Lecture on: Numerical sparse linear algebra and interpolation spaces. June 3, 2014

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1 Lecture on: Numerical sparse linear algebra and interpolation spaces June 3, 2014

2 Finite dimensional Hilbert spaces and IR N 2 / 38 (, ) : H H IR scalar product and u H = (u, u) u H norm.

3 Finite dimensional Hilbert spaces and IR N 2 / 38 (, ) : H H IR scalar product and u H = (u, u) u H norm. {ψ i } i=1,...,n a basis for H u H u = N i=1 u iψ i u i IR i = 1,..., N

4 Finite dimensional Hilbert spaces and IR N 2 / 38 (, ) : H H IR scalar product and u H = (u, u) u H norm. {ψ i } i=1,...,n a basis for H u H u = N i=1 u iψ i u i IR i = 1,..., N Representation of scalar product in IR N. Let u = N i=1 u iψ i and v = N i=1 v iψ i. Then N N (u, v) = u i v j (ψ i, ψ j ) = v T Hu i=1 j=1 where H ij = H ji = (ψ i, ψ j ) and u, v IR N. Moreover, u T Hu > 0 iff u 0 and, thus H SPD.

5 Dual space H 3 / 38 f H : H IR (functional);

6 Dual space H 3 / 38 f H : H IR (functional); f (αu + βv) = αf (u) + βf (v) u, v H

7 Dual space H 3 / 38 f H : H IR (functional); f (αu + βv) = αf (u) + βf (v) u, v H H is the space of the linear functionals on H f f (u) H = sup u 0 u H

8 Dual space H 3 / 38 f H : H IR (functional); f (αu + βv) = αf (u) + βf (v) u, v H H is the space of the linear functionals on H f f (u) H = sup u 0 u H If H finite dimensional and u = N i=1 u iψ i, then f (u) = N i=1 u if (ψ i ) = f T u

9 Dual space H 3 / 38 f H : H IR (functional); f (αu + βv) = αf (u) + βf (v) u, v H H is the space of the linear functionals on H f f (u) H = sup u 0 u H If H finite dimensional and u = N i=1 u iψ i, then f (u) = N i=1 u if (ψ i ) = f T u Dual vector Let u H, u 0, then f u H such that (Hahn-Banach). f u (u) = u H

10 Dual space H 4 / 38 Let H be a Hilbert finite dimensional space and H the real N N matrix identifying the scalar product.

11 Dual space H 4 / 38 Let H be a Hilbert finite dimensional space and H the real N N matrix identifying the scalar product. f u (u) = f T u = (u T Hu) 1/2 The dual vector of u has the following representation:

12 Dual space H 4 / 38 Let H be a Hilbert finite dimensional space and H the real N N matrix identifying the scalar product. f u (u) = f T u = (u T Hu) 1/2 The dual vector of u has the following representation: f = Hu u H and f u 2 H = ut Hu = f T H 1 f

13 Dual space basis 5 / 38 The general definitions of a dual basis for H is { 1 i = j φ j (ψ i ) = 0 i j

14 Dual space basis 5 / 38 The general definitions of a dual basis for H is { 1 i = j φ j (ψ i ) = 0 i j The φ i are linearly independent: N N β i φ i (u) = 0 u H= β i φ i (ψ i ) = 0= β i = 0. i=1 i=1

15 Dual space basis 5 / 38 The general definitions of a dual basis for H is { 1 i = j φ j (ψ i ) = 0 i j The φ i are linearly independent: N N β i φ i (u) = 0 u H= β i φ i (ψ i ) = 0= β i = 0. i=1 i=1 f (ψ i ) = γ i and f (u) = f ( N i=1 u iψ i ) = N i=1 γ iu i N φ i (u) = φ( u i ψ i ) = u i = f = i=1 N α i φ i i=1

16 Linear operator 6 / 38 A : H V where H and V finite dimensional Hilbert spaces. H and V are the SPD matrices of the scalar products

17 Linear operator 6 / 38 A : H V where H and V finite dimensional Hilbert spaces. H and V are the SPD matrices of the scalar products A H,V = max u 0 A u V u H = V 1/2 AH 1/2 2

18 Linear operator 6 / 38 A : H V where H and V finite dimensional Hilbert spaces. H and V are the SPD matrices of the scalar products A H,V = max u 0 A u V u H = V 1/2 AH 1/2 2 The result follows from the generalized eigenvalue problem in IR N A T VAu = λhu

19 Linear operator 6 / 38 A : H V where H and V finite dimensional Hilbert spaces. H and V are the SPD matrices of the scalar products A H,V = max u 0 A u V u H = V 1/2 AH 1/2 2 The result follows from the generalized eigenvalue problem in IR N A T VAu = λhu κ H (M) = M H,H 1 M 1 H 1,H.

