Simple formula for integration of polynomials on a simplex

08/19/2019 ∙ by Jean Lasserre, et al. ∙ 0

We show that integrating a polynomial of degree t on an arbitrary simplex (with respect to Lebesgue measure) reduces to evaluating t homogeneous polynomials of degree j = 1, 2,. .. , t, each at a unique point ξ j of the simplex. This new and very simple formula can be exploited in finite (and extended finite) element methods, as well as in other applications where such integrals are needed.

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1. Introduction

We consider the problem of integrating a polynomial on an arbitrary simplex of and with respect to the Lebesgue measure. As (where is the degree of and each is homogeneous of degree ), after an affine transformation this problem is completely equivalent to integrating a related polynomial of degree on the canonical simplex where .

Background

In addition to being a mathematical problem of independent interest, integrating a polynomial on a polytope has important applications in e.g. computational geometry, in approximation theory (constructing splines), in finite element methods, to cite a few; see e.g. the discussion in Baldoni et al. [2]. In particular, because of applications in finite (and extended finite) element methods and also for volume computation in the Natural Element Method (NEM), there has been a recent renewal of interest in developing efficient integration numerical schemes for polynomials on convex and non-convex polytopes

For instance the HNI (Homogeneous Numerical Integration) technique developed in Chin et al. [4] and based on [11], has been proved to be particular efficient in some finite (and extended finite) element methods; see e.g. Antonietti et al. [1], Chin and Sukumar [5], Nagy and Benson [13] for intensive experimentation, Zhang et al. [14] for NEM, Frenning [6] for DEM (Discrete Element Method), Leoni and Shokef [12] for volume computation. For exact volume computation of polytopes in computational geometry, the interested reader is also referred to Büeler et al. [3] and references therein.

For integrating a polynomial on a polytope, one possible route is to use the HNI method developed in [11, 4] and also extended in [1], without partitioning the polytope in simplices. Another direction is to consider efficient numerical schemes for simplices since quoting Baldoni et al. [2]among all polytopes, simplices are the fundamental case to consider for integration since any convex polytope can be triangulated into finitely many simplices”. In [2] the authors analyze the computational complexity of the latter case and describe several formulas; in particular they show that the problem is NP-hard in general. However, if the number of variables is fixed then one may integrate polynomials of linear forms efficiently (with Straight-Line program for evaluation) and if the degree is fixed one may integrate any polynomial efficiently. They also describe several formulas in closed form for integrating powers of linear forms [2, Corollary 12] and also arbitrary homogeneous polynomials [2, Proposition 18] and [8], all stated in terms of a summation over vertices of the simplex.

Our main result

With , is associated the polynomial :

(1.1)

We establish the following simple formula:

Theorem 1.1.

Let be the canonical simplex. If is a polynomial of degree then with :

(1.2)

where and each is homogeneous of degree .

Theorem 1.1 states that integrating a polynomial of degree on the canonical simplex can be done by evaluating each at a unique point . In addition all points are on a line between the origin and the point on the boundary of . To the best of our knowledge, and despite its simplicity, formula (1.2) is new.

Similarly, integrating a polynomial of degree on an arbitrary simplex can be done by evaluating related polynomials of degree , each at a certain point of . Indeed an arbitrary (full-dimensional) simplex can be mapped to the canonical simplex by an affine transformation for some real nonsingular matrix

and vector

. Therefore (1.2) translates to a similar formula on with ad-hoc polynomials and aligned points .

Hence formula (1.2) is much simpler than those in [2, 8]. In particular it is valid for an arbitrary polynomial and there are only points involved if the degree of the polynomial is . In contrast, for integrating a -power of a linear form, the formula in [2] requires a summation at the vertices and in [8], integrating a form of degree requires evaluations of of a multilinear form at the vertices.

We would like to emphasize that (1.2) resembles a cubature formula but is not. A cubature formula is of the form

(1.3)

for some integer and points with associated weights , . However Theorem 1.1 suggests that as long as polynomials are concerned, the simpler alternative formula (1.2) could be preferable to cubature formulas involving many points. The point of view is different. Instead of evaluating the single polynomial of degree at several points in (1.3), in (1.2) one evaluates other polynomials of degree , (simply related to ); each polynomial is evaluated at a single point only.

Our technique of proof is relatively simple. It uses (i) Laplace transform technique and homogeneity to reduce integration on with respect to (w.r.t.) Lebesgue measure to integration on the positive orthant w.r.t. exponential density; this technique was already advocated in [9] for computing certain multivariate integrals and in [10] for volume computation of polytopes. Then (ii) integration of monomials w.r.t. exponential density can be done in closed-form and results in a simple formula in closed form.

Interestingly and somehow related, recently Kozhasov et al. [7] have considered integration of a “monomial” (with ) with respect to exponential density on the positive orthant. In [7] the density is called the Riesz kernel of the monomial . The Riesz kernel offers a certificate of positivity for a function to be completely monotone (a strong positivity property of functions on cones).

