Kernel approximation, elliptic PDE and boundary effects

Transcription

1 Kernel approximation, elliptic PDE and boundary effects Thomas Hangelbroek University of Hawaii at Manoa Schloss Rauischholzhausen, 2016 work supported by: NSF DMS

2 Surface Spline Approximation in R d Surface splines: φ m (x) := { x 2m d log x, x 2m d, d even d odd.

3 Surface Spline Approximation in R d Surface splines: φ m (x) := { x 2m d log x, x 2m d, d even d odd. Fundamental solution: for f C (R d ) with compact support, f (x) = C m,d R d m f (α)φ m (x α) dα

4 Surface Spline Approximation in R d Surface splines: φ m (x) := { x 2m d log x, x 2m d, d even d odd. Fundamental solution: for f C (R d ) with compact support, f (x) = C m,d R d m f (α)φ m (x α) dα Approximation scheme: φ m k(x, α) = ξ Ξ a(α, ξ)φ m(x ξ) T Ξ f (x) = C m,d R m f (α) a(α, ξ)φ m (x ξ) dα d ξ Ξ T Ξ f = ξ Ξ A ξφ m ( ξ) span ξ Ξ φ m ( ξ) =: S(φ m, Ξ)

5 Replacing kernel Replace kernel φ m (x α) k(x, α) = ξ Ξ a(α, ξ)φ m(x ξ) by using a local polynomial reproduction so that ( p P L ) a(α, ξ)p(ξ) = p(α) and supp a(α, ξ)δ ξ B(α, H). ξ Ξ ξ Ξ H can be proportional to fill-distance h := max x supp(f ) min ξ Ξ x ξ [Wu-Schaback] and [Devore-Ron] permit H to depend on α

6 Replacing kernel Replace kernel φ m (x α) k(x, α) = ξ Ξ a(α, ξ)φ m(x ξ) by using a local polynomial reproduction so that ( p P L ) a(α, ξ)p(ξ) = p(α) and supp a(α, ξ)δ ξ B(α, H). ξ Ξ ξ Ξ Error is small and bell shaped ([Dyn-Levin-Rippa], [Rabut]) Err(x, α) = φ(x α) k(x, ( α) H 2m d 1 + x α H ) 2m d L

7 Replacing kernel Replace kernel φ m (x α) k(x, α) = ξ Ξ a(α, ξ)φ m(x ξ) by using a local polynomial reproduction so that ( p P L ) a(α, ξ)p(ξ) = p(α) and supp a(α, ξ)δ ξ B(α, H). ξ Ξ ξ Ξ Error is small and bell shaped ([Dyn-Levin-Rippa], [Rabut]) Err(x, α) = φ(x α) k(x, ( α) H 2m d 1 + x α H ) 2m d L Pointwise error: f (x) T Ξ f (x) m f (α) Err(x, α)dα =: (E m f )(x) R d

8 Replacing kernel Replace kernel φ m (x α) k(x, α) = ξ Ξ a(α, ξ)φ m(x ξ) by using a local polynomial reproduction so that ( p P L ) a(α, ξ)p(ξ) = p(α) and supp a(α, ξ)δ ξ B(α, H). ξ Ξ ξ Ξ Error is small and bell shaped ([Dyn-Levin-Rippa], [Rabut]) Err(x, α) = φ(x α) k(x, ( α) H 2m d 1 + x α H ) 2m d L Pointwise error: f (x) T Ξ f (x) m f (α) Err(x, α)dα =: (E m f )(x) R d E 1 1 E H 2m d R d (1 + x H )2m d L dx H 2m.

