Motivation
A Line Search Multigrid Method
Numerical Results
What’s New?
Multilevel/Multigrid Methods in “Siam Conference on
Optimization 2008”
One plenary talk: “Multiscale Optimization”
One dedicated session
Four sessions on PDE-Based problems which are mainly
handled by multigrid methods
Almost 20 invited and contributed talks
Wen, Goldfarb Line Search MG
Motivation
A Line Search Multigrid Method
Numerical Results
Statement of Problem
Previous Work
Statement of Problem
Consider solving problem min
u∈V
F(u)
Infinite-dimensional problem: F is a functional
F has a closely related representations {f
h
} on a
hierarchical discretization levels h.
Discretize-then-Optimize scheme: min f
h
Solutions of {min f
h
} might have similar structures
Figure: Solution Structure of F(u) =
R
Ω
p
1 + k∇u(x)k
2
dx
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.05
0.1
0.15
0.2
0.25
MG Solution level =4
(a) Level 4
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.05
0.1
0.15
0.2
0.25
MG Solution level =5
(b) Level 5
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.05
0.1
0.15
0.2
0.25
MG Solution level =6
(c) Level 6
Wen, Goldfarb Line Search MG
Motivation
A Line Search Multigrid Method
Numerical Results
Statement of Problem
Previous Work
Sources of Problems
Applications in nonlinear PDEs, image processing:
min
u∈U
F(u) =
Z
Ω
L(∇u, u, x) dx
PDE-constrained optimization: optimal control problems
and inverse Problems. Example: finding a local volatility
σ(t, x) such that the prices C(T , S) from the Black-Scholes
PDEs match the observed prices on the market.
min J(C, σ) :=
X
I
|C(S
i
, T
i
) − z(S
i
, T
i
)|
2
+ αJ
r
(σ)
s.t. ∂
τ
C −
σ
2
K
2
2
∂
2
KK
C + (r − q)K ∂
K
C + qC = 0,
C(0, K ) = (S − K )
+
, K > 0, τ ∈ (0, +∞),
where J
r
(σ) is the regularization term.
Wen, Goldfarb Line Search MG
Motivation
A Line Search Multigrid Method
Numerical Results
Statement of Problem
Previous Work
Mesh-refinement Method
Finest Level Problem
Prolongation
Finer Level Problem
Prolongation
... .. ... .
Coarser Level Problem
Prolongation
Coarsest Level Problem
Finest Level Problem
Restriction
Prolongation
Finer Level Problem
Restriction
Prolongation
... .. ... .
Coarser Level Problem
Restriction
Prolongation
Coarsest Level Problem
Wen, Goldfarb Line Search MG
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