Numerical solution of surface PDEs with Radial Basis Functions

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1 Numerical solution of surface PDEs with Radial Basis Functions Andriy Sokolov Institut für Angewandte Mathematik (LS3) TU Dortmund TU Dortmund June 1, 2017

2 Outline 1 Radial Basis Functions (RBFs) 2 FD-RBF approximation of differential operators (Finite Difference Radial Basis Functions) 3 FD-RBF level set method for surface PDEs 4 Outlook

3 Outline 1 Radial Basis Functions (RBFs) 2 FD-RBF approximation of differential operators (Finite Difference Radial Basis Functions) 3 FD-RBF level set method for surface PDEs 4 Outlook

4 Outline 1 Radial Basis Functions (RBFs) 2 FD-RBF approximation of differential operators (Finite Difference Radial Basis Functions) 3 FD-RBF level set method for surface PDEs 4 Outlook

5 Outline 1 Radial Basis Functions (RBFs) 2 FD-RBF approximation of differential operators (Finite Difference Radial Basis Functions) 3 FD-RBF level set method for surface PDEs 4 Outlook

6 Problem Figure: Scattered data interpolation

7 Idea 1: polynomial interpolation Given points a < x 1,x 2,...,x N < b and data values f 1,f 2,...,f N construct a polynomial p of degree N 1 s.t. p(x i) = f i.

8 Idea 1: polynomial interpolation Given points a < x 1,x 2,...,x N < b and data values f 1,f 2,...,f N construct a polynomial p of degree N 1 s.t. p(x i) = f i.

9 Idea 1: polynomial interpolation Given points a < x 1,x 2,...,x N < b and data values f 1,f 2,...,f N construct a polynomial p of degree N 1 s.t. p(x i) = f i. Figure: Runge phenomenon

10 Idea 2: splines Cubic splines: S 3(X) = {s C 2 [a,b] : s [xi P3(R), 0 i N}.,x i+1 ] dims 3(X) = N +4, but N interpolation conditions s(x i) = f i.

11 Idea 2: splines Cubic splines: S 3(X) = {s C 2 [a,b] : s [xi P3(R), 0 i N}.,x i+1 ] dims 3(X) = N +4, but N interpolation conditions s(x i) = f i. Natural splines: N 3(X) = {s S 3(X) : s,s [a,x1 ] P1(R)}. [xn,b] Here, the initial interpolation problem has a unique solution

12 Idea 2: splines, 2D case Figure: Stanford Bunny In the subdivision of a region by triangles, the dimension of the spline space is in general unknown Even if great progresses have been made in the 2D setting, the method is not suited for general dimensions

13 Idea 2: splines, 2D case Figure: Stanford Bunny In the subdivision of a region by triangles, the dimension of the spline space is in general unknown Even if great progresses have been made in the 2D setting, the method is not suited for general dimensions

14 Idea 2: splines, 2D case Figure: Stanford Bunny In the subdivision of a region by triangles, the dimension of the spline space is in general unknown Even if great progresses have been made in the 2D setting, the method is not suited for general dimensions

15 Interpolation with RBFs Idea: Let us search interpolant as a linear combination of basis functions {ϕ j} N j=1: N s(x) = λ jϕ j(x). j=1

16 Interpolation with RBFs Idea: Let us search interpolant as a linear combination of basis functions {ϕ j} N j=1: s(x) = Applying interpolation conditions N λ jϕ j(x). j=1 s(x i) = f i, i = 1,...,N we obtain a system of linear equations ϕ 1(x 1)... ϕ N(x 1) λ 1 f =.... ϕ 1(x N)... ϕ N(x N) λ N f N }{{} =A

17 Interpolation with RBFs Idea: Let us search interpolant as a linear combination of basis functions {ϕ j} N j=1: s(x) = Applying interpolation conditions N λ jϕ j(x). j=1 s(x i) = f i, i = 1,...,N we obtain a system of linear equations ϕ 1(x 1)... ϕ N(x 1) λ 1 f =.... ϕ 1(x N)... ϕ N(x N) λ N f N }{{} =A Is this SLE solvable?

