Markov inequalities, Dubiner distance, norming meshes and polynomial optimization on convex bodies
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Citations
Near optimal polynomial regression on norming meshes
Worst-case examples for Lasserre's measure--based hierarchy for polynomial optimization on the hypercube
Worst-case examples for Lasserre's measure-based hierarchy for polynomial optimization on the hypercube
Spectral norms in spaces of polynomials
References
NIST Handbook of Mathematical Functions
Shapes and Geometries
Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization
Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, Second Edition
Related Papers (5)
A note on total degree polynomial optimization by Chebyshev grids
Manifold Optimization Over the Set of Doubly Stochastic Matrices: A Second-Order Geometry
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 ⊂