Isogeometric Analysis for the Fractional Laplacian

Transcription

1 Isogeometric Analysis for the Fractional Laplacian Kailai Xu & Eric Darve CME500 May 14, / 26

2 Outline Fractional Calculus Isogeometric Analysis Numerical Experiment Regularity for the Fractional Laplacian 2 / 26

3 Long, Long Ago (1695) 3 / 26

4 Euler s Solution Euler observed d n x m dx n m n = m(m 1)... (m n + 1)x Γ(m + 1) = m(m 1)... (m n + 1)Γ(m n + 1) Taking m = 1, n = 1 2, we have d n x m dx n = Γ(m + 1) Γ(m n + 1) x m n d 1 2 x dx 1 2 = 2 π x 1/2 4 / 26

5 Modern Definition: Fractional Integral J 0 f (t) = f (t) J 1 f (t) = f (s)ds 0 J 2 f (t) =... t t s 0 0 f (τ)dτds = 0 t (t s)f (s)ds J n t f (t) =... s f (τ)dτ... ds = 1 t 0 0 (n 1)! (t 0 s)n 1 f (s)ds We generalize it to all α R +, J α f (t) = 1 Γ(α) t 0 (t τ)α 1 f (τ)dτ 5 / 26

6 Modern Definition: Fractional Derivative Here we only consider the left side derivatives. Assume n 1 < α n, n N. Definition (Riemann-Liouvill Derivative) 0Dt α f (t) = d n 1 d n dt n Jn α f (t) = Γ(n α) dt n 0 t f (u) du (t u) 1 n+α Definition (Caputo Derivative) C 0 Dt α f (t) = J n α d n dt n f (t) = 1 t Γ(n α) dt f (u) n du 0 (t u) 1 n+α d n 6 / 26

7 Fractional Laplacian We want to generalize the fractional derivative to higher dimension in a isotropic sense. The Fourier transform of the Laplacian operator F(( )f )(ξ) = ξ 2 Ff (ξ) Definition (Fractional Laplacian) Let f L 2, the fractional Laplacian operator ( ) α is defined by F(( )f )(ξ) = ξ 2α Ff (ξ) Fractional Laplacian describe anomalous diffusion, which has heavy tails compared to normal diffusion. 7 / 26

8 Fractional Laplacian: Integral Form It can be proved that the fractional Laplacian has the following representation for smooth enough u(x) ( ) α u(x) = c 2α,d P.V. R d u(x) u(y) dy x y d+2α where P.V. stands for the Cauchy principal value. c α,d = 2α 1 Γ ( d+α 2 ) π d Γ (1 α 2 ) 8 / 26

9 Fractional Laplacian on the Bounded Domain It is still an open question how the fractional Laplacian should be defined on a bounded domain. Physically, diffusion takes place locally; numerically, we can only handle essentially bounded domains. Probablistic interpretation: fractional Laplacian is a generator of the symmetric α-stable process (a special case of Lévy process). Figure 1: Brownian Motion vs. Lévy Process 9 / 26

10 Fractional Laplacian on the Bounded Domain There are three inequivalent definition (or more) of the fractional Laplacian on the bounded domain. Table 1: Integral, Spectral and Regional fractional Laplacian Operator Description ( ) α I Symmetric α-stable process killed upon leaving Ω ( ) α S Subordinate killed Brownian Motion ( ) α R Censored/Reflected α-stable process A short note on my website for the above connection 10 / 26

11 The Fractional Poisson Problem Since the fractional Poisson equation is the building block for many fractional PDE problems, we study ( ) α u(x) = f (x) u(x) = 0 x Ω x Ω c where Ω is a bounded domain with Lipschitz boundary. 11 / 26

12 An Example ( ) α u(x) = f (x) u(x) = 0 x Ω x Ω c Let Ω = B 0 (1) be the unit ball in R d, u(x) in the ball ( ) α u in the ball (1 x 2 ) α 4 α Γ (α + 1) Γ ( d+2α 2 ) Γ( d 2 ) 1 (1 x 2 ) α+1 4 α Γ (α + 2) Γ ( d+2α 2 ) Γ( d 2 ) 1 [1 (1 + 2α d ) x 2 ] Takeaway: Even though Ω and f are smooth, the solution may only have Hölder continuity (C 0,α ). 12 / 26

13 Numerical Methods Overview Method Reference Finite Difference (Huang and Oberman, 2014; Minden and Ying, 2018; Duo et al., 2018) Finite Element (Ainsworth and Glusa, 2017b; Acosta and Borthagaray, 2017; Ainsworth and Glusa, 2017a) Monte Carlo (Kyprianou et al., 2016; Shardlow, 2018) Meshless (Lischke et al., 2018) 13 / 26

14 Singularity Subtraction Idea: Separate the hyper-singularity kernel into an analytical singular part and a continuous part. u(x) u(y) x y 2+2α = u(x) u(y) + σ( x y )g x (y) σ( x y )g x (y) x y 2+2α x y 2+2α Continous: Numerical Quadrature Evaluate Analytically g x(y) = u 1(x)v 1 + u 2(x)v 2 + u 11(x) v u22(x) v u12(x)v1v2 + u v (x) 6 + u112(x) v 1 2 v u v1v (x) 2 + u222(x) v / 26

