APPROXIMATING PDE’s IN L1
Veselin Dobrev
Jean-Luc Guermond
Bojan Popov
Department of Mathematics
Texas A&M University
NONLINEAR APPROXIMATION TECHNIQUES USING L1
Texas A&M
May 16-18, 2008
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Outline
1
Ill-posed transport equations
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Outline
1
Ill-posed transport equations
2
Steady Hamilton Jacobi equations in L1 (Ω)
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Outline
1
Ill-posed transport equations
2
Steady Hamilton Jacobi equations in L1 (Ω)
3
Terrain data reconstruction
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
motivation
Viscosity solutions
Why L1 for viscosity solutions?
The theory
Numerics
Outline
1
Ill-posed transport equations
2
Steady Hamilton Jacobi equations in L1 (Ω)
3
Terrain data reconstruction
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
motivation
Viscosity solutions
Why L1 for viscosity solutions?
The theory
Numerics
Motivation
Advection dominated problem:
u + β·∇u − ∇2 u = f ;
u|∂Ω = 0
Approximation on coarse mesh kβkL∞ h .
Standard discretization ⇒ Spurious oscillations
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
motivation
Viscosity solutions
Why L1 for viscosity solutions?
The theory
Numerics
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Fri May 21 16:08:40 1999
Terrain data reconstruction
PLOT
Galerkin/Finite Elements
PLOT
1
1
Y−Axis
Z−Axis
1
0
−1
−1
0
0
0
1
X−Axis
Y−AxisP
Mesh
Galerkin
1

2

 ∂x u − 0.002∇ u = 0,
u(0, y ) = 0, u(1, y ) = 1,


∂y u(x, 0) = 0, ∂y u(x, 1) = 0.
Z
X
Dobrev/Guermond/Popov
Y
APPROXIMATING PDE’s IN L1
1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
motivation
Viscosity solutions
Why L1 for viscosity solutions?
The theory
Numerics
Viscosity solutions of PDE’s
In Ω ∈ Rd consider:
αu + ∇·f (u) = g ;
u|∂Ω = u0 .
Ill-posed in standard sense, but notion of viscosity solution applies
(Bardos, Leroux, and Nédélec (1979), Kružkov ...)
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
motivation
Viscosity solutions
Why L1 for viscosity solutions?
The theory
Numerics
Viscosity solutions of PDE’s
The viscosity solution can be interpreted as the limit→0 of
αu + ∇·f (u) − ∇2 u = g ;
Dobrev/Guermond/Popov
u|∂Ω = u0 .
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
motivation
Viscosity solutions
Why L1 for viscosity solutions?
The theory
Numerics
Viscosity solutions of PDE’s
The viscosity solution can be interpreted as the limit→0 of
αu + ∇·f (u) − ∇2 u = g ;
u|∂Ω = u0 .
One tries to solve the ill-posed pb when kf 0 kL∞ h.
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
motivation
Viscosity solutions
Why L1 for viscosity solutions?
The theory
Numerics
Viscosity solutions of PDE’s
The viscosity solution can be interpreted as the limit→0 of
αu + ∇·f (u) − ∇2 u = g ;
u|∂Ω = u0 .
One tries to solve the ill-posed pb when kf 0 kL∞ h.
A good scheme should be able to
approximate the viscosity solution!
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
motivation
Viscosity solutions
Why L1 for viscosity solutions?
The theory
Numerics
Why L1 for viscosity solutions?
Let uvisc be viscosity solution of
0
vvisc + vvisc
= f , in (0, 1), vvisc (0) = 0,
Dobrev/Guermond/Popov
vvisc (1) = 0.
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
motivation
Viscosity solutions
Why L1 for viscosity solutions?
The theory
Numerics
Why L1 for viscosity solutions?
Let uvisc be viscosity solution of
0
vvisc + vvisc
= f , in (0, 1), vvisc (0) = 0,
vvisc (1) = 0.
Let f˜ be the zero extension of f on (−∞, +∞).
Let ũ solve v + v 0 = f˜ − uvisc (1)δ(1), in (−∞, +∞), and
v (−∞) = 0, v (+∞) = 0.
