Showing papers on "Asymptotic analysis published in 2010"

Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory

Abstract: In regular statistical models, the leave-one-out cross-validation is asymptotically equivalent to the Akaike information criterion. However, since many learning machines are singular statistical models, the asymptotic behavior of the cross-validation remains unknown. In previous studies, we established the singular learning theory and proposed a widely applicable information criterion, the expectation value of which is asymptotically equal to the average Bayes generalization loss. In the present paper, we theoretically compare the Bayes cross-validation loss and the widely applicable information criterion and prove two theorems. First, the Bayes cross-validation loss is asymptotically equivalent to the widely applicable information criterion as a random variable. Therefore, model selection and hyperparameter optimization using these two values are asymptotically equivalent. Second, the sum of the Bayes generalization error and the Bayes cross-validation error is asymptotically equal to $2\lambda/n$, where $\lambda$ is the real log canonical threshold and $n$ is the number of training samples. Therefore the relation between the cross-validation error and the generalization error is determined by the algebraic geometrical structure of a learning machine. We also clarify that the deviance information criteria are different from the Bayes cross-validation and the widely applicable information criterion.

An asymptotic analysis of the mean first passage time for narrow escape problems: part ii: the sphere ∗

Abstract: The mean first passage time (MFPT) is calculated for a Brownian particle in a spherical domain in $\mathbb{R}^3$ that contains N small nonoverlapping absorbing windows, or traps, on its boundary. For the unit sphere, the method of matched asymptotic expansions is used to derive an explicit three-term asymptotic expansion for the MFPT for the case of N small locally circular absorbing windows. The third term in this expansion, not previously calculated, depends explicitly on the spatial configuration of the absorbing windows on the boundary of the sphere. The three-term asymptotic expansion for the average MFPT is shown to be in very close agreement with full numerical results. The average MFPT is shown to be minimized for trap configurations that minimize a certain discrete variational problem. This variational problem is closely related to the well-known optimization problem of determining the minimum energy configuration for N repelling point charges on the unit sphere. Numerical results, based on globa...

An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part I: Two-Dimensional Domains

Abstract: The mean first passage time (MFPT) is calculated for a Brownian particle in a bounded two-dimensional domain that contains N small nonoverlapping absorbing windows on its boundary. The reciprocal o...

Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS

Abstract: Micro- and nanoelectromechanical systems (MEMS and NEMS), which combine electronics with miniature-size mechanical devices, are essential components of modern technology It is the mathematical model describing 'electrostatically actuated' MEMS that is addressed in this monograph Even the simplified models that the authors deal with still lead to very interesting second- and fourth-order nonlinear elliptic equations (in the stationary case) and to nonlinear parabolic equations (in the dynamic case) While nonlinear eigenvalue problems - where the stationary MEMS models fit - are a well-developed field of PDEs, the type of inverse square nonlinearity that appears here helps shed a new light on the class of singular supercritical problems and their specific challenges Besides the practical considerations, the model is a rich source of interesting mathematical phenomena Numerics, formal asymptotic analysis, and ODE methods give lots of information and point to many conjectures However, even in the simplest idealized versions of electrostatic MEMS, one essentially needs the full available arsenal of modern PDE techniques to do the required rigorous mathematical analysis, which is the main objective of this volume This monograph could therefore be used as an advanced graduate text for a motivational introduction to many recent methods of nonlinear analysis and PDEs through the analysis of a set of equations that have enormous practical significance

A new model for gas flow in pipe networks

Abstract: We introduce a new model for gas dynamics in pipe networks by asymptotic analysis. The model is derived from the isothermal Euler equations. We present the derivation of the model as well as numerical results illustrating the validity and its properties. We compare the new model with existing models from the mathematical and engineering literature. We further give numerical results on a sample network. Copyright © 2009 John Wiley & Sons, Ltd.

Asymptotic Behavior of the Solutions of a Class of Rational Difference Equations

Abstract: In this paper we study the asymptotic behavior of the positive solutions of certain rational difference equations. AMS Subject Classifications: 39A10.

