A WEIGHTED RANDOMIZED KACZMARZ METHOD FOR SOLVING LINEAR SYSTEMS

The Kaczmarz method for solving a linear system Ax = b interprets such a system as a collection of equations < a(i), x > = b(i), where a(i) is the i-th row of A. It then picks such an equation and corrects x(k+1) = x(k)+lambda a(i) where lambda is chosen so that the i-th equation is satisfied. Convergence rates are difficult to establish. Strohmer & Vershynin established that if the order of equations is chosen randomly (with likelihood proportional to the size of parallel to a(i)parallel to(2)(l2)), then E parallel to x(k) - x parallel to(l2) converges exponentially. We prove that if the i-th row is selected with likelihood proportional to vertical bar < a(i), x(k)> - b(i)vertical bar(p), where 0 < p < infinity, then E parallel to x(k) - x parallel to(l2) converges faster than the purely random method. As p -> infinity, the method de-randomizes and explains, among other things, why the maximal correction method works well.