On Multi-Domain Polynomial Interpolation Error Bounds S AMUEL M UTUA - PowerPoint PPT Presentation

Aim Error bound theorems Numerical experiment Results Conclusions On Multi-Domain Polynomial Interpolation Error Bounds S AMUEL M UTUA , P ROF . S.S. M OTSA The 40 th South African Symposium of Numerical and Applied Mathematics 22 - 24 March 2016

Aim Error bound theorems Numerical experiment Results Conclusions Outline Aim 1 Error bound theorems 2 Univariate polynomial interpolation Multi-variate polynomial interpolation Multi-domain Numerical experiment 3 Results 4 Conclusions 5

Aim Error bound theorems Numerical experiment Results Conclusions Aim To state and prove theorems that govern error bounds in polynomial interpolation. To investigate why the Gauss-Lobatto grids points are preferably used in spectral based collocation methods of solution for solving differential equations. To highlight on some benefits of multi-domain approach to polynomial interpolation and its application. To apply piecewise interpolating polynomial in approximating solution of a differential equation.

Aim Error bound theorems Numerical experiment Results Conclusions Function of one variable Theorem 1 If y N ( x ) is a polynomial of degree at most N that interpolates y ( x ) at ( N + 1 ) distinct grid points { x j } N j = 0 ∈ [ a , b ] , and if the first ( N + 1 ) -th derivatives of y ( x ) exists and are continuous, then, ∀ x ∈ [ a , b ] there exist a ξ x [1] for which N 1 � ( N + 1 )! y ( N + 1 ) ( ξ x ) E ( x ) ≤ ( x − x j ) . (1) j = 0

Aim Error bound theorems Numerical experiment Results Conclusions Equispaced Grid Points j = 0 = a + jh , h = b − a { x j } N N Theorem 2 The error bound when equispaced grid points { x j } N j = 0 ∈ [ a , b ] , are used in univariate polynomial interpolation is given by ( h ) N + 1 4 ( N + 1 ) y ( N + 1 ) ( ξ x ) . E ( x ) ≤ (2)

Aim Error bound theorems Numerical experiment Results Conclusions Proof Fix x between two grid points, x k and x k + 1 so that x k ≤ x ≤ x k + 1 and 1 show that | x − x k | | x − x k + 1 | ≤ 1 4 h 2 . N � The product term w ( x ) = ( x − x j ) is bounded above by 2 j = 0 N | x − x j | ≤ 1 � 4 h N + 1 N ! . j = 0 Substitute in equation (1) to complete the proof. 3

Aim Error bound theorems Numerical experiment Results Conclusions Gauss Lobatto (GL) Grid Points � b − a � j π � b + a { x j } N � � � j = 0 = + cos 2 N 2 Theorem 3 The error bound when GL grid points { x j } N j = 0 ∈ [ a , b ] , are used in univariate polynomial interpolation is given by � b − a � N + 1 K N ( N + 1 )! y ( N + 1 ) ( ξ x ) , 2 (3) E ( x ) ≤ where � 2 � ( 2 N )! � � N K N = . N + 1 2 N ( N !) 2

Aim Error bound theorems Numerical experiment Results Conclusions Proof The Gauss-Lobatto nodes are roots of the polynomial 1 x 2 ) P ′ L N + 1 (ˆ x ) = ( 1 − ˆ N (ˆ x ) = − N ˆ xP N (ˆ x ) + NP N − 1 (ˆ x ) = ( N + 1 )ˆ xP N (ˆ x ) − ( N + 1 ) P N + 1 (ˆ x ) . The polynomial L N + 1 (ˆ x ) in the interval ˆ x ∈ [ − 1 , 1 ] is bounded above by 2 x ≤ 1 | L N + 1 (ˆ x ) | ≤ 2 ( N + 2 ) . max − 1 ≤ ˆ Express L N + 1 (ˆ x ) as a monic polynomial 3 L N + 1 (ˆ x ) 2 ( N + 1 ) = 1 (ˆ x − ˆ x 0 )(ˆ x − ˆ x 1 ) . . . (ˆ x − ˆ x N ) . K N Here 4 � 2 � ( 2 N )! � N � K N = . N + 1 2 N ( N !) 2 Substitute in equation (1) to complete the proof. 5

