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Symmetry Preserving Interpolation Erick Rodriguez Bazan, Evelyne Hubert

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Symmetry Preserving Interpolation Erick Rodriguez Bazan, Evelyne Hubert Symmetry Preserving Interpolation Erick Rodriguez Bazan, Evelyne Hubert To cite this version: Erick Rodriguez Bazan, Evelyne Hubert. Symmetry Preserving Interpolation. ISSAC 2019 - International Symposium on Symbolic and Algebraic Computation, Jul 2019, Beijing, China. 10.1145/3326229.3326247. hal-01994016 HAL Id: hal-01994016 https://hal.archives-ouvertes.fr/hal-01994016 Submitted on 25 Jan 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Symmetry Preserving Interpolation Erick Rodriguez Bazan Evelyne Hubert Université Côte d’Azur, France Université Côte d’Azur, France Inria Méditerranée, France Inria Méditerranée, France [email protected] [email protected] ABSTRACT general concept. An interpolation space for a set of linear forms is The article addresses multivariate interpolation in the presence of a subspace of the polynomial ring that has a unique interpolant for symmetry. Interpolation is a prime tool in algebraic computation each instantiated interpolation problem. We show that the unique while symmetry is a qualitative feature that can be more relevant interpolants automatically inherit the symmetry of the problem to a mathematical model than the numerical accuracy of the pa- when the interpolation space is invariant (Section 3). rameters. The article shows how to exactly preserve symmetry A canonical interpolation space, the least interpolation space, was in multivariate interpolation while exploiting it to alleviate the introduced in [3–5]. We shall review that it is invariant as soon as computational cost. We revisit minimal degree and least interpo- the space of linear forms is. In floating point arithmetics though, lation with symmetry adapted bases, rather than monomial bases. the computed interpolation space might fail to be exactly invariant. This allows to construct bases of invariant interpolation spaces in Yet, in mathematical modeling, symmetry is often more relevant blocks, capturing the inherent redundancy in the computations. than numerical accuracy. We shall remedy this flaw and further We show that the so constructed symmetry adapted interpolation exploit symmetry to mitigate the cost and numerical sensitivity of bases alleviate the computational cost of any interpolation problem computing a minimal degree or least interpolation space. and automatically preserve any equivariance of this interpolation As other minimal degree interpolation spaces, the least inter- problem might have. polation space can be constructed by Gaussian elimination in a multivariate Vandermonde (or collocation) matrix. The columns of KEYWORDS the Vandermonde matrix are indexed by monomials. We show how any other graded basis of the polynomial ring can be used. In partic- Interpolation, Symmetry, Representation Theory, Group Action ular there is a two fold gain in using a symmetry adapted basis. On one hand, the computed interpolation space will be exactly invari- 1 INTRODUCTION ant independently of the accuracy of the data for the interpolation Preserving and exploiting symmetry in algebraic computations problem. On the other hand, the new Vandermonde matrix is block is a challenge that has been addressed within a few topics and, diagonal so that Gaussian elimination can be performed indepen- mostly, for specific groups of symmetry [2, 7, 8, 10, 11, 13–16, 18, dently on smaller size matrices, with better conditioning. Further 19, 22]. The present article addresses multivariate interpolation in computational savings result from identical blocks being repeated the presence of symmetry. Due to its relevance in approximation according to the degree of the related irreducible representations theory and geometrical modeling, interpolation is a prime topic in of the group. Symmetry adapted bases also plaid a prominent role algebraic computation. Among the several problems in multivariate in [2, 11, 19] where it allowed the block diagonalisation of a multi- interpolation [9, 17], we focus on the construction of a polynomial variate Hankel matrix. interpolation space for a given set of linear forms. Assuming the In Section 2 we define minimal degree and least interpolation space generated by the linear forms is invariant under a group space and review how to compute a basis of it with Gaussian elimi- action, we show how to, not only, preserve exactly the symmetry, nation. In Section 3 we make explicit how symmetry is expressed but also, exploit it so