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
motivation
Viscosity solutions
Why L1 for viscosity solutions?
The theory
Numerics
Why L1 for viscosity solutions?
Let uvisc be viscosity solution of
0
vvisc + vvisc
= f , in (0, 1), vvisc (0) = 0,
vvisc (1) = 0.
Let f˜ be the zero extension of f on (−∞, +∞).
Let ũ solve v + v 0 = f˜ − uvisc (1)δ(1), in (−∞, +∞), and
v (−∞) = 0, v (+∞) = 0.
Lemma
ũ|[0,1) = uvisc .
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
motivation
Viscosity solutions
Why L1 for viscosity solutions?
The theory
Numerics
Why L1 for viscosity solutions?
Let uvisc be viscosity solution of
0
vvisc + vvisc
= f , in (0, 1), vvisc (0) = 0,
vvisc (1) = 0.
Let f˜ be the zero extension of f on (−∞, +∞).
Let ũ solve v + v 0 = f˜ − uvisc (1)δ(1), in (−∞, +∞), and
v (−∞) = 0, v (+∞) = 0.
Lemma
ũ|[0,1) = uvisc .
ũ solves well-posed pb with RHS bounded measure (≈ L1 (R))
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
motivation
Viscosity solutions
Why L1 for viscosity solutions?
The theory
Numerics
Viscosity solutions: Theory
Consider the 1D pb:
u + β(x)u 0 = f in (0, 1),
Dobrev/Guermond/Popov
u(0) = 0,
u(1) = 0.
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
motivation
Viscosity solutions
Why L1 for viscosity solutions?
The theory
Numerics
Viscosity solutions: Theory
Consider the 1D pb:
u + β(x)u 0 = f in (0, 1),
u(0) = 0,
u(1) = 0.
Assume 0 < inf β, sup β 0 < 1 and f ∈ L1 .
Use piecewise linear polynomials.
Use midpoint rule to approximate the integrals.
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
motivation
Viscosity solutions
Why L1 for viscosity solutions?
The theory
Numerics
Viscosity solutions: Theory
Consider the 1D pb:
u + β(x)u 0 = f in (0, 1),
u(0) = 0,
u(1) = 0.
Assume 0 < inf β, sup β 0 < 1 and f ∈ L1 .
Use piecewise linear polynomials.
Use midpoint rule to approximate the integrals.
Theorem (Guermond-Popov (2007))
1,1
The best L1 -approximation converges in Wloc
([0, 1)) to the
viscosity solution, and the boundary layer is always located in the
last mesh cell.
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
motivation
Viscosity solutions
Why L1 for viscosity solutions?
The theory
Numerics
Ill-posed transport in 2D
P1 elements + preconditioned interior point method.
(
√
u + ∂x u + 2∂y u = 1
u|∂Ω = 0.
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Theory
Discretization
Numerical tests
Outline
1
Ill-posed transport equations
2
Steady Hamilton Jacobi equations in L1 (Ω)
3
Terrain data reconstruction
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Theory
Discretization
Numerical tests
Theory
Let Ω be a smooth domain in R2 .
Consider
H(x, u, ∇u) = 0,
u|∂Ω = α.
H is convex (kξk ≤ c(1 + |H(x, v , ξ)| + |v |)).
H is Lipschitz.
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
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 vc ∈ W 1,∞ and
a function w ∈ W 2,p so that v = vc + w .
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Theory
Discretization
Numerical tests
Discretization
{Th }h>0 regular mesh family.
Xhαh = {vh ∈ C 0 (Ω); vh |K ∈ Pk , ∀K ∈ Th , vh |∂Ω = αh }.
Take p > 2 (p > 1 in one space dimension).
Define
X Z
2−p
Jh (vh ) = kH(·, vh , ∇vh )kL1 (Ω) + h
({−∂n vh }+ )p .
F ∈Fhi
Dobrev/Guermond/Popov
F
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Theory
Discretization
Numerical tests
Discretization
{Th }h>0 regular mesh family.
Xhαh = {vh ∈ C 0 (Ω); vh |K ∈ Pk , ∀K ∈ Th , vh |∂Ω = αh }.
