1 Level Sets for Inverse Problems and Optimization I Martin Burger Johannes Kepler University Linz SFB Numerical-Symbolic-Geometric Scientific Computing Radon Institute for Computational & Applied Mathematics

Level Set Methods for Inverse Problems San Antonio, January Collaborations Benjamin Hackl (Linz) Wolfgang Ring, Michael Hintermüller (Graz)

Level Set Methods for Inverse Problems San Antonio, January Outline Introduction Shape Gradient Methods Framework for Level Set Methods Examples Levenberg-Marquardt Methods

Level Set Methods for Inverse Problems San Antonio, January Introduction Many 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

Level Set Methods for Inverse Problems San Antonio, January 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 San Antonio, January 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 San Antonio, January 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 San Antonio, January 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 ,Crandall- Ishii-Lions 1991)

Level Set Methods for Inverse Problems San Antonio, January Level Set Methods Geometric primitives can expressed via derivatives of the level set function Normal Mean curvature

Level Set Methods for Inverse Problems San Antonio, January 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 San Antonio, January Shape Optimization Calculus on shapes by the speed method: Natural variations are normal velocities

Level Set Methods for Inverse Problems San Antonio, January Shape Derivatives Derivatives can be computed by the level set method Example: Formal computation:

Level Set Methods for Inverse Problems San Antonio, January Shape Derivatives Formal application of co-area formula

Level Set Methods for Inverse Problems San Antonio, January 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 San Antonio, January Shape Optimization Equivalent equation for velocity V n Update by motion of shape in normal direction for a small time , new shape Expansion

Level Set Methods for Inverse Problems San Antonio, January 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 San Antonio, January Inverse Obstacle Problem Identify obstacle from partial measurements f of solution on

Level Set Methods for Inverse Problems San Antonio, January Inverse Obstacle Problem Shape derivative Adjoint method

Level Set Methods for Inverse Problems San Antonio, January Inverse Obstacle Problem Shape derivative Simplest choice of velocity space Velocity

Level Set Methods for Inverse Problems San Antonio, January Example: 5% noise - Norm - Norm

Level Set Methods for Inverse Problems San Antonio, January Example: 5% noise Residual

Level Set Methods for Inverse Problems San Antonio, January Example: 5% noise - error

Level Set Methods for Inverse Problems San Antonio, January 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)

Level Set Methods for Inverse Problems San Antonio, January Tomography-Type Problem Identify obstacle from boundary measurements z of solution on

Level Set Methods for Inverse Problems San Antonio, January Tomography, Single Measurement - Norm - Norm

Level Set Methods for Inverse Problems San Antonio, January Tomography Residual

Level Set Methods for Inverse Problems San Antonio, January Tomography - error

Level Set Methods for Inverse Problems San Antonio, January 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 San Antonio, January Levenberg-Marquardt Method Inverse problems with least-squares functional Choose variable scalar product Variational characterization

Level Set Methods for Inverse Problems San Antonio, January Levenberg-Marquardt Method Example 1: where, and denotes the indicator function of.

Level Set Methods for Inverse Problems San Antonio, January Levenberg-Marquardt Method 1% noise,  =10 -7, Iterations 10 and 15

Level Set Methods for Inverse Problems San Antonio, January Levenberg-Marquardt Method 1% noise,  =10 -7, Iterations 20 and 25

Level Set Methods for Inverse Problems San Antonio, January Levenberg-Marquardt Method 4% noise,  =10 -7, Iterations 10 and 20

Level Set Methods for Inverse Problems San Antonio, January Levenberg-Marquardt Method 4% noise,  =10 -7, Iterations 30 and 40

Level Set Methods for Inverse Problems San Antonio, January Levenberg-Marquardt Method Residual and L 1 -error

Level Set Methods for Inverse Problems San Antonio, January Levenberg-Marquardt Method Example 2: where and denotes the indicator function of.

Level Set Methods for Inverse Problems San Antonio, January Levenberg-Marquardt Method No noise Iterations 2,4,6,8

Level Set Methods for Inverse Problems San Antonio, January Levenberg-Marquardt Method Residual and L 1 -error

Level Set Methods for Inverse Problems San Antonio, January Levenberg-Marquardt Method Residual and L 1 -error

Level Set Methods for Inverse Problems San Antonio, January Levenberg-Marquardt Method 0.1 % noise Iterations 5,10,20,25

Level Set Methods for Inverse Problems San Antonio, January Levenberg-Marquardt Method 1% noise 2% noise 3% noise 4% noise

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