as to reduce the computational cost. and the main ingredient to preserve it. In Section 4 we review symmetry adapted bases and show how the Vandermonde matrix For a set ofr points ξ1;:::; ξr inn-space, andr valuesη1;:::; ηr the basic interpolation problem consists in finding a n-variate poly- becomes block diagonal in these. This is applied to provide an algorithm for the computation of invariant interpolation spaces nomial function p such that p(ξi ) = ηi , for 1 ≤ i ≤ r. The evalu- in Section 5 together with a selection of relevant invariant and ations at the points ξi form a basic example of linear forms. The space they generate is invariant under a group action when the set equivariant interpolation problems. of points is a union of orbits of this group action. A first instance of symmetry is invariance. The above interpolation problem is in- variant if ηi = ηj whenever ξi and ξj belong to the same orbit. It is then natural to expect an invariant polynomial as interpolant. Yet, contrary to the univariate case, there is no unique interpolant of minimal degree and the symmetry of the interpolation problem 2 POLYNOMIAL INTERPOLATION may very well be violated (compare Figure 2 and 1). We review in this section the definitions and constructions of in- In this article we shall consider a general set of linear forms, in- terpolation spaces of minimal degree. By introducing general dual variant under a group action, and seek to compute interpolants that polynomial bases we generalize the construction of least interpola- respect the symmetry of the interpolation problem. We mentioned tion spaces. We shall then be in a position to work with adapted invariance as an instance of symmetry, but equivariance is the more bases to preserve and exploit symmetry. Pδ 2.1 Interpolation space WL . They index a maximal set of linearly independent columns Hereafter, K denotes either C or R. K[x] = K[x ;:::; x ] denotes Pδ 1 n of WL . the ring of polynomials in the variables x1;:::; xn with coefficients in K; K[x]≤δ and K[x]δ the K−vector spaces of polynomials of 2.3 Minimal degree degree at most δ and the space of homogeneous polynomials of It is desirable to build an interpolation space such that the degree degree δ respectively. of the interpolating polynomials be as small as possible. We shall The dual of K[x], the set of K−linear forms on K[x], is denoted by ∗ use the definition of minimal degree solution for an interpolation K[x] . A typical example of a linear form on K[x] is the evaluation problem defined in [4, 5, 20]. e n ξ at a point ξ of K . It is defined by e Definition 2.1. An interpolation space P for Λ is of minimal de- ξ : K[x] ! K p 7! p(ξ ): gree if for any other interpolation space Q for Λ Other examples of linear forms on K[x] are given by compositions dim(Q \ K[x]≤δ ) ≤ dim(P \ K[x]≤δ ); 8δ 2 N: of evaluation and differentiation Λ : K[x] ! K We say that a countable set of homogeneous polynomials P = 7! Pr e ◦ @ ; fp1;p2;:::g is ordered by degree if i ≤ j implies that degpi ≤ degpj . p j=1 ξj qj ( )(p) n α @ @ with ξ 2 K ;q 2 K[x] and @ = α ::: α . Proposition 2.2. Let L be a basis of Λ. Let Pδ , δ > 0, be a j j 1 @x n @x1 n Pδ n homogeneous basis of K[x]≤ ordered by degree, such that W has Let ξ1;:::; ξr be a finite set of points in K . Lagrange interpola- δ L 2 Pδ tion consists in finding, for any η1;::: ηr K, a polynomial p such full row rank. Let j1;:::; jr be the echelon sequence of WL obtained e (p) = η ≤ j ≤ r interpolation problem that ξj j , 1 . More generally an by Gauss elimination with partial pivoting. Then P := hpj1 ;:::;pjr i is a pair (Λ; ϕ) where Λ is a finite dimensional linear subspace of is a minimal degree interpolation space for Λ. K[x]∗ and ϕ : Λ −! K is a K-linear map. An interpolant, i.e., a ; ::: solution to the interpolation problem, is a polynomial p such that Proof. LetQ be another interpolation space for Λ. Letq1 q2 qm be a basis of Q \ K[x]≤d with d ≤ δ. Since Pδ is a homogeneous λ(p) = ϕ(λ) for any λ 2 Λ: (1) basis of K[x]≤δ , any qi can be written as a linear combination of P elements of P \ K[x]≤ . Considering qi = ajipj we get that P δ d j An interpolation space for Λ is a polynomial subspace P of K[x] λ(qi ) = j aji λ(pj ) for any λ 2 Λ. such that Equation (1) has a unique solution in P for any map ϕ. Let fp ;p ;::: p g be the elements of P that form a basis of ji1 ji2 jin Pδ P \ K[x]≤d . Gauss elimination on WL ensures that λ(b) is a linear 2.2 Vandermonde matrix 2 P \K 2 combination of λ(pji );::: λ(pjin ) for any b δ [x]≤d and λ 1 Pn For P = fp1;p2;:::;pm g and L = fλ1; λ2;:::; λr g linearly inde- Λ.
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