Probabilist set inversion using pseudo-intervals arithmetic

Probabilist set inversion using pseudo-intervals arithmetic

A. Kenoufi

Scientific Consulting for Research and Engineering, University of Haute-Alsace, Mulhouse, France. E-mail: [e-mail]

ABSTRACT

In this paper, one presents how to use a new interval arithmetic framework based on free algebra construction, called pseudo-intervals, which is associative and distributive and permits to build well-defined inclusion function for interval semi-group and for its associated vector space. One introduces the ψ-algorithm (Probabilist Set Inversion), which performs set inversion of functions and exhibits some numerical examples.

Keywords: Free algebra, pseudo-interval and interval arithmetic, set inversion, probability

RESUMO

Neste artigo, apresenta-se como usar um novo arcabouço de aritmética intervalar com base na construção de álgebra livre, chamado pseudo intervalos, que é associativa e distributiva e permite a construção da inclusão de função bem definida para semi-grupo intervalar e para seu espaço vetorial associado. Apresenta-se o ψ-algoritmo (Inversão Probabilística de Conjuntos), que realiza a inversão de funções e exibe-se alguns exemplos numéricos.

Palavras-chave: Álgebra livre, Pseudo-intervalos e aritmética intervalar, conjunto de inversão, probabilidade

1 SET INVERSION

One of the most recurrent problem arising in sciences and engineering is to perform adjustments of a system in order to get the desired performances. For example, how to set-up a car engine so that some polluting gases ratio are less than a certain amount, or how to settle a robot to make it moving toward a desired target. Such kind of problem are dealing with the inversion of the relation between adjustments and desired performances.

Let us note ⊂ ⁿ the set of feasible adjustments, and ⊂ ^p the set of desired performance of a system. The mathematical modelling of the problem consists of the computation of S = ƒ ^-1 () ∩ , as shown on Figure (1), where ƒ: n→ ^p is the function giving performances from adjustments. Since real number sets can be written as union of intervals, one has to perform this set inversion within the interval semi-group [1]. Some powerful set inversion methods are have been developed those last years, such as SIvIA [18] (Set Inversion via Interval Arithmetic) which is based on interval arithmetic [2, 6, 7, 8, 9, 10, 11, 12, 32, 33, 34, 35, 36].

The first mathematician who has used intervals was the famous Archimedes from Syracuse (287-212 b.C). He has proposed a two-sides bounding of π : 3 + < π < 3 + using polygons and a systematic method to improve it. In the beginning of the twentieth century, the mathematician and physicist Wiener, published two papers [3, 4], and used intervals to give an interpretation to the position and the time of a system. More papers on the subject were written [5, 6, 7, 36] only after Second World War. Nowadays, we consider R.E. Moore [8, 9, 10, 11, 12] as the first mathematician who has proposed a framework for interval arithmetic and analysis. The interval arithmetic, or interval analysis has been introduced to compute very quickly range bounds (for example if a data is given up to an incertitude). Now interval arithmetic is a computing system which permits to perform error analysis by computing mathematics bounds. The extensions ofthe areas of applications are important: non linear problems, PDE, inverse problems. It finds a large place of applications in controllability, automatism, robotics, embedded systems, biomedical, haptic interfaces, form optimization, analysis of architecture plans, ...

Interval calculations are used nowadays as a powerful tool for global optimization and set inversion [8, 9, 10, 11, 13, 18, 36]. Several groups have developed some software and libraries to perform those new approaches such as INTLAB [19], INTOPT90 and GLOBSOL [20], Numerica [21]. But their Achille's heel is the construction of the inclusion function from the natural one due to the lack of distributivity. Some approaches have developed methods to circumvent it with using boolean inclusion tests, series or limited expansions of the natural function where the derivatives are computed at a certain point of the intervals. Nevertheless, those transfers from real functions to functions defined on intervals are not systematic and not given by a formal process. This yields to the fact that the inclusion function definition has to be adapted to each problem with the risk to miss the primitive scope. Moreover, differential calculus and linear algebra need to be performed in the framework of vector space theory and not within semi-group one.

