APPROXIMATING PDE s IN L 1

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1 APPROXIMATING PDE s IN L 1 Veselin Dobrev Jean-Luc Guermond Bojan Popov Department of Mathematics Texas A&M University NONLINEAR APPROXIMATION TECHNIQUES USING L 1 Texas A&M May 16-18, 2008

2 Outline 1

3 Outline 1 2

4 Outline 1 2 3

5 Outline motivation Viscosity solutions Why L 1 for viscosity solutions? The theory Numerics 1 2 3

6 Motivation motivation Viscosity solutions Why L 1 for viscosity solutions? The theory Numerics Advection dominated problem: u + β u ɛ 2 u = f ; u Ω = 0 Approximation on coarse mesh β L h ɛ. Standard discretization Spurious oscillations

7 Fri May 21 16:08: Galerkin/Finite Elements 1 PLOT 1 motivation Viscosity solutions Why L 1 for viscosity solutions? The theory Numerics PLOT 1 Y Axis 0 ZZ Axis 0 1 X Axis Mesh Galerkin Y AxisP 1 X Y x u u = 0, u(0, y) = 0, u(1, y) = 1, y u(x, 0) = 0, y u(x, 1) = 0. 1

8 Viscosity solutions of PDE s motivation Viscosity solutions Why L 1 for viscosity solutions? The theory Numerics In Ω R d consider: αu + f (u) = g; u Ω = u 0. Ill-posed in standard sense, but notion of viscosity solution applies (Bardos, Leroux, and Nédélec (1979), Kružkov...)

9 Viscosity solutions of PDE s motivation Viscosity solutions Why L 1 for viscosity solutions? The theory Numerics The viscosity solution can be interpreted as the limit ɛ 0 of αu + f (u) ɛ 2 u = g; u Ω = u 0.

10 Viscosity solutions of PDE s motivation Viscosity solutions Why L 1 for viscosity solutions? The theory Numerics The viscosity solution can be interpreted as the limit ɛ 0 of αu + f (u) ɛ 2 u = g; u Ω = u 0. One tries to solve the ill-posed pb when ɛ f L h.

11 Viscosity solutions of PDE s motivation Viscosity solutions Why L 1 for viscosity solutions? The theory Numerics The viscosity solution can be interpreted as the limit ɛ 0 of αu + f (u) ɛ 2 u = g; u Ω = u 0. One tries to solve the ill-posed pb when ɛ f L h. A good scheme should be able to approximate the viscosity solution!

12 Why L 1 for viscosity solutions? motivation Viscosity solutions Why L 1 for viscosity solutions? The theory Numerics Let u visc be viscosity solution of v visc + v visc = f, in (0, 1), v visc(0) = 0, v visc (1) = 0.

13 Why L 1 for viscosity solutions? motivation Viscosity solutions Why L 1 for viscosity solutions? The theory Numerics Let u visc be viscosity solution of v visc + v visc = f, in (0, 1), v visc(0) = 0, v visc (1) = 0. Let f be the zero extension of f on (, + ). Let ũ solve v + v = f u visc (1)δ(1), in (, + ), and v( ) = 0, v(+ ) = 0.

14 Why L 1 for viscosity solutions? motivation Viscosity solutions Why L 1 for viscosity solutions? The theory Numerics Let u visc be viscosity solution of v visc + v visc = f, in (0, 1), v visc(0) = 0, v visc (1) = 0. Let f be the zero extension of f on (, + ). Let ũ solve v + v = f u visc (1)δ(1), in (, + ), and v( ) = 0, v(+ ) = 0. Lemma ũ [0,1) = u visc.

15 Why L 1 for viscosity solutions? motivation Viscosity solutions Why L 1 for viscosity solutions? The theory Numerics Let u visc be viscosity solution of v visc + v visc = f, in (0, 1), v visc(0) = 0, v visc (1) = 0. Let f be the zero extension of f on (, + ). Let ũ solve v + v = f u visc (1)δ(1), in (, + ), and v( ) = 0, v(+ ) = 0. Lemma ũ [0,1) = u visc. ũ solves well-posed pb with RHS bounded measure ( L 1 (R))

16 Viscosity solutions: Theory motivation Viscosity solutions Why L 1 for viscosity solutions? The theory Numerics Consider the 1D pb: u + β(x)u = f in (0, 1), u(0) = 0, u(1) = 0.

