Geometric Weakly Admissible Meshes, Discrete Least Squares Approximations and Approximate Fekete Points^††thanks: Supported by the “ex-$60\%$” funds of the University of Padova, by the INdAM GNCS and NSERC, Canada.

In a recent paper [14], Calvi and Levenberg have developed the theory of “admissible meshes” for uniform polynomial approximation over multidimensional compact sets. Their theory is for $\mbox{\BBB C}^{d}$, but here we restrict our attention to $\mbox{\BBB R}^{d}$, the relevant definitions and results being easily adaptable to the general case. We begin with introducing some notation.

Suppose that $K\subset\mbox{\BBB R}^{d}$ is compact. We will require that $K$ not be too small, that is, that it is polynomial determining, i.e., if a polynomial $p$ is zero on $K,$ then it is identically zero. This will certainly be the case if $K$ contains some open ball, as will be the case for all our examples.

We will let $\mbox{\BBB P}_{n}^{d}$ denote the space of polynomials of degree at most $n$ in $d$ real variables.

Then, a Weakly Admissible Mesh (WAM) of $K$ is a sequence of discrete subsets $A_{n}\subset K$ such that

where ${\rm card}(A_{n})$ and the constants $C(A_{n})$ have (at most) polynomial growth in $n,$ i.e.,

and

Throughout the paper, $\|p\|_{X}=\max_{\boldsymbol{x}\in X}{|p(\boldsymbol{x})|}$. When $\{C(A_{n})\}$ is bounded, $C(A_{n})\leq C$, we speak of an Admissible Mesh (AM). It is easy to see that (W)AMs satisfy the following properties (cf. [14]):

• •

  for affine mappings of $K$ the image of a WAM is also a WAM, with the same constant $C(A_{n})$

• •

  any sequence of sets containing $A_{n}$, with polynomially growing cardinalities, is a WAM with the same constants $C(A_{n})$

• •

  any sequence of unisolvent interpolation sets whose Lebesgue constant grows at most polynomially with $n$ is a WAM, $C(A_{n})$ being the Lebesgue constant itself

• •

  a finite union of WAMs is a WAM for the corresponding union of compacts, $C(A_{n})$ being the maximum of the corresponding constants

• •

  a finite cartesian product of WAMs is a WAM for the corresponding product of compacts, $C(A_{n})$ being the product of the corresponding constants.

As shown in [14], such meshes are very useful for polynomial approximation in the max-norm on $K$. In fact, given the classical least-square polynomial approximant on $A_{n}$ to a function $f\in C(K)$, say $\mathcal{L}_{n}f\in\mbox{\BBB P}_{n}^{d}$, we have that

see [14, Thm. 1]. Moreover, Fekete points (points that maximize the Vandermonde determinant) extracted from a WAM have a Lebesgue constant with the bound

that is $C(A_{n})$ times the classical bound for the Lebesgue constant of true (continuum) Fekete points; see [14, §4.4]. Recently, a new algorithm has been proposed for the computation of Approximate Fekete Points, using only standard tools of numerical linear algebra such as the QR factorization of Vandermonde matrices, cf. [7, 30].

These results show that in computational applications it is important to construct WAMs with low cardinalities and slowly increasing constants $C(A_{n})$. We recall that is always possible to construct easily an AM on compact sets that admit a Markov polynomial inequality, as is shown in the key result [14, Thm. 5]. There are wide classes of such compacts, for example convex bodies, and more generally sets satisfying an interior cone condition, but the best known bounds for the cardinality of the resulting meshes grow like $\mathcal{O}(n^{rd})$, where $r$ is the exponent of the Markov inequality (typically $r=2$ in the real case). This means that, for example, for a 3-dimensional cube or ball one should work with $\mathcal{O}(n^{6})$ points, a number that becomes practically intractable already for relatively small values of $n$ (recall that the number of points determines the number of rows of the Least Squares (non-sparse) matrices). But already in dimension two, working with $\mathcal{O}(n^{4})$ points makes the construction of polynomial approximants at moderate values of $n$ computationally rather expensive.

The properties of WAMs listed above (concerning finite unions and products) suggest an alternative: we can obtain good meshes, even on complicated geometries, if we are able to compute WAMs of low cardinality on standard compact “pieces”. For example, it is immediate to get a $\mathcal{O}(n^{d})$ WAM with $C(A_{n})=\mathcal{O}(\log^{d}{(n)})$ for the $d$-dimensional cube, as the tensor-product of 1-dimensional Fekete (or Chebyshev) points.

In this paper we introduce the notion of geometric WAM, that is a WAM obtained by a geometric transformation of a suitable low-cardinality discretization mesh on some reference compact set, like the $d$-dimensional cube. Such geometric WAMs, as well as those obtained by finite unions and products, can be used directly for discrete Least Squares approximation, as well as for the extraction of good interpolation points by means of the Approximate Fekete Points algorithm described in [7, 30], on compact sets with various geometries.