This article reminds first the definition and characteristics of the intervals semi-group and the construction of its associated vector space . After that it is explained how to get an associative and distributive arithmetic of intervals, called pseudo-intervals arithmetic, by embedding the vector space into a free algebra [2]. After that, one proposes a clear and simple scheme to build inclusion functions from the natural one for the semi-group and the vector space.

This permits to present a set inversion scheme, the ψ-algorithm (Probabilist Set Inversion), which is a SIvIA inspired scheme, and using probability calculations. One ends with numerical applications examples for set inversion in order to show how the pseudo-interval arithmeticefficient is.

2 AN ALGEBRAIC APPROACH FOR INTERVALS

An interval X is defined as a non-empty, closed and connected set of real numbers. One writes real numbers as intervals with same bounds, ∀a ∈ , a ≡ [a, a]. We denote by =1 the set of intervals of . The arithmetic operations on intervals, called Minkowski or classical operations, are defined such that the result of the corresponding operation on elements belonging to operand intervals belongs to the resulting interval. That is, if ◊ denotes one of the usual operations +, -, *, /, we have, if X and Y are closed intervals of ,

Although, is provided with a pseudo-inverse operation, it does not satisfy X - X = 0, and hence a subtraction in the usual sense cannot be obtained. In many problems using interval arithmetic, that is the set with the Minkowski operations, there exists an informal transfers principle which permits, to associate with a real function ƒ a function define on the set of intervals which coincides with ƒ on the interval reduced to a point. But this transferred function is not unique. For example, if we consider the real function ƒ (x) = x² + x = x (x + 1), we associate naturally the functions ₁, 2 : → given by ₁ (X) = X (X + 1) and 2 (X) = X² + X. These two functions do not coincide. Usually this problem is removed considering the mostinteresting transfers. But the qualitative "interesting" depends of the studied model and it is not given by a formal process.

In this section, we determine a natural extension of provided with a vector space structure. The vectorial subtraction X \ Y does not correspond to the semantic difference of intervals and the interval \ X has no real interpretation. But these "negative" intervals have a computational role.

An algebraic extension of the classical interval arithmetic, called generalized interval arithmetic [13, 36] has been proposed first by M. Warmus [6, 7]. It has been followed in the seventies by H.-J. Ortolf & E. Kaucher [37, 42, 43, 44, 45]. In this former interval arithmetic, the intervals form a group with respect to addition and a complete lattice with respect to inclusion. In order to adapt it to semantic problems, Gardenes et al. have developed an approach called modal interval arithmetic [46, 47, 48, 49, 50, 51]. S. Markov and others investigate the relation between generalized intervals operations and Minkowski operations for classic intervals and propose the so-called directed interval arithmetic, in which Kaucher's generalized intervals can be viewed as classic intervals plus direction, hence the name directed interval arithmetic [32, 33]. In this arithmetic framework, proper and improper intervals are considered as intervals with sign [34]. Interesting relations and developments for proper and improper intervals arithmetic and for applications can be found in literature [38, 39, 40].

Our approach [2], that we remind below in this article, is similar to the previous ones in the sense that intervals are extended to generalized intervals; intervals and anti-intervals correspond respectively to the proper and improper ones. However we use a construction which is more canonical and based on the semi-group completion into a group, which permits then to build the associated real vector space, and to get an analogy with directed intervals.

In this section we present the set of intervals as a normed vector space with a Banach structure.

2.1 Interval semi-group

Let be the set of intervals. It is in one to one correspondence and can be represented as a point in the half-plane of ², ₁ = {(a, b) ∈ ², a < b}. The set ₂ = {(a, b) ∈ ², a < b} is the set of anti-intervals. is closed for the addition and endowed with a regular semi-group structure. The subtraction on , which is not the symmetric operation of +, corresponds to the following operation on ₁:

(a, b) - (c, d) = (a, b) + s_Δ ○ s₀ (c, d),

where s₀ is the symmetry with respect to 0, and s_Δ with respect to Δ. The multiplication * is not globally defined. Consider the following subset of ₁:

Figure 2

We have the following cases:

1. If (a, b), (c, d) ∈ _{1, 1} the product is written (a, b) * (c, d) = (ac, bd). The vectors e₁ = (1, 1) and e₂ = (0, 1) generate _{1, 1} that is any (x, y) in _{1, 1}, can be decomposed as

(x, y) = xe₁ + (y - x) e₂, with x > 0 and y - x > 0.