17 Viscosity solutions: Theory motivation Viscosity solutions Why L 1 for viscosity solutions? The theory Numerics Consider the 1D pb: u + β(x)u = f in (0, 1), u(0) = 0, u(1) = 0. Assume 0 < inf β, sup β < 1 and f L 1. Use piecewise linear polynomials. Use midpoint rule to approximate the integrals.

18 Viscosity solutions: Theory motivation Viscosity solutions Why L 1 for viscosity solutions? The theory Numerics Consider the 1D pb: u + β(x)u = f in (0, 1), u(0) = 0, u(1) = 0. Assume 0 < inf β, sup β < 1 and f L 1. Use piecewise linear polynomials. Use midpoint rule to approximate the integrals. Theorem (Guermond-Popov (2007)) The best L 1 -approximation converges in W 1,1 loc ([0, 1)) to the viscosity solution, and the boundary layer is always located in the last mesh cell.

19 Ill-posed transport in 2D motivation Viscosity solutions Why L 1 for viscosity solutions? The theory Numerics P 1 elements + preconditioned interior point method. { u + x u + 2 y u = 1 u Ω = 0.

20 Theory Discretization Numerical tests Outline 1 2 3

21 Theory Discretization Numerical tests Theory Let Ω be a smooth domain in R 2. Consider H(x, u, u) = 0, u Ω = α. H is convex ( ξ c(1 + H(x, v, ξ) + v )). H is Lipschitz.

22 Theory Discretization Numerical tests Theory Assume that Ω, f and α are smooth enough for a viscosity solution to exist u W 1, (Ω), u BV (Ω) and u p-semi-concave. Definition v is p-semi-concave if there is a concave function v c W 1, and a function w W 2,p so that v = v c + w.

23 Theory Discretization Numerical tests Discretization {T h } h>0 regular mesh family. X α h h = {v h C 0 (Ω); v h K P k, K T h, v h Ω = α h }. Take p > 2 (p > 1 in one space dimension). Define J h (v h ) = H(, v h, v h ) L 1 (Ω) + h 2 p F F i h F ({ n v h } + ) p.

24 Theory Discretization Numerical tests Discretization {T h } h>0 regular mesh family. X α h h = {v h C 0 (Ω); v h K P k, K T h, v h Ω = α h }. Take p > 2 (p > 1 in one space dimension). Define J h (v h ) = H(, v h, v h ) L 1 (Ω) + h 2 p F F i h F ({ n v h } + ) p. Compute u h X α h h s.t., J(u h ) = min J h (v h ). v h X α h h

25 Theory Discretization Numerical tests Discretization: convergence Theorem (Guermond-Popov (2008) 1D, and (200?) 2D) u h converges to the viscosity solution strongly in W 1,1 (Ω) (the result holds for arbitrary polynomial degree provided an additional volume entropy is added)

26 Theory Discretization Numerical tests Discretization: algorithms 1D Optimal complexity algorithm developed in Guermond-Marpeau-Popov (2008). Algorithm ũ h Theorem (Guermond-Popov (200?) 1D) ũ h converges to the viscosity solution strongly in W 1,1 (Ω) and complexity of the algorithm is O(1/h).

27 Theory Discretization Numerical tests Discretization: algorithms 1D Optimal complexity algorithm developed in Guermond-Marpeau-Popov (2008). Algorithm ũ h Theorem (Guermond-Popov (200?) 1D) ũ h converges to the viscosity solution strongly in W 1,1 (Ω) and complexity of the algorithm is O(1/h). The algorithm is similar to the fast marching and fast sweeping methods (instead of choosing maximal solution, choose the entropy-minimizing solution).

28 Theory Discretization Numerical tests 1D convergence tests Ω = [0, 1], u + 1 π (u ) 2 = f (x), u(0) = u(1) = 1.

29 Theory Discretization Numerical tests 1D convergence tests Ω = [0, 1], u + 1 π (u ) 2 = f (x), u(0) = u(1) = 1. Data: f (x) = cos(πx) + sin 2 (πx),

30 Theory Discretization Numerical tests 1D convergence tests Ω = [0, 1], u + 1 π (u ) 2 = f (x), u(0) = u(1) = 1. Data: f (x) = cos(πx) + sin 2 (πx), Exact solution: u(x) = cos(πx).

