Abstract
<p>In the iterative solution of $n$ linear algebraic equations ${\bf B}{\bf x}={\bf b}$<br />by using the steepest descent method, i.e.,<br />${\bf x}_{k+1}={\bf x}_k-\alpha_k {\bf B}^{\mbox{\scriptsize T}}{\bf r}_k$,<br />it is known that the steplength<br />$\alpha_k:={\bf r}_k^{\mbox{\scriptsize T}} {\bf A}{\bf r}_k/\|{\bf A}{\bf r}_k\|^2$ causes<br />a slow convergence, where ${\bf r}={\bf B}{\bf x}-{\bf b}$ is the residual vector and ${\bf A}={\bf B}{\bf B}^{\mbox{\scriptsize T}}$.<br />In this paper we study the residual symmetry of the residual dynamics<br />for a scaled residual vector ${\bf y}\in {\mathbb S}^{n-1}_{\|{\bf r}_0\|}$, which<br />as expressed in the augmented space is a nonlinear Lorentzian dynamical system, and is endowed with a cone structure in the Minkowski<br />space with the Lorentz group $SO_o(n,1)$ being the internal symmetry group. Consequently,<br />we can modify the steplength to $\alpha_k={\bf y}_k^{\mbox{\scriptsize T}} {\bf A}{\bf y}_k/\|{\bf A}{\bf y}_k\|^2$<br />with ${\bf y}_k$ being computed by a Lorentz group algorithm (LGA)<br />based on $SO_o(n,1)$,<br />which can significantly improve the convergence speed and enhance the stability.<br />Several linear inverse problems are used to assess the numerical performance of the LGA.</p>
Highlights
In this paper we study the internal symmetry of the residual dynamics, abbreviated as residual symmetry, which is defined in terms of the residual vector: r(x) = Bx − b, (1)for a system of linear algebraic equations (LAEs): Bx = b, (2)where x ∈ Rn is an unknown vector to be determined from a given coefficient matrix B ∈ Rn×n and the input vector b ∈ Rn
In this paper we study the residual symmetry of the residual dynamics for a scaled residual vector y ∈ Sn∥r−01∥, which as expressed in the augmented space is a nonlinear Lorentzian dynamical system, and is endowed with a cone structure in the Minkowski space with the Lorentz group S Oo(n, 1) being the internal symmetry group
In this paper we try to answer these problems and propose a theoretical foundation to modify the steplength in Equation (5) from the Lorentz-group symmetry which is acting on a future cone in the Minkowski space
Summary
To solve the ill-posed linear problems, there are several methods developed after the pioneering work of Tikhonov & Arsenin (1977). To account of the sensitivity to noise it is often used a regularization method to solve the ill-posed problem [Kunisch & Jou (1998); Wang & Xiao (2001); Xie & Jou (2002); Resmerita (2005)], where a suitable regularization parameter is used to depress the bias in the computed solution by a better balance of approximation error and propagated data error. In this paper we try to answer these problems and propose a theoretical foundation to modify the steplength in Equation (5) from the Lorentz-group symmetry which is acting on a future cone in the Minkowski space.
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