Local Approximation with Kernels


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1 Local Approximation with Kernels Thomas Hangelbroek University of Hawaii at Manoa 5th International Conference Approximation Theory, 26 work supported by: NSF DMS43726
2 A cubic spline example Consider breakpoints t = < t < t n = with local mesh ratio t i+ t i M = max. i j t j+ t j Marsden ( 74): For M > (3 + 5)/2, there are knot sequences so that (periodic) cubic spline interpolation does not converge f C([, ]) L n f f as max(t j+ t j ).
3 A cubic spline example Consider breakpoints t = < t < t n = with local mesh ratio t i+ t i M = max. i j t j+ t j Marsden ( 74): For M > (3 + 5)/2, there are knot sequences so that (periodic) cubic spline interpolation does not converge f C([, ]) L n f f as max(t j+ t j ). Set t k+ = t k + M(t k t k ) Let H i := t i+ t i. Then H k = M k H. Lagrange function at t =. t Growth restriction: M = max i+ t i i j t j+ t ( j H j C M H i + t i t j H i ) cubic Lagrange function with 9 intervals, m=
4 Local density Consider a set of points Ξ Ω R d with a local density function H : Ω (, ) admitting local polynomial reproduction a : Ξ Ω R : (ξ, α) a(ξ, α).8.6 p Π l, ξ Ξ a(ξ, α)p(ξ) = p(α) α, suppa(, α) B(α, H(α)).4.2 α, ξ Ξ a(ξ, α) K
5 Local density Consider a set of points Ξ Ω R d with a local density function H : Ω (, ) H(.4,,6) =.6 admitting local polynomial reproduction a : Ξ Ω R : (ξ, α) a(ξ, α).8.6 p Π l, ξ Ξ a(ξ, α)p(ξ) = p(α) α, suppa(, α) B(α, H(α)) α, ξ Ξ a(ξ, α) K Local density at α = (.4,.6) admitting LPR with l =
6 Local density Consider a set of points Ξ Ω R d with a local density function H : Ω (, ) H(.2,.2) =.22 admitting local polynomial reproduction a : Ξ Ω R : (ξ, α) a(ξ, α).8.6 p Π l, ξ Ξ a(ξ, α)p(ξ) = p(α) α, suppa(, α) B(α, H(α)) α, ξ Ξ a(ξ, α) K Local density at α = (.2,.2) admitting LPR with l =
7 Local density Consider a set of points Ξ Ω R d with a local density function H : Ω (, ) H(.2,.2) =.22 admitting local polynomial reproduction a : Ξ Ω R : (ξ, α) a(ξ, α).8.6 p Π l, ξ Ξ a(ξ, α)p(ξ) = p(α) α, suppa(, α) B(α, H(α)) α, ξ Ξ a(ξ, α) K Local density at α = (.2,.2) admitting LPR with l = 2. [WuSchaback, 93] Suppose φ C s (R d ) is positive definite and l s. For f N φ, the RBF interpolant I Ξ f span ξ Ξ φ( ξ) satisfies.4.2 f (x) I Ξ f (x) C ( H(x) ) s/2 f Nφ
8 Surface spline approximation Surface splines: φ(x) := { x 2m d log x, x 2m d, d even C 2m d (R d ) d odd.
9 Surface spline approximation Surface splines: φ(x) := { x 2m d log x, x 2m d, d even C 2m d (R d ) d odd. Fundamental solution: for f C C (R d ), f (x) = m f φ(x)
10 Surface spline approximation Surface splines: φ(x) := { x 2m d log x, x 2m d, d even C 2m d (R d ) d odd. Fundamental solution: for f CC (R d ), f (x) = m f φ(x) Localized by LPR: If a : Ξ R d R reproduces Π l, then φ(x α) ( ) 2m d l a(ξ, α)φ(x ξ) ξ Ξ C(H(α)) 2m d x α + H(α) Kernel centered at (.4,,6) H(.4,,6) =.6 Kernel replacement error at (.4,,6) m = 2, d = 2, l =
11 Surface spline approximation Surface splines: φ(x) := { x 2m d log x, x 2m d, d even C 2m d (R d ) d odd. Fundamental solution: for f CC (R d ), f (x) = m f φ(x) Localized by LPR: If a : Ξ R d R reproduces Π l, then φ(x α) ( ) 2m d l a(ξ, α)φ(x ξ) ξ Ξ C(H(α)) 2m d x α + H(α) Kernel centered at (.2,,2) H(.2,.2) =.22 Kernel replacement error at (.2,,2) m = 2, d = 2, l =
12 DeVoreRon : Replace φ(x α) by ξ Ξ a(ξ, α)φ(x ξ) f = m f φ T Ξ f (x) = m f (α) a(α, ξ)φ(x ξ) dα R d ξ Ξ Error in terms of majorant: Suppose H : R d (, ) is a local density function admitting a LPR a : Ξ R d R of order l, and ( H(x) := sup H(α) + α R d ) r x α, < r < l 2m H(α) 2m d then f (x) T Ξ f (x) ( H(x) ) σ f C σ (R d ), < σ 2m
13 Local densities with slow growth A local density H : R d R + has slow growth < ɛ if x, α ( H(α) CH(x) + ) ɛ x α H(x) h(i) For knot sequences in R, H(i) = t i+ t i, M = max i j h(j), ( m < iff H(j) CH(i) + t ) i t j. H(i) By Marsden s example, this is not always enough. The majorant of DeVoreRon satisfies this. In fact ( H(α) + with ɛ = r+ ) r ( x α H(x) iff H(α) H(x) + H(α) ) ɛ x α H(x)
14 Main Result Suppose Ω Ω L, Ξ L Ω L, Ξ := Ξ L Ω. For < ɛ there is ν > so that if H : Ω L (, ) has ɛ slow growth, is a local density admitting a LPR of degree l > (2m d)/ɛ, satisfies H(ξ) q(ξ) M where q(ξ) := min ζ Ξ\{ξ} dist(ξ, ζ), then for ξ Ξ, χ ξ S(φ, Ξ L ) χ ξ (x) M m d/2 exp [ ν ( ) ɛ ] dist(ξ, x), for x Ω. H(ξ)
15 Main Result Suppose Ω Ω L, Ξ L Ω L, Ξ := Ξ L Ω. For < ɛ there is ν > so that if H : Ω L (, ) has ɛ slow growth, is a local density admitting a LPR of degree l > (2m d)/ɛ, satisfies H(ξ) q(ξ) M where q(ξ) := min ζ Ξ\{ξ} dist(ξ, ζ), then for ξ Ξ, χ ξ S(φ, Ξ L ) χ ξ (x) M m d/2 exp [ ν ( ) ɛ ] dist(ξ, x), for x Ω. H(ξ) The penalized Lebesgue constant L Ξ,σ := sup x Ω ξ Ξ χ ξ(x) ( + x ξ H(x) ) σ is bounded (in terms of M, m, d, ɛ, σ).
