Analysis of the stability and accuracy of multivariate polynomial approximation by discrete least squares with evaluations in random or low-discrepancy point sets Giovanni Migliorati MATHICSE-CSQI, ´ Ecole Polytechnique F´ ed´ erale de Lausanne Analysis with random points: joint work with Fabio Nobile (EPFL), Raul Tempone (KAUST), Albert Cohen (UPMC), Abdellah Chkifa (UPMC) and Erik von Schwerin (KTH). Analysis with low-discrepancy points: joint work with Fabio Nobile. epfl-mox-logo Providence - September 23th, 2014 1 G.Migliorati (EPFL) ICERM - Brown University
Outline Discrete least squares on multivariate polynomial spaces 1 Stability and accuracy with evaluations in random points 2 Stability and accuracy with evaluations in low-discrepancy point sets 3 Conclusions 4 epfl-mox-logo Providence - September 23th, 2014 2 G.Migliorati (EPFL) ICERM - Brown University
Discrete least squares on multivariate polynomial spaces Discrete least squares on multivariate polynomial spaces 1 Stability and accuracy with evaluations in random points 2 Stability and accuracy with evaluations in low-discrepancy point 3 sets Conclusions 4 epfl-mox-logo Providence - September 23th, 2014 3 G.Migliorati (EPFL) ICERM - Brown University
Discrete least squares on multivariate polynomial spaces Notation and definitions For any d ≥ 1, Γ := [ − 1 , 1] d and any real numbers α, β > − 1, define d � ρ ( y ) := B ( α, β ) − d (1 − y i ) α (1 + y i ) β , y ∈ Γ , i =1 � M � � f 1 , f 2 � M := 1 � f 1 , f 2 � L 2 ρ (Γ) := f 1 ( y ) f 2 ( y ) ρ ( y ) dy , f 1 ( y m ) f 2 ( y m ) , M Γ m =1 ρ := �· , ·� 1 / 2 � · � M := �· , ·� 1 / 2 � · � L 2 ρ , M , L 2 with y 1 , . . . , y M being any points in Γ, either realizations of i.i.d. random i.i.d. ∼ ρ or deterministically given (e.g. low-discrepancy variables Y 1 , . . . , Y M point sets). Given univariate L 2 ρ -orthonormal polynomials ( ϕ k ) k ≥ 0 and a multi-index set Λ ⊂ N d 0 , for any ν ∈ Λ we define d � ψ ν ( y ) := ϕ ν i ( y i ) , y ∈ Γ , i =1 epfl-mox-logo P Λ := span { ψ ν : ν ∈ Λ } . Providence - September 23th, 2014 4 G.Migliorati (EPFL) ICERM - Brown University
Discrete least squares on multivariate polynomial spaces Markov and Nikolskii inequalities for multivariate polynomials with downward closed multi-index sets Definition (Downward closed multi-index set) ν ′ ≤ ν ) ⇒ ν ′ ∈ Λ . Λ is downward closed if ( ν ∈ Λ and Lemma (M. 2014) In any dimension, for any Λ downward closed and any α, β ∈ N 0 it holds � u � 2 L ∞ (Γ) ≤ (#Λ) 2 max { α,β } +2 � u � 2 ρ (Γ) , ∀ u ∈ P Λ (Γ) . L 2 Lemma (M. 2014) In any dimension and for any Λ downward closed, when α = β = 0 (Legendre polynomials), it holds � � 2 � ∂ d � � � ≤ 4 − d (#Λ) 4 � u � 2 u ρ (Γ) , ∀ u ∈ P Λ (Γ) . � � epfl-mox-logo L 2 ∂ y 1 · · · ∂ y d L 2 ρ (Γ) Providence - September 23th, 2014 5 G.Migliorati (EPFL) ICERM - Brown University
Discrete least squares on multivariate polynomial spaces Discrete least squares on polynomial spaces For any smooth (analytic) real-valued (or Hilbert-valued) function φ : Γ → R , we define its continuous and discrete L 2 projections over P Λ as Π M Π Λ φ := argmin � φ − v � L 2 ρ , Λ φ := argmin � φ − v � M . v ∈ P Λ v ∈ P Λ Algebraic formulation: design matrix [ D ] ij = ψ j ( y i ), right-hand side [ b ] i = φ ( y i ), for any i = 1 , . . . , M and j = 1 , . . . , #Λ. Normal equations: D ⊤ D β = D ⊤ b , Λ φ = � with β containing the coefficients of the expansion Π M ν ∈ Λ β ν ψ ν . We define also the matrix G := D ⊤ D / M . epfl-mox-logo Providence - September 23th, 2014 6 G.Migliorati (EPFL) ICERM - Brown University
Discrete least squares on multivariate polynomial spaces Optimality of discrete least squares in the L 2 ρ norm In any dimension, with any index set Λ and any ρ with bounded support: Proposition (M., Nobile, von Schwerin and Tempone, FoCM 2014) For any (random or deterministic) choice of M points in Γ it holds � � � � φ − Π M ||| G − 1 ||| Λ φ � L 2 ρ ≤ 1 + v ∈ P Λ � φ − v � L ∞ . inf Proof Theorem (M., Nobile, von Schwerin and Tempone, FoCM 2014) Given M points in Γ , being realizations of random variables independent and identically distributed w.r.t. ρ , it holds M → + ∞ ||| G − 1 ||| = lim M → + ∞ ||| G ||| = 1 , lim almost surely . Proposition (M., Nobile, von Schwerin and Tempone, FoCM 2014) epfl-mox-logo cond ( G ) = ||| G ||| ||| G − 1 ||| . Providence - September 23th, 2014 7 G.Migliorati (EPFL) ICERM - Brown University