20 Linear operator 6 / 38 A : H V where H and V finite dimensional Hilbert spaces. H and V are the SPD matrices of the scalar products A H,V = max u 0 A u V u H = V 1/2 AH 1/2 2 The result follows from the generalized eigenvalue problem in IR N A T VAu = λhu κ H (M) = M H,H 1 M 1 H 1,H. The interesting case is κ H (M) independent of N

21 Interpolation spaces 7 / 38 Then S self-adjoint such that H = (IR N, (u, v) H = u T Hv) M = (IR N, (u, v) M = u T Mv) (u, v) H = (u, S v) M = (S u, v) M

22 Interpolation spaces 7 / 38 Then S self-adjoint such that H = (IR N, (u, v) H = u T Hv) M = (IR N, (u, v) M = u T Mv) i.e. (u, v) H = (u, S v) M = (S u, v) M (u, v) H = (u, Sv) M = (Su, v) M where S = M 1 H S (self-adjoint in the good scalar product!)

23 Interpolation spaces 7 / 38 Then S self-adjoint such that i.e. H = (IR N, (u, v) H = u T Hv) M = (IR N, (u, v) M = u T Mv) (u, v) H = (u, S v) M = (S u, v) M (u, v) H = (u, Sv) M = (Su, v) M where S = M 1 H S (self-adjoint in the good scalar product!) { } Sx = µx Hx = µmx µ = δ 2 > 0 W s.t. M = W T W, H = W T 2 W, diagonal 0

24 Interpolation spaces 8 / 38 Λ = W 1 W Λ 1/2 = W 1 1/2 W

25 Interpolation spaces 8 / 38 Λ = W 1 W Λ 1/2 = W 1 1/2 W S = M 1 H = W 1 W T W T 2 W = W 1 WW 1 W = Λ 2

26 Interpolation spaces 8 / 38 Λ = W 1 W Λ 1/2 = W 1 1/2 W S = M 1 H = W 1 W T W T 2 W = W 1 WW 1 W = Λ 2 MΛ = W T WW 1 W T W T W = Λ T M = (u, Λv) M = (Λu, v) M

27 Interpolation spaces 8 / 38 Λ = W 1 W Λ 1/2 = W 1 1/2 W S = M 1 H = W 1 W T W T 2 W = W 1 WW 1 W = Λ 2 MΛ = W T WW 1 W T W T W = Λ T M = (u, Λv) M = (Λu, v) M and (Λ 1/2 u, Λ 1/2 u) M = (u, Λu) M

28 Interpolation spaces 9 / 38 [ { ) 1/2 } H, M ]ϑ = u IR N ; ((u, u) M + (u, S 1 ϑ u) M

29 Interpolation spaces 9 / 38 [ { ) 1/2 } H, M ]ϑ = u IR N ; ((u, u) M + (u, S 1 ϑ u) M [ { ) 1/2 } H, M ]1/2 = u IR N ; ((u, u) M + (u, Λu) M

30 Interpolation spaces 9 / 38 [ { ) 1/2 } H, M ]ϑ = u IR N ; ((u, u) M + (u, S 1 ϑ u) M ( v 2 ϑ,h = v 2 H ϑ,h = v T M + MS 1 ϑ) v ( H ϑ,h = M I + S 1 ϑ) ( = W T I + 2(1 ϑ)) W (Bessel)

31 Interpolation spaces [ { ) 1/2 } H, M ]ϑ = u IR N ; ((u, u) M + (u, S 1 ϑ u) M ( v 2 ϑ,h = v 2 H ϑ,h = v T M + MS 1 ϑ) v ( H ϑ,h = M I + S 1 ϑ) ( = W T I + 2(1 ϑ)) W (Bessel) Let us drop one of the M { } u IR N ; (u, S 1 ϑ u) 1/2 M ( v 2 ϑ = v 2 H ϑ = v T MS 1 ϑ) v ( H ϑ = M S 1 ϑ) ( = W T 2(1 ϑ)) W (Riesz) 9 / 38 H ϑ H ϑ,h

