By a discrete version of Tchakaloff Theorem on positive quadrature formulas, we prove that any real multidimensional compact set admitting a Markov polynomial inequality with exponent 2 possesses a near optimal polynomial mesh. This improves for example previous results on general convex bodies and starlike bodies with Lipschitz boundary, being applicable to any compact set satisfying a uniform interior cone condition. We also discuss two algorithmic approaches for the computation of near optimal Tchakaloff meshes in low dimension

Vianello, M.

English

Vianello M.

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Special issue dedicated to Norm Levenberg on the occasion of his 60th birthday, Volume 11 · 2018 · Pages 79–83Near optimal Tchakaloff meshes for compact sets with Markovexponent 2Marco Vianello aCommunicated by F. PiazzonAbstractBy a discrete version of Tchakaloff Theorem on positive quadrature formulas, we prove that any realmultidimensional compact set admitting a Markov polynomial inequality with exponent 2 possesses anear optimal polynomial mesh. This improves for example previous results on general convex bodiesand starlike bodies with Lipschitz boundary, being applicable to any compact set satisfying a uniforminterior cone condition. We also discuss two algorithmic approaches for the computation of near optimalTchakaloff meshes in low dimension.2010 AMS subject classification: 41A10, 41A17, 65D32.Keywords: near optimal polynomial meshes, Markov inequality, Tchakaloff Theorem, uniform interior cone condition, Lipschitz boundary, convexbodies, NonNegative Least Squares.1 IntroductionLet K ⊂ Rd (or Cd) be a polynomial determing compact set (i.e., polynomials vanishing on K vanish everywhere). We recall thata polynomial mesh on K is a sequence of finite norming sets Xn ⊂ K , such that‖p‖K ≤ C ‖p‖Xn , ∀p ∈ Pdn , card(Xn) =O(ns) , (1)for some constant C ≥ 1 and s ≥ d, where Pdn denotes the subspace of polynomials of total-degree not exceeding n with dimensionN = N(n) = dim(Pdn) = n+dd, and ‖p‖Y the uniform norm on a continuous or discrete compact set Y ).When s = d the mesh is called “optimal” in the literature, since necessarily necessarily card(Xn)≥ N ∼ nd/d!, n→∞, sothat it has the lowest possible order of growth with respect to n, whereas it is called near optimal when a logarithmic factor in nmultiplies nd , such as O(nd logk n), k ≤ d.Polynomial meshes, that are ultimately good discrete models of compact sets when polynomials are involved, have beenplaying an important role in multivariate polynomial approximation during the last decade, from both the theoretical and thecomputational point of view. The latter is witnessed by the role of polynomial meshes in interpolation (Fekete-like subsets) andleast squares, Bernstein-Markov measures and pluripotential numerics, and more recently in polynomial optimization. We mayrefer the reader for example to [2, 5, 8, 11, 15, 16, 20, 19], with the references therein.We shall focus here on compact sets in Rd . It is well-known by the fundamental construction of Calvi and Levenberg [5, Thm.5] that any real compact set admitting a Markov polynomial inequality with exponent r, i.e. there exists a constant M> 0 suchthat‖∇p(x)‖2 ≤Mnr ‖p‖K , ∀p ∈ Pdn , (2)possesses a polynomial mesh with O(nrd) points.On the other hand, optimal polynomial meshes have been constructed on several classes of compact sets, such as for examplestarlike and more general bodies with smooth boundary [11, 12, 15], bidimensional general convex bodies [13], general polytopes[11], and suitable sections of disk, sphere, ball and torus [8, 24]. Near optimal meshes are known on Cα