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With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students

Large-Scale Methods in Inverse Problems Per Christian Hansen Informatics and Mathematical Modelling Technical University of Denmark. With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students

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With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students

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  1. Large-Scale Methods in Inverse Problems • Per Christian Hansen • Informatics and Mathematical Modelling • Technical University of Denmark • With contributions from: • Michael Jacobsen, Toke Koldborg Jensen - PhD students • Line H. Clemmensen, Iben Kraglund, Kristine Horn,Jesper Pedersen, Marie-Louise H. Rasmussen - Master students Large-Scale Methods in Inverse Problems

  2. Overview of Talk • A survey of numerical methods for large-scale inverse problems • Some examples. • The need for regularization algorithms. • Krylov subspace methods for large-scale problems. • Preconditioning for regularization problems. • Signal subspaces and (semi)norms. • GMRES as a regularization method. • Alternatives to spectral filtering. • Many details are skipped, to get the big picture!!! Large-Scale Methods in Inverse Problems

  3. Related Work • Many people work on similar problems and algorithms: • Åke Björck, Lars Eldén, Tommy Elfving • Martin Hanke, James G. Nagy, Robert Plemmons • Misha E. Kilmer, Dianne P. Oleary • Daniela Calvetti, Lothar Reichel, Brian Lewis • Gene H. Golub, Urs von Matt • Uri Asher, Eldad Haber, Douglas Oldenburg • Jerry Eriksson, Mårten Gullikson, Per-Åke Wedin • Marielba Rojas, Trond Steihaug • Tony Chan, Stanley Osher, Curtis R. Vogel • Jesse Barlow, Raymond Chan, Michael Ng • Recent Matlab software packages: • Restore Tools (Nagy, Palmer, Perrone, 2004) • MOORe Tools (Jacobsen, 2004) • GeoTools (Pedersen, 2005) Large-Scale Methods in Inverse Problems

  4. Inverse Geomagnetic Problems Large-Scale Methods in Inverse Problems

  5. Inverse Acoustic Problems Oticon/ Rhinometrics Large-Scale Methods in Inverse Problems

  6. Image Restoration Problems blurring deblurring Io (moon of Saturn) You cannot depend on your eyes when your imagination is out of focus – Mark Twain Large-Scale Methods in Inverse Problems

  7. Model Problem and Discretization Vertical component of magnetic field from a dipole Large-Scale Methods in Inverse Problems

  8. The Need for Regularization Regularization: keep the “good” SVD components and discard the noisy ones! Large-Scale Methods in Inverse Problems

  9. Regularization – TSVD & Tikhonov Large-Scale Methods in Inverse Problems

  10. Singular Vectors (Always) Oscillate Large-Scale Methods in Inverse Problems

  11. Large-Scale Aspects (the easy case) Large-Scale Methods in Inverse Problems

  12. Large-Scale Aspects (the real problems) Toeplitz matrix-vector multiplication flop count. Large-Scale Methods in Inverse Problems

  13. Large-Scale Tikhonov Regularization Large-Scale Methods in Inverse Problems

  14. Difficulties and Remedies I Large-Scale Methods in Inverse Problems

  15. Difficulties and Remedies II Large-Scale Methods in Inverse Problems

  16. The Art of Preconditioning Large-Scale Methods in Inverse Problems

  17. Explicit Subspace Preconditiong Large-Scale Methods in Inverse Problems

  18. Krylov Signal Subspaces Smiley Crater, Mars Large-Scale Methods in Inverse Problems

  19. Pros and Cons of Regularizing Iterations Large-Scale Methods in Inverse Problems

  20. Projection, then Regularization Large-Scale Methods in Inverse Problems

  21. Bounds on “Everything” Large-Scale Methods in Inverse Problems

  22. A Dilemma With Projection + Regular. Large-Scale Methods in Inverse Problems

  23. Better Basis Vectors! Large-Scale Methods in Inverse Problems

  24. Considerations in 2D … … Large-Scale Methods in Inverse Problems

  25. Good Seminorms for 2D Problems Large-Scale Methods in Inverse Problems

  26. Seminorms and Regularizing Iterations Large-Scale Methods in Inverse Problems

  27. Krylov Implementation Large-Scale Methods in Inverse Problems

  28. Avoiding the Transpose: GMRES Large-Scale Methods in Inverse Problems

  29. GMRES and CGLS Basis Vectors Large-Scale Methods in Inverse Problems

  30. CGLS and GMRES Solutions Large-Scale Methods in Inverse Problems

  31. The “Freckles’’ DCT spectrum spatial domain Large-Scale Methods in Inverse Problems

  32. Preconditioning for GMRES Large-Scale Methods in Inverse Problems

  33. A New and Better Approach Large-Scale Methods in Inverse Problems

  34. (P)CGLS and (P)GMRES Large-Scale Methods in Inverse Problems

  35. Away From 2-Norms Io (moon of Saturn) q = 1.1 q = 2 Large-Scale Methods in Inverse Problems

  36. Functionals Defined on Sols. to DIP Large-Scale Methods in Inverse Problems

  37. Large-Scale Algorithm MLFIP Large-Scale Methods in Inverse Problems

  38. Confidence Invervals with MLFIP Large-Scale Methods in Inverse Problems

  39. Many Topics Not Covered … • Algorithms for other norms (p and q≠ 2). • In particular, total variation (TV). • Nonnegativity constraints. • General linear inequality constraints. • Compression of dense coefficient matrix A. • Color images (and color TV). • Implementation aspects and software. • The choice the of regularization parameter. Large-Scale Methods in Inverse Problems

  40. “Conclusions and Further Work” • I hesitate to give any conclusion – • the work is ongoing; • there are many open problems, • lots of challenges (mathematical and numerical), • and a multitude of practical problems waiting to be solved. Large-Scale Methods in Inverse Problems

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