Inverse Problem-- LeaksInverse Problem-- Leaks

Carrie Nugent

December 6th, 2007

Inverse Theory

Carrie Nugent

December 6th, 2007

Inverse Theory

BackgroundBackground Collaboration with Gretel

Greene and Sailor of UT Austin’s ACTLab

“We situate our work at the hotly contested intersections where technology, art, and culture collide.”

http://home.actlab.utexas.edu/

Collaboration with Gretel Greene and Sailor of UT Austin’s ACTLab

“We situate our work at the hotly contested intersections where technology, art, and culture collide.”

http://home.actlab.utexas.edu/

This problem inspired by the idea of a leaking iron lung

This problem inspired by the idea of a leaking iron lung

ModelModel 2-D model of perforated barrier Sites (m vector) are set as “leaking” (1) or “not leaking” (0) Measurement is taken at distance, problem is solved to find

leaking sites. Note that you can’t have a negative leak, and leaks should not

be large (1.5 or more)

2-D model of perforated barrier Sites (m vector) are set as “leaking” (1) or “not leaking” (0) Measurement is taken at distance, problem is solved to find

leaking sites. Note that you can’t have a negative leak, and leaks should not

be large (1.5 or more)

HIGH PRESSURE

LOW PRESSUREDETECTOR

LEAKS!

ModelLocalized leaks

ModelLocalized leaks

QuickTime™ and aTIFF (Uncompressed) decompressor

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QuickTime™ and aTIFF (Uncompressed) decompressor

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Least Squares Fit and Tikhonov Regularization for localized leak arrangement.

Large singular values prevented good least squares fit

0.0027 0.0027 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0027 0.0027 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0027 0.0027 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0027 0.0027 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0027 0.0027 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0027 0.0027 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0027 0.0027 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0027 0.0027 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0027 0.0027

G matrix 0

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Tikhonov RegularizationTikhonov Regularization

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (Uncompressed) decompressor

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m vector d vector

All d vectors fit actual d vector within error bars.All d vectors fit actual d vector within error bars.

New case-- alternating New case-- alternating Most difficult case-- detectors are equidistant from leaks

Fitting must rely on unique edges Most difficult case-- detectors are equidistant from leaks

Fitting must rely on unique edges

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (Uncompressed) decompressor

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m vector d vector

Least squares falls apart completelyTikhonov captures pattern if not magnitude

Least squares falls apart completelyTikhonov captures pattern if not magnitude

RecapRecap

Least squares fit not suitable (too many singular values)

Tikhonov regularization works well for localized case, alright in alternating case

What happens when we add noise?

Least squares fit not suitable (too many singular values)

Tikhonov regularization works well for localized case, alright in alternating case

What happens when we add noise?

Addition of NoiseAddition of Noise Very small amount of noise- std of 0.01, average of 0 Analogous to detector errors or very very slight cross-

breezes

Very small amount of noise- std of 0.01, average of 0 Analogous to detector errors or very very slight cross-

breezes

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (Uncompressed) decompressor

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Bad news, folks, bad newsBad news, folks, bad news

Would it be practical to use a pressure detector to find leaks on an iron lung?Would it be practical to use a pressure detector to find leaks on an iron lung?

[1] Baisch, F. J., R. Gerzer, Breathing Assistance by the Iron Lung Increases Sympathetic Tone and Modifies Fluid Excretion, IAF abstracts, 34th COSPAR Scientific Assembly, The Second World Space Congress, (2002), Houston, TX

*Authors noted a pressure difference of 15 cm H2O, which I converted to approximately 0.01449 atm

[1] Baisch, F. J., R. Gerzer, Breathing Assistance by the Iron Lung Increases Sympathetic Tone and Modifies Fluid Excretion, IAF abstracts, 34th COSPAR Scientific Assembly, The Second World Space Congress, (2002), Houston, TX

*Authors noted a pressure difference of 15 cm H2O, which I converted to approximately 0.01449 atm

Assume pressure drops off linearly Model: pressure difference by factor of 2 Actual: pressure difference approx. 1.45% [1] Model: max distance ~ 10x leak spacing Actual: ~20 inches

2 /0.0145 = 20 inches/ x inches Maximum distance of ~.145 inches (about a third

of a centimeter) without noise

Assume pressure drops off linearly Model: pressure difference by factor of 2 Actual: pressure difference approx. 1.45% [1] Model: max distance ~ 10x leak spacing Actual: ~20 inches

2 /0.0145 = 20 inches/ x inches Maximum distance of ~.145 inches (about a third

of a centimeter) without noise

Extra- 1952 Polio outbreakExtra- 1952 Polio outbreak