Tensor structured Markov chains are part of stochastic models of many practical applications, e.g., in the description of complex production or telephone networks. The most interesting question in Markov chain models is the determination of the stationary distribution as a description of the long term behavior of the system. This involves the computation of the eigenvector corresponding to the dominant eigenvalue or equivalently the solution of a singular linear system of equations. Due to the tensor structure of the models the dimension of the operators grows rapidly and a direct solution without exploiting the tensor structure becomes infeasible. Algebraic multigrid methods have proven to be efficient when dealing with Markov chains without using tensor structure. In this work we present an approach to adapt the algebraic multigrid framework to the tensor frame, not only using the tensor structure in matrix-vector multiplications, but also tensor structured coarse-grid operators and tensor representations of the solution vector.
Multigrid methods combined with low-rank approximation for tensor structured Markov chains
Institut für Mathematik, Universität Kassel（卡塞尔大学数学研究所）；Fakultät für Mathematik und Naturwissenschaften, Bergische Universität Wuppertal（伍珀塔尔大学自然科学学院）；EPF Lausanne, SB-MATHICSE-ANCHP（洛桑联邦理工学院数学与应用计算中心）；IST, Alameda Campus（里斯本科技大学阿拉梅达校区）
Hierarchical Gaussian Random Fields for Multilevel Markov Chain Monte Carlo: Coupling Stochastic Partial Differential Equation and The Karhunen-Loève Decomposition