18 Interpolation with RBFs The answer: in general NO

19 Interpolation with RBFs The answer: in general NO Definition Let the finite-dimensional linear function space Φ C(Ω), have a basis {φ 1,φ 2,...,φ N}. Then Φ is a Haar space on Ω if det(a) 0 for any set of distinct points x 1,...,x N Ω. Here, A = {φ j(x i)} ij=1,n.

20 Interpolation with RBFs The answer: in general NO Definition Let the finite-dimensional linear function space Φ C(Ω), have a basis {φ 1,φ 2,...,φ N}. Then Φ is a Haar space on Ω if det(a) 0 for any set of distinct points x 1,...,x N Ω. Here, A = {φ j(x i)} ij=1,n. Theorem (Mairhuber-Curtis) Let Ω R d, d 2 contains an interior point, then there exist no Haar space of continuous functions except for the 1-dimensional case.

21 Interpolation with RBFs The answer: in general NO Definition Let the finite-dimensional linear function space Φ C(Ω), have a basis {φ 1,φ 2,...,φ N}. Then Φ is a Haar space on Ω if det(a) 0 for any set of distinct points x 1,...,x N Ω. Here, A = {φ j(x i)} ij=1,n. Theorem (Mairhuber-Curtis) Let Ω R d, d 2 contains an interior point, then there exist no Haar space of continuous functions except for the 1-dimensional case. Let φ i be dependent on x i!

22 Interpolation with RBFs Approach shift the function ϕ j(x) to the point x j ϕ j( x x j ) Choose a radially symmetric function ϕ(r) = ϕ( x )

23 Interpolation with RBFs Approach shift the function ϕ j(x) to the point x j ϕ j( x x j ) Choose a radially symmetric function ϕ(r) = ϕ( x ) Remark 1: Every natural spline s has the representation N s(x) = a jϕ( x x j )+p(x), x R j=1 where ϕ(r) = r 3, r 0 and p P 1(R).

24 Interpolation with RBFs Radial Basis Functions (RBFs) N s(x) = λ jϕ( x x j ), j=1

25 Interpolation with RBFs Radial Basis Functions (RBFs) s(x) = N λ jϕ( x x j ), j=1 where λ j s are calculated to satisfy s(x i) = f i: ϕ( x 1 x 1 ) ϕ( x 1 x 2 )... ϕ( x 1 x N ) λ 1 f 1 ϕ( x 2 x 1 ) ϕ( x 2 x 2 )... ϕ( x 2 x N ).. λ = f ϕ( x N x 1 ) ϕ( x N x 2 )... ϕ( x N x N ) λ N f N }{{} =A distance matrix

26 Interpolation with RBFs Radial Basis Functions (RBFs) s(x) = dimension independent, sufficiently smooth, n λ jϕ( x x j ), j=1 easy to find (multiple-) derivatives, the corresponding SLE is well-posed, i.e. det(a) 0, meshfree, other advantages to discuss later.

27 Interpolation with RBFs Linear vs Multiquadratic RBF: N N s(x) = λ j x x j s(x) = λ j c 2 + x x j 2 j=1 j=1

28 Interpolation with RBFs Types of RBFs: Infinitely smooth RBFs ϕ(r) (r 0) Gaussian (GA) Inverse quadratic (IQ) Inverse multiquadratic (IMQ) e (εr)2 1 1+(εr) (εr) 2 Multiquadratic (MQ) Piecewise smooth RBFs Linear 1+(εr) 2 r Cubic r 3 Thin plate spline (TPS) r 2 log r

29 Interpolation with RBFs

30 Interpolation with RBFs Solvability of the SLE : ϕ( x 1 x 1 ) ϕ( x 1 x 2 )... ϕ( x 1 x N ) λ 1 f 1 ϕ( x 2 x 1 ) ϕ( x 2 x 2 )... ϕ( x 2 x N ). λ = f 2... ϕ( x N x 1 ) ϕ( x N x 2 )... ϕ( x N x λ N ) N f N }{{} =A distance matrix

31 Interpolation with RBFs Solvability of the SLE : ϕ ε( x 1 x 1 ) ϕ ε( x 1 x 2 )... ϕ ε( x 1 x N ) λ 1 f 1 ϕ ε( x 2 x 1 ) ϕ ε( x 2 x 2 )... ϕ ε( x 2 x N ). λ = f 2... ϕ ε( x N x 1 ) ϕ ε( x N x 2 )... ϕ ε( x N x N ) λ N f N }{{} =A(ε) distance matrix

