Orthogonal functions with a skew-Hermitian differentiation matrix - PowerPoint PPT Presentation

Orthogonal functions with a skew-Hermitian differentiation matrix Adhemar Bultheel Dept. Computer Science, KU Leuven ETNA25, Recent Advances in Scientific Computation Santa Margherita di Pula, Sardinia, May 27-30, 2019 http://nalag.cs.kuleuven.be/papers/ade/ETNA25 Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 1 / 28

Motivation: stability of methods for time-dependent PDE Example (diffusion eq.) ∂ u ∂ t = ∂ � a ( x ) ∂ u � , x ∈ [ − 1 , 1] , t > 0 , a ( x ) > 0 u (0) = u 0 . ∂ x ∂ x Discretisation ( D = differentiation matrix): u ′ = DAD u , t > 0 A > 0 , u (0) = u 0 . Stability: if D ⊤ = −D then d � u � 2 1 = u ⊤ u ′ = u ⊤ DAD u = ( D ⊤ u ) ⊤ A ( D u ) ≤ 0 2 d t Challenge: find (orthogonal) basis Φ = [ φ 0 , φ 1 , . . . ] ⊤ , u ( x ) = � u k φ k such that Φ ′ = D Φ with D ⊤ = −D . Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 2 / 28

Example: Divided differences 0 1 0 · · ·   ... − 1 0 1 1     D = ... ...   2∆ x 0 − 1     . ... ... ... . . Note: only first order! Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 3 / 28

Example: Hermite functions ( − 1) n 2 n n ! n 1 / 4 e − x 2 / 2 H n ( x ) , φ n ( x ) = √ n = 0 , 1 , . . . , x ∈ R � φ m ( x ) φ n ( x ) d x = δ m , n , m , n ∈ Z + R � � n n + 1 φ ′ n ( x ) = − 2 φ n − 1 ( x ) + φ n +1 ( x ) , n ∈ Z + 2 0 b 0 0 · · ·   ... − b 0 0 b 1 �   n + 1 Φ ′ = D Φ ,   D = , b n = ... ...   2 0 − b 1     . ... ... ... . . Note: φ n eigenfunction of Fourier operator. Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 4 / 28

How to arrive at such a system? OPS { p n } dense in L 2 ( R , d µ ), d µ ( x ) = w ( x ) d x ξ p n ( ξ ) = β n − 1 p n − 1 ( ξ ) + δ n p n ( ξ ) + β n p n +1 ( ξ ) , δ n ∈ R , β n > 0 skew Hermitian × i n +1 i ξ { i n p n ( ξ ) } = − β n − 1 { i n − 1 p n − 1 ( ξ ) } + i δ n { i n p n ( ξ ) } + β n { i n +1 p n +1 ( ξ ) } get derivative with Fourier transform � ∞ i n e i x ξ p n ( ξ ) | w ( ξ ) | 1 / 2 d ξ φ n ( x ) := √ 2 π −∞ φ ′ n ( x ) = − β n − 1 φ n − 1 ( x ) + i δ n φ n ( x ) + β n φ n +1 ( x ) The { φ n } is orthogonal in L 2 ( R ). Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 5 / 28

How to arrive at such a system? OPS { p n } dense in L 2 ( R , d µ ), d µ ( x ) = w ( x ) d x ξ p n ( ξ ) = β n − 1 p n − 1 ( ξ ) + δ n p n ( ξ ) + β n p n +1 ( ξ ) , δ n ∈ R , β n > 0 skew Hermitian × i n +1 i ξ { i n p n ( ξ ) } = − β n − 1 { i n − 1 p n − 1 ( ξ ) } + i δ n { i n p n ( ξ ) } + β n { i n +1 p n +1 ( ξ ) } get derivative with Fourier transform � ∞ i n e i x ξ p n ( ξ ) | w ( ξ ) | 1 / 2 d ξ φ n ( x ) := √ 2 π −∞ φ ′ n ( x ) = − β n − 1 φ n − 1 ( x ) + i δ n φ n ( x ) + β n φ n +1 ( x ) The { φ n } is orthogonal in L 2 ( R ). Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 5 / 28