In Section 2 we illustrate the idea of geometric WAMs by several examples in $\mbox{\BBB R}^{2}$, the disk, triangles, trapezoids, and polygons. In Section 3 we prove a general result on the asymptotics of approximate Fekete points extracted from WAMs by the greedy algorithm in [7, 30]. Finally, in Section 4 we present some numerical results concerning discrete Least Squares approximation and interpolation at Approximate Fekete Points on geometric WAMs.

Let $K$ and $Q$ be compact subsets of $\mbox{\BBB R}^{d}$,

and $\{S_{n}\}$ a sequence of discrete subsets of $Q$. A geometric WAM of $K$ is a sequence

where the defining properties of a WAM, (1)-(3) stem from the “geometric structure” of $K$, $Q$, $\boldsymbol{t}$, and $S_{n}$. To make more precise this still somewhat vague notion, we give some illustrative examples in $\mbox{\BBB R}^{2}$, where the reference compact $Q$ is a rectangle.

In this section we go to the general setting of polynomial interpolation in several complex variables. Suppose that $K\subset\mbox{\BBB C}^{d}$ is a compact polynomially determining set. If ${\cal B}_{n}=\{p_{1},p_{2},\cdots,p_{N}\}$ is a basis for $\mbox{\BBB P}_{n}^{d},$ where $N:={\rm dim}(\mbox{\BBB P}_{n}^{d})$ and $Z_{n}\subset K$ is a discrete subset of cardinality $N,$ then

is called the corresponding Vandermonde determinant. If the determinant is non-zero then we may form the so-called fundamental Lagrange interpolating polynomials

Indeed, then for any $f\,:\,K\to\mbox{\BBB C}$ the polynomial (of degree $n$)

interpolates $f$ at the points of $Z_{n},$ i.e., $\Pi_{n}(\boldsymbol{a})=f(\boldsymbol{a}),$ $\boldsymbol{a}\in Z_{n}.$ A set of points $F_{n}\subset K$ which maximize ${\rm vdm}(Z_{n};{\cal B}_{n})$ as a function of $Z_{n},$ are called Fekete points of degree $n$ for $K$ and have the special property that $\|\ell_{\boldsymbol{a}}\|_{K}=1$ (and hence Lebesgue constant bounded by $N$) and provide a very good (often excellent) set of interpolation points for $K.$ However, they are typically very difficult to compute, even for moderate values of $n.$

For any $\mbox{\BBB P}_{n}^{d}$ determining subset $A_{n}\subset K$ (thought of as a sufficiently good discrete model of $K$) the algorithm introduced by Sommariva and Vianello in [30] and studied by Bos and Levenberg in [7] selects, in a simple and efficient manner, a subset ${\cal F}_{n}\subset A_{n}$ of Approximate Fekete Points, and hence provides a practical alternative to true Fekete points, $F_{n}.$ The optimization problem is nonlinear, and large-scale already for moderate values of $n$, but the algorithm is able to give an approximate solution using only standard tools of numerical linear algebra.

Some remarks on the polynomial basis ${\cal B}_{n}$ are in order. First note that if ${\cal B}^{\prime}_{n}:=\{q_{1},\cdots,q_{N}\}$ is some other basis of $\mbox{\BBB P}_{n}^{d}$ then there is a transition matrix $T_{N}\in\mbox{\BBB C}^{N\times N}$ so that ${\cal B}^{\prime}_{n}={\cal B}_{n}T_{N}$. It is easy to verify that then

Hence true Fekete points do not depend on the basis used and also the Lagrange polynomials $\ell_{\boldsymbol{a}}$ of (15) are independent of the basis. Moreover, if $A_{n}$ and $B_{n}$ are two point sets for which

for some constant $C,$ then the same inequality holds using the basis ${\cal B}^{\prime}_{n},$ i.e.,

since both sides scale by the same factor.

The greedy Algorithm described above is in general affected by the basis. But this not withstanding, in the theorem below we show that if the initial set $A_{n}\subset K$ is a WAM, then the so selected approximate Fekete points, using any basis, and the true Fekete points for $K$ both have the same asymptotic distribution.

By

we will mean the Vandermonde determinant computed using the standard monomial basis.

Note also that a set of true Fekete points $F_{n}$ is also a WAM and hence we may take $A_{n}=F_{n},$ in which case the algorithm will select $B_{n}=F_{n}$ (there is no other choice) and so the true Fekete points must necessarily also have these two properties.

The final statement, that $\mu_{n}$ converges weak-* to $d\mu_{K}$ then follows by the main result of Berman and Bouksom [1] (see also [4]). $\square$

In this section, we present a suite of numerical tests concerning discrete least squares approximation on geometric WAMs and polynomial interpolation at Approximate Fekete Points extracted from them. The tests concern the 2-dimensional compact sets discussed in Section 2, that are the unit disk, the unit simplex, a linear and a cubic trapezoid, a convex and a nonconvex polygon; see Figures 1-4. All the tests have been done in Matlab (see [25]), by an Intel-Centrino Duo T-2400 processor with 1 Gb RAM.

