Markov inequalities, Dubiner distance, norming meshes and polynomial optimization on convex bodies
Summary (2 min read)
1 Introduction
- Sampling methods, typically on suitable grids, are one of the possible approaches in the vast literature on polynomial optimization theory, cf., e.g., [10, 11, 36] with the references therein.
- All these notions can be given more generally for K ⊂ Polynomial meshes were formally introduced in the seminal paper [9] as a tool for studying the uniform convergence of discrete least squares polynomial approximation, and then studied from both the theoretical and the computational point of view throughout a series of papers.
- This opens the way for a computational use of polynomial meshes in the framework of polynomial optimization, in view of the general elementary estimate given below.
- A similar approach, though essentially in a tensor-product framework, was adopted also in [36].
3 General convex bodies
- The bound (6) is clearly an overestimate, that is attained only in special cases, for example with K = [0, L]d.
- From (13) and (15) the authors finally obtain the approximate cardinality bound card(An(ε)) .
- Indeed, a deep result of convex geometry (Leichtweiss inequality [20]) asserts that, given the Loewner minimal volume ellipsoid enclosing a given convex body K, and considering the regular affine transformation, say ψ, that maps the ellipsoid into the unit Euclidean ball, then diam(K ′)/w(K ′) ≤ √ d , where K ′ = ψ(K) , (17) cf. also [15].
- For an overview on the computation of Loewner ellipsoids the authors quote e.g. [30], with the references therein.
- The authors stress that the cardinality estimate does not depend on the shape of the convex body.
4 Smooth convex bodies
- The norming meshes constructed in the previous sections by standard Markov inequalities are ultimately related to (affinely mapped) uniform grids.
- By tangential Markov inequalities and estimates of the Dubiner distance, obtaining nonuniform norming meshes of much lower cardinality.the authors.
- It can be proved that good interpolation points for degree n on some standard real compact sets ar sp ced proportionally to 1/n in such a distance, like the Morrow-Patterson and the Padua interpolation points on the square [8], or the Fekete points on the cube or ball (in any dimension), cf. [6, 5] and reference therein.
- Unfortunately, the Dubiner distance is known analytically ([5] and references therein) only on the d-dimensional cube, ball and on the sphere Sd−1 (where it turns out to be the geodesic distance).
- More recently it has been computed in the case of univariate trigonometric polynomials (even on subintervals of the period); cf. [6, 34].
5 A numerical example
- The advantage of using the Dubiner distance is that the mesh constant becomes 1/ cos(θ(ε)), which ensures an error ε (relative to the polynomial range) in mesh-based polynomial optimization by O(n2/ε) samples (notice also that for d = 2 using the general approach of Proposition 3 the authors would use O(n4/ε2) samples).
- The authors see that the error behavior is consistent with Proposition 6 and quite satisfactory.
- As expected, it scales linearly with ε, and moreover is below the estimate ε by at least two orders of magnitude (the latter phenomenon has been already observed in other numerical examples on polynomial optimization by norming meshes, cf. [28, 33]).
- On the other hand, it could be useful not only by its direct application, but also to generate starting guesses for more sophisticated optimization procedures.
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Cites background from "Markov inequalities, Dubiner distan..."
...This result is stated in the following Lemma 2 Let K ⊂ Rd be a compact set of the form (21)-(22)....
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...By Lemma 1 we can now prove the following proposition on near G-optimality by polynomial meshes constructed via the Dubiner distance....
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...Lemma 1 Let Yn = Yn(α), n ≥ 1, be a sequence of finite sets of a compact set K ⊂ Rd, whose covering radius with respect to the Dubiner distance does not exceed α/n, where α ∈ (0, π/2), i.e. r(Yn) = max x∈K dubK(x, Yn) = max x∈K min y∈Yn dubK(x, y) ≤ α n ....
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...By Lemma 2 we get immediately the following proposition....
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...(19) The proof follows essentially the lines of that of Proposition 1, with Y2n(π/(2m)) replacing X2mn, observing that by Lemma 1 for every p ∈ Pd2n(K) we have ‖p‖K ≤ cm ‖p‖Y2n(π/(2m))....
