applications and adaptations of a globally convergent numerical method by: aubrey rhoden advisor:...

Applications and Adaptations ofa Globally Convergent Numerical Method

By: Aubrey Rhoden

Advisor: Dr. Jianzhong Su

The Inverse Problem: Locating Blood Clots or other Inclusionswithin a domain given boundary information about

heat or light intensity

CAT Scan

The arrows show a case of an Inflamed appendix indicatingAppendicitis.

Globally Convergent Method

Given this type of domain where the sourceposition runs along the line B the attempt is to reconstruct the coefficient a(x) fromthe information on the boundary Ω. Two inclusions are located within this domainthat through the coefficient a(x) will representan inclusion inside of the brain such as a stroke, blood clot, or tumor. The forwardand inverse problem will be approximatedusing a bi-quadratic serendipitous finiteelement method.

X

Y

6 8 10 12 14

6

8

10

12

14W

37.237.1837.1637.1437.1237.137.0837.0637.0437.023736.9836.9636.9436.92

X

Y

6 8 10 12 14

6

8

10

12

14W

37.1837.1637.1437.1237.137.0837.0637.0437.023736.9836.9636.9436.92

Uniform (No Inclusions) a(x)=0.001 inside two inclusions

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6 8 10 12 14

6

8

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14W

37.1837.1637.1437.1237.137.0837.0637.0437.023736.9836.9636.9436.9236.9

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Y

6 8 10 12 14

6

8

10

12

14W

37.1637.1437.1237.137.0837.0637.0437.023736.9836.9636.9436.9236.936.8836.86

a(x)=0.003 inside two inclusionsa(x)=0.002 inside two inclusions

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6 8 10 12 14

6

8

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14W

37.1237.137.0837.0637.0437.023736.9836.9636.9436.9236.936.8836.8636.8436.8236.836.7836.7636.74

a(x)=0.007 inside two inclusions a(x)=0.016 inside two inclusions

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6 8 10 12 14

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14W

37.053736.9536.936.8536.836.7536.736.6536.636.5536.5

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6 8 10 12 14

6

8

10

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14W

36.836.736.636.536.436.336.236.13635.9

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6 8 10 12 14

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8

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14W

3736.9536.936.8536.836.7536.736.6536.636.5536.536.4536.436.35

a(x)=0.02 inside two inclusions a(x)=0.038 inside two inclusions

UNT’s Medical Center performed the experiments on the mice so that our forward model correctly

matched the physical setting of a mouse with two blood clots located in its brain.

Exact Solutionfor a(x).

Reconstructionwith the additionof noise that was10% of the total difference intemperature

Schematic for theSteady-State Problem

Schematic for theTime-Dependent Problem

Forward heat solution:The heat source is kept lowso that we don’t cook the animal

Forward optic solution:The light source has to berather strong since the light intensity decays in the tissue.

Exact Solutions Reconstructions