16 Main Result Suppose Ω Ω L, Ξ L Ω L, Ξ := Ξ L Ω. For < ɛ there is ν > so that if H : Ω L (, ) has ɛ slow growth, is a local density admitting a LPR of degree l > (2m d)/ɛ, satisfies H(ξ) q(ξ) M where q(ξ) := min ζ Ξ\{ξ} dist(ξ, ζ), then for ξ Ξ, χ ξ S(φ, Ξ L ) χ ξ (x) M m d/2 exp [ ν ( ) ɛ ] dist(ξ, x), for x Ω. H(ξ) The penalized Lebesgue constant L Ξ,σ := sup x Ω ξ Ξ χ ξ(x) ( + x ξ ) σ H(x) is bounded (in terms of M, m, d, ɛ, σ). The penalized Lebesgue constant controls the operator norm of I Ξ : I Ξ f /H σ L Ξ,σ f /H σ
17 Main Result Suppose Ω Ω L, Ξ L Ω L, Ξ := Ξ L Ω. For < ɛ there is ν > so that if H : Ω L (, ) has ɛ slow growth, is a local density admitting a LPR of degree l > (2m d)/ɛ, satisfies H(ξ) q(ξ) M where q(ξ) := min ζ Ξ\{ξ} dist(ξ, ζ), then for ξ Ξ, χ ξ S(φ, Ξ L ) χ ξ (x) M m d/2 exp [ ν ( ) ɛ ] dist(ξ, x), for x Ω. H(ξ) The penalized Lebesgue constant L Ξ,σ := sup x Ω ξ Ξ χ ξ(x) ( + x ξ ) σ H(x) is bounded (in terms of M, m, d, ɛ, σ). The penalized Lebesgue constant controls the operator norm of I Ξ : I Ξ f /H σ L Ξ,σ f /H σ supp(f ) Ω = f (x) I Ξ f (x) ( H(x) )σ f C σ (R d ), < σ 2m.
18 Boundary effects for Lagrange functions A Lagrange function centered in the interior of [, ] 2. Lagrange function centered at (.44,.497 )
19 Boundary effects for Lagrange functions A Lagrange function centered in the interior of [, ] 2. Lagrange function centered at (.2,.56 )
20 Boundary effects for Lagrange functions A Lagrange function centered in the interior of [, ] 2. Lagrange function centered at (.87,.499 )
21 Boundary effects for Lagrange functions A Lagrange function centered in the interior of [, ] 2. Lagrange function centered at (.6,.495 )
22 Boundary effects for Lagrange functions A Lagrange function centered in the interior of [, ] 2. Lagrange function centered at (.29,.52 )
23 Boundary effects for Lagrange functions A Lagrange function centered in the interior of [, ] 2. Lagrange function centered at (.,.5 )
24 How important is it to be boundaryfree? For compact Ω R d, and R, Ω R := {x Ω dist(x, Ω) R}. Decompose Ω en annuli away from boundary There exist positive constants C, h and ν depending only on Ω and m so that for h < h we have ( ) χ ξ (x) Ce ν max dist(x, Ω) dist(ξ, Ω), h. Question: For ξ Ω, does χ ξ (x) decay along boundary?
25 A final experiment 378 centers h =.57 Leb Const =.766 Lagrange function centered at (,), 378 centers relatively equispaced centers, and the (logscaled) Lagrange function centered at the origin.
26 A final experiment 26 centers h =.57, h2 =.33 Leb Const = Lagrange function centered at (,), 26 centers Adding three rings of centers near the boundary, with spacing h 2 =.33.
27 A final experiment 4336 centers h =.57, h2 =.33 Leb Const = 2.52 Lagrange function centered at (,), 4336 centers Replacing these with nicely varying centers (created with DistMesh), having.33 h(x).57, and satisfying (i.e., ɛ = 5/2). h(x) h(y)( + x y h(y) )7/2