Aim Error bound theorems Numerical experiment Results Conclusions Chebyshev Grid Points [5] � b − a � b + a � � 2 j + 1 { x j } N � � j = 0 = + cos 2 N + 2 π 2 2 Theorem 4 The error bound when Chebyshev grid points { x j } N j = 0 ∈ [ a , b ] , are used in univariate polynomial interpolation is given by � b − a � N + 1 2 N ( N + 1 )! y ( N + 1 ) ( ξ x ) . 2 (4) E ( x ) ≤

Aim Error bound theorems Numerical experiment Results Conclusions Proof The leading coefficient of ( N + 1 ) -th degree Chebyshev polynomial is 1 2 N . Take 2 � � x ) = 1 1 � ≤ 1 � � w (ˆ 2 N T N + 1 (ˆ x ) , where 2 N T N + 1 (ˆ x ) 2 N , � � � to be the monic polynomial whose roots are the Chebyshev nodes. Substitute in equation (1) to complete the proof. 3 We note that for N > 3, � b − a � N + 1 ( b − a ) N + 1 ( b − a ) N + 1 N 4 ( N + 1 ) > K N ( 2 ) N + 1 ( N + 1 )! > 2 ( 4 ) N ( N + 1 )! .

Aim Error bound theorems Numerical experiment Results Conclusions Function of many variables Theorem 5 Let u ( x , t ) ∈ C N + M + 2 ([ a , b ] × [ 0 , T ]) be sufficiently smooth such that at least the ( N + 1 ) -th partial derivative with respect to x , ( M + 1 ) -th partial derivative with respect to t and ( N + M + 2 ) -th mixed partial derivative with respect to x and t exists and are all continuous, then there exists values ξ x , ξ ′ x ∈ ( a , b ) , and ξ t , ξ ′ t ∈ ( 0 , T ) , [2] such that N M E ( x , t ) ≤ ∂ N + 1 u ( ξ x , t ) ( x − x i )+ ∂ M + 1 u ( x , ξ t ) � � ( t − t j ) ∂ x N + 1 ( N + 1 )! ∂ t M + 1 ( M + 1 )! i = 0 j = 0 (5) N M ∂ N + M + 2 u ( ξ ′ x , ξ ′ t ) � � − ( x − x i ) ( t − t j ) . ∂ x N + 1 ∂ t M + 1 ( N + 1 )!( M + 1 )! i = 0 j = 0

Aim Error bound theorems Numerical experiment Results Conclusions Equispaced Theorem 6 The error bound when equispaced grid points { x i } N i = 0 ∈ [ a , b ] and { t j } M j = 0 ∈ [ 0 , T ] , in x -variable and t -variable, respectively, are used in bivariate polynomial interpolation is given by � b − a � T � N + 1 � M + 1 N M E ( x , t ) = | u ( x , t ) − U ( x , t ) | ≤ C 1 4 ( N + 1 ) + C 2 4 ( M + 1 ) (6) � b − a � N + 1 � T � M + 1 N M + C 3 4 2 ( N + 1 )( M + 1 ) .

Aim Error bound theorems Numerical experiment Results Conclusions Gauss Lobatto Theorem 7 The error bound when GL grid points { x i } N i = 0 ∈ [ a , b ] , in x -variable and { t j } M j = 0 ∈ [ 0 , T ] , in t -variable are used in bivariate polynomial interpolation is given by ( b − a ) N + 1 ( T ) M + 1 E ( x , t ) ≤ C 1 2 N + 1 K N ( N + 1 )! + C 2 2 M + 1 K M ( M + 1 )! (7) ( b − a ) N + 1 ( T ) M + 1 + C 3 ( 2 ) ( N + M + 2 ) K N K M ( N + 1 )!( M + 1 )! , where � 2 � ( 2 N )! � � N K N = . N + 1 2 N ( N !) 2

Aim Error bound theorems Numerical experiment Results Conclusions Chebyshev Theorem 8 The error bound for Chebyshev grid points { x i } N i = 0 ∈ [ a , b ] and { t j } M j = 0 ∈ [ 0 , T ] , in x -variable and t -variable, respectively, in bivariate polynomial interpolation is given by ( b − a ) N + 1 ( T ) M + 1 E ( x , t ) ≤ C 1 2 ( 4 ) N ( N + 1 )! + C 2 2 ( 4 ) M ( M + 1 )! (8) ( b − a ) N + 1 ( T ) M + 1 + C 3 2 2 ( 4 ) N + M ( N + 1 )!( M + 1 )! .

