Numerical Approximation of the Integral Fractional Laplacian Wenyu - PowerPoint PPT Presentation

Numerical Approximation of the Integral Fractional Laplacian Wenyu Lei Department of Mathematics Texas A&M University Joint work with Andrea Bonito and Joseph Pasciak July 26, 2018 • deal.II Workshop • SISSA

Introduction Algorithm Implementation Results Application Conclusion Outline Introduction Numerical Algorithm Implementation in deal.II Results Application: an obstacle problem for a class of integro-differential operators Conclusion Approximation of fractional Laplaican Wenyu Lei

Introduction Algorithm Implementation Results Application Conclusion Outline Introduction Numerical Algorithm Implementation in deal.II Results Application: an obstacle problem for a class of integro-differential operators Conclusion Approximation of fractional Laplaican Wenyu Lei

Introduction Algorithm Implementation Results Application Conclusion Integral Fractional Laplacian ( − ∆) s • Let s ∈ (0 , 1) and η : R d → R be in the Schwarz class. • Definition via integral representation: � η ( x ) − η ( y ) (( − ∆) s η )( x ) = c d,s P.V. | x − y | d +2 s dy, R d where P.V. stands for the principle value and c d,s is a normalization constant. • An equivalent definition via Fourier transform is given by F (( − ∆) s η )( ζ ) = | ζ | 2 s F ( η )( ζ ) . ( ∗ ) Here F is the Fourier transform. • ( ∗ ) defines the unbounded operator ( − ∆) s on L 2 ( R d ) with domain of definition D (( − ∆) s ) := { f ∈ L 2 ( R d ) : | ζ | 2 s F ( f ) ∈ L 2 ( R d ) } ⊂ H s ( R d ) . • Applications: s -stable L´ evy process, electroconvection and the surface quasigeostrophic models [I. Held et al., 1995]. Approximation of fractional Laplaican Wenyu Lei

Introduction Algorithm Implementation Results Application Conclusion Boundary Value Problem • Ω : open, bounded domain in R d with Lipschitz boundary; • f ∈ L 2 (Ω) ; • We consider the stationary problem ( − ∆) s � u | Ω = f, in Ω . Here � u means zero extension of u . • It is a nonlocal problem. • The test space: H s (Ω) := { f ∈ L 2 (Ω) : � � � f � H s ( R d ) < ∞} . • For any η, θ ∈ � H s (Ω) , define the bilinear form � a ( η, θ ) := (( − ∆) s/ 2 � η, ( − ∆) s/ 2 � R d | ζ | 2 s F ( � η ) F ( � θ ) L 2 ( R d ) = θ ) dζ. • Therefore, the weak formulation is: find u ∈ � H s (Ω) satisfying � for all v ∈ � H s (Ω) . a ( u, v ) = fv dx, Ω Approximation of fractional Laplaican Wenyu Lei

Introduction Algorithm Implementation Results Application Conclusion Finite Element Approximations of the Boundary Value Problem Finite element approaches: [Acosta & Borthagaray, 2017]: Directly approximate each entry in the stiffness matrix using the integral form � � a ( u, φ ) = c d,s 1 u ( x ) − � u ( x ) − ( � R d ( � φ ( y ))( � φ ( y )) | x − y | d +2 s dy dx 2 R d together with a certain mesh setting and special quadrature formulas (boundary element approach). [D’ Elia & Gunzburger, 2013]: Approximate the above integral in a bounded truncated domain. Approximate the bilinear form based on its Dunford-Taylor integral representation. Approximation of fractional Laplaican Wenyu Lei

Introduction Algorithm Implementation Results Application Conclusion A Dunford-Taylor Integral Representation of a ( · , · ) An equivalent representation � ∞ � � � � a ( η, θ ) = 2 sin( πs ) t 1 − 2 s ( − ∆)( I − t 2 ∆) − 1 � η θ dx dt =: I π R d 0 Proof • Parseval’s Theorem: � � | ζ | 2 � � � ( − ∆)( I − t 2 ∆) − 1 � η )( ζ ) F ( � η θ dx = 1 + t 2 | ζ | 2 F ( � θ )( ζ ) dζ. R d R d • Apply the Fubini’s Theorem and use the change of variable y = t | ζ | : � � ∞ t 1 − 2 s | ζ | 2 − 2 s I = 2 sin( πs ) R d | ζ | 2 s F ( η )( ζ ) F ( θ )( ζ ) 1 + t 2 | ζ | 2 dt dζ π 0 � R d | ζ | 2 s F ( η )( ζ ) F ( θ )( ζ ) dζ. = Note that �� ∞ � − 1 y 1 − 2 s = 2 sin( πs ) c s := 1 + y 2 dy . π 0 Approximation of fractional Laplaican Wenyu Lei

Introduction Algorithm Implementation Results Application Conclusion Game Plan � ∞ � � � t − 1 − 2 s ( − t 2 ∆)( I − t 2 ∆) − 1 � for η, θ ∈ � H s (Ω) . a ( η, θ ) = c s η θ dx dt � �� � 0 Ω w ( t ) η − ( I − t 2 ∆) − 1 � Here w ( t ) = � η := � η + v ( t ) with v ( t ) satisfying � � � R d v ( t ) φ dx + t 2 for all φ ∈ H 1 ( R d ) , R d ∇ v ( t ) · ∇ φ dx = − ηφ dx, Ω • Discretize the outer integral with a quadrature spacing k : � � a k ( η, θ ) = Ck t − 1 − 2 s w ( t j ) θ dx. j Ω j • Approximate w ( t ) or ( v ( t ) ) in a bounded domain B M ( t ) : � � t − 1 − 2 s a k,M ( η, θ ) = Ck w M ( t j ) θ dx. j Ω j • Approximate w M ( t ) (or v M ( t ) ) using the finite element method: � � a k,M t − 1 − 2 s w M ( η h , θ h ) = Ck h ( t j ) θ h dx. j h Ω j Approximation of fractional Laplaican Wenyu Lei