The multiplication corresponds in this case to the following associative commutative algebra:

2. Assume that (a, b) ∈ _{1, 1} and (c, d) ∈ _{1, 2} so c < 0 and d > 0. Thus we obtain (a, b) * (c, d) = (bc, bd) and this product does not depend of a. Then we obtain the same result for any a < b. The product (a, b) * (c, d) = (bc, bd) corresponds to

This algebra is not commutative and it is different from the previous.

3. If (a, b) ∈ _{1, 1} and (c, d) ∈ _{1, 3} then a > 0, b > 0 and c < 0, d < 0 and we have (a, b) * (c, d) = (bc, ad). Let e₁ = (1, 1), e₂ = (0, 1). This product corresponds to the following associative algebra:

This algebra is not associative because (e₂e₁) e₁≠ e₂(e₁e₁). We have similar results for the cases (∈ _{1, 2}, _{1, 2}), (∈ _{1, 2}, ∈ _{1, 3}) and (∈ _{1, 3}, ∈ _{1, 3}).

An objective of this paper is to present an associative algebra which contains all these results.

2.2 The real vector space

We recall briefly the construction proposed by Markov [14] to define a structure of abeliangroup. As (, +) is a commutative and regular semi-group, the quotient set, denoted by , associated with the equivalence relations:

(A, B) ~ (C, D) ↔ A + D = B + C,

for all A, B, C, D ∈ , is provided with a structure of abelian group for the natural addition:

where is the equivalence class of (A, B). We denote by \ the inverse of for the interval addition.

We have \ = . If X = [a, a], a ∈ , then = where -X = [-a, -a], and \ = . In this case, we identify X = [a, a] with a and we denote always by the subset of intervals of type [a, a].

Naturally, the group is isomorphic to the additive group ² by the isomorphism → (a - c, b - d) (Fig. 2). We find the notion of generalized interval and this yields immediately to the resulting result:

Proposition 1. Let X = be in . Thus

(1) If l (Y) < l (X), there is an unique A ∈ \ such that = ,

(2) If l (Y) > l (X), there is an unique A ∈ \ such that = = ,

(3) If l (Y) = l (X), there is an unique A = α ∈ such that = = .

Any element = with A ∈ - is said positive and we write > 0. Any element = with A ∈ - is said negative and we write < 0. We write > ' if \ ' > 0. For example if and ' are positive, > ' ↔ l () > l ('). The elements with a ∈ * are neither positive nor negative.

In [14], one defines on the abelian group , a structure of quasi linear space. Our approach is a little bit different. We propose to construct a real vector space structure. We consider the external multiplication:

·: × →

defined, for all A ∈ , by

for all α > 0. If α < 0 we put β = -α. So we put:

We denote α instead of α · . This operation satisfies

1. For any α ∈ and ∈ we have:

For all α, β ∈ , and for all , ' ∈ , we have

Theorem 1. The triplet (, +, ·) is a real vector space and the vectors ₁ = and ₂ = of determine a basis of . So = 2.

Proof. We have the following decompositions:

The linear map

φ : → ²

defined by

is a linear isomorphism and is canonically isomorphic to ². The following map

with respectively l(X) and c(X) the width and the center of the interval X, is obviously a norm. Since is isomorphic to ² which is complete, this yields to the fact that this norm endows with a Banach space structure. Thus, it is possible to perform differential calculus in [22].

3 A 4-dimensional free algebra associated with

We define in this section a four-dimensional associative and distributive free algebra in which the real vector space is embedded.

3.1 Definition of

₄

In introduction, we have observed that the semi-group is identified to _{1, 1} ∪ _{1, 2}∪ _{1, 3}. Let us consider the following vectors of ²:

They correspond to the intervals [1, 1], [0, 1], [-1, 0], and [-1, -1]. Any point of

_{1, 1} ∪ _{1, 2}∪ _{1, 3} admits the decomposition

(a, b) = α₁e₁ + α₂e₂ + α₃e₃ + α₄e₄

with α_i> 0. The dependence relations between the vectors e_i are

Thus there exists a unique decomposition of (a, b) in a chosen basis such that the coefficients are non negative. These basis are {e₁, e₂} for _{1, 1} , {e₂, e₃} for _{1, 2}, {e₃, e₄} for _{1, 3} . Let us consider the free algebra of basis {e₁, e₂, e₃, e₄} whose products correspond to the Minkowski products. The multiplication table is

This algebra is associative and its elements are called pseudo-intervals.

