Linear approximation

About: Linear approximation is a research topic. Over the lifetime, 3901 publications have been published within this topic receiving 74764 citations.

Lyapunov exponents for systems with unbounded coefficients

Abstract: In this article we propose a generalization of the concept of Lyapunov exponents for discrete linear systems which may be used in the case of unbounded coefficients. We show some simplest properties of this generalization and apply it to define a generalization of regular system. Finally, we discuss the problem of stability by linear approximation. † This article was originally published with an error. This version has been corrected. Please see Corrigendum (http://dx.doi.org/10.1080/14689367.2012.756700)

Relativistic epicycles: another approach to geodesic deviations

Abstract: We solve the geodesic deviation equations for the orbital motions in the Schwarzschild metric which are close to a circular orbit. It turns out that in this particular case the equations reduce to a linear system, which after diagonalization describes just a collection of harmonic oscillators with two characteristic frequencies. The new geodesic obtained by adding this solution to the circular one, describes not only the linear approximation of Kepler's laws but also gives the right value of the perihelion advance (in the limit of almost circular orbits). We also derive the equations for higher order deviations and show how these equations lead to better approximations including the non-linear effects. The approximate orbital solutions are then inserted into the quadrupole formula to estimate the gravitational radiation from non-circular orbits.

Dyna-Style Planning with Linear Function Approximation and Prioritized Sweeping

Abstract: We consider the problem of efficiently learning optimal control policies and value functions over large state spaces in an online setting in which estimates must be available after each interaction with the world. This paper develops an explicitly model-based approach extending the Dyna architecture to linear function approximation. Dynastyle planning proceeds by generating imaginary experience from the world model and then applying model-free reinforcement learning algorithms to the imagined state transitions. Our main results are to prove that linear Dyna-style planning converges to a unique solution independent of the generating distribution, under natural conditions. In the policy evaluation setting, we prove that the limit point is the least-squares (LSTD) solution. An implication of our results is that prioritized-sweeping can be soundly extended to the linear approximation case, backing up to preceding features rather than to preceding states. We introduce two versions of prioritized sweeping with linear Dyna and briefly illustrate their performance empirically on the Mountain Car and Boyan Chain problems.