Recursive recovery of Markov transition probabilities from boundary value data
- Univ. of California, Berkeley, CA (United States)
In an effort to mathematically describe the anisotropic diffusion of infrared radiation in biological tissue Gruenbaum posed an anisotropic diffusion boundary value problem in 1989. In order to accommodate anisotropy, he discretized the temporal as well as the spatial domain. The probabilistic interpretation of the diffusion equation is retained; radiation is assumed to travel according to a random walk (of sorts). In this random walk the probabilities with which photons change direction depend upon their previous as well as present location. The forward problem gives boundary value data as a function of the Markov transition probabilities. The inverse problem requires finding the transition probabilities from boundary value data. Problems in the plane are studied carefully in this thesis. Consistency conditions amongst the data are derived. These conditions have two effects: they prohibit inversion of the forward map but permit smoothing of noisy data. Next, a recursive algorithm which yields a family of solutions to the inverse problem is detailed. This algorithm takes advantage of all independent data and generates a system of highly nonlinear algebraic equations. Pluecker-Grassmann relations are instrumental in simplifying the equations. The algorithm is used to solve the 4 x 4 problem. Finally, the smallest nontrivial problem in three dimensions, the 2 x 2 x 2 problem, is solved.
- Research Organization:
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
- Sponsoring Organization:
- USDOE; Department of Defense (DOD); National Aeronautics and Space Administration (NASA); National Science Foundation (NSF)
- DOE Contract Number:
- AC03-76SF00098
- OSTI ID:
- 10150977
- Report Number(s):
- LBL-35495; ON: DE94011914; CNN: Contract FDF-49620-92-J-0067-11792; Grant NAG3-1143; Grant DMS89-02831
- Resource Relation:
- Other Information: TH: Thesis (Ph.D.); PBD: Apr 1994
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
73 NUCLEAR PHYSICS AND RADIATION PHYSICS
97 MATHEMATICS AND COMPUTING
TISSUES
RADIATION TRANSPORT
BOUNDARY-VALUE PROBLEMS
DIFFUSION
ANISOTROPY
INFRARED RADIATION
TRANSITION AMPLITUDES
ALGORITHMS
TOMOGRAPHY
RECURSION RELATIONS
560101
663600
990200
DOSIMETRY AND MONITORING
RADIATION PHYSICS
MATHEMATICS AND COMPUTERS