Smooth Interpolation Arie Israel Courant Institute June 18, 2012 Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 1 / 45
Contributions from Whitney (1930’s) Glaeser (1950’s) Brudnyi-Shvartsman (1980’s-present) Bierstone-Milman-Pawlucki (2000’s-present) Fefferman/Fefferman-Klartag (2003-present) Fefferman-I-Luli (2010-present) Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 2 / 45
Notation Let F : R n → R be sufficiently smooth. For any multi-index α = ( α 1 , . . . , α n ), ∂ α F ( x ) := ∂ α 1 1 · · · ∂ α n n F ( x ); | α | := α 1 + · · · + α n . For k ≥ 1, ∂ α F ( x ) ∇ k F ( x ) := � � | α | = k . Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 3 / 45
Notation Let F : R n → R be sufficiently smooth. For m ≥ 1, x ∈ R n |∇ m F ( x ) | . � F � C m := sup Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 4 / 45
The Problem Given: Finite subset E ⊂ R n with cardinality N ; Function f : E → R . Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 5 / 45
The Problem Given: Finite subset E ⊂ R n with cardinality N ; Function f : E → R . Compute a C -optimal interpolant: F : R n → R with (a) F = f on E ; (b) � F � C m ≤ C · � G � C m whenever G = f on E . Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 5 / 45
The Problem Given: Finite subset E ⊂ R n with cardinality N ; Function f : E → R . Compute a C -optimal interpolant: F : R n → R with (a) F = f on E ; (b) � F � C m ≤ C · � G � C m whenever G = f on E . Side Questions: Estimate the nearly minimal norm � F � C m . How long do these computations take? Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 5 / 45
Theorem (Fefferman-Klartag (’09)) Can construct C 1 -optimal interpolants in time C 2 N log( N ) . Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 6 / 45
A Variant Problem For m ≥ 1 and p ≥ 1, let � 1 / p �� x ∈ R n |∇ m F ( x ) | p dx � F � L m , p := . Compute a C -optimal Sobolev interpolant: F : R n → R with F = f on E ; � F � L m , p ≤ C · � G � L m , p whenever G = f on E . Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 7 / 45
Theorem (Fefferman-I-Luli (’11)) Can construct C-optimal Sobolev interpolants. Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 8 / 45
Theorem (Fefferman-I-Luli (’11)) Can construct C-optimal Sobolev interpolants. Plausible running-time bound is O m , n , p ( N log(∆) r ), where ∆ := max {| x − y | : x , y ∈ E } min {| x − y | : x , y ∈ E } Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 8 / 45
Theorem (Fefferman-I-Luli (’11)) Can construct C-optimal Sobolev interpolants. Plausible running-time bound is O m , n , p ( N log(∆) r ), where ∆ := max {| x − y | : x , y ∈ E } min {| x − y | : x , y ∈ E } Can we prove this? Can we achieve O ( N log( N ))? Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 8 / 45
Example I Given: t 1 , . . . , t N ∈ R p 1 , . . . , p N ∈ R Construct p : R → R with (a) p ( t 1 ) = p 1 , · · · , p ( t N ) = p N ; (b) sup t ∈ R | p ′ ( t ) | ≤ sup t ∈ R | q ′ ( t ) | , for any other interpolant q . Estimate: | p ′ ( t ) | . M = sup t ∈ R Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 9 / 45
p (t8,p8) (t7,p7) (t4−6,p4−6) (t3,p3) (t2,p2) (t1,p1) t Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 10 / 45
p (t8,p8) (t7,p7) (t4−6,p4−6) (t3,p3) (t2,p2) (t1,p1) t Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 11 / 45
� p 2 − p 3 � � (1) sup | p ′ ( t ) | = � . � � t 2 − t 3 ⇒ The competitor q interpolates the data, so MVT = q ′ ( t ∗ ) = p 2 − p 3 (2) ∃ t ∗ ∈ [ t 2 , t 3 ] with . t 2 − t 3 Finally, (1) and (2) = ⇒ (3) sup | p ′ ( t ) | ≤ C sup | q ′ ( t ) | . Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 12 / 45
Example II Given: t 1 , . . . , t N ∈ R p 1 , . . . , p N ∈ R Construct p : R → R with (a) p ( t 1 ) = p 1 , · · · , p ( t N ) = p N ; (b) sup t ∈ R | p ′′ ( t ) | ≤ sup t ∈ R | q ′′ ( t ) | , for any other interpolant q . Estimate: | p ′′ ( t ) | . M = sup t ∈ R Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 13 / 45
p (t8,p8) (t7,p7) (t4−6,p4−6) (t3,p3) (t2,p2) (t1,p1) t Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 14 / 45
