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The Concept of Best Vector Used to Solve Ill-Posed Linear Inverse Problems

Chein-Shan Liu

Computer Modeling in Engineering & Sciences 2012, 83(5), 499-526.


The iterative algorithms based on the concept of best vector are proposed to solve an ill-conditioned linear system: Bx-b=0, which might be a discretization of linear inverse problem. In terms of r:=Bx-b and a monotonically increasing positive function Q(t) of a time-like variable t, we define a future cone in the Minkowski space, wherein the discrete dynamics of the proposed algorithm is evolved. We propose two methods to approximate the best vector B-1r, and obtain three iterative algorithms for solving x, which we label them as the steepest-descent and optimal vectors iterative algorithm (SOVIA), the mixed optimal iterative algorithm (MOIA), as well as the optimal vector iterative algorithm (OVIA). These algorithms are compared with the relaxed steepest descent method (RSDM), the conjugate gradient method (CGM) and an optimal iterative algorithm with an optimal descent vector (OIA/ODV) by testing several ill-posed linear inverse problems.


Linear inverse problems, Ill-conditioned linear system, Steepest-descent and optimal vector iterative algorithm (SOVIA), Mixed optimal iterative algorithm (MOIA), Optimal vector iterative algorithm (OVIA), Future cone, Invariant-manifold

Cite This Article

Liu, C. (2012). The Concept of Best Vector Used to Solve Ill-Posed Linear Inverse Problems. CMES-Computer Modeling in Engineering & Sciences, 83(5), 499–526.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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