9 Replacing kernel Replace kernel φ m (x α) k(x, α) = ξ Ξ a(α, ξ)φ m(x ξ) by using a local polynomial reproduction so that ( p P L ) a(α, ξ)p(ξ) = p(α) and supp a(α, ξ)δ ξ B(α, H). ξ Ξ ξ Ξ Error is small and bell shaped ([Dyn-Levin-Rippa], [Rabut]) Err(x, α) = φ(x α) k(x, ( α) H 2m d 1 + x α H ) 2m d L Pointwise error: f (x) T Ξ f (x) m f (α) Err(x, α)dα =: (E m f )(x) R d E 1 1 E H 2m d R d (1 + x H )2m d L dx H 2m. Basic Error Estimate : f T Ξ f p E( m f ) p CH 2m m f p

10 General Strategy Represent functions with an integral using kernel Replace kernel in integral with combination of scattered shifts

11 General Strategy Represent functions with an integral using kernel Replace kernel in integral with combination of scattered shifts { Example: On S d (1 x y) m d/2 d odd, k(x, y) = (1 x y) m d/2 log(1 x y) d even m f (x) = Lf (y)k(x, y)dy + p(x), L = ( r j ). S d Replace k(x, y) by a(y, ξ)k(x, ξ) using a local spherical harmonic reproduction ( ) d 1 a(y, ξ)δ ξ. Err(x, y) H 2m d 1 + dist(x,y) T Ξ f = ξ Ξ ( S d Lf (y)a(y, ξ)dy j=1 ) k(, ξ) + p has approx. order 2m H

12 General Strategy Represent functions with an integral using kernel Replace kernel in integral with combination of scattered shifts ( ) m 3/2 Example [H-Schmid, 11]: On SO(3) k(x, y) = sin( ω(x,y) 2 ) f (x) = SO(3) Lf (y)k(x, y)dy + p(x), L = m ( r j ). Replace k(x, y) by a(y, ξ)k(x, ξ) using a local eigenfunction reproduction ( ) 4 a(y, ξ)δ ξ : Err(x, y) H 2m d 1 + dist(x,y) T Ξ f = ( ) Lf (y)a(y, ξ)dy k(, ξ) + p has approx. order 2m ξ Ξ SO(3) j=1 H

13 Kernel approximation on manifolds M - compact, d-dimensional Riemannian manifold, without boundary. For Ω M, the Sobolev space W m 2 (Ω): f, g W m 2 (Ω) := m k f, k g x dx. k=0 Locally metric-equivalent to Sobolev spaces on R d via exponential map: W m 2 (B) W m 2 (exp x (B)): W2 m (Ω) is a reproducing kernel Hilbert space when m > d/2. f (x) = f, k(x, ) W m 2 (M) = Lf (y)k(x, y)dy where L = m k=0 ( k ) k. How to replace kernel? k(x, y) k(x, y) = ξ Ξ a ξk(x, ξ) Ω M

14 Kernel approximation on manifolds For Ξ M, ξ Ξ, let χ ξ = ζ Ξ A ξ,ζk(x, ζ), { 1 ζ = ξ χ ξ (ζ) = 0 ζ Ξ \ ξ

15 Kernel approximation on manifolds For Ξ M, ξ Ξ, let χ ξ = ζ Ξ A ξ,ζk(x, ζ), { 1 ζ = ξ χ ξ (ζ) = 0 ζ Ξ \ ξ [H-Narcowich-Ward, 10] There is ν > 0 so that dist(x,ξ) ν( χ ξ (x) e h ). h := max x M min ξ Ξ x ξ A ξ,ζ = χ ξ, χ ζ W m 2 (M) q d 2m dist(ξ,ζ) ν( e h ). q = min ζ η dist(η, ζ). To replace: use Ξ {y} to construct χ y. Replace k(x, y) with k(x, y) := A y,ζ ζ Ξ\{y} A y,y k(x, ζ). Then Err(x, y) = k(x, y) + A y,ζ χy (x) ζ Ξ\{y} A y,y k(x, ζ) = A y,y h 2m d dist(x,y) ν( e h ) T Ξ f = ( Lf (y) A ) y,ξ dy k(, ξ) has approx. order 2m ξ Ξ M A y,y

16 Approximation on bounded regions Consider Ξ Ω, where Ω R d is bounded, with smooth boundary. Approximate with surface splines { x 2m d log x, d even φ m (x) := x 2m d, d odd. Theorem [Johnson (98)] When centers cannot coalesce near Ω, then approximation order is saturated at m + 1/p.