p (t8,p8) (t7,p7) (t4−6,p4−6) (t3,p3) (t2,p2) (t1,p1) t Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 15 / 45
p (t8,p8) (t7,p7) (t4−6,p4−6) (t3,p3) (t2,p2) (t1,p1) t Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 16 / 45
Higher Dimensions Given: Finite subset E ⊂ [0 , 1] 2 ; Function f : E → R Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 17 / 45
Higher Dimensions Given: Finite subset E ⊂ [0 , 1] 2 ; Function f : E → R There’s a Competitor: G : R 2 → R with G = f on E ; |∇ 2 G | ≤ 1 on R 2 . Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 17 / 45
Higher Dimensions Given: Finite subset E ⊂ [0 , 1] 2 ; Function f : E → R There’s a Competitor: G : R 2 → R with G = f on E ; |∇ 2 G | ≤ 1 on R 2 . Goal: Construct F : [0 , 1] 2 → R with F = f on E ; |∇ 2 F | ≤ C on [0 , 1] 2 . Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 17 / 45
Two Examples (a) E contained in a line. (b) E contained in a smooth curve. (a) (b) Figure: Sets with 1D structure Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 18 / 45
The Straight Line Suppose that E = { (0 , y 1 ) , . . . , (0 , y N ) } ; f : E → R . Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 19 / 45
The Straight Line Suppose that E = { (0 , y 1 ) , . . . , (0 , y N ) } ; f : E → R . Step 1: Let g : R → R be the cubic spline with g ( y k ) = f (0 , y k ) for k = 1 , . . . , N , and | g ′′ ( y ) | ≤ C . Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 19 / 45
The Straight Line Suppose that E = { (0 , y 1 ) , . . . , (0 , y N ) } ; f : E → R . Step 1: Let g : R → R be the cubic spline with g ( y k ) = f (0 , y k ) for k = 1 , . . . , N , and | g ′′ ( y ) | ≤ C . Step 2: Define F ( x , y ) := g ( y ). Then |∇ 2 F ( x , y ) | = | g ′′ ( y ) | ≤ C for all ( x , y ) . Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 19 / 45
The Smooth Curve Suppose that | φ ′′ | ≤ 1 . E ⊂ { ( φ ( y ) , x ) } , where (a) (b) Figure: Sets with 1D structure Consider the diffeomorphism Φ : R 2 → R 2 : Φ( x , y ) = ( x − φ ( y ) , y ) . Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 20 / 45
The Smooth Curve Suppose that | φ ′′ | ≤ 1 . E ⊂ { ( φ ( y ) , x ) } , where (c) (d) Figure: Sets with 1D structure Consider the diffeomorphism Φ : R 2 → R 2 : Φ( x , y ) = ( x − φ ( y ) , y ) . Note that Φ maps E onto a line segment. Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 20 / 45
The Smooth Curve Suppose that | φ ′′ | ≤ 1 . E ⊂ { ( φ ( y ) , x ) } , where (e) (f) Figure: Sets with 1D structure Consider the diffeomorphism Φ : R 2 → R 2 : Φ( x , y ) = ( x − φ ( y ) , y ) . Note that Φ maps E onto a line segment. There is a 1 − 1 correspondence between interpolation problems on E and on Φ( E ). Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 20 / 45
Some Notation S ( x , δ ) := square with center x and sidelength δ. δ ( S ) := sidelength of the square S . A · S := A -dilate of S about its center . Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 21 / 45
Definition (Neat Squares) A square S is neat if 3 S ∩ E lies on the graph of a function h with | h ′′ | ≤ δ ( S ) − 1 uniformly . Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 22 / 45
Definition (Neat Squares) A square S is neat if 3 S ∩ E lies on the graph of a function h with | h ′′ | ≤ δ ( S ) − 1 uniformly . Equivalently, S neat when δ ( S ) − 1 · (3 S ∩ E ) lies on the graph of a function h with | h ′′ | ≤ 1 uniformly . Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 22 / 45
Definition (Neat Squares) A square S is neat if 3 S ∩ E lies on the graph of a function h with | h ′′ | ≤ δ ( S ) − 1 uniformly . Equivalently, S neat when δ ( S ) − 1 · (3 S ∩ E ) lies on the graph of a function h with | h ′′ | ≤ 1 uniformly . Small enough squares are neat. If S is neat and S ′ ⊂ S then S ′ is neat. Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 22 / 45
Lemma Suppose that S is neat. Then we can construct F : 3 S → R with F = f on E ∩ 3 S and |∇ 2 F | ≤ C on 3 S. (a) A Neat S ... (b) Rescaled Arie Israel (Courant Institute) Smooth Interpolation June 18, 2012 23 / 45
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