Interval Algorithms

Due to the dependency problem, a straightforward adaption of an algorithm designed for floating-point arithmetic is usually unsuitable, typically delivering excessively wide result intervals. For example, an unmodified interval Gaussian elimination for the solution of a system of linear equations with interval coefficients will generally quickly fail due to division by the interval values for pivots containing zero, cf. Subsection 2.3.3. In general, therefore, floating-point algorithms have to be adapted in order to minimise the effect of the dependency problem. In some cases, completely new algorithms have to be devised.

2.3.1 Branch-and-Bound Methods

Branch-and-bound algorithms are a general scheme most often employed for the solution of optimisation problems, but also for systems of nonlinear equations and other types of problems. Required is a starting finite search space — in most formulations, a box — which is successively subdivided into smaller sub-regions until they can be excluded from the search. The search space is structured in the form of a tree and the sub-regions are usually (but not always) defined as boxes (see Definition 2.2).

It should be noted that the branch-and-bound method is not an interval method per se, however in the common case where boxes and sub-boxes are used, it naturally lends itself to interval approaches, which are often (but by no means always) employed for the bounding step (see below).

The first branch-and-bound methods, not utilising interval analysis, appeared the in

2 Interval Analysis

also described in [RR88]. Thereafter, interval branch-and-bound methods were further de-veloped by H. Ratschek and J. Rokne [RR88] and E. R. Hansen [Han80, Han03], amongst others.

There are two key steps to a branch-and-bound approach:

• Asubdivision (or branching, splitting) step, which partitions the current (sub-)region under investigation into two (or more) smaller sub-regions. Typically, such a region consists of an n-dimensional box, and a subdivision (often a bisection) is performed in one direction.

• A bounding (or pruning) step, where sub-regions are safely discarded as candidates for containing a solution. The character of this step depends on whether a system of equations or an optimisation problem is being solved. In the former case, bounds on the ranges of the component functions may be computed; where one such function can be guaranteed to exclude zero, the sub-region may be discarded. In the latter case, which is canonically presented as a minimisation problem, the sub-region may likewise be tested for feasibility, but a lower bound for the objective function is also computed. This is compared against a global variable which records the minimum upper bound for sub-regions so far; where the lower bound for a sub-region exceeds this, it can safely be discarded. This step is also called ‘pruning’, because branches of the search tree are removed.

Also required is a search strategy for the order in which sub-regions are processed and a termination criterion which stipulates when sub-regions become satisfactorily small. At the end, zero or more sub-regions of this small size remain, which enclose the solution set (in the case of a system of equations) or contain the minimiser and provide an upper bound on the corresponding value of the objective function (in the case of an optimisation problem).

Heuristics and tests may be employed to accelerate the algorithm, by minimising the total number of sub-regions to be processed [RR88, Section 3.11]. These may include tree search strategies, heuristics for selection of subdivision direction, e.g. [RC95], monotonicity tests, and box contraction techniques, e.g. [VHMK97].

2.3.2 Interval Newton Methods

The interval Newton method is obtained by replacing the real (floating-point) variables in the standard Newton iteration by intervals. It is discussed in further detail in Subsection 5.3.3.

2.3.3 Matrix Methods

An interval matrix is one whose entries are intervals, instead of real numbers. Repetetive operations on such matrix entries or rows, where each interval variable often appears mul-tiple times, make them very susceptible to the dependency problem. A common example is the classic Gaussian elimination, used to solve a system of linear equations or find a matrix

2 Interval Analysis

inverse; a naive interval version of this method will fail for all but the simplest of prob-lems, with the excessive widths of the entries used as pivots rapidly causing a division by zero. Such methods therefore need to be adapted in order to circumvent or mitigate these problems, e.g. with a scheme for the tightening of interval pivots [Gar09].

2.3.4 Other Interval Methods

Many other interval methods for a number of different application areas exist. A good overview is given in [Int, Kea08] and a few are listed below:

• Taylor models with interval arithmetic, commonly used for the verified solution of ordinary differential equations,

• Methods for automated theorem proving (a notable example is a proof of the Kepler conjecture, concerning sphere packing in three dimensions, by T. Hales [Hal05]; see [Rum05] for further examples),

• Constraint propagation, applied to global optimisation or constraint satisfaction prob-lems, e.g. [Gra07],

• Various methods for the solution of finite element truss structure models with uncer-tain parameters, e.g. [Zha05, GPS12].

