Browsing by Author "Buchholz, P."

Now showing 1 - 6 of 6

Open Access
Block SOR for Kronecker structured representations
(Elsevier, 2004) Buchholz, P.; Dayar, Tuğrul
The Kronecker structure of a hierarchical Markovian model (HMM) induces nested block partitionings in the transition matrix of its underlying Markov chain. This paper shows how sparse real Schur factors of certain diagonal blocks of a given partitioning induced by the Kronecker structure can be constructed from smaller component matrices and their real Schur factors. Furthermore, it shows how the column approximate minimum degree (COLAMD) ordering algorithm can be used to reduce fill-in of the remaining diagonal blocks that are sparse LU factorized. Combining these ideas, the paper proposes three-level block successive over-relaxation (BSOR) as a competitive steady state solver for HMMs. Finally, on a set of numerical experiments it demonstrates how these ideas reduce storage required by the factors of the diagonal blocks and improve solution time compared to an all LU factorization implementation of the BSOR solver. © 2004 Elsevier Inc. All rights reserved.
Open Access
Compact representation of solution vectors in Kronecker-based Markovian analysis
(Springer, 2016-08) Buchholz, P.; Dayar, Tuğrul; Kriege, J.; Orhan, M. Can
It is well known that the infinitesimal generator underlying a multi-dimensional Markov chain with a relatively large reachable state space can be represented compactly on a computer in the form of a block matrix in which each nonzero block is expressed as a sum of Kronecker products of smaller matrices. Nevertheless, solution vectors used in the analysis of such Kronecker-based Markovian representations still require memory proportional to the size of the reachable state space, and this becomes a bigger problem as the number of dimensions increases. The current paper shows that it is possible to use the hierarchical Tucker decomposition (HTD) to store the solution vectors during Kroneckerbased Markovian analysis relatively compactly and still carry out the basic operation of vector-matrix multiplication in Kronecker form relatively efficiently. Numerical experiments on two different problems of varying sizes indicate that larger memory savings are obtained with the HTD approach as the number of dimensions increases. © Springer International Publishing Switzerland 2016.
Open Access
Comparison of multilevel methods for kronecker-based Markovian representations
(Springer, 2004) Buchholz, P.; Dayar T.
The paper presents a class of numerical methods to compute the stationary distribution of Markov chains (MCs) with large and structured state spaces. A popular way of dealing with large state spaces in Markovian modeling and analysis is to employ Kronecker-based representations for the generator matrix and to exploit this matrix structure in numerical analysis methods. This paper presents various multilevel (ML) methods for a broad class of MCs with a hierarchcial Kronecker structure of the generator matrix. The particular ML methods are inspired by multigrid and aggregation-disaggregation techniques, and differ among each other by the type of multigrid cycle, the type of smoother, and the order of component aggregation they use. Numerical experiments demonstrate that so far ML methods with successive over-relaxation as smoother provide the most effective solvers for considerably large Markov chains modeled as HMMs with multiple macrostates.
Open Access
Efficient transient analysis of a class of compositional fluid stochastic petri nets
(IEEE, 2018) Buchholz, P.; Dayar, Tuğrul
Fluid Stochastic Petri Nets (FSPNs) which have discrete and continuous places are an established model class to describe and analyze several dependability problems for computer systems, software architectures or critical infrastructures. Unfortunately, their analysis is faced with the curse of dimensionality resulting in very large systems of differential equations for a sufficiently accurate analysis. This contribution introduces a class of FSPNs with a compositional structure and shows how the underlying stochastic process can be described by a set of coupled partial differential equations. Using semi discretization, a set of linear ordinary differential equations is generated which can be described by a (hierarchical) sum of Kronecker products. Based on this compact representation of the transition matrix, a numerical solution approach is applied which also represents transient solution vectors in compact form using the recently developed concept of a Hierarchical Tucker Decomposition. The applicability of the approach is presented in a case study analyzing a degrading software system with rejuvenation, restart, and replication.
Open Access
On compact solution vectors in Kronecker-based Markovian analysis
(Elsevier, 2017) Buchholz, P.; Dayar T.; Kriege, J.; Orhan, M. C.
State based analysis of stochastic models for performance and dependability often requires the computation of the stationary distribution of a multidimensional continuous-time Markov chain (CTMC). The infinitesimal generator underlying a multidimensional CTMC with a large reachable state space can be represented compactly in the form of a block matrix in which each nonzero block is expressed as a sum of Kronecker products of smaller matrices. However, solution vectors used in the analysis of such Kronecker-based Markovian representations require memory proportional to the size of the reachable state space. This implies that memory allocated to solution vectors becomes a bottleneck as the size of the reachable state space increases. Here, it is shown that the hierarchical Tucker decomposition (HTD) can be used with adaptive truncation strategies to store the solution vectors during Kronecker-based Markovian analysis compactly and still carry out the basic operations including vector–matrix multiplication in Kronecker form within Power, Jacobi, and Generalized Minimal Residual methods. Numerical experiments on multidimensional problems of varying sizes indicate that larger memory savings are obtained with the HTD approach as the number of dimensions increases. © 2017 Elsevier B.V.
Open Access
On the convergence of a class of multilevel methods for large sparse Markov chains
(Society for Industrial and Applied Mathematics, 2007) Buchholz, P.; Dayar, T.
This paper investigates the theory behind the steady state analysis of large sparse Markov chains with a recently proposed class of multilevel methods using concepts from algebraic multigrid and iterative aggregation- disaggregation. The motivation is to better understand the convergence characteristics of the class of multilevel methods and to have a clearer formulation that will aid their implementation. In doing this, restriction (or aggregation) and prolongation (or disaggregation) operators of multigrid are used, and the Kronecker-based approach for hierarchical Markovian models is employed, since it suggests a natural and compact definition of grids (or levels). However, the formalism used to describe the class of multilevel methods for large sparse Markov chains has no influence on the theoretical results derived. © 2007 Society for Industrial and Applied Mathematics.