starlike bodies withα = 2−2/d (in particular on planar Lipschitz starlike bodies, [12]), and on the general class of fat real subanalytic sets (essentially,finite unions of analytic images of boxes, cf. [20]). It should also be recalled that near optimal polynomial meshes are known toexist on any compact set in Cd (cf. [1, 3], and also [4]), but such results are essentially based on Fekete interpolation sets ofsuitable degree, that are explicitly available only in very few instances and are extremely hard to compute.aDepartment of Mathematics, University of Padova, ItalyVianello 80On the contrary, in this note we show that any real compact set satisfying a Markov polynomial inequality (2) with exponentr = 2 possesses a near optimal mesh, in view of a discrete version of Tchakaloff Theorem on positive quadrature, and that such amesh can be computed by standard Linear and Quadratic Programming algorithms (at least in low dimension and for moderatedegrees). Such a class includes for example any convex body [26], and more generally any compact body satisfying a uniforminterior cone condition (that is with locally Lipschitz boundary), cf. [9].We recall as a Lemma a discrete version of Tchakaloff Theorem on the existence of positive multivariate quadrature formulasexact on polynomial spaces. Originally proved by V. Tchakaloff in 1957 for absolutely continuous measures [23], it has then beenextended to any measure with finite polynomial moments, cf. e.g. [7].Lemma 1.1. Let µ be a multivariate discrete measure supported at a finite set X = {x i} ⊂ Rd , with correspondent positive weights(masses) λ= {λi}, i = 1, . . . , M.Then, there exist a quadrature formula with nodes Tn = {t j} ⊆ X , that we may term the “Tchakaloff points” of (X ,µ), and positiveweights w = {w j}, 1≤ j ≤ m≤ N = dim(Pdn), such that∫Xp(x) dµ=M∑i=1λi f (pi) =m∑j=1w j p(t j) , ∀p ∈ Pdn . (3)Proof. We recall also the proof (cf. e.g. [17]), since it gives the base for a numerical algorithm to compute Tchakaloff points andweights. Let {p1, . . . , pN} be a basis of Pdn , and V = (vi j) = (p j(x i)) the Vandermonde-like matrix of the basis computed at thesupport points. If M > N (otherwise there is nothing to prove), existence of a positive quadrature formula for µ with cardinalitynot exceeding N can be immediately translated into existence of a nonnegative solution with at most N nonvanishing componentsto the underdetermined linear systemV t u = b , u ≥ 0 , (4)whereb = V tλ=∫Xp j(x) dµ, 1≤ j ≤ N , (5)is the column vector of µ-moments of the basis {p j}.Existence then holds by the well-known Caratheodory Theorem applied to the columns of V t , which asserts that a conic(i.e., with positive coefficients) combination of any numer of vectors in RN can be rewritten as a conic combination of at most N(linearly independent) of them; cf. [6]. We can now state and prove our main result.Proposition 1.2. Let K ⊂ Rd be a compact set admitting a Markov polynomial inequality like (2) with exponent r = 2.Then, K possesses a polynomial mesh Zn with cardinality O(n2d). Moreover, the Tchakaloff points T2kn extracted from Zkn withunit mass measure, where kn = n`n, `n = [log n] + 1, form a near optimal polynomial mesh for K with card(T2kn) =O((n log n)d).Proof. The first part is Calvi-Levenberg construction in [5, Thm. 5]. Let L be the maximal length of the convex hulls of theprojections of K on the cartesian axes. Since polynomial meshes are affinely invariant, we may assume up to a translation thatK ⊆ [0, L]d . Fix θ ∈ (0, 1) and define ν=  pdMLn2θ exp(−pd θ )£. Consider in [0, L]d a uniform grid with stepsize h= L/ν. For every boxof the grid which intersects K choose a