TY - EJOU AU - Liu, Chein-Shan TI - The Concept of Best Vector Used to Solve Ill-Posed Linear Inverse Problems T2 - Computer Modeling in Engineering \& Sciences PY - 2012 VL - 83 IS - 5 SN - 1526-1506 AB - 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. KW - Linear inverse problems KW - Ill-conditioned linear system KW - Steepest-descent and optimal vector iterative algorithm (SOVIA) KW - Mixed optimal iterative algorithm (MOIA) KW - Optimal vector iterative algorithm (OVIA) KW - Future cone KW - Invariant-manifold DO - 10.3970/cmes.2012.083.499