Estimation of Spectral Bounds in Gradient Algorithms
We consider the solution of linear systems of equations Ax=b, with A a symmetric positive-definite matrix in ℝn×n, through Richardson-type iterations or, equivalently, the minimization of convex quadratic functions (1/2)(Ax,x)−(b,x) with a gradient algorithm. The use of step-sizes asymptotically distributed with the arcsine distribution on the spectrum of A then yields an asymptotic rate of convergence after k<n iterations, k→∞, that coincides with that of the conjugate-gradient algorithm in the worst case. However, the spectral bounds m and M are generally unknown and thus need to be estimated to allow the construction of simple and cost-effective gradient algorithms with fast convergence. It is the purpose of this paper to analyse the properties of estimators of m and M based on moments of probability measures νk defined on the spectrum of A and generated by the algorithm on its way towards the optimal solution. A precise analysis of the behavior of the rate of convergence of the algorithm is also given. Two situations are considered: (i) the sequence of step-sizes corresponds to i.i.d. random variables, (ii) they are generated through a dynamical system (fractional parts of the golden ratio) producing a low-discrepancy sequence. In the first case, properties of random walk can be used to prove the convergence of simple spectral bound estimators based on the first moment of νk. The second option requires a more careful choice of spectral bounds estimators but is shown to produce much less fluctuations for the rate of convergence of the algorithm.
KeywordsEstimation of leading eigenvalues Arcsine distribution Gradient algorithms Conjugate gradient Fibonacci numbers
Mathematics Subject Classification65F10 65F15
- 6.Hall, P., Heyde, C.C.: Martingale Limit Theory and Its Applications. Academic Press, New York (1980) Google Scholar
- 18.Saad, Y.: Iterative Methods for Sparse Linear Systems. Society for Industrial and Applied Mathematics, Philadelphia (2008) Google Scholar
- 21.van der Vorst, H.A.: Iterative Methods for Large Linear Systems. Utrecht University, Utrecht (2000) Google Scholar