Aim Error bound theorems Numerical experiment Results Conclusions Generalized multi-variate polynomial interpolation If U ( x 1 , x 2 , . . . , x n ) approximates u ( x 1 , x 2 , . . . , x n ) , ( x 1 , x 2 , . . . , x n ) ∈ [ a 1 , b 1 ] × [ a 2 , b 2 ] × . . . × [ a n , b n ] , and suppose that there are N i , i = 1 , 2 , . . . , n grid points in x i -variable, then the error bound in the best approximation is ( b 1 − a 1 ) N 1 + 1 ( b 2 − a 2 ) N 2 + 1 E c ≤ C 1 2 ( 4 ) N 1 ( N 1 + 1 )! + C 2 2 ( 4 ) N 2 ( N 2 + 1 )! ( b n − a n ) N n + 1 + . . . + C n (9) 2 ( 4 ) N n ( N n + 1 )! ( b 1 − a 1 ) N 1 + 1 ( b 2 − a 2 ) N 2 + 1 . . . ( b n − a n ) N n + 1 + C n + 1 2 n ( 4 ) ( N 1 + N 2 + ... + N n ) ( N 1 + 1 )!( N 2 + 1 )! . . . ( N n + 1 )! . � � ∂ ( N 1 + N 2 + ... + N n + n ) u ( x 1 , x 2 , x 3 , . . . , x n ) � � C n + 1 = � . max (10) � � ∂ x N 1 + 1 ∂ x N 2 + 1 . . . ∂ x N n + 1 � � [ x 1 , x 2 ,..., x n ] ∈ Ω n � 1 2

Aim Error bound theorems Numerical experiment Results Conclusions Illustration of the concept of multi-domain [3] Let t ∈ Γ where Γ ∈ [ 0 , T ] . The domain Γ is decomposed into p non-overlapping subintervals as Γ k = [ t k − 1 , t k ] , t k − 1 < t k , t 0 = 0 , t p = T , k = 1 , 2 , . . . , p . STRATEGY Perform interpolation on each subinterval. Define the interpolating polynomial over the entire domain in piece-wise form.

Aim Error bound theorems Numerical experiment Results Conclusions Equispaced Theorem 9 The error bound when equispaced grid points { x i } N i = 0 ∈ [ a , b ] for x -variable and { t ( k ) } M j = 0 ∈ [ t k − 1 , t k ] , k = 1 , 2 , . . . , p , for the decomposed domain in j t -variable, are used in bivariate polynomial interpolation is given by � b − a � T � N + 1 � M + 1 � M � 1 N M E ( x , t ) ≤ C 1 4 ( N + 1 ) + C 2 p 4 ( M + 1 ) (11) � b − a � N + 1 � T � M + 1 � M � 1 N M + C 3 4 2 ( N + 1 )( M + 1 ) . p

Aim Error bound theorems Numerical experiment Results Conclusions Proof Each subinterval � � � T � T � M + 1 1 M � M + 1 � M + 1 � � � 1 ≤ 1 ( t − t ( k ) � � � ) M ! = M ! . � � j 4 pM p 4 M � � j = 0 � � � T � M + 1 p � M + 1 M + 1 � 1 Break C 2 ( T M ) � C ( k ) M 4 ( M + 1 ) into 4 ( M + 1 ) . 2 p k = 1 where ∂ M + 1 u ( x , t ) ∂ M + 1 u ( x , ξ k ) � � � � � ≤ C ( k ) � � � � � = t ∈ [ t k − 1 , t k ] . max 2 , � � � � ∂ t M + 1 ∂ t M + 1 ( x , t ) ∈ Ω � � Multi-Domain � T � T � M + 1 � M + 1 p � M + 1 � M � 1 � 1 � C ( k ) M M 4 ( M + 1 ) ≤ C 2 4 ( M + 1 ) . (12) 2 p p k = 1 � M N + 1 ( T M + 1 C 3 ( b − a N ) M ) � 1 Similarly, last term in equation ( 6 ) reduces to 4 2 ( N + 1 )( M + 1 ) . p