INVERSE PROBLEMS

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TMA 4180 Optimeringsteori
INVERSE PROBLEMS AND OPTIMIZATION PART 2
Harald E. Krogstad, IMF, Spring. 2007
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• THE KEY PROBLEMS
• How to select the best/most probable/optimal
  solution
• How to dampen ill-conditionness
• (regularize - but not exaggerate!)

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The effect of Tikhonov Regularization for the
test case
Solution
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1.5
)
s
1
f(
0.5
0
s
0
m1/2
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FREDHOLM INTEGRAL EQUATIONS
K(t,t) is called the kernel
Discretized Fredholm Integral Equations are
generally ill-conditioned
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L-CURVE ANALYSIS
(Per Chr. Hansen, DTU)
Plot of the two terms in the Tikhonov objective
function
decreasing
m
Penalty
increasing
Error
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ITERATIVE METHODS
The basic iterative method for a linear
equation is (fix-point iteration)
is called a relaxation parameter. Convergence if
is chosen so that
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Fix point iteration is used for inverse
problems as a general technique where the
number of iterations is the regularization!
Nonlinear filter
Input
Observation
Common name van Cittert iteration
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• DE-CONVOLUTION OF SPECTRAL DATA
• Astronomy
• Mass spectrography
• Optics
• Nuclear Magnetic Resonance

Measurement should consist of narrow spectral
lines, but the instrument blurs the
lines De-blurring is needed!
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2000 iterations!
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A very common situation in image processing
I Image BI Blurred image fPS Point
Spread Function
(NB Should have some idea about fPS !
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vanCittert/Landweber iteration is simple and fast!