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Frequently Asked Questions (16)
Q2. What is the key ingredient for the approximation algorithms proposed in [24]?
the authors mention that polynomial meshes are the key ingredient for the approximation algorithms proposed in [24], where the numerical approximation of the main quantities of pluripotential theory (a non linear potential theory in Cd, d > 1) is studied.
Q3. What is the direct application of polynomial meshes?
Since polynomial meshes have first been introduced in the framework of discrete least squares, their most direct application is in the approximation of functions and data.
Q4. what is the euclidean norm of ddimensional vectors?
Given positive scalars r,M > 0, a compact set K is said to admit a Markov Inequality of exponent r and constant M if, for every n ∈ N , the authors have‖∇p‖K ≤Mnr‖p‖K , ∀p ∈ Pdn , (3)where ‖∇p‖K = maxx∈K ‖∇p(x)‖2, ‖ · ‖2 denoting the euclidean norm of ddimensional vectors.
Q5. What is the role of the norming mesh in multivariate approximation theory?
Polynomial inequalities based on the notion of norming mesh mesh have been recently playing a relevant role in multivariate approximation theory, as well in its computational applications.
Q6. What is the sup-norm of a polynomial?
The authors recall that a polynomial (norming) mesh of a polynomial determining compact set K ⊂ Rd (i.e., a polynomial vanishing on K vanishes everywhere), is a sequence of finite subsets
Q7. What is the simplest way to construct a polynomial mesh?
Polynomial meshes have been constructed by different analytical and geometrical techniques on various classes of compact sets, such as Markov and subanalytic sets, polytopes, convex and starlike bodies; the authors refer the reader, e.g., to [3, 9, 16, 23, 25, 29] and the references therein, for a comprehensive view of construction methods.
Q8. What is the property of being a polynomial mesh?
Among their features, the authors recall for example that the property of being a polynomial mesh is stable under invertible affine transformations and small perturbations (see [13, 25]).
Q9. What is the advantage of using the Dubiner distance?
The advantage of using the Dubiner distance is that the mesh constant becomes 1/ cos(θ(ε)), which ensures an error ε (relative to the polynomial range) in mesh-based polynomial optimization by O(n2/ε) samples (notice also that for d = 2 using the general approach of Proposition 3 the authors would use O(n4/ε2) samples).
Q10. What is the simplest way to estimate the cardinality of a convex body?
In this section the authors modify and improve the construction on smooth convex bodies, by tangential Markov inequalities and estimates of the Dubiner distance, obtaining nonuniform norming meshes of much lower cardinality.
Q11. How many points does the mesh An( ) have?
It has about 19000 points, whereas the Dubiner-like (i.e., constructed by Proposition 6) mesh An( ), n = 4 and ε = 0.2, consists of about 1100 points.
Q12. How many points does the Dubiner-like mesh have?
If the authors move to the case ε = 0.01 keeping n fixed, the grid-based mesh of Proposition 3 has more than 5 millions points, whereas the Dubiner-like one about 23000.
Q13. What is the UIBC code for generating a boundary geodesic mesh?
The latter has been obtained by a Matlab code for polynomial mesh generation on smooth 2-dimensional convex bodies, that computes numerically the boundary curve length and curvature (the rolling ball radius ρ is the reciprocal of the maximal curvature), and then uses an approximate arclength parametrization to compute a geodesic grid with the required density; the code is available at [13].
Q14. What is the equivalence class of convex bodies?
The authors can then search, in the equivalence class of convex bodies generated from K by invertible affine transformations, a representative K ′ with bounded aspect ratio diam(K ′)/w(K ′).
Q15. What is the simplest way to solve the non linear equations?
Note that this algorithm can be generalized to higher dimension d > 2, however this requires to solve O((n2/ )d−1) non linear equations as n2/ → ∞.
Q16. What is the radial projection of a convex body?
Note that, using the Minkowski functional, one can define the radial projection onto ∂K by settingx′ := xφK(x) ∈ ∂K, ∀x ∈ Rd. (32)Proposition 5. Let K ⊂