Take p > 2 (p > 1 in one space dimension).
Define
X Z
2−p
Jh (vh ) = kH(·, vh , ∇vh )kL1 (Ω) + h
({−∂n vh }+ )p .
F ∈Fhi
Compute uh ∈ Xhαh s.t.,
Dobrev/Guermond/Popov
F
J(uh ) = minα Jh (vh ).
vh ∈Xh
h
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Theory
Discretization
Numerical tests
Discretization: convergence
Theorem (Guermond-Popov (2008) 1D, and (200?) 2D)
uh 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)
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Theory
Discretization
Numerical tests
Discretization: algorithms 1D
Optimal complexity algorithm developed in
Guermond-Marpeau-Popov (2008).
Algorithm → ũh
Theorem (Guermond-Popov (200?) 1D)
ũh converges to the viscosity solution strongly in W 1,1 (Ω) and
complexity of the algorithm is O(1/h).
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Theory
Discretization
Numerical tests
Discretization: algorithms 1D
Optimal complexity algorithm developed in
Guermond-Marpeau-Popov (2008).
Algorithm → ũh
Theorem (Guermond-Popov (200?) 1D)
ũh 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).
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Theory
Discretization
Numerical tests
1D convergence tests
Ω = [0, 1],
u + π1 (u 0 )2 = f (x),
Dobrev/Guermond/Popov
u(0) = u(1) = −1.
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Theory
Discretization
Numerical tests
1D convergence tests
Ω = [0, 1],
u + π1 (u 0 )2 = f (x),
u(0) = u(1) = −1.
Data: f (x) = −| cos(πx)| + sin2 (πx),
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Theory
Discretization
Numerical tests
1D convergence tests
Ω = [0, 1],
u + π1 (u 0 )2 = f (x),
u(0) = u(1) = −1.
Data: f (x) = −| cos(πx)| + sin2 (πx),
Exact solution: u(x) = −| cos(πx)|.
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
1D convergence tests
Tue May 16 14:10:37 2006
Theory
Discretization
Numerical tests
Tue May 16 14:11:00 2006
PLOT
PLOT
Y−Axis
0
Y−Axis
0
−1
−1
0
1
X−Axis
Odd # points (9,19, 39)
Dobrev/Guermond/Popov
0
1
X−Axis
Even # points (10, 20, 40)
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
1D convergence tests
Mon May 15 17:31:30 2006
Theory
Discretization
Numerical tests
Mon May 15 17:31:14 2006
1.0×100
1.0×10−1
1.0×10−1
1.0×10−2
1.0×10−2
1.0×10−3
1.0×10−3
1.0×10−4
Odd # of points
W11−error, order 1
Max−error, order 1
L1−error, order 2
1.0×10−4
1.0×10−5
Even # of points
W11−error, order 1
Max−error, order 2
L1−error, order 2
1.0×10−5
0.001
0.01
0.1
1
Odd # points
Dobrev/Guermond/Popov
1.0×10−6
0.001
0.01
Even # points
APPROXIMATING PDE’s IN L1
0.1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Theory
Discretization
Numerical tests
1D convergence tests
(u 0 )2 + 3u + 21 x 2 − |x| = 0, in (−0.95, 0.95)
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Theory
Discretization
Numerical tests
1D convergence tests
(u 0 )2 + 3u + 21 x 2 − |x| = 0, in (−0.95, 0.95)
Boundary condition set so that the viscosity solution uvisc is
3
uvisc (x) = − 21 x 2 + 32 |x| 2 , i.e. u(±0.95) = uvisc (±0.95)
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Theory
Discretization
Numerical tests
1D convergence tests
(u 0 )2 + 3u + 21 x 2 − |x| = 0, in (−0.95, 0.95)
Boundary condition set so that the viscosity solution uvisc is
3
uvisc (x) = − 21 x 2 + 32 |x| 2 , i.e. u(±0.95) = uvisc (±0.95)
uvisc is in W 1,∞ (Ω) ∩ W 2,p (Ω) for any p ∈ [1, 2)
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Theory
Discretization
Numerical tests
1D convergence tests
(u 0 )2 + 3u + 21 x 2 − |x| = 0, in (−0.95, 0.95)
Boundary condition set so that the viscosity solution uvisc is
3
uvisc (x) = − 21 x 2 + 32 |x| 2 , i.e. u(±0.95) = uvisc (±0.95)
uvisc is in W 1,∞ (Ω) ∩ W 2,p (Ω) for any p ∈ [1, 2)
uvisc is p-semi-concave for any p ∈ [1, 2)
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Tue Jul 10 14:19:37 2007
Ill-posed transport equations
Theory
Tue Jul 10 15:25:14 2007
Steady Hamilton Jacobi equations in L1 (Ω)
Discretization
Terrain data reconstruction
Numerical tests
1D convergence tests
PLOT
1.0×100
0.2
Y−Axis
1.0×10−1
1.0×10−2
0.1
slope
W11
Max
L1
1.0×10−3
1.0×10−4
0
−0.95
0
1.0×10−5
0.95 0.0001
0.001
0.01
X−Axis
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
0.1
1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Theory
Discretization
Numerical tests
Eikonal equation 2D
Ω = [0, 1]2 ,
k∇uk = 1,
Dobrev/Guermond/Popov
u|∂Ω = 0.
