Consistency Estimates for gFD Methods and Selection of Sets of Influence Oleg Davydov University of Giessen, Germany Localized Kernel-Based Meshless Methods for PDEs ICERM / Brown University 7–11 August 2017 Oleg Davydov Consistency Estimates for gFD 1
Outline Generalized Finite Difference Methods 1 Error of Polynomial and Kernel Numerical Differentiation 2 Numerical Differentiation Polynomial Formulas Kernel-Based Formulas Least Squares Formulas Selection of Sets of Influence 3 Conclusion 4 Oleg Davydov Consistency Estimates for gFD 2
Outline Generalized Finite Difference Methods 1 Error of Polynomial and Kernel Numerical Differentiation 2 Numerical Differentiation Polynomial Formulas Kernel-Based Formulas Least Squares Formulas Selection of Sets of Influence 3 Conclusion 4 Oleg Davydov Consistency Estimates for gFD 3
Generalized Finite Difference Methods Model problem: Poisson equation with Dirichlet boundary ∆ u = f on Ω , u | ∂ Ω = g . Localized numerical differentiation ( Ξ ⊂ Ω , Ξ i ⊂ Ξ small): � ∆ u ( ξ i ) ≈ w i , j u ( ξ j ) for all ξ i ∈ Ξ \ ∂ Ω ξ j ∈ Ξ i Find a discrete approximate solution ˆ u defined on Ξ s.t. � w i , j ˆ u ( ξ j ) = f ( ξ i ) for ξ i ∈ Ξ \ ∂ Ω ξ j ∈ Ξ i u ( ξ i ) = g ( ξ i ) for ξ i ∈ ∂ Ω ˆ Sparse system matrix [ w i , j ] ξ i ,ξ j ∈ Ξ \ ∂ Ω . Oleg Davydov Consistency Estimates for gFD 3
Generalized Finite Difference Methods Sets of influence: Select Ξ i for each ξ i ∈ Ξ \ ∂ Ξ Ξ i ξ i ξ i Ξ i is the ‘star’ or ‘set of influence’ of ξ i Oleg Davydov Consistency Estimates for gFD 4
Generalized Finite Difference Methods “Consistency and Stability = ⇒ Convergence”: � � � � � � � � ˆ u − u | Ξ � ≤ S ∆ u ( ξ i ) − w i , j u ( ξ j ) � � � �� � ξ i ∈ Ξ \ ∂ Ω ξ j ∈ Ξ i solution error � �� � consistency error S := � [ w i , j ] − 1 ξ i ,ξ j ∈ Ξ \ ∂ Ω � − stability constant � · � − a vector norm, e.g. � · � ∞ (max) or quadratic mean (rms), respectively a matrix norm, � · � ∞ or � · � 2 If S is bounded, then the convergence order for a sequence of discretisations Ξ n is determined by the consistency error: � ∆ u ( ξ i ) − w i , j u ( ξ j ) (numerical differentiation error) ξ j ∈ Ξ i Oleg Davydov Consistency Estimates for gFD 5
Outline Generalized Finite Difference Methods 1 Error of Polynomial and Kernel Numerical Differentiation 2 Numerical Differentiation Polynomial Formulas Kernel-Based Formulas Least Squares Formulas Selection of Sets of Influence 3 Conclusion 4 Oleg Davydov Consistency Estimates for gFD 6
Outline Generalized Finite Difference Methods 1 Error of Polynomial and Kernel Numerical Differentiation 2 Numerical Differentiation Polynomial Formulas Kernel-Based Formulas Least Squares Formulas Selection of Sets of Influence 3 Conclusion 4 Oleg Davydov Consistency Estimates for gFD 6
Numerical Differentiation Given a finite set of points X = { x 1 , . . . , x N } ⊂ R d and function values f j = f ( x j ) , we want to approximate the values Df ( z ) at arbitrary points z , where D is a linear differential operator � a α ( z ) ∂ α f ( z ) Df ( z ) = α ∈ Z d + , | α |≤ k k is the order of D , | α | := α 1 + · · · + α d , a α ( z ) ∈ R . Oleg Davydov Consistency Estimates for gFD 6
Numerical Differentiation Approximation approach Df ( z ) ≈ Dp ( z ) , where p is an approximation of f , e.g., least squares fit from a finite dimensional space P partition of unity interpolant moving least squares fit RBF / kernel interpolant � m If p = a i φ i and the coefficients a i depend linearly on i = 1 f ( x j ) , i.e. a = Af | X , then p = φ a = φ Af | X , N � Dp ( z ) = D φ ( z ) A f | X = w j f ( x j ) . � �� � j = 1 w This leads to a numerical differentiation formula N � Df ( z ) ≈ w j f ( x j ) , w : a weight vector . j = 1 Oleg Davydov Consistency Estimates for gFD 7
Numerical Differentiation Exactness approach Require exactness of the numerical differentiation formula for all elements of a space P : N � Dp ( z ) = w j p ( x j ) for all p ∈ P . j = 1 Notation: w ⊥ D P . E.g., exactness for polynomials of certain order q : P = Π d q , the space of polynomials of total degree < q in d variables. (Polynomial numerical differentiation.) Example: five point star (exact for Π 2 4 ) x 2 � ∆ u ( x 0 ) ≈ 1 u ( x 1 )+ u ( x 2 )+ u ( x 3 )+ u ( x 4 ) h 2 � x 3 x 0 x 1 − 4 u ( x 0 ) x 4 Oleg Davydov Consistency Estimates for gFD 8