The ∂ Steepest Descent Method and the Asymptotic Behavior of Polynomials Orthogonal on the Unit Circle with Fixed and Exponentially Varying Nonanalytic Weights

Abstract: The steepest descent method for asymptotic analysis of matrix Riemann-Hilbert problems was introduced by Deift and Zhou in 1993 [14]. A matrix Riemann-Hilbert problem is specified by giving a triple (Σ, v,N) consisting of an oriented contour Σ in the complex z-plane, a matrix function v : Σ → SL(N) which is usually taken to be continuous except at self-intersection points of Σwhere a certain compatibility condition is required, and a normalization condition N as z → ∞. If Σ is not bounded, certain asymptotic conditions are required of v in order to have compatibility with the normalization condition. Consider an analytic functionM : C \ Σ → SL(N) taking continuous boundary valuesM+(z) (resp., M−(z)) on Σ from the left (resp., right). The Riemann-Hilbert problem (Σ, v,N) is then to find such a matrix M(z) satisfying the normalization condition N as z → ∞ and the jump condition M+(z) = M−(z)v(z) whenever z is a non-self-intersection point of Σ (so the left and right boundary values are indeed well defined). The steepest descent method of Deift and Zhou applies to certain Riemann-Hilbert problems where the jumpmatrix v(z) depends on an auxiliary control parameter, and is a method for extracting asymptotic properties of the solution M(z) (and indeed proving the existence and

New asymptotic expansion for the Gamma function

Abstract: Using a series transformation, the Stirling-De Moivre asymptotic series approximation to the Gamma function is converted into a new one with better convergence properties. The new formula is being compared with those of Stirling, Laplace, and Ramanujan for real arguments greater than 0.5 and turns out to be, for equal number of “correction” terms, numerically superior to all of them. As a side benefit, a closed-form approximation has turned up during the analysis which is about as good as 3rd order Stirling’s (maximum relative error smaller than 1e − 10 for real arguments greater or equal to 24).

Expansions and Asymptotics for Statistics

Abstract: Introduction Expansions and approximations The role of asymptotics Mathematical preliminaries Two complementary approaches General Series Methods A quick overview Power series Enveloping series Asymptotic series Superasymptotic and hyperasymptotic series Asymptotic series for large samples Generalised asymptotic expansions Notes Pade Approximants and Continued Fractions The Pade table Pade approximations for the exponential function Two applications Continued fraction expansions A continued fraction for the normal distribution Approximating transforms and other integrals Multivariate extensions Notes The Delta Method and Its Extensions Introduction to the delta method Preliminary results The delta method for moments Using the delta method in Maple Asymptotic bias Variance stabilising transformations Normalising transformations Parameter transformations Functions of several variables Ratios of averages The delta method for distributions The von Mises calculus Obstacles and opportunities: robustness Optimality and Likelihood Asymptotics Historical overview The organisation of this chapter The likelihood function and its properties Consistency of maximum likelihood Asymptotic normality of maximum likelihood Asymptotic comparison of estimators Local asymptotics Local asymptotic normality Local asymptotic minimaxity Various extensions The Laplace Approximation and Series A simple example The basic approximation The Stirling series for factorials Laplace expansions in Maple Asymptotic bias of the median Recurrence properties of random walks Proofs of the main propositions Integrals with the maximum on the boundary Integrals of higher dimension Integrals with product integrands Applications to statistical inference Estimating location parameters Asymptotic analysis of Bayes estimators Notes The Saddle-Point Method The principle of stationary phase Perron's saddle-point method Harmonic functions and saddle-point geometry Daniels' saddle-point approximation Towards the Barndorff-Nielsen formula Saddle-point method for distribution functions Saddle-point method for discrete variables Ratios of sums of random variables Distributions of M-estimators The Edgeworth expansion Mean, median and mode Hayman's saddle-point approximation The method of Darboux Applications to common distributions Summation of Series Advanced tests for series convergence Convergence of random series Applications in probability and statistics Euler-Maclaurin sum formula Applications of the Euler-Maclaurin formula Accelerating series convergence Applications of acceleration methods Comparing acceleration techniques Divergent series Glossary of Symbols Useful Limits, Series and Products References Index

