## Sequential Monte Carlo Techniques For The Solution Of Linear Systems (1992)

Venue: | Journal of Scientific Computing |

Citations: | 18 - 1 self |

### BibTeX

@ARTICLE{Halton92sequentialmonte,

author = {John H. Halton},

title = {Sequential Monte Carlo Techniques For The Solution Of Linear Systems},

journal = {Journal of Scientific Computing},

year = {1992},

volume = {9},

pages = {213--257}

}

### OpenURL

### Abstract

Given a linear system Ax = b, where x is an m-vector, direct numerical methods, such as Gaussian elimination, take time O(m 3 ) to find x. Iterative numerical methods, such as the Gauss-Seidel method or SOR, reduce the system to the form x = a + Hx, whence x = år=0 #H r a; and then apply the iterations x 0 = a, x s+1 = a + Hx s , until sufficient accuracy is achieved; this takes time O(m 2 ) per iteration. They generate the truncated sums x s = år=0 s #H r a. The usual plain Monte Carlo approach uses independent "random walks," to give an approximation to the truncated sum x s , taking time O(m) per random step. Unfortunately, millions of random steps are typically needed to achieve reasonable accuracy (say, 1% r.m.s. error). Nevertheless, this is what has had to be done, if m is itself of the order of a million or more. The alternative presented here, is to apply a sequential Monte Carlo method, in which the sampling scheme is iteratively improved. Simply put, if x = y + z, where y...