2. Main result

2.1. Notation, definitions and preliminary result

Let denote the ring of real polynomials in the variable . With the set of natural numbers, a polynomial is written

in the canonical basis of monomials. A polynomial is homogeneous of degree if for all and all . For let . Let denotes the positive orthant of . A function is positively homogeneous of degree if

and a polynomial is homogeneous of degree if

Denote by the canonical simplex where . For let .

With a polynomial ,

we associated the polynomial defined by:

(2.1)

2.2. Main result

After an affine transformation, integrating on an arbitrary full-dimensional simplex reduces to integrating a related polynomial of same degree on the canonical simplex where . Therefore in this section we consider integrals of polynomials (and a certain type of positively homogeneous functions) on the canonical simplex .

Theorem 2.1.

Let and let . Let be a positively homogeneous function of degree on such that and . Then:

(2.2)

If is a homogeneous polynomial of degree (hence ) then

(2.3)

and in particular with :

(2.4)

where and .

Proof.

Let be fixed and let be the function:

Observe that on and in addition, is positively homogeneous of degree , so that (well defined since is finite). As , its Laplace transform is well defined and reads:

(2.5)

On the other hand, for real :

Identifying with (2.5) yields (2.2). Next, to get (2.3) observe that for , for all , and therefore:

for every . Summing up yields the result (2.3) and (2.4) with . Finally, the last equality of (2.4) is obtained by homogeneity of . ∎

As the reader can see, formula (2.4) is extremely simple and only requires evaluating at the unique point . This in contrast to the formula in [8] which requires a sum of terms, each involving evaluations at the vertices of .

Remark 2.2.

Notice that (2.2) can also be interpreted as follows: Let , be positively homogeneous of degree and as in Theorem 2.1. Define the function , by: . Then is the Laplace transform of , or equivalently in the terminology of Kozhasov et al. [7], is completely monotone111A real-valued function is completely monotone if for all and for all index sequences of arbitrary length . (by the Bernstein-Hausdorff-Widder-Choquet theorem; see [7, Theorem 2.5]). Next, let be a cone with dual cone . If , , and

for some Borel measure on , then is called a Riesz measure. In addition if has a density with respect to Lebesgue measure on then is called the Riesz kernel of ; see [7, p. 4].

Hence from Remark 2.2 we obtain:

Proposition 2.3.

For every positively homogeneous of degree as in Theorem 2.1, the function :

(2.6)

is completely monotone. In addition if with then is the Riesz kernel of the function on .

Next given denote by the scalar product and by the vector . Finally let the homogeneous polynomial of degree with all coefficients equal to , that is, .

As a consequence of Theorem 2.1 we obtain:

Corollary 2.4.

(i) Let be polynomial of degree and write where each is homogeneous of degree . Then

(2.7)

where and is as in (2.1).

(ii) For every and :

(2.8)
Proof.

(i) As , use Theorem 2.1 for each and sum up. Next, (ii) is a direct consequence of Theorem 2.1 and the fact that

and therefore . ∎

Hence Corollary 2.4 states that integrating on reduces to evaluate each at the unique point , and sum up. In addition, all points , , are aligned in ; they are between the origin and the point , on the line joining to .

Again formula (2.7) and (2.8) are extremely simple. The former only requires evaluating at ( evaluations) and the latter only requires evaluating the polynomial at the point . This is contrast with [2, Corollary 12, p. 307] which is more complicated (even for a single form ).

Finally, Theorem 2.1 can be extended to a class of homogeneous functions

Proposition 2.5.

Let be a finite set of indices and let be positively homogeneous of degree and of the form

for some real coefficients . Then

(2.9)

where with , and

Proof.

Let with be fixed and let . By Theorem 2.1:

Summing up over all yields

and with one obtains (2.9). Finally, the last equality in (2.9) uses the positively homogeneity of and for all . ∎

Example 1.

For illustration purpose consider the elementary two-dimensional example (i.e., ) where . With one obtains and . The right-hand-side of (2.7) reads:

For instance with , one obtains:

and indeed . In Figure 1 is displayed the -Simplex with the points and .

Figure 1. ; Simplex and points and

2.3. Back to an arbitrary simplex

Let be an arbitrary full-dimensional simplex. Then is mapped to by some affine transformation. In a fixed basis this is obtained by the change of variable where is one non singular real matrix and where is a vertex of . Then:

Next, if is a polynomial of degree :

(2.10)

where has same degree as . Next, writing with homogeneous of degree one has

and therefore with as in Corollary 2.4, and . Therefore combining (2.10) and (2.7) in Corollary 2.4 yields:

the analogue for of (2.7) for .

3. Conclusion

We have provided a very simple closed-form expression for the integral of an arbitrary polynomial on an arbitrary full-dimensional simplex. Remarkably if has degree , it consists of evaluating polynomials (related to ) of degree , respectively, and each is evaluated at a unique point of the simplex. To the best of our knowledge this simple formula is new and potentially useful in all applications where such integrals need to be computed; for instance in finite and extended finite element methods. Therefore it could provide a valuable addition to the arsenal of techniques already available for multivariate integration on polytopes.

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