3.2 Pseudo-intervals product

Let φ :→ ₄ the natural injective embedding, ψ the canonical embedding from ₄ to ₄/ F and φ' = ψ º φ. If we identify an interval with its image in ₄, one has:

The application φ is not bijective. Its image on the elements = = is:

Consider in

₄ the linear subspace F generated by the vectors e₁- e₂+ e₃, e₁+ e₄. As

(e₁ + e₄)(e₁ + e₄) = 2(e₁ + e₄)

(e₁ + e₄)( e₁- e₂+ e₃) = (e₁ + e₄)

(e₁- e₂+ e₃)(e₁- e₂+ e₃) = e₁,

F is not a sub-algebra of ₄. Let us consider the map

: → ₄ / F

defined from φ and the canonical projection on the quotient vector space

₄/F. A vector X = Σ α_ie_i∈ ₄ is equivalent to a vector of ₄ with positive components if and only if

α₂ + α₃> 0.

In this case, all the vectors equivalent to x = Σ α_ie_i with α₂ + α₃> 0 correspond to the interval [α₁- α₃- α₄, α₁+ α₂- α₄] of . Thus we have for any equivalent classes of ₄/F associated with Σ α_ie_i with α₂ + α₃> 0 a pre-image in . The map is injective. In fact, two intervals belonging to pieces _{1, i}, _{1, j} with i ≠ j, have distinguish images. Now if (a, b) and (c, d) belong to the same piece, for example _{1, 1}, thus

(a, b) = {(a + λ + µ, b - a - λ, λ, µ) λ, µ ∈ .}

If (c, d) = (a, b), there are λ, µ ∈ such that (c, d) = (a + λ + µ, b - a - λ, λ, µ). This gives a = c, b = d. We have the same results for all the other pieces. Thus : → ₄/F is bijective on its image, that is the hyperplane of ₄/F corresponding to α₂ + α₃> 0.

Practically the multiplication of two intervals will so be made: let X, Y ∈ . Thus X = Σ α_ie_i, Y = Σ β_ie_i with α_i, β_j> 0 and we have the product

X · Y = (φ^' (X) · φ^' (Y))

this product is well defined because ∈ Im. This product is distributive because

X · (Y + Z) = (φ^' (X) · φ^' (Y + Z))

= (φ^' (X) · φ^' (Y) + φ^' (Z))

= (φ^' (X) · φ^' (Y) + φ^' (X) · φ^' (Z))

= X · Y + X · Z

Remark. We have

(φ^' (X) · φ^' (Y + Z)) ≠ (φ^' (X)) · (φ^' (Z))).

We shall be careful not to return in during the calculations as long as the result is not found. Otherwise we find the semantic problems of the distributivity.

We extend naturally the map φ:→₄ to by for every A ∈ by

for every A ∈ .

Theorem 2 The multiplication

· = (φ^' () · φ^' ())

is distributive with respect the addition.

Proof. This is a direct consequence of the previous computations.

3.3 Pseudo-intervals division

Division between intervals can also be defined with solving X · Y = (1, 0, 0, 0) in ₄ or in a isomorphic algebra. In ₄ we consider the change of basis

This change of basis shows that

₄ is isomorphic to

The unit of is the vector . This algebra is a direct sum of two ideals: = I₁+ I₂ where I₁ is generated by and and I₂ is generated by and . It is not an integral domain, that is, we have divisors of 0. For example · = 0.

Proposition 2. The multiplicative group of invertible elements of ₄ is the set of elements x = (x₁, x₂, x₃, x₄) such that

This means that the invertible intervals do not contain 0. If x ∈ we have:

3.4 Monotony property

Let us compute the product of intervals using the product in

₄ and compare it with theMinkowski product. Let X = [a, b] and Y = [c, d] two intervals.

Lemma 1. If X and Y are not in the same piece _{1, i} , then X·Y corresponds to the Minkowski product.