How to arrive at such a system? OPS { p n } dense in L 2 ( R , d µ ), d µ ( x ) = w ( x ) d x ξ p n ( ξ ) = β n − 1 p n − 1 ( ξ ) + δ n p n ( ξ ) + β n p n +1 ( ξ ) , δ n ∈ R , β n > 0 skew Hermitian × i n +1 i ξ { i n p n ( ξ ) } = − β n − 1 { i n − 1 p n − 1 ( ξ ) } + i δ n { i n p n ( ξ ) } + β n { i n +1 p n +1 ( ξ ) } get derivative with Fourier transform � ∞ i n e i x ξ p n ( ξ ) | w ( ξ ) | 1 / 2 d ξ φ n ( x ) := √ 2 π −∞ φ ′ n ( x ) = − β n − 1 φ n − 1 ( x ) + i δ n φ n ( x ) + β n φ n +1 ( x ) The { φ n } is orthogonal in L 2 ( R ). Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 5 / 28

How to arrive at such a system? OPS { p n } dense in L 2 ( R , d µ ), d µ ( x ) = w ( x ) d x ξ p n ( ξ ) = β n − 1 p n − 1 ( ξ ) + δ n p n ( ξ ) + β n p n +1 ( ξ ) , δ n ∈ R , β n > 0 skew Hermitian × i n +1 i ξ { i n p n ( ξ ) } = − β n − 1 { i n − 1 p n − 1 ( ξ ) } + i δ n { i n p n ( ξ ) } + β n { i n +1 p n +1 ( ξ ) } get derivative with Fourier transform � ∞ i n e i x ξ p n ( ξ ) | w ( ξ ) | 1 / 2 d ξ φ n ( x ) := √ 2 π −∞ φ ′ n ( x ) = − β n − 1 φ n − 1 ( x ) + i δ n φ n ( x ) + β n φ n +1 ( x ) The { φ n } is orthogonal in L 2 ( R ). Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 5 / 28

General form The previous construct is a canonical choice. More generally we have: Theorem (Iserles & Webb, 2019 1 ) With { φ n } as before, also φ n ( x ) = Ae i ( ω x + κ n ) φ n ( Bx + C ) ˜ ω, A , B , C , κ n ∈ R , AB � = 0 satisfies the same properties. 1 A. Iserles & M. Webb, A family of orthogonal rational functions and other orthogonal systems with a skew-Hermitian differentiation matrix , J. Fourier Anal. Appl., 2019, to appear. Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 6 / 28

Paley-Wiener Theorem (Iserles & Webb, 2019 2 ) With { φ n } as before, assume the differentiation matrix is irreducible, then { φ n } is dense in PW Ω ( R ) where Ω = supp ( d µ ) . 2 A. Iserles & M. Webb, Orthogonal systems with a skew-symmetric differentiation matrix , Foundations of Computational Mathematics, 2019, to appear. Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 7 / 28

Some notes and computation If φ ′ n ( x ) = − b n − 1 φ n − 1 ( x ) + i c n φ n ( x ) + b n φ n +1 ( x ) then c n = 0 iff d µ is symmetric: w ( − ξ ) = w ( ξ ). � ∞ 1 1 −∞ e ix ξ | w ( ξ ) | 1 / 2 d ξ � If w ( ξ ) d ξ = 1 then p 0 = 1 and φ 0 = √ √ 2 π 2 π φ 1 = 1 b 0 [ φ ′ 0 − i c 0 φ 0 ] φ 2 = 1 1 b 1 [ φ ′ b 0 b 1 [ ∗ φ 0 + ∗ φ ′ 0 + ∗ φ ′′ 1 + b 0 φ 0 − i c 1 φ 1 ] = 0 ] · · · 0 + · · · + β n , n φ ( n ) 1 b 0 b 1 ··· b n − 1 [ β n , 0 φ 0 + β n , 1 φ ′ φ n = 0 ] β 0 , 0 = β 1 , 1 = 1 , β 1 , 0 = − i c 0 + recurrence for β n ,ℓ if n > 1 1 If p n ( ξ ) = � n ℓ =0 p n ,ℓ ξ ℓ , then 1 n i n i n � � − i d ( − i ) ℓ p n ,ℓ φ ( ℓ ) � φ n = = φ 0 p n 0 p 0 , 0 p 0 , 0 d x ℓ =0 1 A. Iserles & M. Webb, A family of orthogonal rational functions and other orthogonal systems with a skew-Hermitian differentiation matrix , J. Fourier Anal. Appl., 2019, to appear. Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 8 / 28

