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Bounding stationary averages of polynomial diffusions via semidefinite programming

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Title: Bounding stationary averages of polynomial diffusions via semidefinite programming
Authors: Kuntz, J
Ottobre, M
Stan, G-B
Barahona, M
Item Type: Journal Article
Abstract: We introduce an algorithm based on semidefinite programming that yields increasing (resp. decreasing) sequences of lower (resp. upper) bounds on polynomial stationary averages of diffusions with polynomial drift vector and diffusion coefficients. The bounds are obtained by optimising an objective, determined by the stationary average of interest, over the set of real vectors defined by certain linear equalities and semidefinite inequalities which are satisfied by the moments of any stationary measure of the diffusion. We exemplify the use of the approach through several applications: a Bayesian inference problem; the computation of Lyapunov exponents of linear ordinary differential equations perturbed by multiplicative white noise; and a reliability problem from structural mechanics. Additionally, we prove that the bounds converge to the infimum and supremum of the set of stationary averages for certain SDEs associated with the computation of the Lyapunov exponents, and we provide numerical evidence of convergence in more general settings.
Issue Date: 20-Dec-2016
Date of Acceptance: 4-Oct-2016
URI: http://hdl.handle.net/10044/1/41170
DOI: https://dx.doi.org/10.1137/16M107801X
ISSN: 1095-7197
Publisher: Society for Industrial and Applied Mathematics
Start Page: A3891
End Page: A3920
Journal / Book Title: SIAM Journal on Scientific Computing
Volume: 38
Issue: 6
Sponsor/Funder: Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
Funder's Grant Number: EP/I017267/1
EP/I032223/1
EP/M002187/1
EP/N014529/1
Keywords: Science & Technology
Physical Sciences
Mathematics, Applied
Mathematics
stochastic differential equations
stationary measures
semidefinite programming
moment problems
Lyapunov exponents
CONTINUOUS-TIME PROCESSES
DIFFERENTIAL-EQUATIONS
MARKOVIAN PROCESSES
LYAPUNOV EXPONENTS
MOMENT CONDITIONS
LINEAR-SYSTEMS
STABILITY
RELAXATIONS
APPROXIMATION
NOISE
math.PR
math.OC
60H10, 60H35, 90C22, 37M25
0102 Applied Mathematics
0103 Numerical And Computational Mathematics
0802 Computation Theory And Mathematics
Numerical & Computational Mathematics
Publication Status: Published
Open Access location: http://epubs.siam.org/doi/pdf/10.1137/16M107801X
Appears in Collections:Faculty of Engineering
Bioengineering
Mathematics
Applied Mathematics and Mathematical Physics
Faculty of Natural Sciences



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