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Multigrid methods for tensor structured Markov chains with low rank approximation [article]

Matthias Bolten and Karsten Kahl and Sonja Sokolović
2015 arXiv   pre-print
Algebraic multigrid methods have proven to be efficient when dealing with Markov chains without using tensor structure.  ...  Tensor structured Markov chains are part of stochastic models of many practical applications, e.g., in the description of complex production or telephone networks.  ...  Another recent approach for approximating the stationary distribution of tensor structured Markov chains by a low rank tensor was presented in [33] , based on similar techniques for eigenvalue computations  ... 
arXiv:1412.0937v2 fatcat:f6w6bau3w5ahtizobuq2rnnqoy

Multigrid methods combined with low-rank approximation for tensor structured Markov chains [article]

Matthias Bolten, Karsten Kahl, Daniel Kressner, Francisco Macedo, and Sonja Sokolović
2016 arXiv   pre-print
Our algorithm combines a tensorized multigrid method with AMEn, an optimization-based low-rank tensor solver, for addressing coarse grid problems.  ...  In this work, we propose a novel tensor-based algorithm to address such tensor structured Markov chains.  ...  We have proposed a novel combination of two methods, AMEn and Multigrid, for computing the stationary distribution of large-scale tensor structured Markov chains.  ... 
arXiv:1605.06246v1 fatcat:j6y5o3sgdzd4nbhgixam7zitgu

Multigrid methods combined with low-rank approximation for tensor-structured Markov chains

Matthias Bolten, Karsten Kahl, Daniel Kressner, Francisco Macedo, Sonja Sokolović
2018 Electronic Transactions on Numerical Analysis  
Our algorithm combines a tensorized multigrid method with AMEn, an optimization-based low-rank tensor solver, for addressing coarse grid problems.  ...  In this work, we propose a novel tensor-based algorithm to address such tensor-structured Markov chains.  ...  To a certain extent, this can be avoided by reducing each n k with the tensorized ETNA Kent State University and Johann Radon Institute (RICAM) LOW-RANK TENSOR MULTIGRID FOR MARKOV CHAINS 349 multigrid  ... 
doi:10.1553/etna_vol48s348 fatcat:hwpj4q6v4jdsjb66fo72qbdmpu

MATHICSE Technical Report : Multigrid methods combined with low-rank approximation for tensor structured Markov chains

Matthias Bolten, Karsten Kahl, Daniel Kressner, Francisco Santos Paredes Quartin De Macedo, Sonja Sokolović
2019
Our algorithm combines a tensorized multigrid method with AMEn, an optimization-based low-rank tensor solver, for addressing coarse grid problems.  ...  In this work, we propose a novel tensor-based algorithm to address such tensor structured Markov chains.  ...  We have proposed a novel combination of two methods, AMEn and Multigrid, for computing the stationary distribution of large-scale tensor structured Markov chains.  ... 
doi:10.5075/epfl-mathicse-271079 fatcat:hakr4uph75gorknm5a5hyw5com

Finding Steady States of Communicating Markov Processes Combining Aggregation/Disaggregation with Tensor Techniques [chapter]

Francisco Macedo
2016 Lecture Notes in Computer Science  
In this work, we develop algorithms for the approximation of steady states of structured Markov chains that consider tensor train decompositions, combined with wellestablished techniques for this problem  ...  This state space explosion severely impairs the use of standard methods for the numerical analysis of such Markov chains.  ...  I thank Daniel Kressner (EPF Lausanne) for helpful discussions.  ... 
doi:10.1007/978-3-319-46433-6_4 fatcat:tnfjezxklvfdhk2fjfact3o7hm

High-Dimensional Stochastic Optimal Control using Continuous Tensor Decompositions [article]

Alex A. Gorodetsky, Sertac Karaman, Youssef M. Marzouk
2018 arXiv   pre-print
We propose novel dynamic programming algorithms that alleviate the curse of dimensionality in problems that exhibit certain low-rank structure.  ...  This approach realizes substantial computational savings in "compressible" problem instances, where value functions admit low-rank approximations.  ...  ACKNOWLEDGEMENTS We thank Ezra Tal for pointing out the proof for Lemma 1.  ... 
arXiv:1611.04706v2 fatcat:qip42az4uvghrodgxlcx757j4m

An Adaptive Algebraic Multigrid Algorithm for Low-Rank Canonical Tensor Decomposition

Hans De Sterck, Killian Miller
2013 SIAM Journal on Scientific Computing  
A new algorithm based on algebraic multigrid is presented for computing the rank-R canonical decomposition of a tensor for small R.  ...  Transfer operators and coarse-level tensors are constructed in an adaptive setup phase that combines multiplicative correction and bootstrap algebraic multigrid.  ...  methods for Markov chains [4, 36, 39] .  ... 
doi:10.1137/110855934 fatcat:efyw3wdphfccvhiasbeqk4kon4

An adaptive algebraic multigrid algorithm for low-rank canonical tensor decomposition [article]

Hans De Sterck, Killian Miller
2011 arXiv   pre-print
This paper presents a multigrid algorithm for the computation of the rank-R canonical decomposition of a tensor for low rank R.  ...  Transfer operators and coarse-level tensors are constructed in an adaptive setup phase based on multiplicative correction and on Bootstrap algebraic multigrid.  ...  methods for Markov chains [35, 32, 3] .  ... 
arXiv:1111.6091v1 fatcat:hqsxms2m6fe6vgsiz332r5fmfm

