Multivariate polynomial interpolation on lower sets Nira Dyn and - PowerPoint PPT Presentation

Multivariate polynomial interpolation on lower sets Nira Dyn and Michael Floater Department of Mathematics, University of Oslo

Lower set interpolation J. Kuntzmann, M´ ees , 1959. ethodes num´ eriques: interpolation, d´ eriv´ H. Werner, Remarks on Newton type multivariate interpolation for subsets of grids, 1980. G. G. Lorentz and R. A. Lorentz, Solvability problems of bivariate interpolation I, 1986. G. M¨ uhlbach, On multivariate interpolation by generalized polynomials on subsets of grids, 1988. C. de Boor and A. Ron, On multivariate interpolation, 1990. C. de Boor, On the error in multivariate polynomial interpolation, 1992. M. Gasca and T. Sauer, Polynomial interpolation in several variables, 2000. T. Sauer, Lagrange interpolation on subgrids of tensor product grids, 2004. A. Chkifa, A. Cohen, and C. Schwab, High-dimensional adaptive sparse polynomial interpolation and applications to parametric PDEs, 2013.

Sparse grids S.A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, 1963. V. Barthelmann, E. Novak, and K. Ritter, High dimensional polynomial interpolation on sparse grids, 2000. M. Hegland, The combination technique and some generalisations 2007.

Cartesian grid of points Cartesian grid of points in R d , α ∈ N d x α = ( x 1 ,α 1 , x 2 ,α 2 , . . . , x d ,α d ) , 0 , where x j , k , k ∈ N 0 , are distinct for each j ∈ { 1 , . . . , d } . Multi-index notation: α = ( α 1 , α 2 , . . . , α d ) ∈ N d 0 , with | α | := α 1 + · · · + α d , and α ≤ β means that α j ≤ β j for j = 1 , . . . , d .

Lower sets We call a finite set L ⊂ N d 0 a lower set if whenever µ ∈ L and 0 ≤ α ≤ µ then α ∈ L . Let Y L = { x α : α ∈ L } . The set Y L can take on different configurations. x x x 3 2,3 2,3 2,1 x x x 2,3 2 2,2 2,1 x x 2,0 x 2,2 1 2,1 0 x x x 2,0 2,2 2,0 x 1,0 x x x x x x x x 1,0 x x x 0 1 2 3 1,1 1,2 1,3 1,2 1,0 1,1 1,3 1,2 1,3 1,1

Interpolation on lower sets Let P L = span { x α : α ∈ L } , where x α is the monomial x α := x α 1 1 · · · x α d d . For every function f defined on Y L there is a unique polynomial p ∈ P L that interpolates f on Y L , i.e., such that p ( x α ) = f ( x α ) for all α ∈ L . Earliest reference: J. Kuntzmann, 1959.

The Newton form One way of expressing p is in Newton form. For each j = 1 , . . . , d , let ω j , 0 ( y ) = 1 and k − 1 � ω j , k ( y ) = ( y − x j , i ) , k ≥ 1 , y ∈ R , i =0 and define the d -variate polynomial α ∈ N d ω α ( x ) = ω 1 ,α 1 ( x 1 ) · · · ω d ,α d ( x d ) , 0 . Let ∆ α,β f , 0 ≤ α ≤ β , be the the tensor-product divided difference of f over the points µ , α ≤ µ ≤ β . Then we can express p as � x ∈ R d . p ( x ) = ω α ( x )∆ 0 ,α f , (1) α ∈ L We will sometimes write p as p ( L ).

Interpolation in terms of blocks A point β ∈ L is a maximal point if there is no µ ∈ L such that β � = µ and β ≤ µ . Let V ⊂ L be the set of maximal points. Then � L = B β , β ∈ V where B β is the (rectangular) ‘block’ B β = { α ∈ N d 0 : 0 ≤ α ≤ β } , For example, in the figure, L = B 1 , 3 ∪ B 3 , 1 .

Interpolation in terms of blocks Suppose first that L is the union of two blocks: L = B α ∪ B β . Then p ( L ) = p ( B α ) + p ( B β ) − p ( B α ∩ B β ) . Proof. Use the Newton form of p ( L ). Since � x ∈ R d , p ( L )( x ) = ω µ ( x )∆ 0 ,µ f , µ ∈ B α ∪ B β the result follows from the fact that � � � � = + − . α ∈ L α ∈ B α α ∈ B β α ∈ B α ∩ B β

Arbitrary number of blocks Similarly, if L is any lower set and B a block, p ( L ∪ B ) = p ( L ) + p ( B ) − p ( L ∩ B ) . Therefore, if L r = B 1 ∪ B 2 ∪ · · · ∪ B r , then p ( L n ) = p ( L n − 1 ) + p ( B n ) − p ( L n − 1 ∩ B n ) , and we obtain the double sum formula n n � � p ( L n ) = p ( B i ) − p ( L i − 1 ∩ B i ) . i =1 i =2

Two dimensions Suppose B 1 , . . . , B n are blocks, B i = B β i , in R 2 . We can order them so that 0 ≤ β 1 1 < β 2 1 < · · · < β n β 1 2 > β 2 2 > · · · > β n 2 ≥ 0 . 1 , The blocks form a staircase. The double sum formula simplifies to n n � � p ( L n ) = p ( B i ) − p ( B i − 1 ∩ B i ) . i =1 i =2

Example, with n =5 The points β i are black circles, The points ( β i − 1 , β i 2 ) are white 1 circles.

Arbitrary dimension With the shorthand p i 1 ,..., i k := p ( B i 1 ∩ · · · ∩ B i k ) , repeated use of the double sum formula leads to Theorem n � ( − 1) k − 1 � p ( L n ) = p i 1 ,..., i k . k =1 1 ≤ i 1 < i 2 < ··· < i k ≤ n The first few cases are p ( L 2 ) = ( p 1 + p 2 ) − p 12 , p ( L 3 ) = ( p 1 + p 2 + p 3 ) − ( p 12 + p 13 + p 23 ) + p 123 , p ( L 4 ) = ( p 1 + p 2 + p 3 + p 4 ) − ( p 12 + p 13 + p 14 + p 23 + p 24 + p 34 ) + ( p 123 + p 124 + p 134 + p 234 ) − p 1234 .

How can we simplify in arbitrary dimension? The theorem implies there are integer coefficients c α , α ∈ L , such that � p ( L ) = c α p ( B α ) . α ∈ L Let χ ( L ) : N d 0 → { 0 , 1 } be the characteristic function � 1 if α ∈ L ; χ ( L )( α ) = 0 otherwise. Theorem � ( − 1) | ǫ | χ ( L )( α + ǫ ) , c α = α ∈ L . ǫ ∈{ 0 , 1 } d

Example: interpolation of total degree For m ≥ 0, let L = { α ∈ N d 0 : | α | ≤ m } . The set of maximal points is V = { α ∈ N d 0 : | α | = m } , and L is the union of the blocks B α with | α | = m . If d = 2, the staircase formula gives � � p ( L ) = p ( B α ) − p ( B α ) . | α | = m | α | = m − 1

For d = 3, the formula for c α leads to � � � p ( L ) = p ( B α ) − 2 p ( B α ) + p ( B α ) . | α | = m | α | = m − 1 | α | = m − 2