Simple Formula for Integration of Polynomials on a Simplex
Jean Lasserre (LAAS-MAC)

TL;DR
The paper presents a straightforward formula that simplifies the integration of polynomials over a simplex by reducing it to evaluations at specific points, enhancing computational efficiency in finite element methods.
Contribution
It introduces a novel, simple formula for integrating polynomials on a simplex, reducing the problem to evaluating homogeneous polynomials at specific points.
Findings
Simplifies polynomial integration on simplexes.
Reduces integration to polynomial evaluations at specific points.
Applicable in finite element and related methods.
Abstract
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 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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11institutetext: J.B. Lasserre 22institutetext: LAAS-CNRS and Institute of Mathematics, University of Toulouse, France
Tel.: +33-5-61336415
Fax: +33-5-61336936
22email: [email protected]
Simple formula for integration of polynomials on a simplex
††thanks: Research funded by the European Research Council (ERC) under the European’s Union Horizon 2020 research and innovation program (grant agreement 666981 TAMING)
Simple formula for integration
Jean B. Lasserre
(Received: date / Accepted: date)
Abstract
We show that integrating a polynomial of degree on an arbitrary simplex (with respect to Lebesgue measure) reduces to evaluating homogeneous related Bombieri polynomials of degree , each at a unique point of the simplex. This new and very simple formula could be exploited in finite (and extended finite) element methods, as well as in applications where such integrals must be evaluated. A similar result also holds for a certain class of positively homogeneous functions that are integrable on the canonical simplex.
Keywords:
Numerical integration simplex Laplace transform
MSC:
65D30 78M12 44A10
††journal: BIT
1 Introduction
We consider the problem of integrating a polynomial on an arbitrary simplex of and with respect to the Lebesgue measure. After an affine transformation this problem is completely equivalent to integrating a related polynomial of same degree on the canonical simplex where . Therefore the result is first proved on and then transferred back to the original simplex. The result is also extended to positively homogeneous functions of the form where is finite and .
1.1 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. baldoni . 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. chin and based on lass-poly , has been proved to be particular efficient in some finite (and extended finite) element methods; see e.g. Antonietti et al. galerkin , Chin and Sukumar suku , Nagy and Benson nagy for intensive experimentation, Zhang et al. zhang for NEM, Frenning frenning for DEM (Discrete Element Method), Leoni and Shokef leoni for volume computation. For exact volume computation of polytopes in computational geometry, the interested reader is also referred to Büeler et al. bueler and references therein.
For integrating a polynomial on a polytope, one possible route is to use the HNI method developed in lass-poly ; chin and also extended in galerkin , without partitioning the polytope in simplices. Another direction is to consider efficient numerical schemes for simplices since quoting Baldoni et al. baldoni “among all polytopes, simplices are the fundamental case to consider for integration since any convex polytope can be triangulated into finitely many simplices”. In baldoni 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 (baldoni, , Corollary 12) and also arbitrary homogeneous polynomials (baldoni, , Proposition 18) and lass-avra , all stated in terms of a summation over vertices of the simplex.
1.2 Our main result
With , is associated the Bombieri-type polynomial :
[TABLE]
We establish the following simple formula:
Theorem 1.1
Let be the canonical simplex. If is a polynomial of total degree then with :
[TABLE]
where and each is a homogeneous form111 is the unique form of degree which is the sum of all monomials of degree of (with their coefficient). of degree .
Similarly, let be a finite set and let with
[TABLE]
be positively homogeneous of degree (i.e. for all ). Then
[TABLE]
where with , and
[TABLE]
Theorem 1.1 states that integrating a polynomial of degree on the canonical simplex can be done by evaluating each Bombieri form 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 their simplicity, we have not been able to find formula (2) or (4) in the literature, even if they can be obtained in several relatively straightforward manners from previous results in the literature.
The point that we make in (2) is to relate to point evaluation of its Bombieri-polynomial at only very specific points of the simplex ; and similarly for positively homogeneous functions of the form (3).
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 (2) translates to a similar formula on with ad-hoc polynomials and aligned points .
