Showing papers on "Asymptotic analysis published in 2015"

Stochastic Quasi-Fejér Block-Coordinate Fixed Point Iterations with Random Sweeping

Abstract: This work proposes block-coordinate fixed point algorithms with applications to nonlinear analysis and optimization in Hilbert spaces. The asymptotic analysis relies on a notion of stochastic quasi-Fejer monotonicity, which is thoroughly investigated. The iterative methods under consideration feature random sweeping rules to select arbitrarily the blocks of variables that are activated over the course of the iterations and they allow for stochastic errors in the evaluation of the operators. Algorithms using quasi-nonexpansive operators or compositions of averaged nonexpansive operators are constructed, and weak and strong convergence results are established for the sequences they generate. As a by-product, novel block-coordinate operator splitting methods are obtained for solving structured monotone inclusion and convex minimization problems. In particular, the proposed framework leads to random block-coordinate versions of the Douglas--Rachford and forward-backward algorithms and of some of their variant...

On a Fractional Ginzburg–Landau Equation and 1/2-Harmonic Maps into Spheres

Abstract: This paper is devoted to the asymptotic analysis of a fractional version of the Ginzburg-Landau equation in bounded domains, where the Laplacian is replaced by an integro-differential operator related to the square root Laplacian as defined in Fourier space. In the singular Ginzburg-Landau limit, we show that solutions with uniformly bounded energy converge weakly to sphere valued 1/2-harmonic maps, i.e., the fractional analogues of the usual harmonic maps. In addition, the convergence holds in smooth functions spaces away from a (n-1)-rectifiable closed set of finite (n-1)-Hausdorff measure. The proof relies on the representation of the square root Laplacian as a Dirichlet-to-Neumann operator in one more dimension, and on the analysis of a boundary version of the Ginzburg-Landau equation. Besides the analysis of the fractional Ginzburg-Landau equation, we also give a general partial regularity result for stationary 1/2-harmonic maps in arbitrary dimension.

Biaxiality in the asymptotic analysis of a 2D Landau−de Gennes model for liquid crystals

Abstract: We consider the Landaude Gennes variational problem on a bounded, two dimensional domain, subject to Dirichlet smooth boundary conditions. We prove that minimizers are maximally biaxial near the singularities, that is, their biaxiality parameter reaches the maximum value 1. Moreover, we discuss the convergence of minimizers in the vanishing elastic constant limit. Our asymptotic analysis is performed in a general setting, which recovers the Landaude Gennes problem as a specific case.

Vanishing viscosities and error estimate for a Cahn–Hilliard type phase field system related to tumor growth

Abstract: In this paper we perform an asymptotic analysis for two different vanishing viscosity coefficients occurring in a phase field system of Cahn–Hilliard type that was recently introduced in order to approximate a tumor growth model. In particular, we extend some recent results obtained in Colli et al. (2015), letting the two positive viscosity parameters tend to zero independently from each other and weakening the conditions on the initial data in such a way as to maintain the nonlinearities of the PDE system as general as possible. Finally, under proper growth conditions on the interaction potential, we prove an error estimate leading also to the uniqueness result for the limit system.

Asymptotic analysis of the non-steady Navier–Stokes equations in a tube structure. I. The case without boundary-layer-in-time

Abstract: The non-steady Navier–Stokes equations with Dirichlet boundary conditions are considered in thin tube structures. These domains are connected finite unions of thin finite cylinders (in the 2 D case respectively thin rectangles). The complete asymptotic expansion of the solution is constructed in the case without boundary-layer-in-time. The estimates for the difference of the exact solution and its J -th asymptotic approximation is proved. The method of asymptotic partial domain decomposition is formulated and justified for the non-steady Navier–Stokes equations in a tube structure. It gives the asymptotically exact interface conditions of coupling of the 1D and 3D models of the flow. Note that the obtained results hold true for the important in applications case of time periodic flows.