17 Approximation on bounded regions Consider Ξ Ω, where Ω R d is bounded, with smooth boundary. Approximate with surface splines { x 2m d log x, d even φ m (x) := x 2m d, d odd. Theorem [Johnson (98)] When centers cannot coalesce near Ω, then approximation order is saturated at m + 1/p.

18 Surface Spline Approximation Orders 1. For f W2 m(ω), dist(f, S(Ξ)) p = O(h m (d/2 d/p)+ ) s 1. Duchon 77 m d p

19 Surface Spline Approximation Orders 2. For f Wp 2m (R d ), dist(f, S(Ξ)) p = O(h 2m ) s 2m 2. Boundary-free Order 1. Duchon 77 m d p

20 Surface Spline Approximation Orders 3. If dist(f, S(Ξ)) p = o(h m+1/p ) then f is trivial. s 2m 2. Boundary-free Order 3. Upper Bound m 1. Duchon 77 m d p

21 Integral Representation for Bounded Domains Let Ω R d be compact with smooth boundary. For f C 2m (Ω) and x Ω: f (x) = m f (α)φ m (x α) dα Ω m 1 + j=0 Ω N j f (α) λ j,α φ m (x α) dσ(α) + p(x)

22 Integral Representation for Bounded Domains Let Ω R d be compact with smooth boundary. For f C 2m (Ω) and x Ω: f (x) = m f (α)φ m (x α) dα Ω m 1 + j=0 Ω N j f (α) λ j,α φ m (x α) dσ(α) + p(x) Operator of Order j The operators λ j, j = 0... m 1 Dirichlet boundary operators: D n j 1 2, or Tr j 2.

23 Integral Representation for Bounded Domains Let Ω R d be compact with smooth boundary. For f C 2m (Ω) and x Ω: f (x) = m f (α)φ m (x α) dα Ω m 1 + j=0 Ω N j f (α) λ j,α φ m (x α) dσ(α) + p(x) Operator of Order 2m j 1 N j = ψtrb, B a differential operator, Tr trace on the boundary, ψ a pseudodifferential operator on boundary. Order(B) + Order(ψ) 2m j 1.

24 Polyharmonic Dirichlet Problem Find a solution of the Dirichlet Problem { m u(α) = 0, α Ω; λ j u(α) = λ j f (α) α Ω, j = 0,..., m 1; using boundary layer potentials V j g(x) := Ω λ j,αφ(x α)g(α)dα. I.e., of the form m 1 m 1 u(x) = V j g j (x) = j=0 j=0 Ω λ j,α φ(x α)g j (α) dσ(α).

25 Polyharmonic Dirichlet Problem Find a solution of the Dirichlet Problem { m u(α) = 0, α Ω; λ j u(α) = λ j f (α) α Ω, j = 0,..., m 1; using boundary layer potentials V j g(x) := Ω λ j,αφ(x α)g(α)dα. I.e., of the form m 1 m 1 u(x) = V j g j (x) = j=0 j=0 Ω λ j,α φ(x α)g j (α) dσ(α). Equivalently: solve the system of integral equations: g 0 (α) λ 0 f (x) K (x, α). dα =. Ω g m 1 (α) λ m 1 f (x) where K (x, α) = ( λ k,x λ j,α φ(x α) ) j,k.

26 Approximation scheme for Bounded Domains T Ξ f (x) = Ω m 1 m f (α)k(x, α) dα + j=0 Ω N j f (α) k j (x, α) dσ(α) + p(x) Replace φ m (x α) by k(x, α) = ξ Ξ a(α, ξ)φ m(x ξ) Replace: λ j,α φ m (x α) by k j (x, α) = ξ Ξ a j(α, ξ)φ m (x ξ) ( Error kernel: Err j (x, α) H 2m d j 1 + x α Worst case: j = m 1. E j 1 1 = max α Ω Ω E j = max x Ω H ) d 1. Err m 1 (x, α)dx H 2m (m 1) = H m+1 Ω Err m 1 (x, α)dα H 2m m = H m Approximation orders: H h throughout Ω, f T Ξ f p = O(h m+1/p ). If H h 2 near boundary then f T Ξ f p = O(h 2m )

27 END