-1}$ such that:

[TABLE]

In particular:

[TABLE]

For the sake of simplicity, one fixes the value of the initial constant: $r_{0}=1$. Then:

[TABLE]

We set:

[TABLE]

and:

[TABLE]

Since the determination of the harmonic extension of a function appears to be a local problem, on the graph ${\cal F}_{m-1}$, which is linked to the graph ${\cal F}_{m}$ by a similar process as the one that links ${\cal F}_{1}$ to ${\cal F}_{0}$, one deduces, for any strictly positive integer $m$:

[TABLE]

By induction, one gets:

[TABLE]

If $v$ is a real-valued function, defined on $V_{m-1}$, of harmonic extension $\tilde{v}$, we will write:

[TABLE]

For further precision on the construction and existence of harmonic extensions, we refer to [Sab97].

Definition 2.13**.**

**Renormalized energy, for a continuous function $u$, defined on ${\cal F}_{m}$, $m\,\in\,{\mathbb{N}}$

**

Given a natural integer $m$, one defines the normalized energy, for a continuous function $u$, defined on ${\cal F}_{m}$, by:

[TABLE]

Definition 2.14**.**

Normalized energy, for a continuous function $u$, defined on ${\cal F}$

Given a function $u$ defined on

[TABLE]

one defines the normalized energy:

[TABLE]

Definition 2.15**.**

Dirichlet form, for a pair of continuous functions defined on ${\cal F}$

We define the Dirichlet form $\cal{E}$ which, to any pair of real-valued, continuous functions $(u,v)$ defined on the graph ${\cal F}$, associates, subject to its existence:

[TABLE]

Notation**.**

We will denote by:

1. i.

$\text{dom}\,{\cal E}$ the subspace of continuous functions defined on ${\cal F}$, such that:

[TABLE] 2. ii.

$\text{dom}_{0}\,{\cal E}$ the subspace of continuous functions defined on ${\cal F}$, which take the value zero on $V_{0}$, and such that:

[TABLE]

Lemma 2.10**.**

The map:

[TABLE]

defines an inner product on {\text{dom}\,{\cal E}}\big{/}{\text{Constants}}.

Theorem 2.11**.**

\left({\text{dom}\,{\cal E}}\big{/}{\text{Constants}},{\cal E}(\cdot,\cdot)\right)* is a complete Hilbert space.*

Definition 2.16**.**

**Harmonic function

**

A real-valued function $u$, defined on $\displaystyle{V_{\star}=\underset{{i\in{\mathbb{N}}}}{\bigcup}\,V_{i}}$, will be said to be harmonic if, for any natural integer $m$, its restriction $u_{|V_{m}}$ is harmonic:

[TABLE]

Notation**.**

We will denote by $\text{dom}\,{\cal E}$ the subspace of continuous functions $u$ defined on ${\cal F}$, such that:

[TABLE]

Definition 2.17**.**

We will denote by $\text{dom}\,\Delta$ the existence domain of the Laplacian, on ${\cal F}$, as the set of functions $u$ of $\text{dom}\,\mathcal{E}$ such that there exists a continuous function on ${\cal F}$, denoted by $\Delta\,u$, that we will call Laplacian of $u$, such that, $\text{for any }v\,\in\,\text{dom}\,\mathcal{E}\,,\,v_{|{\cal F}_{0}}=0\,$:

[TABLE]

Theorem 2.12**.**

[TABLE]

Notation**.**

i. Given a natural integer $m$, ${\cal S}\left({\cal H}_{0},V_{m}\right)$ denotes the space of spline functions "of level $m$", $u$, defined on ${\cal F}$, continuous, such that, for any word $\cal M$ of length $m$, $u\circ T_{\cal M}$ is harmonic, i.e.:

[TABLE]

ii. ${\cal H}_{0}\subset\text{dom}\,\Delta$ denotes the space of harmonic functions, i.e. the space of functions $u\,\in\,\ \text{dom}\,\Delta$ such that:

[TABLE]

Property 2.13**.**

For any natural integer $m$: ${\cal S}\left({\cal H}_{0},V_{m}\right)\subset\text{dom }{\cal E}$.

Theorem 2.14**.**

**Pointwise formula

**

Let $m$ be a strictly positive integer, $X\,\in\,V_{\star}\setminus V_{0}$, and $\psi_{X}^{m}\,\in\,{\cal S}\left({\cal H}_{0},V_{m}\right)$ a spline function such that:

[TABLE]

1. i.

