Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg...
-
Upload
merilyn-howard -
Category
Documents
-
view
216 -
download
0
Transcript of Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg...
![Page 1: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/1.jpg)
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
![Page 2: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/2.jpg)
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!!!
![Page 3: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/3.jpg)
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)
![Page 4: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/4.jpg)
Large-Scale Methods in Inverse Problems 4
Inverse Geomagnetic Problems
![Page 5: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/5.jpg)
Large-Scale Methods in Inverse Problems 5
Inverse Acoustic Problems
Oticon/Rhinometrics
![Page 6: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/6.jpg)
Large-Scale Methods in Inverse Problems 6
Image Restoration Problems
blurring
deblurring
Io (moon of Saturn)
You cannot depend on your eyes whenyour imagination is out of focus
– Mark Twain
![Page 7: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/7.jpg)
Large-Scale Methods in Inverse Problems 7
Model Problem and Discretization
Vertical component ofmagnetic field from a dipole
![Page 8: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/8.jpg)
Large-Scale Methods in Inverse Problems 8
The Need for Regularization
Regularization:
keep the “good” SVD components
and discard the noisy ones!
![Page 9: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/9.jpg)
Large-Scale Methods in Inverse Problems 9
Regularization – TSVD & Tikhonov
![Page 10: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/10.jpg)
Large-Scale Methods in Inverse Problems 10
Singular Vectors (Always) Oscillate
![Page 11: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/11.jpg)
Large-Scale Methods in Inverse Problems 11
Large-Scale Aspects (the easy case)
![Page 12: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/12.jpg)
Large-Scale Methods in Inverse Problems 12
Large-Scale Aspects (the real problems)
Toeplitz matrix-vectormultiplication flop count.
![Page 13: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/13.jpg)
Large-Scale Methods in Inverse Problems 13
Large-Scale Tikhonov Regularization
![Page 14: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/14.jpg)
Large-Scale Methods in Inverse Problems 14
Difficulties and Remedies I
![Page 15: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/15.jpg)
Large-Scale Methods in Inverse Problems 15
Difficulties and Remedies II
![Page 16: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/16.jpg)
Large-Scale Methods in Inverse Problems 16
The Art of Preconditioning
![Page 17: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/17.jpg)
Large-Scale Methods in Inverse Problems 17
Explicit Subspace Preconditiong
![Page 18: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/18.jpg)
Large-Scale Methods in Inverse Problems 18
Krylov Signal Subspaces
Smiley Crater, Mars
![Page 19: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/19.jpg)
Large-Scale Methods in Inverse Problems 19
Pros and Cons of Regularizing Iterations
![Page 20: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/20.jpg)
Large-Scale Methods in Inverse Problems 20
Projection, then Regularization
![Page 21: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/21.jpg)
Large-Scale Methods in Inverse Problems 21
Bounds on “Everything”
![Page 22: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/22.jpg)
Large-Scale Methods in Inverse Problems 22
A Dilemma With Projection + Regular.
![Page 23: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/23.jpg)
Large-Scale Methods in Inverse Problems 23
Better Basis Vectors!
![Page 24: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/24.jpg)
Large-Scale Methods in Inverse Problems 24
Considerations in 2D
…
…
![Page 25: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/25.jpg)
Large-Scale Methods in Inverse Problems 25
Good Seminorms for 2D Problems
![Page 26: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/26.jpg)
Large-Scale Methods in Inverse Problems 26
Seminorms and Regularizing Iterations
![Page 27: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/27.jpg)
Large-Scale Methods in Inverse Problems 27
Krylov Implementation
![Page 28: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/28.jpg)
Large-Scale Methods in Inverse Problems 28
Avoiding the Transpose: GMRES
![Page 29: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/29.jpg)
Large-Scale Methods in Inverse Problems 29
GMRES and CGLS Basis Vectors
![Page 30: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/30.jpg)
Large-Scale Methods in Inverse Problems 30
CGLS and GMRES Solutions
![Page 31: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/31.jpg)
Large-Scale Methods in Inverse Problems 31
The “Freckles’’
DCT spectrum spatial domain
![Page 32: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/32.jpg)
Large-Scale Methods in Inverse Problems 32
Preconditioning for GMRES
![Page 33: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/33.jpg)
Large-Scale Methods in Inverse Problems 33
A New and Better Approach
![Page 34: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/34.jpg)
Large-Scale Methods in Inverse Problems 34
(P)CGLS and (P)GMRES
![Page 35: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/35.jpg)
Large-Scale Methods in Inverse Problems 35
Away From 2-Norms
Io (
moo
n of
Sat
urn)
q = 2 q = 1.1
![Page 36: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/36.jpg)
Large-Scale Methods in Inverse Problems 36
Functionals Defined on Sols. to DIP
![Page 37: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/37.jpg)
Large-Scale Methods in Inverse Problems 37
Large-Scale Algorithm MLFIP
![Page 38: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/38.jpg)
Large-Scale Methods in Inverse Problems 38
Confidence Invervals with MLFIP
![Page 39: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/39.jpg)
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 …
![Page 40: Large-Scale Methods in Inverse Problems 1 With contributions from: Michael Jacobsen, Toke Koldborg Jensen - PhD students Line H. Clemmensen, Iben Kraglund,](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649e8a5503460f94b8fa09/html5/thumbnails/40.jpg)
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.