Tracy-Widom asymptotics for a random polymer model with gamma-distributed weights

Abstract: We establish Tracy-Widom asymptotics for the partition function of a random polymer model with gamma-distributed weights recently introduced by Seppalainen. We show that the partition function of this random polymer can be represented within the framework of the geometric RSK correspondence and consequently its law can be expressed in terms of Whittaker functions. This leads to a representation of the law of the partition function which is amenable to asymptotic analysis. In this model, the partition function plays a role analogous to the smallest eigenvalue in the Laguerre unitary ensemble of random matrix theory.

Averaged Least-Mean-Squares: Bias-Variance Trade-offs and Optimal Sampling Distributions

Abstract: We consider the least-squares regression problem and provide a detailed asymptotic analysis of the performance of averaged constant-step-size stochastic gradient descent. In the strongly-convex case, we provide an asymptotic expansion up to explicit exponentially decaying terms. Our analysis leads to new insights into stochastic approximation algorithms: (a) it gives a tighter bound on the allowed step-size; (b) the generalization error may be divided into a variance term which is decaying as O(1/n), independently of the step-size , and a bias term that decays as O(1/ 2 n 2 ); (c) when allowing non-uniform sampling of examples over a dataset, the choice of a good sampling density depends on the trade-off between bias and variance: when the variance term dominates, optimal sampling densities do not lead to much gain, while when the bias term dominates, we can choose larger step-sizes that lead to significant improvements.

Non-asymptotic analysis of a new bandit algorithm for semi-bounded rewards

Abstract: In this paper we consider a stochastic multiarmed bandit problem. It is known in this problem that Deterministic Minimum Empirical Divergence (DMED) policy achieves the asymptotic theoretical bound for the model where each reward distribution is supported in a known bounded interval, say [0; 1]. However, the regret bound of DMED is described in an asymptotic form and the performance in finite time has been unknown. We modify this policy and derive a finite-time regret bound for the new policy, Indexed Minimum Empirical Divergence (IMED), by refining large deviation probabilities to a simple nonasymptotic form. Further, the refined analysis reveals that the finite-time regret bound is valid even in the case that the reward is not bounded from below. Therefore, our finite-time result applies to the case that the minimum reward (that is, the maximum loss) is unknown or unbounded. We also present some simulation results which shows that IMED much improves DMED and performs competitively to other state-of-the-art policies.

Asymptotic analysis of the non-steady Navier–Stokes equations in a tube structure. II. General case

Abstract: The non-steady Navier–Stokes equations with Dirichlet boundary conditions are considered in thin tube structures. These domains are connected finite unions of thin finite cylinders (in the 2 D case respectively thin rectangles). The complete asymptotic expansion of the solution is constructed. It contains a regular part and three types of the boundary layer correctors: “in-space”, “in-time” and “in-space-and-in-time”. The estimates for the difference of the exact solution and its J th asymptotic approximation are proved.

Measurement-assisted Landau-Zener transitions

Abstract: Nonselective quantum measurements, i.e., measurements without reading the results, are often considered as a resource for manipulating quantum systems. In this work, we investigate optimal acceleration of the Landau-Zener (LZ) transitions by nonselective quantum measurements. We use the measurements of a population of a diabatic state of the LZ system at certain time instants as control and find the optimal time instants which maximize the LZ transition. We find surprising nonmonotonic behavior of the maximal transition probability with increase of the coupling parameter when the number of measurements is large. This transition probability gives an optimal approximation to the fundamental quantum Zeno effect (which corresponds to continuous measurements) by a fixed number of discrete measurements. The difficulty for the analysis is that the transition probability as a function of time instants has a huge number of local maxima. We resolve this problem both analytically by asymptotic analysis and numerically by the development of efficient algorithms mainly based on the dynamic programming. The proposed numerical methods can be applied, besides this problem, to a wide class of measurement-based optimal control problems.