15 Singularity Subtraction Requirement: Basis functions with continuous derivatives up to order three. 15 / 26

16 B-Spline Basis Functions Knot vectors Ξ = {ξ 1, ξ 2,..., ξ n+p+1 } 1 if ξ i ξ ξ i+1 B i,0 (ξ) = 0 otherwise B i,p = ξ ξ i B i,p 1 (ξ) + ξ i+p+1 ξ B i+1,p 1 (ξ) ξ i+p ξ i ξ i+p+1 ξ i+1 Figure 2: An example of B-spline basis function sets (Hughes et al., 2005) 16 / 26

17 NURBS Basis Functions NURBS (Non-Uniform Rational Basis Spline) Projection B-Spline basis functions from R d+1 to R d. B i,p (ξ)w i N i,p (ξ) = n j=1 B j,p(ξ)w j The basis functions can be arbitrarily smooth. In addition, NURBS can represent many common geometric forms such as cubes, spheres, cones, etc. 17 / 26

18 Isogeometric Analysis Idea: Same basis functions for the solution space and geometry space. Geometry F (u) = n i=1 C in i,p (u), C i R d, u [0, 1] d Solution u(f (u)) = n i=1 c in i,p (u), c i R, u [0, 1] d 18 / 26

19 Result: C solution 1 u(x) = 16 (1 + cos(πx))2 (1 + cos(πy)) 2 ) x [0, 1] 2 0 otherwise 19 / 26

20 Result: C 1,α solution (1 x 2 ) α+1 x B 0 (1) u(x) = 0 otherwise 20 / 26

21 Result: C 0,α solution (1 x 2 ) α x B 0 (1) u(x) = 0 otherwise 21 / 26

22 What goes wrong? Use a smooth NURBS basis functions defined on untuned meshes to fit C 0,α : Runge phenomenon. (1 x 2 ) α x B 0 (1) u(x) = 0 otherwise 22 / 26

23 Regularity Issues One might ask: is C 0,α merely a very special case? The answer is negative. In fact, we have Theorem (Regularity (Ros-Oton and Serra, 2014)) Let Ω be a bounded C 1,1 domain, f L (Ω), u be a solution of the fractional Poisson equation, and δ(x) = dist(x, Ω). Then, u/δ α Ω can be continuously extended to Ω. Moreover, we have u/δ α C s (Ω) and u/δ α C s (Ω) C f L (Ω) for some s > 0 satisfying s < min{α, 1 α}. The constants s and C depend only on Ω and α. 23 / 26

24 When can we do better? Lemma Let Ω R d be a bounded C 1,1 domain which satisfies the exterior ball condition. Assume that n 0, f L (Ω), and for every x 0 Ω we have there exists C 2 C 1 > 0 such that then there exists C 2 C 1 > 0 C 1 δ(x) n f (x) C 2 δ(x) n, x x 0 C 1δ(x) n+α/2 u(x) C 2δ(x) n+α/2, x x 0 Intuition: If f (x) decays fast near the boundary, so does u(x) (the inverse is not true). Conjecture u(x) δ(x) n+α Ω is C s ( Ω) for some s > 0 and near the boundary. 24 / 26

25 Summary Iso-geometric analysis is suitable for smooth functions (C 1,α and above). Non-locality is the spoiler for large computational cost. Regularity issues are the spoiler for slow convergence of numerical methods. Fractional PDEs maybe a novel and breakthrough approach for modeling compared to traditional PDE. 25 / 26

26 References Gabriel Acosta and Juan Pablo Borthagaray. A fractional laplace equation: Regularity of solutions and finite element approximations. SIAM Journal on Numerical Analysis, 55(2): , Mark Ainsworth and Christian Glusa. Hybrid finite element-spectral method for the fractional laplacian: Approximation theory and efficient solver. arxiv preprint arxiv: , 2017a. Mark Ainsworth and Christian Glusa. Towards an efficient finite element method for the integral fractional laplacian on polygonal domains. arxiv preprint arxiv: , 2017b. Siwei Duo, Hans Werner van Wyk, and Yanzhi Zhang. A novel and accurate finite difference method for the fractional laplacian and the fractional poisson problem. Journal of Computational Physics, 355: , Yanghong Huang and Adam Oberman. Numerical methods for the fractional Laplacian: A finite difference-quadrature approach. SIAM Journal on Numerical Analysis, 52(6): , Thomas JR Hughes, John A Cottrell, and Yuri Bazilevs. Isogeometric analysis: Cad, finite elements, nurbs, exact geometry and mesh refinement. Computer methods in applied mechanics and engineering, 194(39-41): , Andreas E Kyprianou, Ana Osojnik, and Tony Shardlow. Unbiased walk-on-spheres monte carlo methods for the fractional laplacian. IMA Journal of Numerical Analysis, Anna Lischke, Guofei Pang, Mamikon Gulian, Fangying Song, Christian Glusa, Xiaoning Zheng, Zhiping Mao, Wei Cai, Mark M Meerschaert, Mark Ainsworth, et al. What is the fractional laplacian? arxiv preprint arxiv: , V. Minden and L. Ying. A simple solver for the fractional Laplacian in multiple dimensions. ArXiv e-prints, February Xavier Ros-Oton and Joaquim Serra. The dirichlet problem for the fractional laplacian: regularity up to the boundary. Journal de Mathématiques Pures et Appliquées, 101(3): , Tony Shardlow. A walk outside spheres for the fractional laplacian: fields and first eigenvalue. arxiv preprint arxiv: , / 26