In Figures 5-7 we display the Approximate Fekete Points of degree $n=8$ extracted form the geometric WAMs of Figures 1-4. The computational advantage of working with a Weakly Admissible Mesh instead of an Admissible Mesh is shown in Table 1, where we show the cardinalities of the relevant discrete sets in the case of the disk. We recall that, following [14, Thm.5], it is always possible to construct an AM for a real $d$-dimensional compact set that admits a Markov polynomial inequality with exponent $r$, by intersection with a uniform grid of stepsize ${\cal O}(n^{-rd})$. In particular, convex compact sets have $r=2$, and it is easily seen from the Markov polynomial inequality $\|\nabla p(x)\|_{2}\leq n^{2}\|p\|_{\infty}$ valid for every $p\in\mbox{\BBB P}_{n}^{2}$ and $x$ in the disk, and the proof of [14, Thm.5], that it is sufficient to take a stepsize $h<1/n^{2}$. The cardinality of the corresponding AM is then ${\cal O}(n^{4})$, to be compared with the ${\cal O}(n^{2})$ cardinality of the geometric WAM (see Section 2.1). For example, at degree $n=30$ an AM for the disk has more than 2 millions points, whereas the geometric WAM has less than 2 thousands points.

In Table 2, we show the Lebesgue constants of the Approximate Fekete Points extracted from the geometric WAMs for the disk at a sequence of degrees, using the algorithm above with different polynomial bases. Such Lebesgue constants have been evaluated numerically on a suitable control mesh, much finer than the extraction mesh. Without refinement iterations, the best results are obtained with the Logan-Shepp basis, which, as is well known, is orthogonal for standard Lebesgue measure (cf. [23]). On the contrary, with the monomial basis we face severe ill-conditioning and even numerical rank deficiency of the Vandermonde matrix, and we get the worst Lebesgue constants. After two refinement iterations, however, we are working in practice with a discrete orthogonal basis, the corresponding Vandermonde matrices are not ill-conditioned, and the Lebesgue constants improve and stabilize. Observe that their growth is much slower than that of the theoretical bound (5).

In Table 3 we compare, for the three test functions below, the errors (in the uniform norm) of discrete least squares approximation on the AM and on the WAM of the disk, and of interpolation at the Approximate Fekete Points extracted from the WAM (with two refinement iterations). The test functions exhibit different regularity: the first is analytic entire, the second is analytic nonentire (a bivariate version of the classical Runge function), the third is $C^{1}$ but has a singularity of the second derivatives at the origin.

Notice that with the AM we have computational failure (“out of memory”) in our computing system already at degree $n=15$, due to the large cardinality of the discrete set; see Table 1. The least squares error on the WAM is close to that on the AM (when comparable), which shows that geometric WAMs are a good choice for polynomial approximation, with a low computational cost. It is worth observing that in the theoretical estimate (4) we even have $C(A_{n})\sqrt{{\rm card}(A_{n})}={\cal O}(n^{2})$ for the AM, and $C(A_{n})\sqrt{{\rm card}(A_{n})}={\cal O}(n\log^{2}{(n)})$ for the WAM, but we recall that these are overestimates, the term $\sqrt{{\rm card}(A_{n})}$ being in some way “artificial” (cf. [14, Thm.2]).

The following tables are devoted to numerical tests on the other domains. First, in Table 4 we show the Lebesgue constants of the Approximate Fekete Points extracted from the geometric WAMs described in Section 2. Again, the growth is much slower than that of the theoretical bound (5).

As for the simplex, the Lebesgue constant of our Approximate Fekete Points points is larger than that of the best points known in the literature. The case of the simplex has been widely studied and several specialized approaches have been proposed for the computation of Fekete or other good interpolation points, due to the relevance in the numerical treatment of PDEs by spectral-element and high order finite-element methods: see e.g. [20, 27, 33, 35] and references therein.

On the other hand, our method for computing Approximate Fekete Points via geometric WAMs is quite general and flexible, since it allows to work on a wide class of compact sets and, differently from other computational approaches, up to reasonably high interpolation degrees. The good quality of the discrete sets used for least squares approximation and for interpolation is evidenced by Tables 5-9. We observe that the singularity of the third test function is in the interior of the domains, apart from the simplex where it is located at a vertex (where the discrete points cluster). This explains the better results with the simplex for this function. In the case of the two polygons, a change of variables is made in order to put the problem in the reference square $[-1,1]^{2}$.

The availability of good interpolation points in compact sets with various geometries has a number of potential applications. One for example is connected to numerical cubature. Indeed, when the moments of the underlying polynomial basis are known [29, 31], cubature weights associated to the Approximate Fekete Points can be computed as a by-product of the algorithm, simply by using the moments vector as right-hand side $b$. This gives an algebraic cubature formula, that can be used directly, or as a starting point towards the computation of a minimal formula, by the method developed in [34].

Another relevant application concerns the numerical treatment of PDEs, where a renewed interest is arising in global polynomial methods, such as collocation and discrete least squares methods, over general domains (see, e.g., [24]). Recently, Approximate Fekete Points have been successfully used for discrete least squares discretization of elliptic equations [37]. Moreover, again in the context of numerical PDEs, Approximate Fekete Points for polygons could play a role in connection with discretization methods on polygonal (non simplicial) meshes (see, e.g., [32] and references therein).