Introduction Algorithm Implementation Results Application Conclusion Outline Introduction Numerical Algorithm Implementation in deal.II Results Application: an obstacle problem for a class of integro-differential operators Conclusion Approximation of fractional Laplaican Wenyu Lei

Introduction Algorithm Implementation Results Application Conclusion Sinc Quadrature • Use the change of variable t = e − y/ 2 ; � ∞ � a ( η, θ ) = c s e sy w ( t ( y )) θ dx dy. 2 −∞ Ω • Use a truncated equally spaced quadrature: let k > 0 and N + and N − are positive integers chosen to be on the order of 1 /k 2 . Define � N + � a k ( η, θ ) := c s k e sy j w ( y j ) θ dx. 2 Ω j = − N − with y j = jk . • Consistency Error: | a ( η, θ ) − a k ( η, θ ) | ≤ Ce − c/k � η � � H δ (Ω) � θ � � H s (Ω) . Here δ ∈ ( s, 2 − s ] . Approximation of fractional Laplaican Wenyu Lei

Introduction Algorithm Implementation Results Application Conclusion Domain Truncation η + v ( t ) where v ( t ) = − ( I − t 2 ∆) − 1 � • w ( t ) = � η . • Let B be a ball enclosing Ω and assume that diam B = 1 , then for a truncation parameter M , we define the dilated domains � { y = (1 + t (1 + M )) x : x ∈ B } , t ≥ 1 B M ( t ) := { y = (2 + M ) x : x ∈ B } , t < 1 . • Truncated solution: w M ( t ) = � η + v M ( t ) and � � � v M ( t ) φ dx + t 2 ∇ v M ( t ) ·∇ φ dx = − for all φ ∈ H 1 0 ( B M ( t )) , ηφ dx, B M ( t ) B M ( t ) Ω • Truncated bilinear form: � N + � a k,M ( η, θ ) := c s k e sy j w M ( y j ) θ dx. 2 Ω j = − N − • Consistency Error: | a k ( η, θ ) − a k,M ( η, θ ) | ≤ Ce − cM � η � L 2 (Ω) � θ � L 2 (Ω) , H s (Ω) . for all η, θ ∈ � Approximation of fractional Laplaican Wenyu Lei

Introduction Algorithm Implementation Results Application Conclusion Two Finite Element Spaces • The red part is the grid of the original domain and it is quasi-uniform. • The red+black part is the grid of the truncated domain. The black part may not be quasi-uniform for implementation. But we assume the quasi-uniformity for numerical analysis. • V h (Ω) ⊂ V h ( B M ( t )) := V M h ( t ) . Approximation of fractional Laplaican Wenyu Lei

Introduction Algorithm Implementation Results Application Conclusion Finite Element Approximation • The discrete solution: w M η + v M h ( t ) and for all φ h ∈ V M ( t ) , h ( t ) = � � � � v M h ( t ) φ h dx + t 2 ∇ v M h ( t ) · ∇ φ h dx = − η h φ h dx. B M ( t ) B M ( t ) Ω • The discrete bilinear form: for η h , θ h ∈ V h (Ω) � N + � ( η h , θ h ) := c s k a k,M e sy j w M h ( y j ) θ h dx. h 2 Ω j = − N − • Consistency Error: let β ∈ ( s, 3 / 2) , | a k,M ( η h , θ h ) − a k,M ( η h , θ h ) | ≤ C (1 + ln( h − 1 )) h β − s � η h � � H β (Ω) � θ h � � H s (Ω) h Approximation of fractional Laplaican Wenyu Lei

Introduction Algorithm Implementation Results Application Conclusion Discrete Problem • The discrete problem: find u h ∈ V h (Ω) satisfying � a k,M ( u h , v h ) = fv h dx for all v h ∈ V h (Ω) . h Ω • V h (Ω) -ellipticity: For a fixed h , choose k small enough so that Ce − c/k h s − δ < 1 / 2 . Then, a k,M ( η h , η h ) > 1 / 2 � η h � 2 H s (Ω) . h � Assume u ∈ � H β (Ω) with β ∈ ( s, 3 / 2) and α = min( s, 1 / 2) . Error estimates Strang’s Lemma: H s (Ω) ≤ C ( e − c/k + e − cM + (1 + ln ( h − 1 )) h β − s ) � u � � � u − u h � � H β (Ω) . Duality argument: if the domain is smooth, � u − u h � L 2 (Ω) ≤ C ln( h − 1 )( e − c/k + e − cM + (1 + ln ( h − 1 )) h β − s + α ) � u � � H β (Ω) . Approximation of fractional Laplaican Wenyu Lei

Introduction Algorithm Implementation Results Application Conclusion Outline Introduction Numerical Algorithm Implementation in deal.II Results Application: an obstacle problem for a class of integro-differential operators Conclusion Approximation of fractional Laplaican Wenyu Lei