Asymptotic analysis for the generalized langevin equation

Abstract: Various qualitative properties of solutions to the generalized Langevin equation (GLE) in a periodic or a confining potential are studied in this paper. We consider a class of quasi-Markovian GLEs, similar to the model that was introduced in \cite{EPR99}. Geometric ergodicity, a homogenization theorem (invariance principle), short time asymptotics and the white noise limit are studied. Our proofs are based on a careful analysis of a hypoelliptic operator which is the generator of an auxiliary Markov process. Systematic use of the recently developed theory of hypocoercivity \cite{Vil04HPI} is made.

Asymptotic and bifurcation analysis of wave-pinning in a reaction-diffusion model for cell polarization

Abstract: We describe and analyze a bistable reaction-diffusion (RD) model for two interconverting chemical species that exhibits a phenomenon of wave-pinning: a wave of activation of one of the species is initiated at one end of the domain, moves into the domain, decelerates, and eventually stops inside the domain, forming a stationary front. The second ("inactive") species is depleted in this process. This behavior arises in a model for chemical polarization of a cell by Rho GTPases in response to stimulation. The initially spatially homogeneous concentration profile (representative of a resting cell) develops into an asymmetric stationary front profile (typical of a polarized cell). Wave-pinning here is based on three properties: (1) mass conservation in a finite domain, (2) nonlinear reaction kinetics allowing for multiple stable steady states, and (3) a sufficiently large difference in diffusion of the two species. Using matched asymptotic analysis, we explain the mathematical basis of wave-pinning, and predict the speed and pinned position of the wave. An analysis of the bifurcation of the pinned front solution reveals how the wave-pinning regime depends on parameters such as rates of diffusion and total mass of the species. We describe two ways in which the pinned solution can be lost depending on the details of the reaction kinetics: a saddle-node or a pitchfork bifurcation.

A new performance index for ICA: properties, computation and asymptotic analysis

Abstract: In the independent component (IC) model it is assumed that the components of the observed p-variate random vector x are linear combinations of the components of a latent p-vector z such that the p components of z are independent. Then x = Ωz where Ω is a full-rank p × p mixing matrix. In the independent component analysis (ICA) the aim is to estimate an unmixing matrix Γ such that Γx has independent components. The comparison of the performances of different unmixing matrix estimates Γ in the simulations is then difficult as the estimates are for different population quantities Γ. In this paper we suggest a new natural performance index which finds the shortest distance (using Frobenius norm) between the identity matrix and the set of matrices equivalent to the gain matrix ΓΩ. The index is shown to possess several nice properties, and it is easy and fast to compute. Also, the limiting behavior of the index as the sample size approaches infinity can be easily derived if the limiting behavior of the estimate Γ is known.

Asymptotic analysis for personalized web search

Abstract: PageRank with personalization is used in Web search as an importance measure for Web documents. The goal of this paper is to characterize the tail behavior of the PageRank distribution in the Web and other complex networks characterized by power laws. To this end, we model the PageRank as a solution of a stochastic equation $R \buildrel\rm D\over= \sum^N_{i=1} A_i R_i + B$, where the $R_i$s are distributed as $R$. This equation is inspired by the original definition of the PageRank. In particular, $N$ models the number of incoming links to a page, and $B$ stays for the user preference. Assuming that $N$ or $B$ are heavy tailed, we employ the theory of regular variation to obtain the asymptotic behavior of $R$ under quite general assumptions on the involved random variables. Our theoretical predictions show good agreement with experimental data.

Multi-robot monitoring in dynamic environments with guaranteed currency of observations

Abstract: In this paper we consider the problem of monitoring a known set of stationary features (or locations of interest) in an environment. To observe a feature, a robot must visit its location. Each feature changes over time, and we assume that the currency, or accuracy of an observation decays linearly with time. Thus, robots must repeatedly visit the features to update their observations. Each feature has a known rate of change, and so the frequency of visits to a feature should be proportional to its rate. The goal is to route the robots so as to minimize the maximum change of a feature between observations. We focus on the asymptotic regime of a large number of features distributed according to a probability density function. In this regime we determine a lower bound on the maximum change of a feature between visits, and develop a robot control policy that, with probability one, performs within a factor of two of the optimal. We also provide a single robot lower bound which holds outside of the asymptotic regime, and present a heuristic algorithm motivated by our asymptotic analysis.