32 Interpolation spaces (duality) 10 / 38 M and H dual spaces of M and H [ ] H, M [M =, H ] ϑ 1 ϑ where H 1 ϑ,h H 1 ϑ,h H 1 ϑ

33 Interpolation spaces (duality) 10 / 38 M and H dual spaces of M and H [ ] H, M [M =, H ] ϑ 1 ϑ where H 1 ϑ,h H 1 ϑ,h H 1 ϑ H 1 ϑ = H 1 (HM 1 ) ϑ = W 1 2(ϑ 1) W T = H 1 ϑ

34 Interpolation spaces ( dimensional case) 11 / 38 X, Y two Hilbert spaces with X Y, X dense and continuously embedded in Y., X,, Y scalar product and X, Y the respective norms. (Riesz representation theory) S : X Y positive and self-adjoint with respect to, Y such that u, v X = u, S v Y. E = S 1/2 : X Y, X = D(E ) with u X u E := ( u 2 Y + E u 2 Y ) 1/2. u θ := ( u 2 Y + E 1 θ u 2 Y ) 1/2. The interpolation space of index θ [X, Y ] θ := D(E 1 θ ), 0 θ 1, with the inner-product u, v θ = u, v Y + u, E 1 θ v is a Hilbert space Y (Lions Magenes 1968). [X, Y ] 0 = X and [X, Y ] 1 = Y. If 0 < θ 1 < θ 2 < 1 then X [X, Y ] θ1 [X, Y ] θ2 Y.

35 Interpolation Theorem 12 / 38 Let X, Y Hilbert spaces X Y with X dense in Y, and with inclusion compact and continuous. Let X, Y satisfy similar properties. Let π L(X; X ) L(Y; Y). Then for all θ (0, 1), π L([X, Y] θ ; [X, Y] θ ).

36 Interpolation Theorem 12 / 38 Let X, Y Hilbert spaces X Y with X dense in Y, and with inclusion compact and continuous. Let X, Y satisfy similar properties. Let π L(X; X ) L(Y; Y). Then for all θ (0, 1), π L([X, Y] θ ; [X, Y] θ ). Let X X h and Y Y h (X h and Y h finite-dimensional spaces) i h : L (X h ; X) L(Y h ; Y) the continuous injection operator i h L([X h, Y h ] θ ; [X, Y] θ ).

37 Interpolation Theorem 12 / 38 Let X, Y Hilbert spaces X Y with X dense in Y, and with inclusion compact and continuous. Let X, Y satisfy similar properties. Let π L(X; X ) L(Y; Y). Then for all θ (0, 1), π L([X, Y] θ ; [X, Y] θ ). u h [X h, Y h ] θ, i h u h θ = u h θ C 1 u h θ,h.

38 Interpolation Theorem 12 / 38 Let X, Y Hilbert spaces X Y with X dense in Y, and with inclusion compact and continuous. Let X, Y satisfy similar properties. Let π L(X; X ) L(Y; Y). Then for all θ (0, 1), π L([X, Y] θ ; [X, Y] θ ). u h [X h, Y h ] θ, i h u h θ = u h θ C 1 u h θ,h. Assume now that there exists an interpolation operator I h such that I h : L(X; X h ) L(Y; Y h ) and I h u = u h for all u h X h. I h L([X, Y] θ ; [X h, Y h ] θ )

39 Interpolation Theorem 12 / 38 Let X, Y Hilbert spaces X Y with X dense in Y, and with inclusion compact and continuous. Let X, Y satisfy similar properties. Let π L(X; X ) L(Y; Y). Then for all θ (0, 1), π L([X, Y] θ ; [X, Y] θ ). u h [X h, Y h ] θ, i h u h θ = u h θ C 1 u h θ,h. u [X, Y] θ, I h u θ,h C 2 u θ.