Proof. i) If X ∈ _{1, 1} and Y ∈ _{1, 2} then φ(X) = (a, b - a, 0, 0) and φ(Y) = (0, d, -c, 0). Thus

φ (X) φ (Y) = (ae₁ + (b) - a)e₂)(de₂ - ce₃)

= bde₂ - cbe₃

= (0, bd, -cb, 0)

= φ([cb, bd]).

ii) If X ∈ _{1, 1} and Y ∈ _{1, 3} then φ(X) = (a, b - a, 0, 0) and φ(Y) = (0, 0, d - c, -d). Thus

φ (X) φ(Y) = (ae₁ + (b - a)e₂)((d - c)e₃ - de₄)

= (ad - bc)e₃ - ade₄

= (0, 0, ad - cb, -ad)

= φ([bc, ad]).

iii) If X ∈ _{1, 2} and Y ∈ _{1, 3} then φ(X) = (0, b, -a, 0) and φ(Y) = (0, 0, d - c, -d). Thus

φ (X) φ(Y) = (be₂ + ae₃)((d - c)e₃ - de₄)

= ace₂ - bce₃

= (0, ac, - cb, -ad)

= φ([bc, ad]).

Lemma 2 If X an Y are boht in the same piece _{1, 1} or _{1, 3}, then the product X · Y corresponds to the Minkowski product. The proof is analogous to the previous.

Let us assume that X = [a, b] and Y = [c, d] belong to _{1, 2}. Thus φ(X) = (0, b, -a, 0) and φ(Y) = (0, d, -c, 0). We obtain

XY = (be₂ - ae₃)(de₂ - ce₃ = (bd + ac)e₂ + (-bc - ad)e₃.

Thus

[a, b][c, d] = [bc + ad, bd + ac].

This result is greater that all the possible results associated with the Minkowski product. However, we have the following property:

Proposition 3Monotony property: Let ₁, ₂∈ . Then

The order relation on

₄ that ones uses here is

Proof. Let us note that the second property is equivalent to the first. It is its translation in . We can suppose that ₁ and ₂ are intervals belonging moreover to _{1, 2}: φ(₁) = (0, b, -a, 0), φ(₂) = (0, d, -c, 0). If φ() = (z₁, z₂, z₃, z₄), then

Thus

But (b - d), -(a - c) £ 0 and z₂, z₃> 0. This implies (₁· ) £ (₂· ).

4 THE ALGEBRAS

_n AND AN BETTER RESULT OF THE PRODUCT

We can refine our result of the product to come closer to the result of Minkowski. Consider the one dimensional extension

₄⊕ e₅ = ₅, where e₅ is a vector corresponding to the interval [-1, 1] of _{1, 2}. The multiplication table of ₅ is

The piece

_{1, 2} is written _{1, 2} = _{1, 2, 1}∪ _{1, 2, 1} where _{1, 2, 1} = {[a, b], -a £ b } and _{1, 2, 2} = {[a, b], -a > b}. If X = [a, b] ∈ _{1, 2, 1} and Y = [c, d] ∈ _{1, 2, 2}, thus

φ(X) · φ(Y) = (0, b + a, 0, 0, –a) · (0, 0,– c – d, 0, d) = (0,– (a + b)(c + d), 0, 0, a(c + d) + bd).

Thus we have

X · Y = [- bd - ac - ad, -bc].

Example. Let X = [-2, 3] and Y = [-4, 2]. We have X ∈ _{1, 2, 1} and Y ∈ _{1, 2, 2}. The product in ₄ gives The product in ₅ gives The Minkowski product is Thus the product in ₅ is better.

Conclusion. Considering a partition of _{1, 2}, we can define an extension of ₄ of dimension n, the choice of n depends on the approach wanted of the Minkowski product. For example, let us consider the vector e₆ corresponding to the interval . Thus the Minkowsky product gives e₆·e₆ = e₇ where e₇ corresponds to . This yields to the fact that ₆ is not an associative algebra but it is the case for ₇ whose table of multiplication is

Example. Let X = [-2, 3] and Y = [-4, 2]. The decomposition on the basis {e₁, ..., e₇} with positive coefficients writes X=e_5+2e_7, Y=2e_6. X Y=(e_5+2e_7)(4e_6)=4e_5+8e_6=[-12, 8].