32 Interpolation with RBFs Solvability of the SLE : ϕ ε( x 1 x 1 ) ϕ ε( x 1 x 2 )... ϕ ε( x 1 x N ) λ 1 f 1 ϕ ε( x 2 x 1 ) ϕ ε( x 2 x 2 )... ϕ ε( x 2 x N ). λ = f 2... ϕ ε( x N x 1 ) ϕ ε( x N x 2 )... ϕ ε( x N x N ) λ N f N }{{} =A(ε) distance matrix

33 Interpolation with RBFs Solvability of the SLE : ϕ ε( x x 1 ) ϕ ε( x x 2 ).. ϕ ε( x x N )

34 Interpolation with RBFs Solvability of the SLE : ϕ ε( x x 1 ) ϕ ε( x x 2 ) ϕ ε( x x N ) = C ε 0 ε 2... ε 4... Y 0 0 (x) Y 1 1 (x) , where C is a matrix with entries of size O(1).

35 Interpolation with RBFs Solvability of the SLE : ϕ ε( x x 1 ) ϕ ε( x x 2 ) ϕ ε( x x N ) = C ε 0 ε 2... ε 4... Y 0 0 (x) Y 1 1 (x) , where C is a matrix with entries of size O(1). Remedy: special QR-decomposition work of Markus Verkely

36 Interpolation with RBFs Error estimation for the RBF interpolation: (Gaussian and multiquadratic-like functions) if f (l) l!m l f s f,x L (Ω) e c/h X,Ω f N(Ω), where if f (l) M l f s f,x L (Ω) e clogh X,Ω/h X,Ω f N(Ω), is the fill distance. h X,Ω = sup min x j X x xj 2 x Ω

37 Interpolation with RBFs exp(log(x)./x) x

38 Outline 1 Radial Basis Functions (RBFs) 2 FD-RBF approximation of differential operators (Finite Difference Radial Basis Functions) 3 FD-RBF level set method for surface PDEs 4 Outlook

39 Finite Difference Radial Basis Function (FD-RBF) Finite difference (notation u(x i) = u i): Forward: (u x) i Backward: (u x) i Central: (u x) i = = = ui+1 ui h ui ui 1 h ui+1 ui 1 2h +R F(h, ) +R B(h, ) +R C(h, )

40 Finite Difference Radial Basis Function (FD-RBF) Finite difference (notation u(x i) = u i): Forward: (u x) i Backward: (u x) i = 1 h = 1 h 1 ui+1 + ui +RF(h, ) h 1 ui + ui 1 +RB(h, ) h Backward: (u x) i = 1 1 ui+1 + ui 1 +RC(h, ) 2h 2h

41 Finite Difference Radial Basis Function (FD-RBF) Finite difference (notation u(x i) = u i): Forward: (u x) i Backward: (u x) i Central: (u x) i = ω i+1u i+1 +ω iu i +R F(h, ) = ω iu i +ω i 1u i 1 +R B(h, ) = ω i+1u i+1 +ω i 1u i 1 +R C(h, )

42 Finite Difference Radial Basis Function (FD-RBF) Finite difference (notation u(x i) = u i): Forward: (u x) i Backward: (u x) i Central: (u x) i = ω iu i +R F(h, ) i Ξ F = ω iu i +R B(h, ) i Ξ B = ω iu i +R C(h, ) i Ξ C

43 Finite Difference Radial Basis Function (FD-RBF) Approximation of the Laplace operator: u = f, Ω R d

44 Finite Difference Radial Basis Function (FD-RBF) Approximation of the Laplace operator: u = f, Ω R d N u(x) s(x) = c iϕ( x x i ) i=1

45 Finite Difference Radial Basis Function (FD-RBF) Approximation of the Laplace operator: u = f, Ω R d s(ζ) ξ Ξ ζ ω ξ u(ξ), Ξ ζ = {ζ,ξ 1,ξ 2,...,ξ K}.

