TMA 4180 Optimeringsteori INVERSE PROBLEMS AND OPTIMIZATION PART

TMA 4180 Optimeringsteori INVERSE PROBLEMS AND OPTIMIZATION PART 1 Harald E. Krogstad, IMF, Spring. 2007 TMA 4180 Optimeringsteori 2007

CONTENTS • What is an inverse problem • Some famous problems • Examples TMA 4180 Optimeringsteori 2007

Jeopardy - The everyday Inverse problem: “It was in 1905” “When did Einstein publish his Theory of Relativity ? ” “When did Robert Koch get the Nobel Price in Medicine? ” “When was my grandmother born? ” “When did Norway and Sweden split up from the union? ” TMA 4180 Optimeringsteori 2007

• The information is incomplete and non-conclusive • Different input leads to the same result • The ”most probable” input depends on circumstances direct problem: given the question, find the answer inverse problem: given the answer, find the question TMA 4180 Optimeringsteori 2007

• To solve an inverse problem is to determine a cause from its effect • Applications in seismology, geosciences, and many other areas of sciences and engineering • Optimization provides methods for inverse problems • The majority of real problems are inverse problems TMA 4180 Optimeringsteori 2007

Well-posed problem: • a solution always exists • there is only one solution • a small change in the problem leads to a small change in the solution Ill-posed problem: • a solution may not exist • there may be more than one solution • a small change in the problem leads to a big change in the solution INVERSE PROBLEMS ARE (ALMOST) ALWAYS ILL-POSED! TMA 4180 Optimeringsteori 2007

SOME FAMOUS INVERSE PROBLEMS • Marc Kac (1966): "Can you hear the shape of a drum? " • Computer Tomography • Seismic Inversion • Image Restoration TMA 4180 Optimeringsteori 2007

NO, you can’t hear the shape of a drum, but you can hear • the area, • the length of the boundary, • and the number of holes! http: //www. ams. org/new-in-math/hap-drum. html TMA 4180 Optimeringsteori 2007

COMPUTER TOMOGRAPHY http: //www. iwr. uni-heidelberg. de/groups/ngg/Tutorial/Tut. CT_121203_Lauritsch. pdf TMA 4180 Optimeringsteori 2007

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Radon transform: Based on line integrals Inverse problem: TMA 4180 Optimeringsteori 2007

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INVERSE SCATTERING TMA 4180 Optimeringsteori 2007

Seismic inversion http: //www. mgnet. org/~douglas/Classes/cs 521 -s 00/asdf. ppt TMA 4180 Optimeringsteori 2007

Given the seismic recordings – compute the sea floor motion Slip Uplift (Bjørn Gjevik, Ui. O. Published by Caltech) TMA 4180 Optimeringsteori 2007

MEDICAL ULTRASOUND TMA 4180 Optimeringsteori 2007

IMAGE RESTORATION Blur Max. Entropy restoration Noise suppres. http: //mip. ups-tlse. fr/~noll/noll_tutorial. html TMA 4180 Optimeringsteori 2007

©ESA A: Original HST photo. C: Ground telescope image. TMA 4180 Optimeringsteori 2007 B: Enlarged section. D: Digitally enhanced image.

Example: The Hilbert Matrix System TMA 4180 Optimeringsteori 2007

The Hilbert matrix system: TMA 4180 Optimeringsteori 2007

Picard-plot (cond. Number = 1. 5 x 1018) TMA 4180 Optimeringsteori 2007

Hilbert problem cont. . The Picard plot shows |aj|: TMA 4180 Optimeringsteori 2007

Hilbert problem cont. . WARNING: A small residual does not mean a good solution for ill-conditioned problems: TMA 4180 Optimeringsteori 2007

The cosine taper TMA 4180 Optimeringsteori 2007

Example: The Hodrick-Prescott Filter TMA 4180 Optimeringsteori 2007

Hodrick-Prescott filter TMA 4180 Optimeringsteori 2007

Data Trend curve N unknowns! TMA 4180 Optimeringsteori 2007

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FREDHOLM INTEGRAL EQUATIONS K(t, t) is called the kernel Discretized Fredholm Integral Equations are generally ill-conditioned TMA 4180 Optimeringsteori 2007

A very common situation in image processing: I BI f. PS BI : Image : Blurred image : Point Spread Function I : ”Deconvolution” The simplest deconvolution is van. Cittert/Landweber iteration: (NB: Should have some idea about f. PS ! TMA 4180 Optimeringsteori 2007

DE-CONVOLUTION TMA 4180 Optimeringsteori 2007

Measurement Spectrum 1 0. 5 0 0 0. 1 0. 2 0. 3 Spectrum 0. 5 0. 6 0. 7 0. 8 0. 9 1 De-convoluted measurement 1 0. 5 0 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 "Ideal" measurement 1 Spectrum 0. 4 0. 5 0 0 0. 1 0. 2 TMA 4180 Optimeringsteori 2007 0. 3 0. 4 0. 5 0. 6 Frequency 0. 7

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van. Cittert/Landweber iteration is simple and fast! TMA 4180 Optimeringsteori 2007