For any function $u$ of $\text{dom}\,\Delta$, such that its Laplacian exists, the sequence

[TABLE]

converges uniformly towards

[TABLE] 2. ii.

Conversely, given a continuous function $u$ on $\cal F$ such that the sequence

[TABLE]

converges uniformly towards a continuous function on $V_{\star}\setminus V_{0}$, one has:

[TABLE]

2.3 Existence of extrema

Definition 2.18**.**

**Extrema

**

Given a continuous function $u$ defined on the fractal set $\cal F$, and $X\,\in\,{\cal F}$, we will say that $u$:

1. i.

has a global minimum (resp. a global maximum) at $X$ if:

[TABLE] 2. ii.

a local minimum (resp. a local maximum) at $X$ if there exist a neighborhood $V$ of $X$ such that

[TABLE]

Theorem 2.15**.**

Given a continuous function $u$ defined on the compact fractal set

[TABLE]

the Weierstrass extreme value theorem ensures the existence of:

[TABLE]

Theorem 2.16**.**

**Laplacian test for fractals [Kig01]

**

Given a continuous, real-valued function $u$ defined on $\cal F$, and belonging to $dom_{\Delta}$:

1. i.

If $u$ admits a local maximum at $X_{0}\,\in\,{\cal F}$, then:

[TABLE] 2. ii.

IIf $u$ admit a local minimum at $X_{0}\,\in\,{\cal F}$, then:

[TABLE]

Proof.

i. If $u$ admits a local maximum at $X_{0}$, then for sufficiently large values of the integer $m$:

[TABLE]

This enables us to conclude that:

[TABLE]

ii. can be proved in a similar way.

∎

3 Numerical algorithm and dynamic programming

In the following, we will present a numerical algorithm, based on discrete gradient, to find a local maximizer (resp. local minimizer) of a function continuous $u$ on $K$.

3.1 The algorithm

We recall that ${\cal F}_{m}=(V_{m},A_{m})$ is the oriented graph approximation of $\cal F$ of order $m$, where $V_{m}$ is the vertices set and $A_{m}$ is the edge set. We can check that the distance between two connected vertices is of order $2^{-m}$.

In order to find the local maximumizer, we will provide every edge $\{XY\}$ with the weight $D_{XY}=u(Y)-u(X)$. In order to find an appropriate approximation of the maximizer $X^{\star}$, we fix a degree of tolerance $\varepsilon>0$, and the graph ${\cal F}_{m}$ of order $m$ such that

[TABLE]

Starting at an arbitrary point $X_{0}$ in $V_{m}$, we follow the direction of the maximal positive gradient at $X_{0}$, i.e.

[TABLE]

We replace the initial point by $\arg\max_{X_{0}\underset{m}{\sim}Y}\{D_{X_{0}Y}\,|\,D_{X_{0}Y}>0\}$, and we do the same operation until

[TABLE]

In this case, the algorithm stop and we have the approximation of $X^{\star}$. We can can resume the algorithm in the following steps :

Discrete gradient algorithm

Fix a degree of tolerance $\varepsilon>0$. 2. 2.

Build the graph ${\cal F}_{m}$, for $m$ such that:

[TABLE] 3. 3.

Fix $X=X_{0}$ for $X_{0}\,\in\,V_{m}$. 4. 4.

While $\displaystyle\max_{X\underset{m}{\sim}Y}\,\{D_{XY}\,|\,D_{XY}\geqslant 0\}>0$:

Update $X=\displaystyle\arg\max_{X\underset{m}{\sim}Y}\{D_{XY}\,|\,D_{XY}>0\}$. 5. 5.

Return $X$.

3.2 Numerical analysis and dynamic programming

The algorithm presented above can be viewed as a dynamical programming algorithm on the directed graph ${\cal F}_{m}$, $m\,\in\,{\mathbb{N}}^{\star}$ (we refer to [BG15]). Values of the function can be calculated recursively:

[TABLE]

where $\mathcal{B}$ denotes the Bellman operator:

[TABLE]

where $D_{XY}$ is the weight associated to the arrow ${XY}\in A_{m}$ and $D_{XY}=-\infty$ if ${XY}\not\in A_{m}$.