46 Finite Difference Radial Basis Function (FD-RBF) Approximation of the Laplace operator: u = f, Ω R d s(ζ) = N c iϕ( ζ ξ i ) = i=0 N c i ϕ( ζ ξ i ) = i=0 ϕ( ζ ξ i ) }{{} rhs = K ω ju(ξ j), j=0 K ω ju(ξ j), j=0 N K c i ω jϕ( ξ j ξ i ) i=0 K j=0 ω j j=0 }{{} ω Φω = rhs( ) ϕ( ξ j ξ i ) }{{} Φ

47 Finite Difference Radial Basis Function (FD-RBF) Approximation of a differential operator: Lu = f, Ω R d L s(ζ) = N c iϕ( ζ ξ i ) = i=0 N c ilϕ( ζ ξ i ) = i=0 Lϕ( ζ ξ i ) }{{} rhs = K ω ju(ξ j), j=0 K ω ju(ξ j), j=0 N K c i ω jϕ( ξ j ξ i ) i=0 K j=0 ω j j=0 }{{} ω Φω = rhs(l) ϕ( ξ j ξ i ) }{{} Φ

48 Finite Difference Radial Basis Function (FD-RBF) Derivative(s) of the Gaussian RBF: ϕ( x ξ ) = ϕ(r(x)) = e ε2 r 2 (x), }{{} r(x)

49 Finite Difference Radial Basis Function (FD-RBF) Derivative(s) of the Gaussian RBF: ϕ( x ξ ) = ϕ(r(x)) = e ε2 r 2 (x), }{{} r(x) s(x) = r2 (x) 2 ϕ xi (s(x)) = ϕ s }{{} 2ε 2 ϕ ϕ(s(x)) = e 2ε2 s(x), s xi = 2ε 2 ϕ[x i ξ i], }{{} (x i ξ i ) ϕ xi x i (s(x)) = ϕ ss s xi s xi +ϕ ss xixi }{{}}{{} (x i ξ i ) 2 1 ϕ xi x j (s(x)) = ϕ ss(x i ξ i)(x j ξ j), since s xi x j = 0.

50 Finite Difference Radial Basis Function (FD-RBF) Consistency + Stability convergence

51 Finite Difference Radial Basis Function (FD-RBF) Consistency + Stability convergence Consistency: Error Bounds for Kernel-Based Numerical Differentiation by O. Davydov and R. Schaback, Numer. Math., 132 (2016), Stability: Error Analysis of Nodal Meshless Methods by R. Schaback, Numerical Analysis, arxiv: (2016)... +???

52 Finite Difference Radial Basis Function (FD-RBF) Poisson equation: ( u(x)) = f(x) in Ω = [0,1] 2, u analyt = sin(πx)sin(πy).

53 Finite Difference Radial Basis Function (FD-RBF) Poisson equation: ( u(x)) = f(x) in Ω = [0,1] 2, u analyt = sin(πx)sin(πy). E l2 EOC(l 2 ) E max EOC(max) h=1/ h=1/ , ,947 h=1/ , ,949 h=1/ , ,000 h=1/80 5,87 1,970 11,89 1,983 Figure: Numerical solution, h=1/20. Table: 5-points stencil (including the main node).

54 Finite Difference Radial Basis Function (FD-RBF) Anisotropic diffusion: (A u(x)) = f(x) in Ω = [0,1] 2, where ( )( )( ) cosφ sinφ 1 0 cosφ sinφ A =. sinφ cosφ 0 ε sinφ cosφ

55 Finite Difference Radial Basis Function (FD-RBF) Anisotropic diffusion: where Figure: Numerical solution, h=1/20. (A u(x)) = f(x) in Ω = [0,1] 2, ( )( )( ) cosφ sinφ 1 0 cosφ sinφ A =. sinφ cosφ 0 ε sinφ cosφ E l2 EOC(l 2) E max EOC(max) Stencil=5, ε = 10 6, φ = π/6 h=1/ h=1/ diverges diverges h=1/ diverges diverges Stencil=9, ε = 10 6, φ = π/6 h=1/ h=1/ h=1/ Stencil=25, ε = 10 6, φ = π/6 h=1/ h=1/ h=1/ Table: Numerical experiments.

56 Finite Difference Radial Basis Function (FD-RBF) Transport equation (the solid body rotation): u t +v u = 0 in Ω = [0,1] 2. Initial solution

57 Finite Difference Radial Basis Function (FD-RBF) Transport equation (the solid body rotation): u t +v u u = 0 in Ω = [0,1] 2. Initial solution

58 Finite Difference Radial Basis Function (FD-RBF) Transport equation (the solid body rotation): u t +v u u = 0 in Ω = [0,1] 2. Stabilization: Stabilization of RBF-generated finite difference methods for convective PDEs by Bengt Fornberg and Erik Lehto, Journal of Computational Physics, 2010.

