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Level Sets for Inverse Problems and Optimization I
Martin Burger
Johannes Kepler University LinzSFB Numerical-Symbolic-Geometric Scientific ComputingRadon Institute for Computational & Applied Mathematics
Level Set Methods for Inverse Problems
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Collaborations
Benjamin Hackl (Linz)
Wolfgang Ring, Michael Hintermüller (Graz)
Level Set Methods for Inverse Problems
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Outline Introduction
Shape Gradient Methods
Framework for Level Set Methods
Examples
Levenberg-Marquardt Methods
Level Set Methods for Inverse Problems
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IntroductionMany applications deal with the reconstruction and optimization of geometries (shapes, topologies), e.g.:
Identification of piecewise constant parameters in PDEs Inverse obstacle scattering Inclusion / cavity detection Topology optimization Image segmentation
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Introduction In such applications, there is no natural a-
priori information on shapes or topological structures of the solution (number of connected components, star-shapedness, convexity, ...)
Flexible representations of the shapes needed!
Level Set Methods for Inverse Problems
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Level Set Methods Osher & Sethian, JCP 1987 Sethian, Cambridge Univ. Press 1999 Osher & Fedkiw, Springer, 2002
Based on dynamic implicit shape representation
with continuous level-set function
Level Set Methods for Inverse Problems
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Level Set Methods Change of the front is translated to a change of the level set function
Automated treatment of topology change
Level Set Methods for Inverse Problems
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Level Set Flows
Geometric flow of the level sets of can be translated into nonlinear differential equation for („level set equation“)
Appropriate solution concept: Viscosity solutions (Crandall, Lions 1981-83,Crandall-Ishii-Lions 1991)
Level Set Methods for Inverse Problems
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Level Set Methods
Geometric primitives can expressed via derivatives of the level set function
Normal
Mean curvature
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Shape Optimization The typical setup in shape optimization
and reconstruction is given by
where is a class of shapes (eventually with additional constraints).
For formulation of optimality conditons and solution, derivatives are needed
Level Set Methods for Inverse Problems
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Shape Optimization Calculus on shapes by the speed method: Natural variations are normal velocities
Level Set Methods for Inverse Problems
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Shape Derivatives Derivatives can be computed by the level set
methodExample:
Formal computation:
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Shape Derivatives Formal application of co-area formula
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Shape Optimization Framework to construct gradient-based methods for shape design problems (MB, Interfaces and Free Boundaries 2004)
After choice of Hilbert space norm for normal velocities, solve variational problem
Level Set Methods for Inverse Problems
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Shape Optimization
Equivalent equation for velocity Vn
Update by motion of shape in normal direction for a small time , new shape
Expansion
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Shape Optimization From definition (with )
Descent method, time step can be chosen by standard optimization rules (Armijo-Goldstein)
Gradient method independent of parametrization, can change topology, but but only by splitting Level set method used to perform update step
Level Set Methods for Inverse Problems
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Inverse Obstacle Problem Identify obstacle from partial measurements f
of solution on
Level Set Methods for Inverse Problems
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Inverse Obstacle Problem Shape derivative
Adjoint method
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Inverse Obstacle Problem Shape derivative
Simplest choice of velocity space
Velocity
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Example: 5% noise
- Norm - Norm
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Example: 5% noise
Residual
Level Set Methods for Inverse Problems
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Example: 5% noise
- error
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Inverse Obstacle Problem Weaker Sobolev space norm H-1/2 for velocity
yields faster method
Easy to realize (Neumann traces, DtN map)
For a related obstacle problem (different energy functional), complete convergence analysis of level set method with H-1/2 norm (MB-Matevosyan 2006)
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Tomography-Type Problem Identify obstacle from boundary
measurements z of solution on
Level Set Methods for Inverse Problems
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Tomography, Single Measurement
- Norm - Norm
Level Set Methods for Inverse Problems
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Tomography
Residual
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Tomography
- error
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Fast Methods Framework can also be used to construct Newton-type methods for shape design problems (Hintermüller-Ring 2004, MB 2004)
If shape Hessian is positive definite, choose
For inverse obstacle problems, Levenberg- Marquardt level set methods can be constructed in the same way
Level Set Methods for Inverse Problems
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Levenberg-Marquardt Method Inverse problems with least-squares functional
Choose variable scalar product
Variational characterization
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Levenberg-Marquardt Method Example 1:
where , and denotes the indicator function of .
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Levenberg-Marquardt Method 1% noise, =10-7, Iterations 10 and 15
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Levenberg-Marquardt Method 1% noise, =10-7, Iterations 20 and 25
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Levenberg-Marquardt Method 4% noise, =10-7, Iterations 10 and 20
Level Set Methods for Inverse Problems
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Levenberg-Marquardt Method 4% noise, =10-7, Iterations 30 and 40
Level Set Methods for Inverse Problems
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Levenberg-Marquardt Method Residual and L1-error
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Levenberg-Marquardt Method Example 2:
where and denotes the indicator function of .
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Levenberg-Marquardt Method No noise
Iterations2,4,6,8
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Levenberg-Marquardt Method Residual and L1-error
Level Set Methods for Inverse Problems
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Levenberg-Marquardt Method Residual and L1-error
Level Set Methods for Inverse Problems
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Levenberg-Marquardt Method 0.1 % noise
Iterations
5,10,20,25
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Levenberg-Marquardt Method 1% noise 2% noise
3% noise 4% noise
Level Set Methods for Inverse Problems
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Download and Contact
Papers and Talks:
www.indmath.uni-linz.ac.at/people/burger
e-mail: martin.burger@jku.at
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