3 Bernstein Expansion

Bernstein expansion refers to the process of rewriting a polynomial in Bernstein form, a procedure which can be employed in function approximation and bounding the ranges of polynomials. A polynomial in Bernstein form is expressed as a linear combination of Bernstein basis polynomials. These polynomials were first defined by S. N. Bernstein in 1912 [Ber12] in a constructive proof of the Weierstrass approximation theorem, which states that a continuous function may be uniformly approximated over a finite interval arbitrarily closely by a polynomial.

In the case of the real numbersRthe theorem may be stated as follows:

Theorem 3.1 (Weierstrass Approximation Theorem). Let f be a continuous real-valued function defined over an interval [a, b]. For any ε > 0 there exists a polynomial p:R→R such that

|f(x)−p(x)| < ε, ∀x∈[a, b].

Bernstein explicitly provided a sequence of polynomials which converge uniformly to a given continuous function over the unit intervalI= [0,1]. It should firstly be observed that we can consider the unit interval without loss of generality. It is readily apparent that an affine transformation of a polynomial to or fromIand a non-degenerate interval[a, b]yields a polynomial of the same degree and that the same transformation preserves the continuity of an arbitrary function.

Given a function f which is continuous over I, for a given degree l Bernstein’s approxi-mation is

f_l(x) = f(0)(1−x)^l+lf(1

l)x(1−x)^l−1+. . .+ l

i

f(i

l)xⁱ(1−x)^l−i+. . .+f(1)x^l. (3.1) Asl → ∞, f_l(x) tends uniformly to f(x) overI. The proof [Ber12] relies upon the obser-vation that f_l(x) (3.1) is a weighted average of f(0), . . . , f(ⁱ_l), . . . , f(1). In particular, as l becomes large, only those terms for which ⁱ_l ≈x make a significant contribution, since the values of the weights depend uponx. The result is obtained by equating this to a probability distribution and employing a bound provided by Chebyshev’s inequality.

It can be seen that each of the weights (or multipliers) for thef values appearing in (3.1) is a degreel polynomial inx; these are the Bernstein basis polynomials and f_l is a degreel polynomial presented in Bernstein form.

There is an early monograph on Bernstein polynomials [Lor53]; thereafter the Bernstein expansion was first applied to the range of univariate polynomials in [CS66] and then [Riv70]

and [Rok77]; an extension to the multivariate case appears in [Gar86]. More recent treat-ment appears in [NA07], [BCR08] and [RN09], and a comprehensive survey paper has just been completed [Far12]. An alternative explanation of the theory of Bernstein polynomials is presented in [Zum08, Chapter 4].

3 Bernstein Expansion

3.1 Fundamentals

In this section the fundamental theory of Bernstein polynomials and Bernstein expansion is introduced.

3.1.1 Bernstein Basis Polynomials

Theith Bernstein basis polynomial of degree lis given by B_i^l(x) =

l i

xⁱ(1−x)^l−i, i= 0, . . . , l. (3.2) Where required, we shall adopt the convention that B_i^l(x) = 0 for all x if i < 0 or i > l.

For l up to and including 4, the Bernstein basis polynomials are given explicitly in Table 3.1 and, for l= 4, depicted in Figure 3.1.

l i= 0 i= 1 i= 2 i= 3 i= 4

0 1

1 1−x x

2 1−2x+x² 2x−2x² x²

3 1−3x+ 3x²−x³ 3x−6x²+ 3x³ 3x²−3x³ x³

4 1−4x+ 6x²−4x³+x⁴ 4x−12x²+ 12x³−4x⁴ 6x²−12x³+ 6x⁴ 4x³−4x⁴ x⁴ Table 3.1: The Bernstein basis polynomials B_i^l(x) forl up to4.

B (x)⁴_i

i = 0 i = 4

i = 1 i = 3

i = 2

0 1

x 0.5

0 1

3 Bernstein Expansion

Here, the term ‘Bernstein polynomial’ is used to mean ‘Bernstein basis polynomial’, and nota polynomial which is presented in Bernstein form (see Subsection 3.1.2). Some authors use this latter meaning, but this would appear to be redundant since everypolynomial is a Bernstein polynomial in this sense — any polynomial can be rewritten in Bernstein form, as stated below.

Basic Properties

It can readily be seen that the Bernstein basis polynomials satisfy the following properties for all l∈N,i= 0, . . . , l:

• B_i^l(x)≥0, ∀x∈I.

