Transcript Level Set Methods For Inverse Obstacle Problems

Nonlocal
Geometric
Motion
Martin Burger
University of California
Los Angeles
University Linz, Austria
Geometric Motion
Joint work with:
- Marc Droske, Duisburg  INV
- Bo Su, MPI Leipzig  GBM
- Günther Bauer, Linz
Motivated by GBM I Discussions with:
- Stan Osher, UCLA
- Omar Lakkis, IACM Crete
- Xiaobing Feng, Tennessee
Nonlocal Geometric Motion
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Geometric Motion
1980s and 1990s: Understanding and
computing local geometric motion (in particular
mean curvature flow and related flows)
Osher & Sethian 1987
Brakke 1980
Evans & Spruck 1991-1994
Almgren, Taylor, Wang 1993
Chen, Giga, Goto 1993
Ley 2001
Osher & Shu 1989
Sussman, Smereka & Osher
1993
Deckelnick & Dziuk 1999-2002
Chambolle 2002
Nonlocal Geometric Motion
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Geometric Motion
But which geometric motion
is local?
Nonlocal Geometric Motion
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Geometric Motion
In typical applications, part of the
velocity is determined by a nonlocal (physical) effect!
Nonlocal Geometric Motion
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Geometric Motion
Nonlocal parts:
 Surface Diffusion
 Bulk Diffusion
 Heat Transfer
 Bulk Elasticity
 Velocity from Shape Optimization /
Inverse Problem
 Curvature Dependent Energies
Nonlocal Geometric Motion
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Geometric Motion
> 2000: Understanding and
computing nonlocal geometric
motion?
Nonlocal Geometric Motion
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Surface Diffusion
Motion law for normal velocity
:
... Diffusion coefficient
... Mean curvature
4-th order PDE, no maximum principle
Nonlocal Geometric Motion
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Surface Diffusion
„Challenge: Devise proofs and
computations for motion by
surface diffusion, including
through topological changes
and other singularities“
Jean Taylor, Some Mathematical Challenges in
Material Science, Bull AMS, 2002
Nonlocal Geometric Motion
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Surface Diffusion
Analysis for graph-like infinite curve:
Baras, Duchon, Robert 1983
Existence for smooth curves:
Elliott, Garcke 1996
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Surface Diffusion
Computation for Graphs/Parametrizations:
Bänsch, Morin, Nocchetto 2003
Computation for Level Sets:
Chopp, Sethian 2000 (explicit FD)
Smereka 2002 (semi-implicit FD)
Nonlocal Geometric Motion
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Surface Diffusion
Level Set Formulation
Weak Form
Nonlocal Geometric Motion
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Surface Diffusion
Finite Element Simulation
Nonlocal Geometric Motion
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Surface Diffusion
But
...
That‘s not enough!
Nonlocal Geometric Motion
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Surface Diffusion
We really want:
... Bulk energy term
... Deposition term
Nonlocal Geometric Motion
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Surface Diffusion
Introduce a new variable
„Chemical potential concentration“, also
used by Droske, Rumpf (2003) for
Willmore flow
Weak form:
Nonlocal Geometric Motion
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Surface Diffusion
Semi-implicit approximation, piecewise
linear finite elements:
Solution of discretized problem with
multigrid-preconditioned GMRES
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Surface Diffusion
Still Open:
-Efficient computation of signed distance
functions and velocity extension on
triangular grids
-Coupling with bulk effects.
-Analysis
Nonlocal Geometric Motion
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Surface Diffusion
Future: couple with elasticity
Butterfly shape transition
Colin, Grilhé,
Junqua, 1998
Nonlocal Geometric Motion
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Surface Diffusion
Coupled with bulk diffusion and elasticity:
Quantum Dots
PbSe/PbEuTe
Springholz et. Al. 2001
Nonlocal Geometric Motion
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Surface Diffusion
Bulk term
given by strain energy density
with misfit strain
Elastic strain and displacement
from
with standard relation
Nonlocal Geometric Motion
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Surface Diffusion
SiGe Quantum Dots:
height small compared to width (10nm/150nm)
graph representation
Bauer et. Al., 1999
Nonlocal Geometric Motion
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Surface Diffusion
Anisotropic Surface Diffusion for graphs:
4th-order PDE for height
Bulk is given by
Nonlocal Geometric Motion
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Surface Diffusion
Single initial nucleus, no deposition
Nonlocal Geometric Motion
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Surface Diffusion
Random initial surface, random deposition
Nonlocal Geometric Motion
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Surface Diffusion
Random initial surface, random deposition
Nonlocal Geometric Motion
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Surface Diffusion
Random (rough) surface, no deposition
Nonlocal Geometric Motion
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Surface Diffusion
Later time
Nonlocal Geometric Motion
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Surface Diffusion
3D
Nonlocal Geometric Motion
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Surface Diffusion
Open Problems:
- No analysis at all for the model (only
weak results for the equilibrium state)
- Bulk term is like a third order term. So
far treated explicitely  extremely small
time steps. Semi-implicit method
coupling bulk solver and interface
solvers in future ?
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Solidification
... temperature, satisfies
... latent heat
... indicator function of
Nonlocal Geometric Motion
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Solidification
Isotactic Polypropylene
Eder, 1997
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Solidification
Analysis: 1D or based on regularity for
single object
For
(nonequilibrium Stefan problem)
weak solution concept (varifolds):
Soner 1995
Nonlocal Geometric Motion
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Solidification
Polymer Crystals:
, (Eder 1997)
, smooth
Analysis:
Friedman & Velazquez (2002): smooth single
crystal, short time existence, 3D
mb (2003): multiple crystals, 1D
Clain (2003): similar chemical attack model, 1D
mb, Su (2004): multiple crystals, 2D, 3D
Nonlocal Geometric Motion
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Solidification
Existence proof: use fixed point argument
Velocity positive, but discontinuous.
Need
-solutions for level set equation
(weaker than viscosity-solution) Chen, Su 2000
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Solidification
Simulation with same technique:
Nonlocal Geometric Motion
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Solidification
Simulation with stochastic nucleation:
Nonlocal Geometric Motion
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Inverse Problems
Miminize least-square functional
level set method.
Evaluating
involves solving PDE
dependent on
.
Choose
from
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Inverse Problems
Arising problems similar to other nonlocal
geometric motions.
Want to have existence and uniqueness
of solution regularizing properties!
Similar computational issues
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Conclusions
Analysis:



Need new solution concepts
Must deal with strong nonlinearities
Lose maximum principles (no global level set
methods)
Numerics:

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Need efficient coupling with elliptic/parabolic
PDE solvers
Need efficient solvers for equations on implicit interfaces Bertalmio,Cheng,Osher,Sapiro
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