Asymptotic Analysis of Upwind Discontinuous Galerkin Approximation of the Radiative Transport Equation in the Diffusive Limit

Abstract: We revisit some results from M. L. Adams [Nucl. Sci. Engrg., 137 (2001), pp. 298-333]. Using functional analytic tools we prove that a necessary and sufficient condition for the standard upwind discontinuous Galerkin approximation to converge to the correct limit solution in the diffusive regime is that the approximation space contains a linear space of continuous functions, and the restrictions of the functions of this space to each mesh cell contain the linear polynomials. Furthermore, the discrete diffusion limit converges in the Sobolev space $H^1$ to the continuous one if the boundary data is isotropic. With anisotropic boundary data, a boundary layer occurs, and convergence holds in the broken Sobolev space $H^s$ with $s<\frac{1}{2}$ only.

An Eigenvalue Perturbation Approach to Stability Analysis, Part I: Eigenvalue Series of Matrix Operators

Abstract: This two-part paper is concerned with stability analysis of linear systems subject to parameter variations, of which linear time-invariant delay systems are of particular interest. We seek to characterize the asymptotic behavior of the characteristic zeros of such systems. This behavior determines, for example, whether the imaginary zeros cross from one half plane into another, and hence plays a critical role in determining the stability of a system. In Part I of the paper we develop necessary mathematical tools for this study, which focuses on the eigenvalue series of holomorphic matrix operators. While of independent interest, the eigenvalue perturbation analysis has a particular bearing on stability analysis and, indeed, has the promise to provide efficient computational solutions to a class of problems relevant to control systems analysis and design, of which time-delay systems are a notable example.

Multi-Robot Monitoring in Dynamic Environments with Guaranteed Currency of Observations Stephen L. Smith Daniela Rus

Abstract: In this paper we consider the problem of monitor- ing a known set of stationary features (or locations of interest) in an environment. To observe a feature, a robot must visit its location. Each feature changes over time, and we assume that the currency, or accuracy of an observation decays linearly with time. Thus, robots must repeatedly visit the features to update their observations. Each feature has a known rate of change, and so the frequency of visits to a feature should be proportional to its rate. The goal is to route the robots so as to minimize the maximum change of a feature between observations. We focus on the asymptotic regime of a large number of features distributed according to a probability density function. In this regime we determine a lower bound on the maximum change of a feature between visits, and develop a robot control policy that, with probability one, performs within a factor of two of the optimal. We also provide a single robot lower bound which holds outside of the asymptotic regime, and present a heuristic algorithm motivated by our asymptotic analysis.

Towards a Better Understanding of Large Scale Network Models

Abstract: Connectivity and capacity are two fundamental properties of wireless multi-hop networks. The scalability of these properties has been a primary concern for which asymptotic analysis is a useful tool. Three related but logically distinct network models are often considered in asymptotic analyses, viz. the dense network model, the extended network model and the infinite network model, which consider respectively a network deployed in a fixed finite area with a sufficiently large node density, a network deployed in a sufficiently large area with a fixed node density, and a network deployed in $\Re^{2}$ with a sufficiently large node density. The infinite network model originated from continuum percolation theory and asymptotic results obtained from the infinite network model have often been applied to the dense and extended networks. In this paper, through two case studies related to network connectivity on the expected number of isolated nodes and on the vanishing of components of finite order k>1 respectively, we demonstrate some subtle but important differences between the infinite network model and the dense and extended network models. Therefore extra scrutiny has to be used in order for the results obtained from the infinite network model to be applicable to the dense and extended network models. Asymptotic results are also obtained on the expected number of isolated nodes, the vanishingly small impact of the boundary effect on the number of isolated nodes and the vanishing of components of finite order k>1 in the dense and extended network models using a generic random connection model.