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Theory
Discretization
Numerical tests
Eikonal equation 2D
Ω = [0, 1]2 ,
k∇uk = 1,
u|∂Ω = 0.
Exact solution: u(x) = dist(x, ∂Ω)
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Theory
Tue May 16 16:00:06
2006
Steady Hamilton Jacobi equations in L1 (Ω)
Discretization
Terrain data reconstruction
Numerical tests
Eikonal equation 2D: P1 finite elements
Tue May 16 15:48:16 2006
PLOT
PLOT
0
1
1
−0.1
−0.2
Y−Axis
Z−Axis
−0.3
−0.4
−0.5
−0.6
−0.7
−0.8
X−
0
0
Ax
0
1
−0.9
1
is
0
1
0
X−Axis
Y−Axis
Z
iso-lines
graph
Y
X
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
−1
Fri Jan 18 17:12:09 2008
Fri Jan 18 17:13:09 2008
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Theory
Discretization
Numerical tests
Eikonal equation
2D: P1 finite elements PLOT
PLOT
1
1
0.904
−10
0.909
0
−1
−1
0
0
1
1
−1
0
0
1
1
Figure: Pentagon: Aligned unstructured mesh (left); Non-aligned
unstructured mesh (right).
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Fri Jan 25 19:15:22 2008
Fri Jan 25 19:01:14 2008
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Theory
Discretization
Numerical tests
Eikonal equation 2D: P1 finite elements
PLOT
PLOT
1
Y−Axis
Y−Axis
1
0
0
−1
−1
−1
0
X−Axis
1
−1
0
X−Axis
Figure: L-shaped domain: Unstructured mesh (left); Iso-lines of
approximate minimizer (right).
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Theory
Discretization
Numerical tests
Algorithms in 2D
Computation done using Newton+regularization (similar
flavor to interior point).
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Theory
Discretization
Numerical tests
Algorithms in 2D
Computation done using Newton+regularization (similar
flavor to interior point).
Conjecture
Fast sweeping/marching techniques produce L1 almost minimizers.
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
Outline
1
Ill-posed transport equations
2
Steady Hamilton Jacobi equations in L1 (Ω)
3
Terrain data reconstruction
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
L1 splines (J. Lavery et al. ARO and NCSU)
Problem: Given a set of data in Rd , (d = 1, 2), construct a spline
approximation
1
From J. Lavery, Computer Aided Geometric Design, 23 (2006) 276:296
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
L1 splines (J. Lavery et al. ARO and NCSU)
Problem: Given a set of data in Rd , (d = 1, 2), construct a spline
approximation
Solution: Minimize the Lp -norm of second derivative.
1
From J. Lavery, Computer Aided Geometric Design, 23 (2006) 276:296
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
L1 splines (J. Lavery et al. ARO and NCSU)
Problem: Given a set of data in Rd , (d = 1, 2), construct a spline
approximation
Solution: Minimize the Lp -norm of second derivative.