Hence formula (2) is much simpler than those in baldoni ; lass-avra . 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 baldoni requires a summation at the vertices and in lass-avra , integrating a form of degree requires evaluations of a multilinear form at the vertices.
We would like to emphasize that (2) resembles a cubature formula but is not. A cubature formula is of the form:
[TABLE]
for some integer and points with associated weights , . However Theorem 1.1 suggests that as long as polynomials are concerned, the simpler alternative formula (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 (5), in (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 nonnegative orthant w.r.t. exponential density; this technique was already advocated in lass-zeron for computing certain multivariate integrals and in volume 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. sturmfels have considered integration of a “monomial” (with ) with respect to exponential density on the positive orthant. In sturmfels 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
[TABLE]
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
[TABLE]
and a polynomial is homogeneous of degree if
[TABLE]
Denote by the canonical simplex where . For let .
With a polynomial in (6) is associated the “Bombieri” polynomial:
[TABLE]
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:
[TABLE]
If is a homogeneous polynomial of degree (hence ) then
[TABLE]
and in particular with :
[TABLE]
where and .
Proof
Let be fixed and let be the function:
[TABLE]
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:
[TABLE]
On the other hand, for real :
[TABLE]
Identifying with (11) yields (8). Next, to get (9) observe that for , for all , and therefore:
[TABLE]
for every . Summing up yields the result (9) and (10) with . Finally, the last equality of (10) is obtained by homogeneity of .
As the reader can see, formula (10) is extremely simple and only requires evaluating at the unique point . This in contrast to the formula in lass-avra which requires a sum of terms, each involving evaluations at the vertices of .
Remark 1
Notice that (8) 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 multidimensional Laplace transform of , or equivalently in the terminology of Kozhasov et al. sturmfels , is completely monotone222A 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 (sturmfels, , Theorem 2.5)). Next, let be an open cone with dual cone . If , , and
[TABLE]
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 (sturmfels, , p. 4).
Hence from Remark 1, with and , we obtain:
Proposition 1
For every positively homogeneous of total degree as in Theorem 2.1, the function :
[TABLE]
is completely monotone. In addition if with then is the Riesz kernel of the function on .
Proposition 1 is in the spirit of sturmfels [Proposition 2.7]. 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 1
(i) Let be polynomial of degree and write where each is homogeneous of degree . Then
[TABLE]
where and is as in (7).
(ii) For every and :
[TABLE]
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 (using ),
[TABLE]
and therefore .
Hence Corollary 1 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 [math] and the point , on the line joining [math] to . Again formula (13) and (14) 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 (baldoni, , 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
Let be a finite set of indices and let be positively homogeneous of degree and of the form
[TABLE]
for some real coefficients . Then
[TABLE]
where with , and
[TABLE]
Proof
Let with be fixed and let . By Theorem 2.1:
[TABLE]
Summing up over all yields
[TABLE]
and with one obtains (15). Finally, the last equality in (15) uses the positive 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 (13) reads:
[TABLE]
For instance with , one obtains:
[TABLE]
and indeed . In Figure 1 is displayed the -Simplex with the points and .
2.3 Back to an arbitrary simplex
Let be an arbitrary full-dimensional simplex (its -dimensional Lebesgue volume is strictly positive). 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:
[TABLE]
Next, if is a polynomial of degree :
[TABLE]
where has same degree as . Next, writing with homogeneous of degree , one has
[TABLE]
and therefore with as in Corollary 1, and . Therefore combining (16) and (13) in Corollary 1 yields:
[TABLE]
the analogue for of (13) for .
In addition, let be homogeneous of degree with Waring-like decomposition333In number theory, the Waring problem consists of writing any positive integer as a sum of a fixed number of th powers of integers, where depends only on . It generalizes to forms as a generic form of degree can be written as a sum of -powers of linear forms; is called the Waring rank of the form.
[TABLE]
for finitely many and , . Then letting :
[TABLE]
where we have used Newton binomial formula and Corollary 1(ii) (and recall that ). In particular, if and now with , , it simplifies to
[TABLE]
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, each 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.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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