point in the intersection, and denote with Zn the (finite) set of such points.Observe that, by the estimate |q(z)| ≤ exp(dMn2δ)‖q‖K , valid for every q ∈ Pdn and for every z ∈ Rd such that dist∞(z, K)≤ δ(cf. [5, Lemma 6]), applied to the components of ∇p = (∂1p, . . . ,∂d p), we get‖∇p(z)‖2 ≤ edMn2δ ‖∇p‖K , ∀z ∈ Rd : dist∞(z, K)≤ δ . (6)Now, for every x ∈ K we can choose y ∈ Zn such that δ = ‖x − y‖∞ ≤ h ≤ θ/(pd Mn2)exp(−pd θ) < θ/(pd Mn2). Bythe mean value theorem, for every x , y ∈ K we have|p(x)− p(y)| ≤ ‖∇p(ξ)‖2 ‖x − y‖2for a suitable ξ in the segment [x , y]. Then by (6) with z = ξ, together with ‖x − y‖2/pd ≤ ‖x − y‖∞ ≤ h, we get|p(x)− p(y)| ≤ edMn2δMn2 ‖x − y‖2 ‖p‖K < epd θpd h‖p‖K ≤ θ ‖p‖K ,and thus|p(x)| ≤ |p(y)|+ |p(x)− p(y)| ≤ ‖p‖Zn + θ ‖p‖K ,from which (1) follows with C = 1/(1− θ ). Notice that card(Zn) does not exceed the fraction of grid boxes intersecting K , andthus it is bounded by the overall number of grid boxescard(Zn)≤ νd ≤ cd n2d , cd = pd LMθ exp(−pd θ )d. (7)In the case of convex bodies, the proof can use simply the mean value theorem, so that the factor exp(−pd θ ) in the denominatoris dropped, in both the definition of ν and (7) (we omit the details for brevity).Dolomites Research Notes on Approximation ISSN 2035-6803Vianello 81Concerning the second part, first observe that for every p ∈ Pdn the polynomial p`n is in Pdkn and thus‖p‖`nK = ‖p`n‖K ≤ C ‖p`n‖Zkn .Now, consider on X = Zkn the discrete measure µ with unit masses. By Lemma 1 we get for every q ∈ Pdkn‖q‖2Zkn ≤ ‖q‖2`2(Zkn )= ‖q‖2`2w(T2kn )=m∑j=1w j q2(t j)≤m∑j=1w j‖q‖2T2kn = µ(Zkn)‖q‖2T2kn= card(Zkn)‖q‖2T2kn.Then we can write‖p`n‖K ≤ Cqcard(Zkn)‖p`n‖T2kn ≤ Cqcd k2dn ‖p`n‖T2knand thus‖p‖K ≤ kdn Cpcd1/`n ‖p‖T2kn =O(1) ‖p‖T2knsince kdn Cpcd1/`n = explog kdn`n Cpcd1/`n= expdlog n+ log`n`n Cpcd1/`n≤ expd1+log`n`n Cpcd1/`n ∼ ed , n→∞ .Notice finally that card(T2kn) =O((n log n)d) as n→∞, sincecard(T2kn)≤ dim(Pd2kn) =2kn + dd∼(2n log n)dd!, n→∞ . (8)The class of compact sets covered by Proposition 1 is very wide. IndeedCorollary 1.3. Any compact domain (the closure of a bounded open set) in Rd satisfying a uniform interior cone condition (eachpoint of K is the vertex of a suitably rotated fixed cone contained in K) possesses a near optimal polynomial mesh. This holds inparticular for any convex body.In fact, such a property implies the fulfillement of a Markov inequality with exponent 2, which is inherited from the cone, cf.e.g. [25]. This is valid on any compact domain with (locally) Lipschitz boundary, the latter property implying the fulfillement of auniform interior cone condition [9].In particular, Proposition 1 is valid on any convex body, where one can prove that a Markov inequality holds with r = 2 and Mproportional to the reciprocal of the body width (the minimum distance between parallel supporting hyperplanes) by a factor 4(or 2 on centrally symmetric bodies), cf. [26]. This improves the previous results for general convex bodies and starlike Lipschitzbodies in dimension d > 2, where the best known cardinality for polynomial meshes was O(n2d−2), cf. [11, 12]. It can also beseen as a further step towards the proof of the Conjecture: “Every convex body in Rd possesses an optimal polynomial mesh”, cf.