59 Finite Difference Radial Basis Function (FD-RBF) Transport equation (the solid body rotation): u t +v u u = 0 in Ω = [0,1] 2. Stabilization: Stabilization of RBF-generated finite difference methods for convective PDEs by Bengt Fornberg and Erik Lehto, Journal of Computational Physics, or???

60 Finite Difference Radial Basis Function (FD-RBF) Reaction-Convection-Diffusion Equation: u t +v u (A(x) u(x)) = f in Ω R 2.

61 Finite Difference Radial Basis Function (FD-RBF) Reaction-Convection-Diffusion Equation: u t +v u (A(x) u(x)) = f in Ω R 2. More? work of Ufuk Erkul

62 Outline 1 Radial Basis Functions (RBFs) 2 FD-RBF approximation of differential operators (Finite Difference Radial Basis Functions) 3 FD-RBF method for surface PDEs 4 Outlook

63 FD-RBF for surface PDEs Laplace-Beltrami: Γ u = u(x)+f on Γ R d, where Γ is sufficiently smooth, closed hypersurface.

64 FD-RBF for surface PDEs Laplace-Beltrami: Γ ( Γ u(x)) = u(x)+f on Γ R d, where Γ is sufficiently smooth, closed hypersurface. How to treat the surface? How to treat Γ?

65 FD-RBF for surface PDEs Laplace-Beltrami: Γ ( Γ u(x)) = u(x)+f on Γ R d, where Γ is sufficiently smooth, closed hypersurface. How to treat the surface? How to treat Γ?

66 FD-RBF for surface PDEs Laplace-Beltrami: Γ ( Γ u(x)) = u(x)+f on Γ R d, where Γ is sufficiently smooth, closed hypersurface. How to treat the surface? How to treat Γ?

67 FD-RBF for surface PDEs The phase-field method: u t ( Γ u(x)) = u(x)+f on Γ R d,

68 FD-RBF for surface PDEs The phase-field method: B(φ)u t (B(φ) u(x)) = B(φ)(u(x)+f) in Ω ε R d, Initial solution

69 FD-RBF for surface PDEs The level set based method: D Γ(t) ( Γ(t) u ) = f(u), on Γ(t)

70 FD-RBF for surface PDEs The level set based method: D Γ(t) ( Γ(t) u ) = f(u), on Γ(t) Introducing the level-set function dist(x,γ) if x is inside Γ φ(x) = 0 if x Γ dist(x,γ) if x is outside Γ

71 FD-RBF for surface PDEs The level set based method: D Γ(t) ( Γ(t) u ) = f(u), on Γ(t) Introducing the level-set function dist(x,γ) if x is inside Γ φ(x) = 0 if x Γ dist(x,γ) if x is outside Γ we obtain P Γ u = ( I φ φ φ ) u. φ

72 FD-RBF for surface PDEs The level set based method: Γ(t) u = (ex n x n) (e y n y n) u = px p y u = Gx G y u (e z n z n) p z G z

73 FD-RBF for surface PDEs The level set based method: Γ(t) u = (ex n x n) (e y n y n) u = px p y u = Gx G y u (e z n z n) p z G z Γ(t) u = Γ(t) Γ(t) u = Γ(t) = ( G x G y G z) Gx G y G z Gx G y G z u u = ( G x G x +G y G y +G z G z) u

74 FD-RBF for surface PDEs The level set based method: (G x I φ u(x)) x=xi = = N c j (G x φ(r j (x))) x=xi j=1 N c j [ ((1 n x i nx i )(x i x j ) j=1 n y i ny i (y i y j ) n z i nz i (z i z j )) φ r (r j(x i )) r j (x i ) ].

75 FD-RBF for surface PDEs The level set based method: u(x) Γ u(x) = f(x) on Γ = {x : x = 1}. u analyt = 1 x 5 26 x 2 x (x5 1 10x 3 1x x 1 x 4 2) f = 26 x 5(x5 1 10x3 1 x2 2 +5x 1x 4 2 ).