As shown in the first section, $\#A_{m}=\mathcal{O}(N^{m})$ and $\#V_{m}=\mathcal{O}(N^{m})$. The number of possible transitions is at most of order $\#A_{m}=\mathcal{O}(N^{m})$. Since

[TABLE]

one can deduce that the calculation time of the maximum is of order $\mathcal{O}(N^{2\,m})$.

3.2.1 Sierpiński simplices

Sierpiński simplices are sparse graph. Thus, the calculation time required for the gradient algorithm can be optimized: each vertex has a finite number of neighbors, which ensures a calculation time at step $m\,\in\,{\mathbb{N}}^{\star}$ which is of order $\mathcal{O}(N^{m})$.

Moreover, computations can be simplified using the fact that every vertex $X$ has two adresses:

[TABLE]

where $\left({\cal W}_{1},{\cal W}_{2}\right)\,\in\,\Sigma^{2}$ and $\left(P_{i},\,P_{j}\right)\,\in\,V_{0}^{2}$, with $i\neq j$. Thus, the neighbors of $X$ are given by:

[TABLE]

3.2.1.1 The Sierpiński Gasket

In the case of Sierpiński Gasket, one can optimize the calculation time of the gradient algorithm. Since, for any natural integer $m$:

[TABLE]

and given the fact that every vertex $X\,\in\,V_{m}\setminus V_{0}$ has four neighbors, it follows that the calculation time of the maximum at step $m\,\in\,{\mathbb{N}}^{\star}$ is thus of order $\mathcal{O}(3^{m})$.

In the sequel (see figures 1 to 4), we present results of our algorithm in the case of Sierpiński Gasket with vertices:

[TABLE]

for the value $m=6$.

The color function is related to the gradient of the one at stake, high values ranging from red to blue.

For this first example, the algorithm starts with $X_{0}=\displaystyle\left(\frac{5}{8},\frac{\sqrt{3}}{8}\right)$, following the largest gradient (red points), the algorithm converges to the local maximum $1$ at $(1,0)$.

In the second example, the algorithm starts with $X_{0}=\displaystyle\left(\frac{3}{4},\frac{\sqrt{3}}{4}\right)$, and converges to the global maximum [math] at $\displaystyle\left(\frac{1}{2},\frac{\sqrt{3}}{4}\right)$.

3.2.1.2 The Sierpiński Tetrahedron

The Sierpiński Tetrahedron requires a calculation time for the maximum at step $m\,\in\,{\mathbb{N}}^{\star}$ which is of order $\mathcal{O}(4^{m})$.

In the sequel (see figures 5 and 6), we present results of our algorithm in the case of Sierpiński Tetrahedron, with vertices:

[TABLE]

for the value $m=6$.

The color function is related to the gradient of the one at stake, high values ranging from red to blue.

For this first example, the algorithm start with $X_{0}=\displaystyle\left(\frac{41}{64},\frac{1}{64}+\frac{\sqrt{3}}{8},\frac{1}{64}\right)$, following the largest gradient (red points), the algorithm converges to the local maximum $1$ at $(1,0,0)$.

3.2.2 Self-similar curves

Self-similar curves require a calculation time which is of order $\mathcal{O}(N^{m})$, due to the fact that every vertex has only two neighbors. In such cases: $V_{0}=\{P_{0},P_{1}\}$. Thus, every vertex $X$ has exactly two adresses:

[TABLE]

where $\left({\cal W}_{i},{\cal W}_{j}\right)\,\in\,\Sigma^{2}$. The neighbors of $X$ are thus given by:

[TABLE]

where ${\cal W}_{i-1}$ (resp. ${\cal W}_{j+1}$) is the next (resp. the past) adress of ${\cal W}_{i}$ (resp. ${\cal W}_{j}$) in the lexicographical order.

In the sequel (see figures 7 and 8), we present results of our algorithm in the case of the Minkowski curve, with $V_{0}=\left\{(0,0);(1,0)\right\}$, for the value $m=3$.

The color function is related to the gradient of the one at stake, high values ranging from red to blue.

For this example, the algorithm starts with $X_{0}=\displaystyle\left(0,0\right)$ ; following the largest gradient (red points), the algorithm converges to the local maximum $\displaystyle\left(\frac{5}{4096}\right)$ at $\displaystyle\left(\frac{1}{32},\frac{1}{64}\right)$.