Some notes and computation If φ ′ n ( x ) = − b n − 1 φ n − 1 ( x ) + i c n φ n ( x ) + b n φ n +1 ( x ) then c n = 0 iff d µ is symmetric: w ( − ξ ) = w ( ξ ). � ∞ 1 1 −∞ e ix ξ | w ( ξ ) | 1 / 2 d ξ � If w ( ξ ) d ξ = 1 then p 0 = 1 and φ 0 = √ √ 2 π 2 π φ 1 = 1 b 0 [ φ ′ 0 − i c 0 φ 0 ] φ 2 = 1 1 b 1 [ φ ′ b 0 b 1 [ ∗ φ 0 + ∗ φ ′ 0 + ∗ φ ′′ 1 + b 0 φ 0 − i c 1 φ 1 ] = 0 ] · · · 0 + · · · + β n , n φ ( n ) 1 b 0 b 1 ··· b n − 1 [ β n , 0 φ 0 + β n , 1 φ ′ φ n = 0 ] β 0 , 0 = β 1 , 1 = 1 , β 1 , 0 = − i c 0 + recurrence for β n ,ℓ if n > 1 1 If p n ( ξ ) = � n ℓ =0 p n ,ℓ ξ ℓ , then 1 n i n i n � � − i d ( − i ) ℓ p n ,ℓ φ ( ℓ ) � φ n = = φ 0 p n 0 p 0 , 0 p 0 , 0 d x ℓ =0 1 A. Iserles & M. Webb, A family of orthogonal rational functions and other orthogonal systems with a skew-Hermitian differentiation matrix , J. Fourier Anal. Appl., 2019, to appear. Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 8 / 28

Some notes and computation If φ ′ n ( x ) = − b n − 1 φ n − 1 ( x ) + i c n φ n ( x ) + b n φ n +1 ( x ) then c n = 0 iff d µ is symmetric: w ( − ξ ) = w ( ξ ). � ∞ 1 1 −∞ e ix ξ | w ( ξ ) | 1 / 2 d ξ � If w ( ξ ) d ξ = 1 then p 0 = 1 and φ 0 = √ √ 2 π 2 π φ 1 = 1 b 0 [ φ ′ 0 − i c 0 φ 0 ] φ 2 = 1 1 b 1 [ φ ′ b 0 b 1 [ ∗ φ 0 + ∗ φ ′ 0 + ∗ φ ′′ 1 + b 0 φ 0 − i c 1 φ 1 ] = 0 ] · · · 0 + · · · + β n , n φ ( n ) 1 b 0 b 1 ··· b n − 1 [ β n , 0 φ 0 + β n , 1 φ ′ φ n = 0 ] β 0 , 0 = β 1 , 1 = 1 , β 1 , 0 = − i c 0 + recurrence for β n ,ℓ if n > 1 1 If p n ( ξ ) = � n ℓ =0 p n ,ℓ ξ ℓ , then 1 n i n i n � � − i d ( − i ) ℓ p n ,ℓ φ ( ℓ ) � φ n = = φ 0 p n 0 p 0 , 0 p 0 , 0 d x ℓ =0 1 A. Iserles & M. Webb, A family of orthogonal rational functions and other orthogonal systems with a skew-Hermitian differentiation matrix , J. Fourier Anal. Appl., 2019, to appear. Adhemar Bultheel (KU leuven) OF with skew differentiation matrix 8 / 28