Sharp criteria of Liouville type for some nonlinear systems

Abstract: In this paper, we establish the sharp criteria for the nonexistence of positive solutions to the Hardy-Littlewood-Sobolev (HLS) system of nonlinear equations and the corresponding nonlinear differential systems of Lane-Emden type. These nonexistence results, known as Liouville theorems, are fundamental in PDE theory and applications. A special iteration scheme, a new shooting method and some Pohozaev identities in integral form as well as in differential form are created. Combining these new techniques with some observations and some critical asymptotic analysis, we establish the sharp criteria of Liouville type for our systems of nonlinear equations. Similar results are also derived for the system of Wolff type of integral equations and the system of $\gamma$-Laplace equations. A dichotomy description in terms of existence and nonexistence for solutions with finite energy is also obtained.

Asymptotic analysis of Vlasov-type equations under strong local alignment regime

Abstract: We consider the hydrodynamic limit of a collisionless and non-diffusive kinetic equation under strong local alignment regime. The local alignment is first considered by Karper, Mellet and Trivisa in [On strong local alignment in the kinetic Cucker–Smale model, in Hyperbolic Conservation Laws and Related Analysis with Applications, Springer Proceedings in Mathematics & Statistics, Vol. 49 (Springer, 2014), pp. 227–242], as a singular limit of an alignment force proposed by Motsch and Tadmor in [A new model for self-organized dynamics and its flocking behavior, J. Statist. Phys. 141 (2011) 923–947]. As the local alignment strongly dominates, a weak solution to the kinetic equation under consideration converges to the local equilibrium, which has the form of mono-kinetic distribution. We use the relative entropy method and weak compactness to rigorously justify the weak convergence of our kinetic equation to the pressureless Euler system.

A Convergence and Asymptotic Analysis of the Generalized Symmetric FastICA Algorithm

Abstract: This contribution deals with the FastICA algorithm in the domain of Independent Component Analysis (ICA). The focus is on the asymptotic behavior of the generalized symmetric variant of the algorithm. The latter has already been shown to possess the potential to achieve the Cramer-Rao Bound (CRB) by allowing the usage of different nonlinearity functions in its implementation. Although the FastICA algorithm along with its variants are among the most extensively studied methods in the domain of ICA, a rigorous study of the asymptotic distribution of the generalized symmetric FastICA algorithm is still missing. In fact, all the existing results exhibit certain limitations. Some ignores the impact of data standardization on the asymptotic statistics; others are only based on heuristic arguments. In this work, we aim at deriving general and rigorous results on the limiting distribution and the asymptotic statistics of the FastICA algorithm. We begin by showing that the generalized symmetric FastICA optimizes a function that is a sum of the contrast functions of traditional one-unit FastICA with a correction of the sign. Based on this characterization, we established the asymptotic normality and derived a closed-form analytic expression of the asymptotic covariance matrix of the generalized symmetric FastICA estimator using the method of estimating equation and M-estimator. Computer simulations are also provided, which support the theoretical results.

Asymptotic analysis and sign-changing bubble towers for Lane–Emden problems

Abstract: We consider the semilinear Lane-Emden problem 1uDjuj p 1 u in; uD 0 on@; (Ep) where p > 1 andis a smooth bounded domain of R 2 . The aim of the paper is to analyze the asymptotic behavior of sign-changing solutions of (Ep) as p! 1. Among other results we show, under some symmetry assumptions on , that the positive and negative parts of a family of symmetric solutions concentrate at the same point as p ! 1, and the limit profile looks like a tower of two bubbles given by superposition of a regular and a singular solution of the Liouville problem in R 2 .

On a model of a population with variable motility

Abstract: We study a reaction–diffusion equation with a nonlocal reaction term that models a population with variable motility. We establish a global supremum bound for solutions of the equation. We investigate the asymptotic (long-time and long-range) behavior of the population. We perform a certain rescaling and prove that solutions of the rescaled problem converge locally uniformly to zero in a certain region and stay positive (in some sense) in another region. These regions are determined by two viscosity solutions of a related Hamilton–Jacobi equation.

