1,380 research outputs found
Matrix geometric approach for random walks: stability condition and equilibrium distribution
In this paper, we analyse a sub-class of two-dimensional homogeneous nearest
neighbour (simple) random walk restricted on the lattice using the matrix
geometric approach. In particular, we first present an alternative approach for
the calculation of the stability condition, extending the result of Neuts drift
conditions [30] and connecting it with the result of Fayolle et al. which is
based on Lyapunov functions [13]. Furthermore, we consider the sub-class of
random walks with equilibrium distributions given as series of product-forms
and, for this class of random walks, we calculate the eigenvalues and the
corresponding eigenvectors of the infinite matrix appearing in the
matrix geometric approach. This result is obtained by connecting and extending
three existing approaches available for such an analysis: the matrix geometric
approach, the compensation approach and the boundary value problem method. In
this paper, we also present the spectral properties of the infinite matrix
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
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
The 1999 Heineman Prize Address- Integrable models in statistical mechanics: The hidden field with unsolved problems
In the past 30 years there have been extensive discoveries in the theory of
integrable statistical mechanical models including the discovery of non-linear
differential equations for Ising model correlation functions, the theory of
random impurities, level crossing transitions in the chiral Potts model and the
use of Rogers-Ramanujan identities to generalize our concepts of Bose/Fermi
statistics. Each of these advances has led to the further discovery of major
unsolved problems of great mathematical and physical interest. I will here
discuss the mathematical advances, the physical insights and extraordinary lack
of visibility of this field of physics.Comment: Text of the 1999 Heineman Prize address given March 24 at the
Centenial Meeting of the American Physical Society in Atlanta 20 pages in
latex, references added and typos correcte
Analytic Combinatorics in Several Variables: Effective Asymptotics and Lattice Path Enumeration
The field of analytic combinatorics, which studies the asymptotic behaviour
of sequences through analytic properties of their generating functions, has led
to the development of deep and powerful tools with applications across
mathematics and the natural sciences. In addition to the now classical
univariate theory, recent work in the study of analytic combinatorics in
several variables (ACSV) has shown how to derive asymptotics for the
coefficients of certain D-finite functions represented by diagonals of
multivariate rational functions. We give a pedagogical introduction to the
methods of ACSV from a computer algebra viewpoint, developing rigorous
algorithms and giving the first complexity results in this area under
conditions which are broadly satisfied. Furthermore, we give several new
applications of ACSV to the enumeration of lattice walks restricted to certain
regions. In addition to proving several open conjectures on the asymptotics of
such walks, a detailed study of lattice walk models with weighted steps is
undertaken.Comment: PhD thesis, University of Waterloo and ENS Lyon - 259 page
Rational Krylov and ADI iteration for infinite size quasi-Toeplitz matrix equations
We consider a class of linear matrix equations involving semi-infinite
matrices which have a quasi-Toeplitz structure. These equations arise in
different settings, mostly connected with PDEs or the study of Markov chains
such as random walks on bidimensional lattices. We present the theory
justifying the existence in an appropriate Banach algebra which is
computationally treatable, and we propose several methods for their solutions.
We show how to adapt the ADI iteration to this particular infinite dimensional
setting, and how to construct rational Krylov methods. Convergence theory is
discussed, and numerical experiments validate the proposed approaches
- …