40 Interpolation Theorem 12 / 38 Let X, Y Hilbert spaces X Y with X dense in Y, and with inclusion compact and continuous. Let X, Y satisfy similar properties. Let π L(X; X ) L(Y; Y). Then for all θ (0, 1), π L([X, Y] θ ; [X, Y] θ ). u h [X h, Y h ] θ, i h u h θ = u h θ C 1 u h θ,h. u [X, Y] θ, I h u θ,h C 2 u θ. Since [X h, Y h ] θ [X, Y] θ then 1 C 1 u h θ u h θ,h C 2 u h θ. i.e. u h θ u h θ,h

41 Interpolation spaces ( dimensional case) 13 / 38 Ω IR n open bounded with smooth boundary Γ and let α denote a multi-index of order m where m is a positive integer H m (Ω) = { u : D α u L 2 (Ω), α m } (H 0 (Ω) = L 2 (Ω)) H s (Ω) := [H m (Ω), H 0 (Ω)] 1 s/m

42 Interpolation spaces ( dimensional case) 13 / 38 Ω IR n open bounded with smooth boundary Γ and let α denote a multi-index of order m where m is a positive integer H m (Ω) = { u : D α u L 2 (Ω), α m } (H 0 (Ω) = L 2 (Ω)) H s (Ω) := [H m (Ω), H 0 (Ω)] 1 s/m H s 0 (Ω) completion of C 0 (Ω) in Hm (Ω), where s > 0.

43 Interpolation spaces ( dimensional case) Ω IR n open bounded with smooth boundary Γ and let α denote a multi-index of order m where m is a positive integer 13 / 38 H m (Ω) = { u : D α u L 2 (Ω), α m } (H 0 (Ω) = L 2 (Ω)) For 0 s 2 < s 1 and k integer H s (Ω) := [H m (Ω), H 0 (Ω)] 1 s/m [H s 1 0 (Ω), Hs 2 0 (Ω)] θ = H (1 θ)s 1+θs 2 0 (Ω) if (1 θ)s 1 + θs 2 k + 1/2 [H s 1 0 (Ω), Hs 2 0 (Ω)] θ = H k+1/2 00 (Ω) H k+1/2 0 if (1 θ)s 1 + θs 2 = k + 1/2 If (1 θ)s 1 + θs 2 = 1/2 H s (Ω) = (H s 0(Ω)) s > 0 [H s 1 (Ω), H s 2 (Ω)] θ = ( H 1/2 00 (Ω) ).

44 Finite-element example 14 / 38 H 1/2 00 (Ω) = [H1 0 (Ω), L 2 (Ω)] 1/2. Let X h H 1 0 (Ω), Y h L 2 (Ω). Let {φ i } 1 i n X h be a spanning set for Y h and let L k IR n n denote the Grammian matrices corresponding to the, H k 0 (Ω) -inner product (H0 (Ω) = L 2 (Ω)): (L k ) ij = φ i, φ j H k 0 (Ω). ( ) H = L 1, M = L 0 and H 1/2,h = L 0 I + (L 1 0 L 1) 1/2 Moreover, we have ( 1/2 H 1/2,h H 1/2 = L 0 L 1 0 1) L (Riesz) (Bessel)

45 Interpolation theorem for FEM 15 / 38 Let the assumptions of Interpolation Theorem hold with (X, Y) replaced by ( (X h, Y h ) defined ) above. Let H θ,h = L 0 I + (L 1 0 L 1) 1 θ, H θ = L 0 (L 1 0 L 1) 1 θ. Then there exist constants c, C independent of n such that c u h [X,Y]θ u Hθ,h C u h [X,Y]θ, c u h [X,Y]θ u Hθ C u h [X,Y]θ, for all u h [X h, Y h ] θ and with θ (0, 1), X = L 2 (Ω), and Y = H 1 0 (Ω)

46 Few examples 16 / H θ 1 δ(x 0.5) 1 θ = 1 1 θ = θ =

47 Few examples 17 / H θ 1 δ(x 0.5) 1 θ =

48 Few examples 18 / H θ 1 δ(x 0.5), 1 θ = 0.5 Riesz Bessel

49 Evaluation of H θ z 19 / 38 Generalised Lanczos HV k = MV k T k + β k+1 Mv k+1 e T k, (T k tridiagonal). VT k MV k = I k

50 Evaluation of H θ z 19 / 38 Generalised Lanczos HV k = MV k T k + β k+1 Mv k+1 e T k, (T k tridiagonal). v 0 = z H θ z MV k T 1 θ k e 1 z M and H θ,h z MV k (I k + T 1 θ k )e 1 z M. VT k MV k = I k

51 Evaluation of H θ z 19 / 38 Generalised Lanczos HV k = MV k T k + β k+1 Mv k+1 e T k, (T k tridiagonal). v 0 = z H θ z MV k T 1 θ k e 1 z M and H θ,h z MV k (I k + T 1 θ k )e 1 z M. VT k MV k = I k

52 Evaluation of H θ z 19 / 38 Generalised Lanczos HV k = MV k T k + β k+1 Mv k+1 e T k, (T k tridiagonal). v 0 = z H θ z MV k T 1 θ k e 1 z M and H θ,h z MV k (I k + T 1 θ k )e 1 z M. v 0 = M 1 z H 1 θ z V kt θ 1 k e 1 z M 1 and H 1 θ,h z V k(i k + T 1 θ k ) 1 e 1 z M 1. VT k MV k = I k Alternative: N. Hale, and N. J. Higham and L. N. Trefethen, SIAM J. Numer. Anal.

