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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.
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6 8 10 12 14
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37.237.1837.1637.1437.1237.137.0837.0637.0437.023736.9836.9636.9436.92
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6 8 10 12 14
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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
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37.1837.1637.1437.1237.137.0837.0637.0437.023736.9836.9636.9436.9236.9
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6 8 10 12 14
6
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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
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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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37.053736.9536.936.8536.836.7536.736.6536.636.5536.5
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6 8 10 12 14
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36.836.736.636.536.436.336.236.13635.9
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6 8 10 12 14
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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
