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Approximate Markovian abstractions for linear stochastic systems

M. Lahijanian, S. B. Andersson, C. Belta
2012 2012 IEEE 51st IEEE Conference on Decision and Control (CDC)  
We derive the exact bound of the approximation error and an explicit expression for its growth over time.  ...  In this paper, we present a method to generate a finite Markovian abstraction for a discrete time linear stochastic system evolving in a full dimensional polytope.  ...  ACKNOWLEDGEMENTS The authors would like to thank Boyan Yordanov from Microsoft Research, Cambridge, UK for his help with polyhedral operations and refinement algorithms.  ... 
doi:10.1109/cdc.2012.6426184 dblp:conf/cdc/LahijanianAB12 fatcat:534u4ikrybdandf654bon2frpy

Page 4024 of Mathematical Reviews Vol. , Issue 97F [page]

1997 Mathematical Reviews  
Daniel (1-WI-E; Madison, W1) A characterization of bounded-input bounded-output stability for linear time-invariant systems with distributional inputs. (English summary) SIAM J.  ...  Summary: “We consider linear time-invariant operators defined on the space of distributions with left-bounded support.  ... 

Moment Propagation of Discrete-Time Stochastic Polynomial Systems using Truncated Carleman Linearization [article]

Sasinee Pruekprasert, Toru Takisaka, Clovis Eberhart, Ahmet Cetinkaya, Jérémy Dubut
2021 arXiv   pre-print
We provide upper bounds on the approximation error for each moment and show that, for large enough truncation limits, the proposed method precisely computes moments for sufficiently small degrees and numbers  ...  We propose a method to compute an approximation of the moments of a discrete-time stochastic polynomial system.  ...  to compute even higher moments and more time steps in the future.  ... 
arXiv:1911.12683v2 fatcat:2k2tuzae4vhz3av76sszjmkana

Balanced Truncation for a Class of Stochastic Jump Linear Systems and Model Reduction for Hidden Markov Models

Georgios Kotsalis, Alexandre Megretski, Munther A. Dahleh
2008 IEEE Transactions on Automatic Control  
The approximation error, which is captured by means of the stochastic 2 gain, is bounded from above by twice the sum of singular numbers associated to the truncated states, similar to the case of linear  ...  Index Terms-Balanced truncation, error bound, finite state machines, hidden Markov models, jump systems, model reduction, reduced order systems, stochastic automata, stochastic hybrid systems, stochastic  ...  ACKNOWLEDGMENT The authors would like to thank the reviewers for valuable comments and suggestions and V. Blondel for [14] .  ... 
doi:10.1109/tac.2008.2006931 fatcat:kxvip3rnxjbplezvx6cs2sgvdy

Error Analysis for the Linear Feedback Particle Filter [article]

Amirhossein Taghvaei, Prashant G. Mehta
2017 arXiv   pre-print
It is shown that the error converges to zero even with finite number of particles. The paper also presents propagation of chaos estimates for the finite-N linear filter.  ...  In this paper, the equations for empirical mean and covariance are derived for the finite-N linear FPF.  ...  We begin by presenting the McKean-Vlasov stochastic differential equation (sde) for the linear FPF algorithm. For these models, the state at time t is denoted as Xt .  ... 
arXiv:1710.11008v1 fatcat:454fw3di5fcsvdemoh2ngklmum

Finite-Time Error Bounds for Biased Stochastic Approximation with Applications to Q-Learning

Gang Wang, Georgios B. Giannakis
2020 International Conference on Artificial Intelligence and Statistics  
The resultant finite-time error bound for Q-learning is the first of its kind, in the sense that it holds: i) for the unmodified version (i.e., without making any modifications to the updates), and ii)  ...  For direct comparison with past works, we also demonstrate these bounds by applying them to Qlearning with linear function approximation, under the realistic Markov chain observation model.  ...  Hence, our finite-time error bound in Thm. 2 also holds for Q-learning with linear approximation.  ... 
dblp:conf/aistats/WangG20a fatcat:khgtq2rwhrhppfywqkl2jc4ewm

Moment Propagation Through Carleman Linearization with Application to Probabilistic Safety Analysis [article]

Sasinee Pruekprasert, Jérémy Dubut, Toru Takisaka, Clovis Eberhart, Ahmet Cetinkaya
2022 arXiv   pre-print
We provide efficient online computation methods for this propagation scheme with several error bounds for the approximation.  ...  We then truncate this deterministic system to obtain a finite-dimensional linear system, and use it for moment approximation by iteratively propagating the moments along the finite-dimensional linear dynamics  ...  Moment approximations for stochastic logistic map. 1 Fig. 4. Error bound on moment approximations.  ... 
arXiv:2201.08648v1 fatcat:d2smmzabkzag7lanzcww4huuba

Finite-Time Error Bounds For Linear Stochastic Approximation and TD Learning [article]

