1 / 40

Large-Scale Methods in Inverse Problems

This talk provides an overview of numerical methods for large-scale inverse problems, including regularization algorithms, Krylov subspace methods, preconditioning, and signal subspaces. Examples from various fields are discussed.

sjake
Download Presentation

Large-Scale Methods in Inverse Problems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related