53 Preconditioners for the Steklov-Poincaré operator 20 / 38 Let Ω be an open subset of IR d with boundary Ω and consider the model problem { u = f in Ω, u = 0 on Ω. Given a partition of Ω into two subdomains Ω Ω 1 Ω 2 with common boundary Γ this problem can be equivalently written as { u1 = f in Ω 1, u 1 = 0 on Ω 1 \ Γ, { u2 = f in Ω 2, u 2 = 0 on Ω 2 \ Γ, with the interface conditions { u1 = u 2 u 1 n 1 = u 2 n 2 on Γ

54 Preconditioners for the Steklov-Poincaré operator 21 / 38 Given λ 1, λ 2 H 1/2 00 (Γ), ψ 1, ψ 2 denote the harmonic extensions of λ 1, λ 2 respectively into Ω 1, Ω 2, i.e., for i = 1, 2, ψ i satisfy ψ i = 0 in Ω i, ψ i = λ i on Γ, ψ i = 0 on Ω i \ Γ. The Steklov-Poincaré operator S : H 1/2 00 (Γ) H 1/2 (Γ) S λ 1, λ 2 H 1/2 (Γ) = ψ 1, ψ 2 L 2 (Ω) =: s(λ 1, λ 2 ). c 1 λ 2 H 1/2 (Γ) s(λ, λ) c 2 λ 2 H 1/2 (Γ).

55 Preconditioners for the Steklov-Poincaré operator 22 / 38 (i) (ii) (iii) { { { The resulting solution is u {1} i = f in Ω i, u {1} i = 0 on Ω i, S λ = u{1} 1 n 1 u{1} 2 on Γ, n 2 u {2} i = 0 in Ω i, u {2} i = λ on Ω i. u Ωi = u {1} i + u {2} i.

56 An other problem 23 / 38 { ν u + b u = f in Ω, u = 0 on Ω.

57 Discrete Formulation 24 / 38 V h = V h,r := { w C 0 (Ω) : w t P k t T h } H 1 (Ω) be a finite-dimensional space of piecewise polynomial functions defined on some subdivision T h of Ω into simplices t of maximum diameter h. Let further VI h, Vh B Vh satisfy VI h VB H Vh where VI h = { w V h : w Ω = 0 }. Let X h H0 1 (Γ) denote the space spanned by the restriction of the basis functions of VI h to the internal boundary Γ.

58 Discrete Formulation 25 / 38 (i) A II,i u {1} i = f I,i, (ii) Su B = f B A T IB,1 u{1} 1 A T IB,2 u{2} 2, (iii) A II,i u {2} i = A T IB,1 u B A T IB,2 u B, where S is the Schur complement corresponding to the boundary nodes S = S 1 + S 2, S i = A BB,i A T IB,i A 1 II,i A IB,i. The resulting solution is (u I,1, u I,2, u B ) where u I,i = u {1} i + u {2} i.

59 H 1/2 00 -preconditioners 26 / 38 Let X h = span {φ i, 1 i m} be defined as above and let (L k ) ij = φ i, φ j H k 0 (Γ) for k = 0, 1. Let Then for all λ IR m \ {0} H 1/2 := L 0 (L 1 0 L 1) 1/2. κ 1 with κ 1, κ 2 independent of h. λt Sλ λ T H 1/2 λ κ 2

60 Discrete DD and Preconditioning 27 / 38 ( ) AII A P = IB 0 P S with A II = νl II + N II where L II is the direct sum of Laplacians assembled on each subdomain and N II is the direct sum of the convection operator b assembled also on each subdomain. With P S we denote the approximation of H 1/2 00 by a vector or of H 1/2 by a vector. Then we use FGMRES.