R. Srikant, Lei Ying
2019 arXiv   pre-print
We consider the dynamics of a linear stochastic approximation algorithm driven by Markovian noise, and derive finite-time bounds on the moments of the error, i.e., deviation of the output of the algorithm  ...  As a by-product of our analysis, we also solve the open problem of obtaining finite-time bounds for the performance of temporal difference learning algorithms with linear function approximation and a constant  ...  FINITE-TIME ERROR BOUNDS FOR LINEAR STOCHASTIC APPROXIMATION AND TD LEARNING Appendix E.  ... 
arXiv:1902.00923v3 fatcat:uqupnpuqq5gorluylrpjakepse

State-Based Confidence Bounds for Data-Driven Stochastic Reachability Using Hilbert Space Embeddings [article]

Adam J. Thorpe, Kendric R. Ortiz, Meeko M. K. Oishi
2021 arXiv   pre-print
In this paper, we compute finite sample bounds for data-driven approximations of the solution to stochastic reachability problems.  ...  We present finite sample bounds for point-based approximations of the safety probabilities through construction of probabilistic confidence bounds that are state- and input-dependent.  ...  Conclusion We provided state-and input-based finite sample bounds for the stochastic reachability probability constructed via conditional distribution embeddings.  ... 
arXiv:2010.08036v2 fatcat:b72shw3wb5dpbdzcgnywk5erxu

Optimal, Robust Predictive Control of Nonlinear Systems under Probabilistic Uncertainty using Particles

Lars Blackmore, Brian C. Williams
2007 American Control Conference (ACC)  
The method extends a previous approach for linear systems that approximates the distribution of the predicted system state using a finite number of particles.  ...  Any solution returned by the algorithm is guaranteed to be -close to a local optimum of the nonlinear stochastic control problem.  ...  Linearization approximates the system dynamics and cost function, but since the linearization region is known and bounded, the approximation error can be bounded using standard results.  ... 
doi:10.1109/acc.2007.4282699 dblp:conf/acc/BlackmoreW07 fatcat:vupvswykrrfolfzxaz6cjmgv6e

Stochastic Galerkin reduced basis methods for parametrized linear elliptic partial differential equations [article]

Sebastian Ullmann, Christopher Müller, Jens Lang
2020 arXiv   pre-print
We propose residual-corrected estimates of the parameter-dependent expectation and variance of linear functional outputs and provide respective computable error bounds.  ...  For a given value of the deterministic paremeter, a stochastic Galerkin finite element (SGFE) method can estimate the corresponding expectation and variance of a linear output at the cost of a single solution  ...  An MCRB method for linear elliptic problems with error bounds for the expectation and variance of a linear functional output is derived in [4] . Improved error bounds are provided by [5] .  ... 
arXiv:1812.08519v2 fatcat:2jz55plj5bepzdu3oetv2zuvuy

A TVD Uncertainty Quantification Method with Bounded Error Applied to Transonic Airfoil Flutter

Jeroen A. S. Witteveen, Hester Bijl
2009 Communications in Computational Physics  
certain conditions also in a bounded error in time.  ...  It is proven that the interpolation of oscillatory samples at constant phase in the UASFE method for unsteady problems results in a bounded error as function of the phase for periodic responses and under  ...  The bounded errorε(ϕ,a) as function of phase ϕ also results in a bounded error ε(t,a) in time for the Unsteady Adaptive Stochastic Finite Elements method l 1 based on first degree Newton-Cotes quadrature  ... 
doi:10.4208/cicp.2009.v6.p403 fatcat:e6yrcoixhrfzlmrra4camv77we

Finite-Time Error Bounds For Linear Stochastic Approximation andTD Learning

R. Srikant, Lei Ying
2019 Annual Conference Computational Learning Theory  
We consider the dynamics of a linear stochastic approximation algorithm driven by Markovian noise, and derive finite-time bounds on the moments of the error, i.e., deviation of the output of the algorithm  ...  As a by-product of our analysis, we also solve the problem of obtaining finite-time bounds for the performance of temporal difference learning algorithms with linear function approximation and a constant  ...  stochastic approximation can be converted to finite-time bounds on TD algorithms.  ... 
dblp:conf/colt/SrikantY19 fatcat:kniymjourrgvde5ztie644vrbm

Finite Abstractions of Stochastic Max-Plus-Linear Systems [chapter]

Dieky Adzkiya, Sadegh Esmaeil Zadeh Soudjani, Alessandro Abate
2014 Lecture Notes in Computer Science  
This work investigates the use of finite abstractions to study the finite-horizon probabilistic invariance problem over Stochastic Max-Plus-Linear (SMPL) systems.  ...  SMPL systems are probabilistic extensions of discrete-event MPL systems that are widely employed in the engineering practice for timing and synchronisation studies.  ...  M ij = λ ij for all i, j ∈ IN n . By Theorem 3 and Proposition 4, the bound of the approximation error is then E = N δ(n + 1) i,j λ ij .  ... 
doi:10.1007/978-3-319-10696-0_7 fatcat:d5m7noumcjdqti5ycdwpisz7f4

Page 7175 of Mathematical Reviews Vol. , Issue 96k [page]

1996 Mathematical Reviews  
An approximate finite-dimensional discrete-time model for a distributed parameter system of parabolic type is constructed.  ...  Under certain controllability and observability conditions and conditions on initial error, noise and smoothness of the nonlinear map featuring in the signal model, error bounds for the linearized, undriven  ... 
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