On the construction of general cubature formula by flat - HAL

On the construction of general cubature formula by flat
extensions
Marta Abril Bucero, Chandrajit Bajaj, Bernard Mourrain
To cite this version:
Marta Abril Bucero, Chandrajit Bajaj, Bernard Mourrain. On the construction of general
cubature formula by flat extensions. Linear Algebra and its Applications, Elsevier, 2016, 502,
pp.104-125. .
HAL Id: hal-01158099
https://hal.inria.fr/hal-01158099v3
Submitted on 2 Jun 2015
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ON THE CONSTRUCTION OF GENERAL CUBATURE
FORMULA BY FLAT EXTENSIONS
M. ABRIL BUCERO, C. BAJAJ, AND B. MOURRAIN
Abstract. We describe a new method to compute general cuba-
ture formulae. The problem is initially transformed into the com-
putation of truncated Hankel operators with flat extensions. We
then analyse the algebraic properties associated to flat extensions
and show how to recover the cubature points and weights from
the truncated Hankel operator. We next present an algorithm to
test the flat extension property and to additionally compute the
decomposition. To generate cubature formulae with a minimal
number of points, we propose a new relaxation hierarchy of convex
optimization problems minimizing the nuclear norm of the Han-
kel operators. For a suitably high order of convex relaxation, the
minimizer of the optimization problem corresponds to a cubature
formula. Furthermore cubature formulae with a minimal number
of points are associated to faces of the convex sets. We illustrate
our method on some examples, and for each we obtain a new min-
imal cubature formula.
1. Cubature formula
1.1. Statement of the problem. Consider the integral for a contin-
uous function f,
I[f] = ZΩ w(x)f(x)dx
where Ω⊂Rnand wis a positive function on Ω.
We are looking for a cubature formula which has the form
(1) hσ|fi=
r
X
j=1
wjf(ζj)
where the points ζj∈Cnand the weights wj∈Rare independent of
the function f. They are chosen so that
hσ|fi=I[f],∀f∈V,
where Vis a finite dimensional vector space of functions. Usually, the
vector space Vis the vector space of polynomials of degree ≤d, because
1

2 M. ABRIL BUCERO, C. BAJAJ, AND B. MOURRAIN
a well-behaved function fcan be approximated by a polynomial, so
that Q[f]approximates the integral I[f].
Given a cubature formula (1) for I, its algebraic degree is the largest
degree dfor which I[f] = hσ|fifor all fof degree ≤d.
1.2. Related works. Prior approaches to the solution of cubature
problem can be grouped into roughly two classes. One, where the goal
is to estimate the fewest weighted, aka cubature points possible for sat-
isfying a prescribed cubature rule of fixed degree [9, 25, 27, 30, 31, 34].
The other class, focusses on the determination and construction of
cubature rules which would yield the fewest cubature points possible
[7, 35, 39, 40, 41, 42, 45, 46]. In [35], for example, Radon introduced a
fundamental technique for constructing minimal cubature rules where
the cubature points are common zeros of multivariate orthogonal poly-
nomials. This fundamental technique has since been extended by many,
including for e.g. [34, 42, 46] where notably, the paper [46] use mul-
tivariate ideal theory, while [34] uses operator dilation theory. In this
paper, we propose another approach to the second class of cubature
solutions, namely, constructing a suitable finite dimensional Hankel
matrix and extracting the cubature points using sub operators of the
Hankel matrix [18]. This approach is related to [21, 22, 24] and which
in turn are based on the methods of multivariate truncated moment
matrices, their positivity and extension properties [11, 12, 13].
Applicatons of such algorithms determining cubature rules and cuba-
ture points over general domains occur in isogeometric modeling and
finite element analysis using generalized Barycentric finite elements
[17, 1, 36, 37]. Additional applications abound in numerical integra-
tion for low dimensional (6-100 dimensions) convolution integrals that
appear naturally in computational molecular biology [3, 2], as well in
truly high dimensional (tens of thousands of dimensions) integrals that
occur in finance [33, 8].
1.3. Reformulation. Let R=R[x]be the ring of polynomials in the
variables x= (x1, . . . , xn)with coefficients in R. Let Rdbe the set of
polynomials of degree 6d. The set of linear forms on R, that is, the
set of linear maps from Rto Ris denoted by R∗. The value of a linear
form Λ∈R∗on a polynomial p∈Ris denoted by hΛ|pi. The set R∗
can be identified with the ring of formal power series in new variables
z= (z1, . . . , zn):
R∗→R[[z]]
Λ7→ Λ(z) = X
α∈Nn
hΛ|xαizα.