61 Green functions on wirebasket 28 / 38 Steklov-Poincaré Neumann-Neumann

62 Green functions on wirebasket 28 / 38 Steklov-Poincaré H 1/2

63 Numerical results: Poisson equation 29 / 38 Linear Quadratic #dom n m H 1/2,h H 1/2 Ĥ 1/2 H 1/2,h H 1/2 Ĥ 1/2 45, , , , , , , , ,050,625 14, FGMRES iterations for model problem.

64 Numerical results: an other problem 30 / 38 Linear Quadratic #dom n m ν = 1 ν = 0.1 ν = 0.01 ν = 1 ν = 0.1 ν = , , , , , , , , ,050,625 14, FGMRES iterations for 2nd model problem

65 where n k (x) is the normal to m k at x. A., Kourounis, Loghin IMA J. Num. Anal / 38 Laplace-Beltrami We can extend everything to an interface that is the union of manifolds m k by using the Lapace-Beltrami operator and interpolating between L 2 (Γ) and H Ω 1 (Γ) with the norm using H 1 0 (m k) with ( K 1/2 u H 1 Ω (Γ) = u k 2 H Ω k)) 1 (m. v 2 H 1 0 (m k) = k=1 m k k Γ v 2 ds(mk ) where k Γ denote the tangential gradient of v with respect to m k k Γ v(x) := v(x) n k(x)(n k (x) v(x)),

66 Other Domains: CRYSTAL A., Kourounis, Loghin IMA J. Num. Anal / 38

67 Other Domains: CRYSTAL 32 / 38

68 Other Domains: CRYSTAL 32 / 38 (L, M) 1/2 (L, I ) 1/2 n N = 16 N = 64 N = 256 N = 16 N = 64 N = , ,521, Iterations for the crystal problem with and without the mass matrix.

69 Other Domains: BRAIN 33 / 38

70 Other Domains: BRAIN 33 / 38

71 Other Domains: BRAIN 33 / 38 N = 1024 N = 2048 N = 4096 n = n = n b = n b = it = 22 it = 22 n b = n b = it = 22 it = 23 n b = n b = it = 24 it = 23 Results for reaction-diffusion PDE on Brain (N number of subdomains, n b number of nodes in interface, it FGMRES iteration number, and θ = 0.7 ).

72 Reaction-Diffusion Systems 34 / 38 Rodrigue Kammogne, D. Loghin Proceed. DD 2012 Rodrigue Kammogne, D. Loghin Tech. Rep. in preparation Ω IR 2 { D u + Mu = f in Ω u = 0 on Ω

73 Reaction-Diffusion Systems 34 / 38 Rodrigue Kammogne, D. Loghin Proceed. DD 2012 Rodrigue Kammogne, D. Loghin Tech. Rep. in preparation Ω IR 2 { D u + Mu = f in Ω u = 0 on Ω u = [ u1 u 2 ], M = [ ] α( x, y) β 1 (x, y), f = β 2 (x, y) α 2 (x, y) [ f1 f 2 ] [ ] d1 0, D = 0 d 2 (SPD). f L 2 (Ω) and M satisfies 0 < γ min < ξt Mξ ξ T ξ ξ IR2 \ {} ; and M < γ max.

74 Reaction-Diffusion Systems 34 / 38 Rodrigue Kammogne, D. Loghin Proceed. DD 2012 Rodrigue Kammogne, D. Loghin Tech. Rep. in preparation Ω IR 2 { D u + Mu = f in Ω u = 0 on Ω { 1 if x α 1 = 2 + y 2 < 1/4 100 otherwise { 0.1 if x β 1 = 2 + y 2 < 1/4 1 otherwise { 100 if x ; α 2 = 2 + y 2 < 1/4 1 otherwise { 1 if x β 2 = 2 + y 2 < 1/4 0.1 otherwise

75 Reaction-Diffusion Systems 34 / 38 Rodrigue Kammogne, D. Loghin Proceed. DD 2012 Rodrigue Kammogne, D. Loghin Tech. Rep. in preparation Ω IR 2 { D u + Mu = f in Ω u = 0 on Ω d 1 = 1, d 2 = 0.1 domains= size=

76 Reaction-Diffusion Systems 34 / 38 Rodrigue Kammogne, D. Loghin Proceed. DDRodrigue 2012 Kammogne and Daniel Lo Rodrigue Kammogne, D. Loghin Tech. Rep. in preparation Fig. 1 Solution for Problem 3.

