1. 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

2. Level Set Methods for Inverse Problems San Antonio, January 20052 Collaborations Benjamin Hackl (Linz) Wolfgang Ring, Michael Hintermüller (Graz)

3. Level Set Methods for Inverse Problems San Antonio, January 20053 Outline Introduction Shape Gradient Methods Framework for Level Set Methods Examples Levenberg-Marquardt Methods

4. Level Set Methods for Inverse Problems San Antonio, January 20054 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

5. Level Set Methods for Inverse Problems San Antonio, January 20055 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!

6. Level Set Methods for Inverse Problems San Antonio, January 20056 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

7. Level Set Methods for Inverse Problems San Antonio, January 20057 Level Set Methods Change of the front is translated to a change of the level set function Automated treatment of topology change

8. Level Set Methods for Inverse Problems San Antonio, January 20058 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)

9. Level Set Methods for Inverse Problems San Antonio, January 20059 Level Set Methods Geometric primitives can expressed via derivatives of the level set function Normal Mean curvature

10. Level Set Methods for Inverse Problems San Antonio, January 200510 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

11. Level Set Methods for Inverse Problems San Antonio, January 200511 Shape Optimization Calculus on shapes by the speed method: Natural variations are normal velocities

12. Level Set Methods for Inverse Problems San Antonio, January 200512 Shape Derivatives Derivatives can be computed by the level set method Example: Formal computation:

13. Level Set Methods for Inverse Problems San Antonio, January 200513 Shape Derivatives Formal application of co-area formula

14. Level Set Methods for Inverse Problems San Antonio, January 200514 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

15. Level Set Methods for Inverse Problems San Antonio, January 200515 Shape Optimization Equivalent equation for velocity V n Update by motion of shape in normal direction for a small time , new shape Expansion

16. Level Set Methods for Inverse Problems San Antonio, January 200516 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

17. Level Set Methods for Inverse Problems San Antonio, January 200517 Inverse Obstacle Problem Identify obstacle from partial measurements f of solution on

18. Level Set Methods for Inverse Problems San Antonio, January 200518 Inverse Obstacle Problem Shape derivative Adjoint method

19. Level Set Methods for Inverse Problems San Antonio, January 200519 Inverse Obstacle Problem Shape derivative Simplest choice of velocity space Velocity

20. Level Set Methods for Inverse Problems San Antonio, January 200520 Example: 5% noise - Norm - Norm

21. Level Set Methods for Inverse Problems San Antonio, January 200521 Example: 5% noise Residual

22. Level Set Methods for Inverse Problems San Antonio, January 200522 Example: 5% noise - error

23. Level Set Methods for Inverse Problems San Antonio, January 200523 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)

24. Level Set Methods for Inverse Problems San Antonio, January 200524 Tomography-Type Problem Identify obstacle from boundary measurements z of solution on

25. Level Set Methods for Inverse Problems San Antonio, January 200525 Tomography, Single Measurement - Norm - Norm

26. Level Set Methods for Inverse Problems San Antonio, January 200526 Tomography Residual

27. Level Set Methods for Inverse Problems San Antonio, January 200527 Tomography - error

28. Level Set Methods for Inverse Problems San Antonio, January 200528 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

29. Level Set Methods for Inverse Problems San Antonio, January 200529 Levenberg-Marquardt Method Inverse problems with least-squares functional Choose variable scalar product Variational characterization

30. Level Set Methods for Inverse Problems San Antonio, January 200530 Levenberg-Marquardt Method Example 1: where, and denotes the indicator function of.

31. Level Set Methods for Inverse Problems San Antonio, January 200531 Levenberg-Marquardt Method 1% noise,  =10 -7, Iterations 10 and 15

32. Level Set Methods for Inverse Problems San Antonio, January 200532 Levenberg-Marquardt Method 1% noise,  =10 -7, Iterations 20 and 25

33. Level Set Methods for Inverse Problems San Antonio, January 200533 Levenberg-Marquardt Method 4% noise,  =10 -7, Iterations 10 and 20

34. Level Set Methods for Inverse Problems San Antonio, January 200534 Levenberg-Marquardt Method 4% noise,  =10 -7, Iterations 30 and 40

35. Level Set Methods for Inverse Problems San Antonio, January 200535 Levenberg-Marquardt Method Residual and L 1 -error

36. Level Set Methods for Inverse Problems San Antonio, January 200536 Levenberg-Marquardt Method Example 2: where and denotes the indicator function of.

37. Level Set Methods for Inverse Problems San Antonio, January 200537 Levenberg-Marquardt Method No noise Iterations 2,4,6,8

38. Level Set Methods for Inverse Problems San Antonio, January 200538 Levenberg-Marquardt Method Residual and L 1 -error

39. Level Set Methods for Inverse Problems San Antonio, January 200539 Levenberg-Marquardt Method Residual and L 1 -error

40. Level Set Methods for Inverse Problems San Antonio, January 200540 Levenberg-Marquardt Method 0.1 % noise Iterations 5,10,20,25

41. Level Set Methods for Inverse Problems San Antonio, January 200541 Levenberg-Marquardt Method 1% noise 2% noise 3% noise 4% noise

42. Level Set Methods for Inverse Problems San Antonio, January 200542 Download and Contact Papers and Talks: www.indmath.uni-linz.ac.at/people/burger e-mail: martin.burger@jku.at