Vanishing beyond all orders: Stokes lines in a water-wave model equation

Abstract: We study the existence of solutions homoclinic to a saddle centre in a family of singularly perturbed fourth order differential equations, originating from a water-wave model. Due to a reversibility symmetry, the occurrence of such embedded solitons is a codimension-1 phenomenon. By varying a parameter a countable family of solitary waves is found. We examine the asymptotic frequency at which this phenomenon of persistence in the singular limit occurs, by performing a refined Stokes line analysis. In the limit where the parameter tends to infinity, each Stokes line splits into a pair, and the contributions of these two Stokes lines cancel each other for a countable set of parameter values. More generally, we derive the full leading order asymptotics for the Stokes constant, which governs the (exponentially small) amplitude of the (minimal) oscillations in the tails of nearly homoclinic solutions. True homoclinic trajectories are characterized by the Stokes constant vanishing. This formal asymptotic analysis is supplemented with numerical calculations.

On the Singularities of Generalized Solutions to n-Body-Type Problems

Abstract: The validity of Sundman-type asymptotic estimates for collision solutions is established for a wide class of dynamical systems with singular forces, including the classical N‐body problems with Newtonian, quasi‐homogeneous and logarithmic potentials. The solutions are meant in the generalized sense of Morse (locally ‐in space and time‐ minimal trajectories with respect to compactly supported variations) and their uniform limits. The analysis includes the extension of the Von Zeipel’s Theorem and the proof of isolatedness of collisions. Furthermore, such asymptotic analysis is applied to prove the absence of collisions for locally minimal trajectories.

Asymptotic behavior of solutions to the full compressible Navier-Stokes equations in the half space

Abstract: The one-dimensional motion of compressible viscous and heat-conductive fluid is investigated in the half space. By examining the sign of fluid velocity prescribed on the boundary, initial boundary value problems with Dirichlet type boundary conditions are classified into three cases: impermeable wall problem, inflow problem and outflow problem. In this paper, the asymptotic stability of the rarefaction wave, boundary layer solution, and their combination is established for both the impermeable wall problem and the inflow problem under some smallness conditions. The proof is given by an elementary energy method.

Long Time Behavior of a Two-Phase Optimal Design for the Heat Equation

Abstract: We consider a two-phase isotropic optimal design problem within the context of the transient heat equation. The objective is to minimize the average of the dissipated thermal energy during a fixed time interval $[0,T]$. The time-independent material properties are taken as design variables. A full relaxation for this problem was established in [A. Munch, P. Pedregal, and F. Periago, J. Math. Pures Appl. (9), 89 (2008), pp. 225-247] by using the homogenization method. In this paper, we study the asymptotic behavior as $T$ goes to infinity of the solutions of the relaxed problem and prove that they converge to an optimal relaxed design of the corresponding two-phase optimization problem for the stationary heat equation. Next we study necessary optimality conditions for the relaxed optimization problem under the transient heat equation and use those to characterize the microstructure of the optimal designs, which appears in the form of a sequential laminate of rank at most $N$, the spatial dimension. An asymptotic analysis of the optimality conditions lets us prove that, for $T$ large enough, the order of lamination is, in fact, of at most $N-1$. Several numerical experiments in two dimensions complete our study.

Vibration Analysis of Composite Beams With End Effects via the Formal Asymptotic Method

Abstract: Vibration analysis of composite beams is carried out by using a finite element-based formal asymptotic expansioh method The formulation begins with three-dimensional (3D) equilibrium equations in which cross-sectional coordinates are scaled by the characteristic length of the beam Microscopic two-dimensional and macroscopic one-dimensional (ID) equations obtained via the asymptotic expansion method are discretized by applying a conventional finite element method Boundary conditions associated with macroscopic ID equations are considered to investigate the end effect It is then described how one could form and solve the eigenvalue problems derived from the asymptotic method beyond the classical approximation The results obtained are compared with those of 3D finite element method and those available in the literature for composite beams with solid cross section and thin-walled cross section