Least Squares Ranking on Graphs [article]

Anil N. Hirani, Kaushik Kalyanaraman, Seth Watts
2011 arXiv   pre-print
These connections are to theoretical computer science (spectral graph theory, and multilevel methods for graph Laplacian systems); numerical analysis (algebraic multigrid, and finite element exterior calculus  ...  Another aim is to use our numerical experiments for guidance on selecting methods and exposing the need for further development.  ...  We thank Nathan Dunfield, Xiaoye Jiang, Rich Lehoucq, Luke Olson, Jacob Schroder, and Han Wang for useful discussions.  ... 
arXiv:1011.1716v4 fatcat:bsnlc4teb5bpvb5vlcsld2rcue

Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions

Andrzej Cichocki, Namgil Lee, Ivan Oseledets, Anh-Huy Phan, Qibin Zhao, Danilo P. Mandic
2016 Foundations and Trends® in Machine Learning  
It is therefore timely and valuable for the multidisciplinary research community to review tensor decompositions and tensor networks as emerging tools for large-scale data analysis and data mining.  ...  We provide the mathematical and graphical representations and interpretation of tensor networks, with the main focus on the Tucker and Tensor Train (TT) decompositions and their extensions or generalizations  ...  Multigrid methods for tensor structured Markov chains with low rank approximation. SIAM Journal on Scientific Computing, 38(2):A649–A667, 2016.  ... 
doi:10.1561/2200000059 fatcat:ememscddezeovamsoqrcpp33z4

ECP Math Libraries: Capabilities and Application Engagement [article]

Sherry Li, Lois Curfman McInnes, ECP Math Libraries Community
2021 figshare.com  
The application teams will have an opportunity for conversations with the math library developers to explain their needs, particularly focus on circumstances of applications teams, with goals of advancing  ...  also can customize library capabilities as needed for their projects.  ...  using structured and unstructured data • Statistical analysis • Fast surrogates for multiphysics simulations -Hierarchical data representation for data-reduction and data-mining -Markov Chain Monte Carlo  ... 
doi:10.6084/m9.figshare.14346254.v1 fatcat:5j6utwsevrehhif2gytsraob7m

Scalable and efficient algorithms for the propagation of uncertainty from data through inference to prediction for large-scale problems, with application to flow of the Antarctic ice sheet

Tobin Isaac, Noemi Petra, Georg Stadler, Omar Ghattas
2015 Journal of Computational Physics  
This property can be exploited to construct a low rank approximation of the linearized parameter-to-observable map.  ...  We present efficient and scalable algorithms for this end-to-end, data-to-prediction process under the Gaussian approximation and in the context of modeling the flow of the Antarctic ice sheet and its  ...  The method of choice is Markov chain Monte Carlo (MCMC), which judiciously samples the posterior so that sample statistics can be computed.  ... 
doi:10.1016/j.jcp.2015.04.047 fatcat:fpshnibcz5curhq6wtmznqc7vm

Extreme-scale UQ for Bayesian inverse problems governed by PDEs

Tan Bui-Thanh, Carsten Burstedde, Omar Ghattas, James Martin, Georg Stadler, Lucas C. Wilcox
2012 2012 International Conference for High Performance Computing, Networking, Storage and Analysis  
To overcome the curse of dimensionality of conventional methods, we exploit the fact that the data are typically informative about low-dimensional manifolds of parameter space to construct low rank approximations  ...  We apply the method to the Bayesian solution of an inverse problem in 3D global seismic wave propagation with over one million uncertain earth model parameters, 630 million wave propagation unknowns, on  ...  A RANDOMIZED ALGORITHM FOR LOW-RANK HESSIAN We exploit this structure to construct a low rank approximation ofH misfit using randomized algorithms for approximate matrix decomposition [13] , [14] .  ... 
doi:10.1109/sc.2012.56 dblp:conf/sc/Bui-ThanhBGMSW12 fatcat:epwc2nprxvc55h3j5mfzujoxqu

Author index to volumes 61–80 (1984–1986)

1986 Linear Algebra and its Applications  
) FRYDMAN, HALINA: A Structure of Doubly Stochastic Markov Chains, 67:51 (1985) FUHRMANN, P.  ...  Method with Mesh Ratio Two for Solving Model Prob- lems, 79:23 (1986) :ARLSON, DAVID: What Are Schur Comple- ments, Anyway?  ... 
doi:10.1016/0024-3795(86)90286-7 fatcat:3ivgcj4ikve65dlalfohnzkv2m

Improving the forward solver for the complete electrode model in EIT using algebraic multigrid

M. Soleimani, C.E. Powell, N. Polydorides
2005 IEEE Transactions on Medical Imaging  
Implementing a standard finite element method for this particular forward problem yields a linear system that is symmetric and positive definite and solvable via the conjugate gradient method.  ...  Index Terms-Algebraic multigrid, complete electrode model, electrical impedance tomography, finite element method, forward problem, preconditioning.  ...  ACKNOWLEDGMENT The authors would like to thank the SCI Institute at the University of Utah for providing the head models.  ... 
doi:10.1109/tmi.2005.843741 pmid:15889545 fatcat:5tzwogdkmbbgdlzjf5lydezdum
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