An Overview of Quasi-Monte Carlo Methods in Control Systems

Many control problems are so complex that analytic techniques fail to solve them [2]. Furthermore, even when analytic solutions are available, they may be computationally costly [2] and generally result in very high-order compensators [3]. Due to these reasons, we tend to accept approximate answers which provide us with certain performance guarantees for such problems. Sampling methods thus come into the picture to try and remedy the “cost of solution” problem by drawing samples from an appropriate space, and providing an approximate answer. For many years, random sampling has dominated the afore mentioned arena [8, 11, 4]. Random sample generation, with a uniform underlying distribution, however tends to cluster the samples on the boundary of the sample space in higher dimensions. It is for this reason that we are interested in presenting a method that distributes the points regularly in the sample space while providing deterministic guarantees on the error involved. Recently, deterministic or quasi-Monte Carlo (QMC) methods have proven superior to random methods in several applications such as the calculation of certain integrals [6], financial derivatives [7] and motion planning in robotics [10]. They have also been used for stability analysis of high speed networks [9]. In this work, we provide an overview of such deterministic quasi-Monte Carlo method of sampling, and their applications to control systems analysis and design. We present the basic concepts pertaining to quasi-Monte Carlo deterministic sampling. Such concepts include the following: Indicator functions, performance objective, generation of point sets, total variation, and error bounds

NumGfun: a Package for Numerical and Analytic Computation with D-finite Functions

This article describes the implementation in the software package NumGfun of classical algorithms that operate on solutions of linear differential equations or recurrence relations with polynomial coefficients, including what seems to be the first general implementation of the fast high-precision numerical evaluation algorithms of Chudnovsky & Chudnovsky. In some cases, our descriptions contain improvements over existing algorithms. We also provide references to relevant ideas not currently used in NumGfun

Decay properties of spectral projectors with applications to electronic structure

Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the exponential off-diagonal decay ("nearsightedness") for the density matrix of gapped systems at zero electronic temperature in both orthogonal and non-orthogonal representations, thus providing a firm theoretical basis for the possibility of linear scaling methods in electronic structure calculations for non-metallic systems. We further discuss the case of density matrices for metallic systems at positive electronic temperature. A few other possible applications are also discussed.Comment: 63 pages, 13 figure

Domains of analyticity of Lindstedt expansions of KAM tori in dissipative perturbations of Hamiltonian systems

Many problems in Physics are described by dynamical systems that are conformally symplectic (e.g., mechanical systems with a friction proportional to the velocity, variational problems with a small discount or thermostated systems). Conformally symplectic systems are characterized by the property that they transform a symplectic form into a multiple of itself. The limit of small dissipation, which is the object of the present study, is particularly interesting. We provide all details for maps, but we present also the modifications needed to obtain a direct proof for the case of differential equations. We consider a family of conformally symplectic maps $f_{\mu, \epsilon}$ defined on a $2d$-dimensional symplectic manifold $\mathcal M$ with exact symplectic form $\Omega$; we assume that $f_{\mu,\epsilon}$ satisfies $f_{\mu,\epsilon}^*\Omega=\lambda(\epsilon) \Omega$. We assume that the family depends on a $d$-dimensional parameter $\mu$ (called drift) and also on a small scalar parameter $\epsilon$. Furthermore, we assume that the conformal factor $\lambda$ depends on $\epsilon$, in such a way that for $\epsilon=0$ we have $\lambda(0)=1$ (the symplectic case). We study the domains of analyticity in $\epsilon$ near $\epsilon=0$ of perturbative expansions (Lindstedt series) of the parameterization of the quasi--periodic orbits of frequency $\omega$ (assumed to be Diophantine) and of the parameter $\mu$. Notice that this is a singular perturbation, since any friction (no matter how small) reduces the set of quasi-periodic solutions in the system. We prove that the Lindstedt series are analytic in a domain in the complex $\epsilon$ plane, which is obtained by taking from a ball centered at zero a sequence of smaller balls with center along smooth lines going through the origin. The radii of the excluded balls decrease faster than any power of the distance of the center to the origin