Cubic L1 spline
1
Cubic L2 spline (Least-Squares)1
From J. Lavery, Computer Aided Geometric Design, 23 (2006) 276:296
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
L1 splines (J. Lavery et al. ARO and NCSU)
Cubic L1 spline
2
Cubic L2 spline (Least-Squares)2
From J. Lavery, Computer Aided Geometric Design, 18 (2001) 321:343
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
L1 splines (J. Lavery et al. ARO and NCSU)
Cubic L1 spline
3
Cubic L2 spline (Least-Squares)
3
(16 × 16)
From J. Lavery, Computer Aided Geometric Design, 22 (2005) 818:837
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
L1 splines (J. Lavery et al. ARO and NCSU)
Data (128 × 128)
Cubic L1 spline (16 × 16)
4
Cubic L2 spline (Least-Squares)4
Courtesy from J. Lavery et al.
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
L1 splines (J. Lavery et al. ARO and NCSU)
Observations:
L1 splines are less oscillatory than L2 splines.
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
L1 splines (J. Lavery et al. ARO and NCSU)
Observations:
L1 splines are less oscillatory than L2 splines.
L1 splines can compress data better than L2 splines.
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
C 0 Finite element approach
Ω ⊂ R2
Th mesh composed of triangles/quadrangles.
Vh set of vertices of Th .
Fhi set of interior edges.
Data given at the vertices of the mesh, (dv )v ∈Vh
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
C 0 Finite element approach
Ω ⊂ R2
Th mesh composed of triangles/quadrangles.
Vh set of vertices of Th .
Fhi set of interior edges.
Data given at the vertices of the mesh, (dv )v ∈Vh
Problem
Given data at the nodes of Th , construct a smooth non-oscillatory
representation of the data, (dv )v ∈Vh .
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
C 0 Finite element approach
Define manifold
Xh = {φ ∈ C 0 (Ω); φ|K ∈ P3 /Q3 , ∀K ∈ Th , φ(v ) = dv , ∀v ∈ Vh }.
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
C 0 Finite element approach
Define manifold
Xh = {φ ∈ C 0 (Ω); φ|K ∈ P3 /Q3 , ∀K ∈ Th , φ(v ) = dv , ∀v ∈ Vh }.
Let α be a real positive number. Define functional
X Z
X Z
Jh (u) =
(|uxx | + 2|uxy | + |uyy |) + α
|{∂n u}|
K ∈Th
K
Dobrev/Guermond/Popov
F ∈Fhi
APPROXIMATING PDE’s IN L1
F
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
C 0 Finite element approach
Define manifold
Xh = {φ ∈ C 0 (Ω); φ|K ∈ P3 /Q3 , ∀K ∈ Th , φ(v ) = dv , ∀v ∈ Vh }.
Let α be a real positive number. Define functional
X Z
X Z
Jh (u) =
(|uxx | + 2|uxy | + |uyy |) + α
|{∂n u}|
K ∈Th
K
F ∈Fhi
Problem
u = argminw ∈Xh Jh (w )
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
F
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
Implementation details
α = 0 or small −→ u is P1 /Q1 interpolant.
α large −→ C 1 smoothness.
In practice we take α = 3.
Quadrature must be rich enough (9 to 12 points).
Interior point method
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
Numerical results: 16x16, 3D view
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
Numerical results: 16x16, 3D view
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
Numerical results: 16x16, level sets L2/L1
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
Numerical results: Lavery’s test case; Q1
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
Numerical results: Lavery’s test case; L1
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
Numerical results: Lavery’s test case; L2
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
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
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
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
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
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
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
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
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
Numerical results: The peppers
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
Numerical results: The peppers; zoomed
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
Numerical results: The peppers; zoomed/Photoshop CS3
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
Motivation: L1 splines
C 0 Finite element approach
Implementation details
Numerical results
Numerical results: The peppers; zoomed/L1
Dobrev/Guermond/Popov
APPROXIMATING PDE’s IN L1
Ill-posed transport equations
Steady Hamilton Jacobi equations in L1 (Ω)
Terrain data reconstruction
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