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

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!!!

Large-Scale Methods in Inverse Problems 3

Related WorkMany 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)

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Inverse Geomagnetic Problems

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Inverse Acoustic Problems

Oticon/Rhinometrics

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Image Restoration Problems

blurring

deblurring

Io (moon of Saturn)

You cannot depend on your eyes whenyour imagination is out of focus

– Mark Twain

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Model Problem and Discretization

Vertical component ofmagnetic field from a dipole

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The Need for Regularization

Regularization:

keep the “good” SVD components

and discard the noisy ones!

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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-vectormultiplication flop count.

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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

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Explicit Subspace Preconditiong

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Krylov Signal Subspaces

Smiley Crater, Mars

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Pros and Cons of Regularizing Iterations

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Projection, then Regularization

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Bounds on “Everything”

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A Dilemma With Projection + Regular.

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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

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Preconditioning for GMRES

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A New and Better Approach

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(P)CGLS and (P)GMRES

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Away From 2-Norms

Io (

moo

n of

Sat

urn)

q = 2 q = 1.1

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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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• 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 …

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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.