76 FD-RBF for surface PDEs The level set based method: u(x) Γ u(x) = f(x) on Γ = {x : x = 1}. solutions in band

77 FD-RBF for surface PDEs The level set based method: u(x) Γ u(x) = f(x) on Γ = {x : x = 1}.

78 FD-RBF for surface PDEs The level set based method: u(x) Γ u(x) = f(x) on Γ = {x : x = 1}. E(ε band) EOC(ε band) h=3/ h=3/ h=3/ h=3/ h=3/ Table: 9-points stencil (including the main node), ε-band = 0.1. E(ε band) = u num u analyt l2(ε band) / #nodes

79 FD-RBF for surface PDEs The level set based method: u(x) Γ u(x) = f(x) on Γ = {x : x = 1}. E(ε band) EOC(ε band) h=3/ h=3/ h=3/ h=3/ h=3/ Table: 9-points stencil (including the main node), ε-band = 0.1. E(ε band) = u num u analyt l2(ε band) / #nodes More? work of David Borringo

80 FD-RBF for surface PDEs The level set based method for Γ(t): tu = D Γ(t) u(x,t)+f(x,t) on Γ = Γ(t), where Γ(t) = {x : φ(t,x) = 0} and φ(t,x) = x 0.75+sin(4t)( x 0.5)(1 x ).

81 FD-RBF for surface PDEs The level set based method for Γ(t): t {}} u { ( u t (x,t)+v u+u Γ v = D }{{} Γ(t) Γ(t) u(x,t) ) +f(x,t) on Γ = Γ(t), t u where v = Vn+v S and V = v n = φ t φ.

82 FD-RBF for surface PDEs The level set based method for Γ(t): t {}} u { ( u t (x,t)+v u+u Γ v = D }{{} Γ(t) Γ(t) u(x,t) ) +f(x,t) on Γ = Γ(t), t u where v = Vn+v S and V = v n = φ t φ. The analytical solution is chosen to be u(x,t) = e t/ x 2 x 1 x.

83 FD-RBF for surface PDEs The level set based method for Γ(t): tu = D Γ(t) u(x,t)+f(x,t) on Γ = Γ(t)

84 FD-RBF for surface PDEs The level set based method for Γ(t): tu = D Γ(t) u(x,t)+f(x,t) solutions in band on Γ = Γ(t). numerical analytical

85 FD-RBF for surface PDEs The level set based method for Γ(t): tu = D Γ(t) u(x,t)+f(x,t) on Γ = Γ(t). error in Omega

86 FD-RBF for surface PDEs The level set based method for Γ(t): tu = D Γ(t) u(x,t)+f(x,t) on Γ = Γ(t). E(ε band) EOC(ε band) N= N= N= N= Table: 9-points stencil (including the main node). ε-band = 0.1

87 FD-RBF for surface PDEs The level set based method for Γ(t): tu = D Γ(t) u(x,t)+f(x,t) on Γ = Γ(t). E(ε band) EOC(ε band) N= N= N= N= Table: 9-points stencil (including the main node). ε-band = 0.1 More? work in progress...

88 Conclusion (FD)RBF is a new rapidly developing approach It it possible to use it for real-world applications Mesh adaptivity

89 Conclusion (FD)RBF is a new rapidly developing approach It it possible to use it for real-world applications Mesh adaptivity

90 Conclusion (FD)RBF is a new rapidly developing approach It it possible to use it for real-world applications Mesh adaptivity

91 Conclusion (FD)RBF is a new rapidly developing approach It it possible to use it for real-world applications Mesh adaptivity Lots of new and unexplored fields

92 Conclusion (FD)RBF is a new rapidly developing approach It it possible to use it for real-world applications Mesh adaptivity Lots of new and unexplored fields

93 Acknowledgements Prof. Dr. Oleg Davydov, University of Giessen Prof. Dr. Dmitri Kuzmin, TU Dortmund Prof. Dr. Stefan Turek, TU Dortmund

94 Thank you very much for your attention!

A flux-corrected RBF-FD method for convection dominated problems in domains and on manifolds

A flux-corrected RBF-FD method for convection dominated problems in domains and on manifolds Andriy Sokolov Dmitri Kuzmin, Oleg Davydov and Stefan Turek Institut für Angewandte Mathematik (LS3) TU Dortmund