[11].The proof of Proposition 1 is completely constructive, and easily implementable, at least in low dimension. In particular,differently from other relevant families of points in multivariate polynomial approximation, such as Fekete points or Lebesguepoints, Tchakaloff points can in principle be computed by basic algorithms of Linear and Quadratic Programming.In fact, the discrete version of Tchakaloff Theorem in Lemma 1, requires ultimately to compute a sparse nonnegative solutionto the underdetermined linear system (4)-(5). In the literature on quadrature compression, essentially two approaches have beenused.The Linear Programming (LP) approach consists in minimizing the linear functional c t u for a suitable choice of the vector c,subject to the constraints V t u = b and u ≥ 0. In fact, the solution is a vertex of the polytope defined by the constraints, whichhas (at least) M − N null components, cf. e.g. [21]. Observe that a usual choice of the popular compressed sensing field (BasisPursuit, cf. [10]), namely c = (1, . . . , 1) that is minimizing ‖u‖1 subject to the constraints, is not feasible in the present context,since ‖u‖1 = µ(X ) for any u satisfying (4) by exactness of the quadrature formula on the constants.As an alternative, the Quadratic Programming approach consists in solving the NonNegative Least Squares (NNLS) problemcompute u∗ : ‖V t u∗ − b‖2 =min‖V t u − b‖2 , u ≥ 0 , (9)that can be done by the well-known Lawson-Hanson active set optimization method [14], which automatically seeks a sparsesolution and is implemented for example by the lsqnonneg native algorithm of Matlab. Our limited computational experience,in low dimension (d = 2, 3) and with moderate degrees (polynomial spaces of dimension up to the hundreds), has shown thatin such setting Lawson-Hanson NNLS is more efficient than the most common implementations of LP. A Matlab code for theDolomites Research Notes on Approximation ISSN 2035-6803Vianello 82computation of Tchakaloff points based on NNLS is provided in the software packages quoted in [17, 22], where the reader canfind a more detailed discussion.In order to make an illustrative example, in Figure 1 we display for degree n= 4 the grid-based mesh Zn = Z4 with θ = 1/2(approximately 8800 points) and the near optimal Tchakaloff mesh T2kn = T16 (153 points extracted from the approximately140000 points of Zkn = Z8) on a quarter of a Cassini oval, that isK = {x = (x1, x2) ∈ R2 : ((x1 − a)2 + x22)((x1 + a)2 + x22)≤ b4, x1, x2 ≥ 0} ,with a = 1, b = 2 (the Cassini ovals are convex for b/a ≥p2); the Tchakaloff points have been computed by Lawson-HansonNNLS algorithm.0 0.5 1 1.5 2 2.500.20.40.60.811.21.41.61.8Figure 1: Grid-based polynomial mesh (around 8800 points) and Tchakaloff near optimal mesh (153 points) for degree n= 4 on a quarter of aCassini oval.Acknowledgments. Work partially supported by the DOR funds and the biennial project BIRD163015 of the University of Padova,and by the GNCS-INdAM. This research has been accomplished within the RITA “Research ITalian network on Approximation”.References[1] T. Bloom, L. Bos, J.P. Calvi and N. Levenberg. Polynomial Interpolation and Approximation in Cd . Ann. Polon. Math., 106: 53–81, 2012.[2] T. Bloom, N. Levenberg, F. Piazzon and F. Wielonsky. Bernstein-Markov: a survey. Dolomites Res. Notes Approx. DRNA, 8: 75–91, 2015.[3] A. Brudnyi. On near-optimal admissible meshes. arXiv preprint 1402.2303, 2014.[4] A. Brudnyi. Banach spaces of polynomials as “large” subspaces of `∞-spaces. J. Funct. Anal., 267: 1285–1290, 2014.[5] J.P. Calvi and N. Levenberg. Uniform approximation by discrete least squares polynomials. J. Approx. Theory, 152: 82–100, 2008.[6] C. Caratheodory. Über den Variabilittsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen. Rend. Circ. 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Near optimal Tchakaloff meshes for compact sets with Markov exponent 2

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Near optimal Tchakaloff meshes for compact sets with Markov exponent 2

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