31 Tue May 16 14:10: D convergence tests Theory Discretization Numerical tests Tue May 16 14:11: PLOT PLOT 0 0 Y Axis Y Axis X Axis X Axis Odd # points (9,19, 39) Even # points (10, 20, 40)

32 1D convergence Mon May 15 17:31: tests Theory Discretization Numerical tests Mon May 15 17:31: Odd # of points W11 error, order 1 Max error, order 1 L1 error, order Even # of points W11 error, order 1 Max error, order 2 L1 error, order Odd # points Even # points

33 Theory Discretization Numerical tests 1D convergence tests (u ) 2 + 3u x 2 x = 0, in ( 0.95, 0.95)

34 Theory Discretization Numerical tests 1D convergence tests (u ) 2 + 3u x 2 x = 0, in ( 0.95, 0.95) Boundary condition set so that the viscosity solution u visc is u visc (x) = 1 2 x x 3 2, i.e. u(±0.95) = u visc (±0.95)

35 Theory Discretization Numerical tests 1D convergence tests (u ) 2 + 3u x 2 x = 0, in ( 0.95, 0.95) Boundary condition set so that the viscosity solution u visc is u visc (x) = 1 2 x x 3 2, i.e. u(±0.95) = u visc (±0.95) u visc is in W 1, (Ω) W 2,p (Ω) for any p [1, 2)

36 Theory Discretization Numerical tests 1D convergence tests (u ) 2 + 3u x 2 x = 0, in ( 0.95, 0.95) Boundary condition set so that the viscosity solution u visc is u visc (x) = 1 2 x x 3 2, i.e. u(±0.95) = u visc (±0.95) u visc is in W 1, (Ω) W 2,p (Ω) for any p [1, 2) u visc is p-semi-concave for any p [1, 2)

37 Tue Jul 10 14:19: Theory Discretization Numerical tests Tue Jul 10 15:25: D convergence tests PLOT Y Axis slope W11 Max L X Axis

38 Theory Discretization Numerical tests Eikonal equation 2D Ω = [0, 1] 2, u = 1, u Ω = 0.

39 Theory Discretization Numerical tests Eikonal equation 2D Ω = [0, 1] 2, u = 1, u Ω = 0. Exact solution: u(x) = dist(x, Ω)

40 Tue May 16 16:00: Theory Discretization Numerical tests Tue May 16 15:48: Eikonal equation 2D: P 1 finite elements PLOT 1 PLOT Y Axis Z Axis X Axis iso-lines X Axis Z Y graph Y Axis 1 X

41 Fri Jan 18 17:12: Theory Discretization Numerical tests Eikonal equation 2D: P 1 finite elements PLOT Fri Jan 18 17:13: PLOT Figure: Pentagon: Aligned unstructured mesh (left); Non-aligned unstructured mesh (right).

42 Fri Jan 25 19:15: Theory Discretization Numerical tests Eikonal equation 2D: P 1 finite elements 1 PLOT Fri Jan 25 19:01: PLOT Y Axis 0 Y Axis X Axis X Axis Figure: L-shaped domain: Unstructured mesh (left); Iso-lines of approximate minimizer (right).

43 Theory Discretization Numerical tests Algorithms in 2D Computation done using Newton+regularization (similar flavor to interior point).

44 Theory Discretization Numerical tests Algorithms in 2D Computation done using Newton+regularization (similar flavor to interior point). Conjecture Fast sweeping/marching techniques produce L 1 almost minimizers.

45 Outline Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results 1 2 3

46 Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results L 1 splines (J. Lavery et al. ARO and NCSU) Problem: Given a set of data in R d, (d = 1, 2), construct a spline approximation 1 From J. Lavery, Computer Aided Geometric Design, 23 (2006) 276:296

47 Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results L 1 splines (J. Lavery et al. ARO and NCSU) Problem: Given a set of data in R d, (d = 1, 2), construct a spline approximation Solution: Minimize the L p -norm of second derivative. 1 From J. Lavery, Computer Aided Geometric Design, 23 (2006) 276:296

48 Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results L 1 splines (J. Lavery et al. ARO and NCSU) Problem: Given a set of data in R d, (d = 1, 2), construct a spline approximation Solution: Minimize the L p -norm of second derivative. Cubic L 1 spline Cubic L 2 spline (Least-Squares) 1 1 From J. Lavery, Computer Aided Geometric Design, 23 (2006) 276:296

49 Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results L 1 splines (J. Lavery et al. ARO and NCSU) Cubic L 1 spline Cubic L 2 spline (Least-Squares) 2 2 From J. Lavery, Computer Aided Geometric Design, 18 (2001) 321:343

50 Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results L 1 splines (J. Lavery et al. ARO and NCSU) Cubic L 1 spline Cubic L 2 spline (Least-Squares) 3 (16 16) 3 From J. Lavery, Computer Aided Geometric Design, 22 (2005) 818:837

51 Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results L 1 splines (J. Lavery et al. ARO and NCSU) Data ( ) Cubic L 1 spline (16 16) Cubic L 2 spline (Least-Squares) 4 4 Courtesy from J. Lavery et al.

