Inverse Geomagnetic Problems

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

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!!!

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)

4
Inverse Geomagnetic Problems
5
Inverse Acoustic Problems
Oticon/ Rhinometrics
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
Model Problem and Discretization
Vertical component of magnetic field from a dipole
8
The Need for Regularization
Regularization keep the good SVD components
and discard the noisy ones!
9
Regularization TSVD Tikhonov
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Singular Vectors (Always) Oscillate
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Large-Scale Aspects (the easy case)
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Large-Scale Aspects (the real problems)
Toeplitz matrix-vector multiplication flop count.
13
Large-Scale Tikhonov Regularization
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Difficulties and Remedies I
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Difficulties and Remedies II
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The Art of Preconditioning
17
Explicit Subspace Preconditiong
18
Krylov Signal Subspaces
Smiley Crater, Mars
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Pros and Cons of Regularizing Iterations
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Projection, then Regularization
21
Bounds on Everything
22
A Dilemma With Projection Regular.
23
Better Basis Vectors!
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Considerations in 2D


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Good Seminorms for 2D Problems
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Seminorms and Regularizing Iterations
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Krylov Implementation
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Avoiding the Transpose GMRES
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GMRES and CGLS Basis Vectors
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CGLS and GMRES Solutions
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The Freckles
DCT spectrum
spatial domain
32
Preconditioning for GMRES
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A New and Better Approach
34
(P)CGLS and (P)GMRES
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Away From 2-Norms
Io (moon of Saturn)
q 1.1
q 2
36
Functionals Defined on Sols. to DIP
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Large-Scale Algorithm MLFIP
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Confidence Invervals with MLFIP
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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.

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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.