Solving the Ill-posed linear Problem by Using the Specified Best Vector as a Descent Direction in Iterative Algorithms
Date Issued
2012
Date
2012
Author(s)
Huang, Yu-Fang
Abstract
We define a monotonically increasing function of a time-like variable for solving the ill-conditioned system of linear equations ,and construct a future cone in the Minkowski space, wherein the discrete dynamics of the proposed algorithm is evolved. Then we propose two techniques to approximate the best vector ,and obtain iterative algorithms for solving . The first method is to consider the combination of the descent vector and the weighted residual vector . The parameter is best in the descent vector. In this paper we design a switching system, make the descent direction converted in both directions, and then search for the solution of the problem. The two algorithms that developed with this concept are steepest descent and optimal vector iterative algorithm (SOVIA) and mixed optimal iterative algorithm (MOIA).The second method is to use the best parameter in the imaginary space to find an optimal vector, which is called the an optimal vector iterative algorithm (OVIA).
Finally, the three algorithms are proved by several linear problems such as Hilbert linear problem and Laplace equation, and some linear inverse problems such as the back heat conduction problems, inverse external force problem and the inverse Cauchy problem. The results were compared with the CGM, RSDM, and OIA / ODV.
Subjects
Linear inverse problems
Ill-conditioned system
future cone
Steepest -descent and the optimal vector iterative algorithm(SOVIA)
Mixed optimal iterative algorithm (MOIA)
Optimal vector iterative algorithm(OVIA)
Type
thesis
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