A computational framework for two-dimensional random walks with restarts

The treatment of two-dimensional random walks in the quarter plane leads to Markov processes which involve semi-infinite matrices having Toeplitz or block Toeplitz structure plus a low-rank correction. We propose an extension of the framework introduced in [D. A. Bini, S. Massei, and B. Meini, Math. Comp., 87 (2018), pp. 2811-2830] which allows us to deal with more general situations such as processes involving restart events. This is motivated by the need for modeling processes that can incur in unexpected failures like computer system reboots. We present a theoretical analysis of an enriched Banach algebra that, combined with appropriate algorithms, enables the numerical treatment of these problems. The results are applied to the solution of bidimensional quasi-birth-death processes with infinitely many phases which model random walks in the quarter plane, relying on the matrix analytic approach. The reliability of our approach is confirmed by extensive numerical experimentation on several case studies

A computational framework for two-dimensional random walks with restarts

The treatment of two-dimensional random walks in the quarter plane leads to Markov processes which involve semi-infinite matrices having Toeplitz or block Toeplitz structure plus a low-rank correction. Finding the steady state probability distribution of the process requires to perform operations involving these structured matrices. We propose an extension of the framework of [5] which allows to deal with more general situations such as processes involving restart events. This is motivated by the need for modeling processes that can incur in unexpected failures like computer system reboots. Algebraically, this gives rise to corrections with infinite support that cannot be treated using the tools currently available in the literature. We present a theoretical analysis of an enriched Banach algebra that, combined with appropriate algorithms, enables the numerical treatment of these problems. The results are applied to the solution of bidimensional Quasi-Birth-Death processes with infinitely many phases which model random walks in the quarter plane, relying on the matrix analytic approach. This methodology reduces the problem to solving a quadratic matrix equation with coefficients of infinite size. We provide conditions on the transition probabilities which ensure that the solution of interest of the matrix equation belongs to the enriched algebra. The reliability of our approach is confirmed by extensive numerical experimentation on some case studies

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282