Numerical Differentiation Exactness approach A classical method for computing weights w ⊥ D Π d q is truncation of Taylor expansion of local error f − p near z (as in the Finite Difference Method). Instead, we can look at N � for all p ∈ Π d Dp ( z ) = w j p ( x j ) q . j = 1 as an underdetermined linear system w.r.t. w , and pick solutions with desired properties. Similar to quadrature rules (Gauss formulas), there are special point sets that admit weights with particularly high exactness order for a given N (five point star). Oleg Davydov Consistency Estimates for gFD 9
Numerical Differentiation Joint work with Robert Schaback O. Davydov and R. Schaback, Error bounds for kernel-based numerical differentiation, Numer. Math., 132 (2016), 243-269. O. Davydov and R. Schaback, Minimal numerical differentiation formulas, preprint. arXiv:1611.05001 O. Davydov and R. Schaback, Optimal stencils in Sobolev spaces, preprint. arXiv:1611.04750 Oleg Davydov Consistency Estimates for gFD 10
Outline Generalized Finite Difference Methods 1 Error of Polynomial and Kernel Numerical Differentiation 2 Numerical Differentiation Polynomial Formulas Kernel-Based Formulas Least Squares Formulas Selection of Sets of Influence 3 Conclusion 4 Oleg Davydov Consistency Estimates for gFD 11
Polynomial Formulas: General Error Bound Theorem If w is exact for polynomials of order q > k (the order of D ), then N N � � | w j |� x j − z � q | Df ( z ) − w j f ( x j ) | ≤ | f | ∞ , q , Ω 2 , j = 1 j = 1 � 1 � 1 / 2 � 1 α ! � ∂ α f � 2 where | f | ∞ , q , Ω := . ∞ , Ω q ! | α | = q Oleg Davydov Consistency Estimates for gFD 11
Polynomial Formulas: General Error Bound Theorem If w is exact for polynomials of order q > k (the order of D ), then N N � � | w j |� x j − z � q | Df ( z ) − w j f ( x j ) | ≤ | f | ∞ , q , Ω 2 , j = 1 j = 1 � 1 � 1 / 2 � 1 α ! � ∂ α f � 2 where | f | ∞ , q , Ω := . ∞ , Ω q ! | α | = q Proof: Let R ( x ) := f ( x ) − T q , z f ( x ) be the remainder of the Taylor polynomial or order q . Recall the integral representation � 1 � ( x − z ) α ( 1 − t ) q − 1 ∂ α f ( z + t ( x − z )) dt . R ( x ) = q α ! 0 | α | = q Since q > k , it follows that DR ( z ) = 0. Oleg Davydov Consistency Estimates for gFD 11
Polynomial Formulas: General Error Bound Thus, we have for R ( x ) := f ( x ) − T q , z f ( x ) : DR ( z ) = 0, � 1 � ( x − z ) α ( 1 − t ) q − 1 ∂ α f ( z + t ( x − z )) dt . R ( x ) = q α ! 0 | α | = q Hence � | ( x j − z ) α | � ∂ α f � C (Ω) | R ( x j ) | ≤ α ! | α | = q � � � ∂ α f � 2 � 1 / 2 ( x j − z ) 2 α � C (Ω) ≤ α ! α ! | α | = q | α | = q � 1 � 1 / 2 � 1 = � x j − z � q α ! � ∂ α f � 2 2 C (Ω) q ! | α | = q � �� � = | f | ∞ , q , Ω Oleg Davydov Consistency Estimates for gFD 12
Polynomial Formulas: General Error Bound With R ( x ) := f ( x ) − T q , z f ( x ) , DR ( z ) = 0 and | R ( x j ) | ≤ � x j − z � q 2 | f | ∞ , q , Ω , polynomial exactness implies N N � � | Df ( z ) − w j f ( x j ) | = | DR ( z ) − w j R ( x j ) | j = 1 j = 1 N � ≤ | w j R ( x j ) | j = 1 N � | w j |� x j − z � q = | f | ∞ , q , Ω 2 . j = 1 Oleg Davydov Consistency Estimates for gFD 13
Polynomial Formulas: General Error Bound If w is exact for polynomials of order q > k (the order of D ), then N N � � | w j |� x j − z � q | Df ( z ) − w j f ( x j ) | ≤ | f | ∞ , q , Ω 2 . j = 1 j = 1 Gives in particular an error bound in terms of Lebesgue � w � 1 := � N (stability) constant j = 1 | w j | : N � w j f ( x j ) | ≤ | f | ∞ , q , Ω � w � 1 h q Df ( z ) − z , X , j = 1 where h z , X := max 1 ≤ j ≤ N � x j − z � 2 is the radius of the set of influence. Applicable in particular to polyharmonic formulas. Oleg Davydov Consistency Estimates for gFD 14
Polynomial Formulas: General Error Bound If w is exact for polynomials of order q > k (the order of D ), then N N � � | w j |� x j − z � q | Df ( z ) − w j f ( x j ) | ≤ | f | ∞ , q , Ω 2 . j = 1 j = 1 N � | w j |� x j − z � q The best bound is obtained if is 2 j = 1 minimized over all weights w satisfying the exactness condition Dp ( z ) = � N j = 1 w j p ( x j ) , ∀ p ∈ Π d q . ( w ⊥ D Π d q ) We call them �·� 1 , q -minimal weights. Oleg Davydov Consistency Estimates for gFD 14
Polynomial Formulas: �·� 1 ,µ -minimal weights An �·� 1 ,µ -minimal ( µ ≥ 0) weight vector w ∗ satisfies N N � � j |� x j − z � µ | w j |� x j − z � µ | w ∗ 2 = inf 2 . w ∈ R N j = 1 j = 1 w ⊥ D Π d q Oleg Davydov Consistency Estimates for gFD 15
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