1. Large-Scale Methods in Inverse Problems 1 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 Per Christian Hansen Informatics and Mathematical Modelling Technical University of Denmark

2. Large-Scale Methods in Inverse Problems 2 Overview of Talk A survey of numerical methods for large-scale inverse problems 1.Some examples. 2.The need for regularization algorithms. 3.Krylov subspace methods for large-scale problems. 4.Preconditioning for regularization problems. 5.Signal subspaces and (semi)norms. 6.GMRES as a regularization method. 7.Alternatives to spectral filtering. Many details are skipped, to get the big picture!!!

3. 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) M OO Re Tools (Jacobsen, 2004) GeoTools (Pedersen, 2005)

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

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

6. 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

27. Large-Scale Methods in Inverse Problems 27 Krylov Implementation

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

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

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

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

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

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

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

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

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

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

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

39. Large-Scale Methods in Inverse Problems 39 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. Many Topics Not Covered …

40. 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.