52 Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results L 1 splines (J. Lavery et al. ARO and NCSU) Observations: L 1 splines are less oscillatory than L 2 splines.

53 Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results L 1 splines (J. Lavery et al. ARO and NCSU) Observations: L 1 splines are less oscillatory than L 2 splines. L 1 splines can compress data better than L 2 splines.

54 C 0 Finite element approach Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results Ω R 2 T h mesh composed of triangles/quadrangles. V h set of vertices of T h. F i h set of interior edges. Data given at the vertices of the mesh, (d v ) v Vh

55 C 0 Finite element approach Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results Ω R 2 T h mesh composed of triangles/quadrangles. V h set of vertices of T h. F i h set of interior edges. Data given at the vertices of the mesh, (d v ) v Vh Problem Given data at the nodes of T h, construct a smooth non-oscillatory representation of the data, (d v ) v Vh.

56 C 0 Finite element approach Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results Define manifold X h = {φ C 0 (Ω); φ K P 3 /Q 3, K T h, φ(v) = d v, v V h }.

57 C 0 Finite element approach Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results Define manifold X h = {φ C 0 (Ω); φ K P 3 /Q 3, K T h, φ(v) = d v, v V h }. Let α be a real positive number. Define functional J h (u) = ( u xx + 2 u xy + u yy ) + α K T h K F Fh i F { n u}

58 C 0 Finite element approach Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results Define manifold X h = {φ C 0 (Ω); φ K P 3 /Q 3, K T h, φ(v) = d v, v V h }. Let α be a real positive number. Define functional J h (u) = ( u xx + 2 u xy + u yy ) + α K T h K F Fh i F { n u} Problem u = argmin w Xh J h (w)

59 Implementation details Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results α = 0 or small u is P 1 /Q 1 interpolant. α large C 1 smoothness. In practice we take α = 3. Quadrature must be rich enough (9 to 12 points). Interior point method

60 Numerical results: 16x16, 3D view Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results

61 Numerical results: 16x16, 3D view Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results

62 Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results Numerical results: 16x16, level sets L2/L1

63 Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results Numerical results: Lavery s test case; Q1

64 Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results Numerical results: Lavery s test case; L1

65 Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results Numerical results: Lavery s test case; L2

details Numerical results Numerical results: Austin

66 Steady Hamilton Jacobi equations in L1 (Ω) Motivation: L1 splines C 0 Finite element approach Implementation details Numerical results Numerical results: Austin West DEM Dobrev/Guermond/Popov APPROXIMATING PDE s IN L1

67 Steady Hamilton Jacobi equations in L1 (Ω) Motivation: L1 splines C 0 Finite element approach Implementation details Numerical results Numerical results: Barton creek; Q1 Dobrev/Guermond/Popov APPROXIMATING PDE s IN L1

68 Steady Hamilton Jacobi equations in L1 (Ω) Motivation: L1 splines C 0 Finite element approach Implementation details Numerical results Numerical results: Barton creek; L1 Dobrev/Guermond/Popov APPROXIMATING PDE s IN L1

69 Steady Hamilton Jacobi equations in L1 (Ω) Motivation: L1 splines C 0 Finite element approach Implementation details Numerical results Numerical results: Barton creek; L2 Dobrev/Guermond/Popov APPROXIMATING PDE s IN L1

70 Numerical results: The peppers Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results

71 Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results Numerical results: The peppers; zoomed

72 Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results Numerical results: The peppers; zoomed/photoshop CS3

73 Motivation: L 1 splines C 0 Finite element approach Implementation details Numerical results Numerical results: The peppers; zoomed/l1

74 Steady Hamilton Jacobi equations in L1 (Ω) Motivation: L1 splines C 0 Finite element approach Implementation details Numerical results Numerical results: Barton creek THE END Dobrev/Guermond/Popov APPROXIMATING PDE s IN L1

An efficient filtered scheme for some first order Hamilton-Jacobi-Bellman equat

An efficient filtered scheme for some first order Hamilton-Jacobi-Bellman equations joint work with Olivier Bokanowski 2 and Maurizio Falcone SAPIENZA - Università di Roma 2 Laboratoire Jacques-Louis Lions,