ON THE CONSTRUCTION OF GENERAL CUBATURE FORMULA BY FLAT EXTENSIONS3
The coefficients hΛ|xαiof these series are called the moments of Λ. The
evaluation at a point ζ∈Rnis an element of R, denoted by eζ, and
defined by eζ:f∈R7→ f(ζ)∈R. For any p∈Rand any Λ∈R∗, let
p ? Λ : q∈R7→ Λ(pq).
Cubature problem: Let V⊂Rbe the vector space of polynomials
and consider the linear form ¯ I∈V∗defined by
¯
I:V→R
v7→ I[v]
Computing a cubature formula for Ion Vthen consists in finding a
linear form
σ=
r
X
i=1
wieζi:f7→
r
X
j=1
wjf(ζj).
which coincides on Vwith ¯ I. In other words, given the linear form ¯ I
on Rd, we wish to find a linear form σ=Pr
i=1 wieζiwhich extends ¯
I.
2. Cubature formulae and Hankel operators
To find such a linear form σ∈R∗, we exploit the properties of its
associated bilinear form Hσ: (p, q)∈R×R→ hσ|p qi, or equivalently,
the associated Hankel operator:
Hσ:R→R∗
p7→ p?σ
The kernel of Hσis ker Hσ={p∈R| ∀q∈R, hσ|p qi= 0}. It is an
ideal of R. Let Aσ=R/ ker Hσbe the associated quotient ring.
The matrix of the bilinear form or the Hankel operator Hσassociated
to σin the monomial basis, and its dual are (hΛ|xα+βi)α,β∈Nn. If we
restrict them to a space Vspanned by the monomial basis (xα)α∈Afor
some finite set A⊂Nn, we obtain a finite dimensional matrix [HA,A
σ] =
(hΛ|xα+βi)α,β∈A, and which is a Hankel matrix.. More generally, for
any vector spaces V, V 0⊂R, we define the truncated bilinear form
and Hankel operators: HV,V 0
σ: (v, v0)∈V×V07→ hσ|pqi ∈ Rand
HV,V 0
σ:v∈V7→ v ? σ ∈V0∗. If V(resp. V0) is spanned by a monomial
set xAfor A⊂Nn(resp. xBfor B⊂Nn), the truncated bilinear
form and truncated Hankel operator are also denoted by HA,B
σ. The
associated Hankel matrix in the monomial basis is then [HA,B
σ] =
(hΛ|xα+βi)α∈A,β∈B.
The main property that we will use to characterize a cubature for-
mula is the following (see [23, 20]):

4 M. ABRIL BUCERO, C. BAJAJ, AND B. MOURRAIN
Proposition 2.1. A linear form σ∈R∗can be decomposed as σ=
Pr
i=1 wieζiwith wi∈C\ {0},ζi∈Cniff
•Hσ:p7→ p?σ is of rank r,
•ker Hσis the ideal of polynomials vanishing at the points {ζ1, . . . , ζr}.
This shows that in order to find the points ζiof a cubature formula,
it is sufficient to compute the polynomials p∈Rsuch that ∀q∈R,
hσ|pqi= 0 , and to determine their common zeroes. In section 4 we
describe a direct way to recover the points ζi, and the weights ωifrom
suboperators of Hσ.
In the case of cubature formulae with real points and positive weights,
we already have the following stronger result (see [23, 20]):
Proposition 2.2. Let σ∈R∗.
σ=
r
X
i=1
wieζi
with wi>0,ζi∈Rniff rankHσ=rand Hσ<0.
A linear form σ=Pr
i=1 wieζiwith wi>0,ζi∈Rnis called a r-
atomic measure since it coincides with the weighted sum of the rDirac
measures at the points ζi.
Therefore, the problem of constructing a cubature formula σfor I
exact on V⊂Rcan be reformulated as follows: Construct a linear
form σ∈R∗such that
•rankHσ=r < ∞and Hσ<0.
• ∀v∈V,I[v] = hσ|vi.
The rank rof Hσis given by the number of points of the cubature
formula, which is expected to be small or even minimal.
The following result states that a cubature formula with dim(V)
points, always exist.
Theorem 2.3. [43, 4] If a sequence (σα)α∈Nn,|α|6tis the truncated mo-
ment sequence of a measure mu (i.e. σα=Rxαdmu for |α|6t)
then it can also be represented by an r-atomic measure: for |α|6t,
σα=Pr
i=1 wiξα
iwhere r6st,wi>0,ζi∈supp(mu).
This result can be generalized to any set of linearly independent
polynomials v1, . . . , vr∈R(see the proof in [4] or Theorem 5.9 in
[23]). We deduce that the cubature problem always has a solution with
dim(V)or less points.
Definition 2.4. Let rc(I)be the maximum rank of the bilinear form
HW,W 0
I: (w, w0)∈W×W07→ I[ww0]where W, W 0⊂Vare such that
∀w∈W, ∀w0∈W0,w w0∈V. It is called the Catalecticant rank of I.