We obtain now the Minkowski product.

5 INCLUSION FUNCTIONS

It is necessary for some problems to extend the definition of a function defined for real numbers ƒ: → to function defined for intervals [ ƒ ]: → such as [ ƒ ]([a, a]) = ƒ(a) for any a ∈ . It will be convenient to have the same formal expression for ƒ and [ ƒ ]. Usually the lack of distributivity in Minkowski arithmetic doesn't give the possibility to get the same formal expressions. But with the pseudo-intervals arithmetic we have presented, there is no data dependency any more and one can define easily inclusion functions from the natural one. For example, let's extend to intervals the real functions ƒ₀(x) = x²-2x + 1, ƒ₁(x) = (x - 1)², ƒ₂(x) = x (x - 2) + 1. Usually, with the Minkwoski operations, the three expressions of this same function for theinterval X = [3, 4] are [ ƒ ]₀(X) = [2, 11], [ ƒ ]₁(X) = [4, 9] and [ ƒ ]₂(X) = [6, 12]. Data dependency occurs when the variable appears more than once in the function expression. The deep reason of that is the lack of distributivity in Minkowski arithmetic. But within the arithmeticdeveloped in ₄ or higher dimension free algebras [2], this problem vanishes. For example:with X = [3, 4] and since X ∈ ₁₁,

Since e₁ = (1, 0, 0, 0), e₂ = (0, 1, 0, 0) and with means of product table, one has

and

Thus, [ ƒ ]₀(X) = [ ƒ ]₁(X) = [ ƒ ]₂(X) = [4, 9] and the inclusion function is defined univocally regardless the way to write the original one.

On the other hand, the construction of the inclusion function depends on the type of problem one deals with. If one aims to perform set inversion for example, it has to be done in the semi-group . But, the subtraction is not defined in . This problem can be circumvented by replacing it with an addition and a multiplication with the interval e₄ = [-1, -1]. This maintains the associativity and distributivity of arithmetic and permits to introduce a pseudo-subtraction. For example: if ƒ(x) = x²- x = x(x - 1) for real numbers, one defines [ ƒ ](X) = X² + e₄ · X. One reminds the product [-1, -1] · [a, b] is equal to [-b, -a]. Due to the fact that the arithmetic is now associative and distributive, one doesn't have data dependency anymore and [ ƒ ](X) = X² + e₄ · X = X · (X + e₄). The last term corresponds to the transfer of x (x - 1). Division can be transferred to the semi-group in the same way by replacing = x^-1 with .

Taylor polynomial expansions, differential calculus and linear algebra operations are defined only in a vector space. Therefore the transfer for the vector space is done directly. This permits to get infinitesimal intervals with the subtraction and to compute derivatives. This is of course not allowed and not possible into the semi-group. From to the vector space , ƒ: x -x is transferred to [ ƒ ]: X ≡ \ ≡ \ X. This means that [a, b] subtraction is the anti-interval [-a, -b] addition. One of the most important consequence is that it is possible to transfer some functions directly to the pseudo-intervals. For example, it is easy to prove analytically in that [exp] with means of Taylor expansion.

6 PROBABILIST SET INVERSION: ψ -algorithm

6.1 Flowchart

One presents an efficient set inversion method whose flowchart is very simple. One of the powerful application of interval calculus is the set inversion of a real-valued function defined on real numbers. As mentioned in the first section, the mathematical modelling of this problem is the following as shown on Figure 1: let's note ƒ: ⁿ

^p a function for a physical system, which is required to be surjective only, ⊂ ⁿ the set of adjustments, and ⊂ ^p the set of performance of a system. Set inversion consists of the computation of S = ƒ^-1() ∩ , and one has to perform it within the semi-group . Some interesting and powerful methods using intervals have been developed those last years, especially SIVIA [18], Set Inversion Via Interval Analysis. But the inclusion function being not well defined in the semi-group with the Minkowski arithmetic, SIVIA uses boolean inclusion tests and finds accepted, rejected and "uncertain" domains.