• The set of Bernstein polynomials of degree lform a partition of unity, i.e.

l

• The Bernstein polynomials of degreelform a basis for the vector space of polynomials of degree less than or equal to l. Thus a Bernstein representation exists for every polynomial, and the coefficients are defined in Section 3.1.3.

Degree Elevation and Reduction Properties

• A Bernstein polynomial of degree l can be generated recursively from lower-degree polynomials:

B^l_i(x) = xB_i−1^l−1(x) + (1−x)B_i^l−1(x). (3.3) This can be proven using the recursive relation for binomial coefficients:

B^l_i(x) =

3 Bernstein Expansion

• A Bernstein polynomial of degree l can also be expressed as a linear combination of two Bernstein polynomials of degreel+ 1:

B_i^l(x) = l+ 1−i

l+ 1 B_i^l+1(x) +i+ 1

l+ 1B^l+1_i+1(x). (3.4) This is proven by combining the following two results:

• A Bernstein polynomial of degree lcan further be related to a single Bernstein poly-nomial of degreel+ 1as follows:

(1−x)B_i^l(x) =

• By repeated application of (3.4), a Bernstein polynomial of degreelcan be expressed as a linear combination of Bernstein polynomials of degreel+r, wherer ∈N:

B_i^l(x) =

Here we begin to make use of the abbreviated notation for multipowers and vectors. If n is set to 1, then we have the univariate case. Let us suppose that we have an n-variate polynomialp presented in the usual power form (i.e. in the power basis) as

p(x) =

l

X

i=0

a_ixⁱ, x= (x₁, . . . , x_n). (3.6) It may be rewritten in Bernstein form (i.e. in the Bernstein basis) overI= [0,1]ⁿ as

l

3 Bernstein Expansion

where B_i^l is the ith Bernstein basis polynomial of degree l (3.8) and bi are the so-called Bernstein coefficients. The derivation of these coefficients is given below in Subsection 3.1.3. In the n-dimensional case, theith Bernstein polynomial of degree l= (l₁, . . . , l_n) is given as a straightforward product of univariate Bernstein polynomials (3.2) as follows:

B^l_i(x) = B^l_i¹

1(x₁)B_i^l2

2(x₂)·. . .·B^l_iⁿ

n(x_n). (3.8)

We may allow that the degree ofpis given by somer, wherer < l. In this case, the formulae herein remain in force with the convention thataj = 0 ifj > r.

3.1.3 Basis Conversion

Here we derive formulae for converting between the coefficients{a_i}of a polynomial in the power basis and the coefficients {b_i}in the Bernstein basis.

Conversion From Power Basis to Bernstein Basis

Theorem 3.2. Let p be a multivariate polynomial in power form given as in (3.6). Then its coefficients in Bernstein form (its Bernstein coefficients) are given by

bi =

3 Bernstein Expansion

Conversion From Bernstein Basis to Power Basis

Theorem 3.3. Let p be a multivariate polynomial in Bernstein form given as in (3.7).

Then its coefficients in power form are given by

a_i = {b_i} over [0,1]² calculated according to (3.9) are given in Table 3.2.

i1 = 0 i1= 1 i2 = 2 i3 = 3

i2 = 0 3 3 3 5

i₂ = 1 3 ⁷₃ ¹⁹₁₂ ¹¹₄

i₂ = 2 10 ²⁶₃ ⁴³₆ ¹⁵₂

Table 3.2: Bernstein coefficients bi of p(x1, x2) = 2x³₁−¹₂x²₁x2+ 7x²₂−4x1x2+ 3.

3 Bernstein Expansion

3.1.4 Generalised Bernstein Form

In many cases it is desired to calculate the Bernstein expansion of a polynomial over a general non-degeneraten-dimensional box

X = [x₁, x1]×. . .×[x_n, xn]. (3.11) It is possible to firstly apply the affine transformation which transforms the unit boxI to X and then apply (3.9), using the coefficients of the transformed polynomial. However it is sometimes useful to consider a direct calculation. Here, theith Bernstein polynomial of degree l= (l₁, . . . , l_n) overXmay be written as

The Bernstein coefficientsb_i of a polynomialp (3.6) of degreeloverX (3.11) are given by bi =

Im Dokument Enclosure Methods for Systems of Polynomial Equations and Inequalities (Seite 32-41)