Discrete least squares on multivariate polynomial spaces Norm equivalence on P Λ (case of random points) Find δ ∈ (0 , 1) such that (1 − δ ) � v � 2 ρ ≤ � v � 2 M ≤ (1 + δ ) � v � 2 ρ , ∀ v ∈ P Λ , L 2 L 2 with high probability. Since � v � 2 M = M − 1 � D v , D v � 2 R #Λ = � G v , v � 2 R #Λ and � v � 2 ρ = � v , v � 2 R #Λ , the L 2 matrix G satisfies � v � 2 � v � 2 L 2 M ||| G − 1 ||| = ||| G ||| = sup , sup ρ . � v � 2 � v � 2 v ∈ P Λ \{ v ≡ 0 } v ∈ P Λ \{ v ≡ 0 } L 2 M ρ Hence, norm equivalence on P Λ w.h.p. iff concentration bounds 1 − δ ≤ ||| G ||| ≤ 1 + δ, 1 1 1 + δ ≤ ||| G − 1 ||| ≤ 1 − δ, ||| G − I ||| ≤ δ, epfl-mox-logo again with high probability. Providence - September 23th, 2014 8 G.Migliorati (EPFL) ICERM - Brown University
Stability and accuracy with evaluations in random points Discrete least squares on multivariate polynomial spaces 1 Stability and accuracy with evaluations in random points 2 Stability and accuracy with evaluations in low-discrepancy point 3 sets Conclusions 4 epfl-mox-logo Providence - September 23th, 2014 9 G.Migliorati (EPFL) ICERM - Brown University
Stability and accuracy with evaluations in random points Given any L 2 ρ -orthonormal polynomial basis ( ψ ν ) ν ∈ Λ of P Λ , define �� � � v � 2 L ∞ | ψ ν ( y ) | 2 K (Λ) := sup = sup . � v � 2 y ∈ Γ v ∈ P Λ L 2 ν ∈ Λ ρ Lemma (Chkifa, Cohen, M., Nobile and Tempone, 2013) In any dimension and for any downward closed Λ it holds K (Λ) ≤ (#Λ) ln 3 / ln 2 , with tensorized Chebyshev 1st kind polynomials. Lemma (M. 2014) In any dimension, for any downward closed Λ and any α, β ∈ N 0 it holds K (Λ) ≤ (#Λ) 2 max { α,β } +2 , with tensorized Jacobi polynomials . These bounds are quite general, and set the ground for adaptive epfl-mox-logo polynomial approximation based on discrete least squares. Providence - September 23th, 2014 10 G.Migliorati (EPFL) ICERM - Brown University
Stability and accuracy with evaluations in random points Assume that | φ | ≤ τ almost surely w.r.t. ρ and define Π M � Λ := T τ ( Π M T τ ( t ) := sign ( t ) min { τ, | t |} , Λ ) . Theorem (Chkifa, Cohen, M., Nobile and Tempone, 2013) For any γ> 0 and any downward closed Λ , if M is such that K (Λ) ≤ 0 . 15 M 1 + γ ln M then, for any φ ∈ L ∞ (Γ) with � φ � L ∞ ≤ τ , it holds that Pr ( cond ( G ) ≤ 3) ≥ 1 − 2 M − γ , � � √ ≥ 1 − 2 M − γ , � φ − Π M Pr Λ φ � L 2 ρ ≤ (1 + 2) inf v ∈ P Λ � φ − v � L ∞ � � � � 0 . 6 � φ − � Π M Λ φ � 2 � φ − Π Λ φ � 2 ρ + 8 τ 2 M − γ . ≤ 1 + E L 2 L 2 (1 + γ ) ln M ρ epfl-mox-logo ( δ = 1 / 2 everywhere!) Providence - September 23th, 2014 11 G.Migliorati (EPFL) ICERM - Brown University
Stability and accuracy with low-discrepancy point sets Discrete least squares on multivariate polynomial spaces 1 Stability and accuracy with evaluations in random points 2 Stability and accuracy with evaluations in low-discrepancy point 3 sets Conclusions 4 epfl-mox-logo Providence - September 23th, 2014 12 G.Migliorati (EPFL) ICERM - Brown University
Stability and accuracy with low-discrepancy point sets Discrete least squares with deterministic points: multivariate case with Chebyshev density in [0 , 1] d Deterministic points introduced by Zhou, Narayan and Xu: � 2 π � M ( j , . . . , j d ) ∈ [ − 1 , 1] d , y j = cos j = 1 , . . . , M , asymptotically distributed according to the Chebyshev density. Theorem (Zhou, Narayan and Xu, arXiv 2014) In any dimension d and with the Chebyshev density, if M is a prime number and M ≥ 4 d +1 d 2 (#Λ) 2 then it holds that � � 4 � φ − Π M Λ φ � L 2 ρ ≤ 1 + v ∈ P Λ � φ − v � L ∞ . inf d 2 #Λ epfl-mox-logo The proof uses arguments from number theory. Providence - September 23th, 2014 13 G.Migliorati (EPFL) ICERM - Brown University
Stability and accuracy with low-discrepancy point sets Discrete least squares with deterministic points: the multivariate case with uniform density in [0 , 1] d Given any set of M points y 1 , . . . , y M ∈ [0 , 1] d and any set ∅ � = U ⊆ { 1 , . . . , d } , we define its local discrepancy M � � � ∆ U ( t , 1) := 1 I [0 , t q ] ( y q t q , t ∈ [0 , 1] | U | , i ) − M i =1 q ∈ U q ∈ U and its star-discrepancy D ∗ , U := t ∈ [0 , 1] | U | | ∆ U ( t , 1) | . sup Values of components in { 1 , ..., d } \ U are frozen to 1. epfl-mox-logo Providence - September 23th, 2014 14 G.Migliorati (EPFL) ICERM - Brown University
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