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Level Sets for Inverse Problems and Optimization I

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. Collaborations. Benjamin Hackl (Linz)

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Level Sets for Inverse Problems and Optimization I

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  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. Collaborations • Benjamin Hackl (Linz) • Wolfgang Ring, Michael Hintermüller (Graz) Level Set Methods for Inverse Problems San Antonio, January 2005

  3. Outline • Introduction • Shape Gradient Methods • Framework for Level Set Methods • Examples • Levenberg-Marquardt Methods Level Set Methods for Inverse Problems San Antonio, January 2005

  4. Introduction Many applications deal with the reconstruction and optimization of geometries (shapes, topologies), e.g.: • Identification of piecewise constant parametersin PDEs • Inverse obstacle scattering • Inclusion / cavity detection • Topology optimization • Image segmentation Level Set Methods for Inverse Problems San Antonio, January 2005

  5. Introduction Flexible representations of the shapes needed! Level Set Methods for Inverse Problems San Antonio, January 2005 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, ...)

  6. Level Set Methods Level Set Methods for Inverse Problems San Antonio, January 2005 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 • 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 2005

  8. Level Set Flows Level Set Methods for Inverse Problems San Antonio, January 2005 Geometric flow of the level sets of f can be translated into nonlinear differential equation for f („level set equation“) Appropriate solution concept: Viscosity solutions (Crandall, Lions 1981-83,Crandall-Ishii-Lions 1991)

  9. Level Set Methods Level Set Methods for Inverse Problems San Antonio, January 2005 Geometric primitives can expressed via derivatives of the level set function Normal Mean curvature

  10. 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 2005

  11. Shape Optimization • Calculus on shapes by the speed method: Natural variations are normal velocities Level Set Methods for Inverse Problems San Antonio, January 2005

  12. Shape Derivatives • Derivatives can be computed by the level set methodExample: • Formal computation: Level Set Methods for Inverse Problems San Antonio, January 2005

  13. Shape Derivatives • Formal application of co-area formula Level Set Methods for Inverse Problems San Antonio, January 2005

  14. 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 2005

  15. Shape Optimization • Equivalent equation for velocity Vn • Update by motion of shape in normal direction for a small time t , new shape • Expansion Level Set Methods for Inverse Problems San Antonio, January 2005

  16. Shape Optimization • From definition (with ) • Descent method, time stept 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 2005

  17. Inverse Obstacle Problem • Identify obstacle from partial measurements f of solution on Level Set Methods for Inverse Problems San Antonio, January 2005

  18. Inverse Obstacle Problem • Shape derivative • Adjoint method Level Set Methods for Inverse Problems San Antonio, January 2005

  19. Inverse Obstacle Problem Level Set Methods for Inverse Problems San Antonio, January 2005 Shape derivative Simplest choice of velocity space Velocity

  20. Example: 5% noise - Norm - Norm Level Set Methods for Inverse Problems San Antonio, January 2005

  21. Example: 5% noise Residual Level Set Methods for Inverse Problems San Antonio, January 2005

  22. Example: 5% noise - error Level Set Methods for Inverse Problems San Antonio, January 2005

  23. Inverse Obstacle Problem Level Set Methods for Inverse Problems San Antonio, January 2005 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. Tomography-Type Problem • Identify obstacle from boundary measurements z of solution on Level Set Methods for Inverse Problems San Antonio, January 2005

  25. Tomography, Single Measurement - Norm - Norm Level Set Methods for Inverse Problems San Antonio, January 2005

  26. Tomography Residual Level Set Methods for Inverse Problems San Antonio, January 2005

  27. Tomography - error Level Set Methods for Inverse Problems San Antonio, January 2005

  28. 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 2005

  29. Levenberg-Marquardt Method • Inverse problems with least-squares functional • Choose variable scalar product • Variational characterization Level Set Methods for Inverse Problems San Antonio, January 2005

  30. Levenberg-Marquardt Method • Example 1: • where , and denotes the indicator function of . Level Set Methods for Inverse Problems San Antonio, January 2005

  31. Levenberg-Marquardt Method • 1% noise, b=10-7, Iterations 10 and 15 Level Set Methods for Inverse Problems San Antonio, January 2005

  32. Levenberg-Marquardt Method • 1% noise, b=10-7, Iterations 20 and 25 Level Set Methods for Inverse Problems San Antonio, January 2005

  33. Levenberg-Marquardt Method • 4% noise, b=10-7, Iterations 10 and 20 Level Set Methods for Inverse Problems San Antonio, January 2005

  34. Levenberg-Marquardt Method • 4% noise, b=10-7, Iterations 30 and 40 Level Set Methods for Inverse Problems San Antonio, January 2005

  35. Levenberg-Marquardt Method • Residual and L1-error Level Set Methods for Inverse Problems San Antonio, January 2005

  36. Levenberg-Marquardt Method • Example 2: • where and denotes the indicator function of . Level Set Methods for Inverse Problems San Antonio, January 2005

  37. Levenberg-Marquardt Method • No noise • Iterations2,4,6,8 Level Set Methods for Inverse Problems San Antonio, January 2005

  38. Levenberg-Marquardt Method • Residual and L1-error Level Set Methods for Inverse Problems San Antonio, January 2005

  39. Levenberg-Marquardt Method • Residual and L1-error Level Set Methods for Inverse Problems San Antonio, January 2005

  40. Levenberg-Marquardt Method • 0.1 % noise • Iterations 5,10,20,25 Level Set Methods for Inverse Problems San Antonio, January 2005

  41. Levenberg-Marquardt Method • 1% noise 2% noise • 3% noise 4% noise Level Set Methods for Inverse Problems San Antonio, January 2005

  42. Download and Contact • Papers and Talks: www.indmath.uni-linz.ac.at/people/burger • e-mail: martin.burger@jku.at Level Set Methods for Inverse Problems San Antonio, January 2005

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