With the algebraic arithmetic, one doesn't need boolean tests since the inclusion functionsare well-defined. Thus, we propose the ψ-algorithm (Probabilist Set Inversion) inspired from SIVIA but without boolean tests and with a conditional probability calculation and domain bisections. This yields to accepted or rejected domains only. We are interested to compute the following conditional probability

where mes is the Lebesgue measure in

^p (length, surface, ...). If this probability equals 1 then the set is added to the list of solutions. If it is zero the set is rejected and removed from the list of interval candidates. If the probability is such as p () ∈ ]0, 1[, then is bisected and ψ-algorithm applies the same procedure recursively for the resulting intervals until the size is lower than a fixed size resolution of the intervals or until the sets are accepted or rejected. Since ψ-algorithm creates sequences of decreasing intervals which are compact sets, it is obvious that ψ-algorithm converges to fixed points probabilities which are simply 0 and 1. In fact, we consider a sequence of compact sets satisfying d(K_n) > d(K_{n + 1}) where d is the diameter ofthe compact set. If K_n∩ K_m = , for any n ≠ m, the sequence is convergent and the limit isthe empty set. Then, one has just to consider the sequence of bisected sets.

4.2 Numerical applications

We have developed a numerical library for python environment [27] called yet typhon. It is a pure numerical implementation performing the basic arithmetic presented above [2]. This library aims to give simple and optimized routines to perform interval calculations based on the algebraic arithmetic. One gives in this section some numerical application examples of ψ-algorithm in order to illustrate how it can treat usual inversion problems and build well-defined inclusion functions.

Let's define the non-linear functions ƒ_i: ²→ ², i = 1, 2, with respectively adjustments and performances sets _i, _i:

Those examples have been chosen to give examples of addition, subtraction, product and division transfers from to , and to exhibit the difference between the usual Minkowski arithmeticand the algebraic one [2]. The calculations with the ψ-algorithm are shown on Figures 3, 4 and 5. The convergence to 0 or 1 probabilities only, shows that inclusion functions are well constructed and that the pseudo-interval arithmetic is robust. The following example

presented on Figure 5 shows clearly that the ψ-algorithm implemented in the algebraic arithmetic we use is not data dependant. The variables appear more than once in the formal expression of the function ƒ₃. The CPU time for this inversion is about 255 seconds on a simple 1.67 Ghz Intel processor for a spatial surface resolution of 10^-4.

There is no limitation for the dimensions of the adjustments and performances sets as shown on Figure 6 for the function ƒ₄: ³→ ⁴.

Due to the bisection, the algorithm computational complexity is exponential according to the iterations N, and it is not improved compared to SIVIA one. In our scheme, computational time is defined as

However, if the native function is differentiable on

ⁿ, it is possible to define an adaptive mesh, with bisection spanned only in the space directions where the derivative magnitude is larger than a certain fixed value, because it is not useful to bisect in flat directions. This will obviously decrease the computational complexity constant k. It is possible to decrease the computational time constant with mean of paralleling using domain decomposition [18]. The adjustment set is divided on the first axis, and each processor performs the ψ-algorithm on one of those sub-domains. The master processor collects all the results at the end of the calculations.

7 CONCLUSION

A new algebraic approach for interval arithmetic, called pseudo-interval arithmetic has been proposed. It is based on free algebra build from Minkowski products of basis intervals and with dimension higher or equal to 4. One has identified intervals with the elements of this associative algebra and showed that their product is distributive with respect to their addition. Increasing the dimension will give pseudo-intervals product closer to Minkowski one's.

One has presented also a heuristic way to transfer real functions to inclusion ones depending on the space needed (semi-group or vector space). This permits to define a simple but very efficient algorithm for set inversion, the ψ-algorithm, which uses pseudo-intervals arithmetic and probability calculations. The convergence of this algorithm is guaranteed, and it offers several possibilities of applications, such as solving algebraic equations, differential equations, probability law of random variables calculations (discrete or continuous), topological analysis, numerical Lebesgue integrals computations, data analysis such as principal components analysis, andparameters identification.

ACKNOWLEDGMENTS

The author thanks Michel Gondran from University of Paris-Dauphine and Thierry Socrounfrom Electricity of France for useful and helpful discussions.

Received on November 10, 2012

Accepted on January 28, 2014

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Publication Dates

• Publication in this collection
  10 June 2014
• Date of issue
  Apr 2014

History

• Accepted
  28 Jan 2014
• Received
  10 Oct 2012