77 Other applications: Mathematical Finance 35 / 38 L.Silvestre Communications on Pure and Applied Mathematics 2007 Luis A. Caffarelli, Sandro Salsa, Luis Silvestre Invent. math Let X t be an α-stable Levy process such that X 0 = x for some point x IR n.

78 Other applications: Mathematical Finance 35 / 38 L.Silvestre Communications on Pure and Applied Mathematics 2007 Luis A. Caffarelli, Sandro Salsa, Luis Silvestre Invent. math Let X t be an α-stable Levy process such that X 0 = x for some point x IR n. Let τ be the optimal stopping time that maximises u(x) = sup E[φ(X t ) : τ < + ] τ

79 Other applications: Mathematical Finance 35 / 38 L.Silvestre Communications on Pure and Applied Mathematics 2007 Luis A. Caffarelli, Sandro Salsa, Luis Silvestre Invent. math Let X t be an α-stable Levy process such that X 0 = x for some point x IR n. Let τ be the optimal stopping time that maximises u(x) = sup E[φ(X t ) : τ < + ] τ u(x) φ(x) in IR n, ( ) s u 0 in IR n, ( ) s u(x) = 0 for those x s.t. u(x) > φ(x) lim x + u(x) = 0.

80 Other applications: Mathematical Finance 35 / 38 L.Silvestre Communications on Pure and Applied Mathematics 2007 Luis A. Caffarelli, Sandro Salsa, Luis Silvestre Invent. math Let X t be an α-stable Levy process such that X 0 = x for some point x IR n. Let τ be the optimal stopping time that maximises u(x) = sup E[e λτ φ(x t )] τ u(x) φ(x) in IR n, λu + ( ) s u 0 in IR n, λu + ( ) s u(x) = 0 for those x s.t. u(x) > φ(x) lim x + u(x) = 0.

81 Other applications: Quasi-Geostrophic Equation 36 / 38 L.Silvestre Ann. I. H. Poincare (2010) L. Caffarelli, L. Silvestre, Comm. Partial Differential Equations (2007) L. Caffarelli, A. Vasseur, Ann. of Math., (2012) P. Constantin, J. Wu, Ann. I. H. Poincare Anal. Non Lin. (2008), (2009) P. Constantin, J. Wu, SIAM J. Math. Anal. (1999)

82 Other applications: Quasi-Geostrophic Equation 36 / 38 θ : IR 2 [0, + ) IR t θ(x, t) + w θ(x, t) + ( ) α/2 θ(x, t) = 0, θ(x, 0) = θ 0 and w = (R 2 θ, R 1 θ) where R i are the Riesz transforms (y i x i )θ(y) R i θ(x) = cpv IR 2 y x 3 dy.

83 Summary 37 / 38 Interpolation spaces produce dense matrices i.e. non-local operators BUT we can compute everything using SPARSE LINEAR ALGEBRA

84 Summary 37 / 38 Interpolation spaces produce dense matrices i.e. non-local operators BUT we can compute everything using SPARSE LINEAR ALGEBRA Interpolation spaces are not only useful in DD

85 Summary 37 / 38 Interpolation spaces produce dense matrices i.e. non-local operators BUT we can compute everything using SPARSE LINEAR ALGEBRA Interpolation spaces are not only useful in DD Link with integro-differential operator such as Riemann-Liouville fractional derivative (M.Riesz 1938,1949).

86 Summary 37 / 38 Interpolation spaces produce dense matrices i.e. non-local operators BUT we can compute everything using SPARSE LINEAR ALGEBRA Interpolation spaces are not only useful in DD Link with integro-differential operator such as Riemann-Liouville fractional derivative (M.Riesz 1938,1949). In modelling complex phenomena the use of non-local operators is a new promising subject attracting increasing attention. Other areas of application that are worth to mention include: BEM and image processing (filtering):

87 38 / 38 top-left: original top-right:noised bottom-left: min { Ω u(x) dx + 1/50 Ω (u 0(x) u(x)) 2 dx } Pascal Getreuer (2007) bottom-right: min { u 2 1/2 + 1/50 } Ω (u 0(x) u(x)) 2 dx

Matrix square root and interpolation spaces

Matrix square root and interpolation spaces Mario Arioli and Daniel Loghin m.arioli@rl.ac.uk STFC-Rutherford Appleton Laboratory, and University of Birmingham Sparse Days,CERFACS, Toulouse, 2008 p.1/30