Introduction to Computer Aided Geometric Design (CAGD) for inverse problems.

1. Σ YSTEMS
Introduction to Computer Aided Geometric Design
(CAGD) for Inverse Problems
Dimitrios Papadopoulos
Delta Pi Systems
Thessaloniki, Greece, 3 August 2014

2. Σ YSTEMS
Overview
◮ Problem formulation
◮ A minimization problem with PDEs as constraints
◮ Computational Grids
◮ Approximation and Interpolation
◮ B´ezier and B-spline curves
◮ Least squares approximation
◮ Smoothing and regularization
◮ Parametrization - finding a knot sequence
◮ Tensor product patches

3. Σ YSTEMS
Problem Formulation [Schlegel(2004)]
◮ Dynamic optimization problem
min
u(t),p,tf
Φ = ΦM (y(tf )) +
tf
t0
ΦL(y(t), u(t), p, t)dt (1)
subject to the Differential-Algebraic Equations (DAE)
B(y(t), u(t), p, t) ˙y = f(y(t), u(t), p, t), t ∈ I (2)
0 = l( ˙y(t0), y(y0)), (3)
0 ≥ h(y(t), u(t), p, t), t ∈ I (4)
0 ≥ e(y(tf ) (5)
where
◮ y(t) ∈ Rny
state variables
u(t) ∈ Rnu
control variables
p ∈ Rnp
time-invariant parameters

4. Σ YSTEMS
Problem Formulation [Tr¨oltzsch(2005)]
◮ inverse problem
min
y(x)
J = J(u(x), y(x), p, x) = [S(ω) − ˜S(ω)]2
dω (6)
subject to
F(
∂u
∂x1
, . . . ,
∂u
∂xnx
, u(x), y(x), p, x) = 0 (7)
where
◮ x ∈ Rnx
independent variables
u(x) ∈ Rnu
state variables
y(x) ∈ Rny
control variables
p ∈ Rnp
invariant parameters

5. Σ YSTEMS
Computational Grids
◮ State variables are discretized on the finite element grid
◮ Control variables are on both finite element and optimization grid
◮ Interpolation of the control variables from finite element grid to
optimization grid
◮ Finite element grid is much finer than optimization grid so
less variables enter the optimization procedure

6. Σ YSTEMS
Interpolation
◮ Bernstein polynomials (form a partition of unity)
Bn
i (t) =
n
i
ti
(1 − t)n−i
(8)
or recursively:
Bn
i (t) = (1 − t)Bn−1
i (t) + tBn−1
i−1 (t) (9)
with
B0
0(t) ≡ 1
Bn
j (t) ≡ 0 forj /∈ {0, . . . , n}
◮ B´ezier curve (affine invariant) formed by control points bi
b(t) =
n
i=0
biBn
i (t) (10)

7. Σ YSTEMS
Interpolation (cntd)
◮ B-spline basis functions
N0
i (t) =
1 if τi−1 ≤ t < τi
0 else
, (11)
Nn
j (t) =
t − τj−1
τj+n−1 − τj−1
Nn−1
j (t) +
τj+n − t
τj+n − τj
Nn−1
j+1 (t), n > 1 (12)
◮ B-spline curve formed by control points ci
b(t) =
L
j=0
cjNn
j (t) (13)

8. Σ YSTEMS
Least Squares Approximation
◮ Given m + 1 data points d0, . . . , dm, each di being associated with a
parameter value ti, find a polynomial curve b(t) of a given degree n such
that
minimize f(c1, . . . , cL) =
m
i=0
di −
L
j=0
cjNn
j (ti)
2
(14)
which leads to the following system (in case of L2 norm)
NT
NC = NT
D (15)
◮ The data points are the control variables (concentration) discretized on the
finite element grid and the control points cj are on the optimization grid.

9. Σ YSTEMS
Smoothing - Regularization
◮ minimize f(c1, . . . , cL) =
P
i=0 di −
L
j=0 cjNn
j (ti)
2
+ αS[b(ti)]
◮ second differences
c0 − 2c1 + c2 = 0
... (16)
cL−2 − 2cL−1 + cL = 0
◮ smoothing equations
(1 − α)N
αS
C =
(1 − α)D
0
(17)
◮ α allows a weighting between initial data fitting and smoothing

10. Σ YSTEMS
Parametrization - Finding a knot sequence
◮ uniform or equidistant
(ti − ti−1)/(ti+1 − ti) = 1 (18)
◮ chord length
∆ti
∆ti+1
=
||∆di||
||∆di+1||
(19)
◮ cetripetal
∆ti
∆ti+1
=
||∆di||
||∆di+1||
1/2
(20)
◮ Natural end conditions
¨b(t0) = 0, ¨b(tm) = 0 (21)

11. Σ YSTEMS
Tensor Product Patches [Farin(2002)]
◮ B´ezier surface
bm,n
(u, v) =
m
i=0
n
j=0
bi,jBm
i (u)Bn
j (v) (22)
◮ B-spline surface
bm,n
(u, v) =
m
i=0
n
j=0
ci,jNm
i (u)Nn
j (v) (23)
◮ A surface with the topology of a sphere is not representable as a tensor
product surface, without degeneracies

12. Σ YSTEMS
Parametrization of surfaces
◮ Project the data points into a plane, i.e. drop the z-coordinate and fit into
the unit square.
◮ For each point find a good initial guess, and
perform Newton iteration until convergence is reached.
◮ Can be generalized to 4-dimensional hypersurface for the case of a
3-dimensional problem.
◮ Triangulation of data points based on graph theory.
◮ End conditions involve partial derivatives of the surface.

13. Σ YSTEMS
Synopsis - Outlook
Synopsis
◮ Inverse problem as a minimization problem with PDEs as constaints
◮ Optimization grid much smaller than the finite element grid
◮ Interpolation with B´ezier and B-spline curves
◮ Smoothing
◮ Different parametrization methods
◮ Natural end conditions
Outlook
◮ For a 3D space-time, a 5-dimensional hypersurface would be needed,
◮ tensor product approach is easily generalized,
◮ a projection and Newton iteration can also be generalized.
◮ NURBS curves & patches [Piegl and Tiller(1997)]

14. Σ YSTEMS
Bibliography
Farin, G., 2002. Curves and Surfaces for CAGD: A Practical Guide.
Academic Press.
Piegl, L. A., Tiller, W., 1997. The NURBS Book. Springer Science and
Business Media.
Schlegel, M., 2004. Adaptive discretization methods for the efficient
solution of dynamic optimization problems. Ph.D. thesis, RWTH Aachen.
Tr¨oltzsch, F., 2005. Optimale Steuerung partieller Differentialgleichungen:
Theorie, Verfahren und Anwendungen. Vieweg.