Numerical iterative methods for Markovian dependability and performability models: new results and a comparison

In this paper we deal with iterative numerical methods to solve linear systems arising in continuous-time Markov chain (CTMC) models. We develop an algorithm to dynamically tune the relaxation parameter of the successive over-relaxation method. We give a sufficient condition for the Gauss-Seidel method to converge when computing the steady-state probability vector of a finite irreducible CTMC, an a suffient condition for the Generalized Minimal Residual projection method not to converge to the trivial solution 0 when computing that vector. Finally, we compare several splitting-based iterative methods an a variant of the Generalized Minimal Residual projection method.Postprint (published version

Generalized Forward-Backward Splitting

This paper introduces the generalized forward-backward splitting algorithm for minimizing convex functions of the form $F + \sum_{i=1}^n G_i$, where $F$ has a Lipschitz-continuous gradient and the $G_i$'s are simple in the sense that their Moreau proximity operators are easy to compute. While the forward-backward algorithm cannot deal with more than $n = 1$ non-smooth function, our method generalizes it to the case of arbitrary $n$. Our method makes an explicit use of the regularity of $F$ in the forward step, and the proximity operators of the $G_i$'s are applied in parallel in the backward step. This allows the generalized forward backward to efficiently address an important class of convex problems. We prove its convergence in infinite dimension, and its robustness to errors on the computation of the proximity operators and of the gradient of $F$. Examples on inverse problems in imaging demonstrate the advantage of the proposed methods in comparison to other splitting algorithms.Comment: 24 pages, 4 figure

The Grover algorithm with large nuclear spins in semiconductors

We show a possible way to implement the Grover algorithm in large nuclear spins 1/2<I<9/2 in semiconductors. The Grover sequence is performed by means of multiphoton transitions that distribute the spin amplitude between the nuclear spin states. They are distinguishable due to the quadrupolar splitting, which makes the nuclear spin levels non-equidistant. We introduce a generalized rotating frame for an effective Hamiltonian that governs the non-perturbative time evolution of the nuclear spin states for arbitrary spin lengths I. The larger the quadrupolar splitting, the better the agreement between our approximative method using the generalized rotating frame and exact numerical calculations.Comment: 11 pages, 18 EPS figures, REVTe