Vanishing viscosities and error estimate for a Cahn--Hilliard type phase field system related to tumor growth

Abstract: In this paper we perform an asymptotic analysis for two different vanishing viscosity coefficients occurring in a phase field system of Cahn–Hilliard type that was recently introduced in order to approximate a tumor growth model. In particular, we extend some recent results obtained in Colli et al. (2015), letting the two positive viscosity parameters tend to zero independently from each other and weakening the conditions on the initial data in such a way as to maintain the nonlinearities of the PDE system as general as possible. Finally, under proper growth conditions on the interaction potential, we prove an error estimate leading also to the uniqueness result for the limit system.

Asymptotic expansions and numerical simulations of I-V relations via a steady state Poisson-Nernst-Planck system

Abstract: A steady state Poisson-Nernst-Planck (PNP) system is studied both analytically and numerically with particular attention on I-V relations of ion channels. Assuming the dielectric constant $\varepsilon$ is small, the PNP system can be viewed as a singularly perturbed system. Due to the special structures of the zeroth order inner and outer systems, one is able to derive more explicit expressions of higher order terms in asymptotic expansions. For the case of zero permanent charge, under the assumption of electro-neutrality at both ends of the channel, our result concerning the I-V relation for two oppositely charged ion species is that the third order correction is \textit{cubic} in $V$, and, furthermore (Theorem \ref{3rd order}), up to the third order, the cubic I-V relation has \textit{three distinct real roots} (except for a very degenerate case) which corresponds to the bi-stable structure in the FitzHugh-Nagumo simplification of the Hodgkin-Huxley model. Three numerical experiments are conducted to check the cubic-like feature of the I-V curve, study the boundary value effect on the I-V relation and investigate the permanent charge effect on the I-V curve, respectively.

Analysis of the diffuse-domain method for solving PDEs in complex geometries

Abstract: In recent work, Li et al.\ (Comm.\ Math.\ Sci., 7:81-107, 2009) developed a diffuse-domain method (DDM) for solving partial differential equations in complex, dynamic geometries with Dirichlet, Neumann, and Robin boundary conditions. The diffuse-domain method uses an implicit representation of the geometry where the sharp boundary is replaced by a diffuse layer with thickness $\epsilon$ that is typically proportional to the minimum grid size. The original equations are reformulated on a larger regular domain and the boundary conditions are incorporated via singular source terms. The resulting equations can be solved with standard finite difference and finite element software packages. Here, we present a matched asymptotic analysis of general diffuse-domain methods for Neumann and Robin boundary conditions. Our analysis shows that for certain choices of the boundary condition approximations, the DDM is second-order accurate in $\epsilon$. However, for other choices the DDM is only first-order accurate. This helps to explain why the choice of boundary-condition approximation is important for rapid global convergence and high accuracy. Our analysis also suggests correction terms that may be added to yield more accurate diffuse-domain methods. Simple modifications of first-order boundary condition approximations are proposed to achieve asymptotically second-order accurate schemes. Our analytic results are confirmed numerically in the $L^2$ and $L^\infty$ norms for selected test problems.

Cnoidal Waves on Fermi–Pasta–Ulam Lattices

Abstract: We study a chain of infinitely many particles coupled by nonlinear springs, obeying the equations of motion $$\begin{aligned} \ddot{q}_n = V'(q_{n+1}-q_n) - V'(q_n-q_{n-1}) \end{aligned}$$ with generic nearest-neighbour potential $$V$$ . We show that this chain carries exact spatially periodic travelling waves whose profile is asymptotic, in a small-amlitude long-wave regime, to the KdV cnoidal waves. The discrete waves have three interesting features: (1) being exact travelling waves they keep their shape for infinite time, rather than just up to a timescale of order wavelength $$^{-3}$$ suggested by formal asymptotic analysis, (2) unlike solitary waves they carry a nonzero amount of energy per particle, (3) analogous behaviour of their KdV continuum counterparts suggests long-time stability properties under nonlinear interaction with each other. Connections with the Fermi–Pasta–Ulam recurrence phenomena are indicated. Proofs involve an adaptation of the renormalization approach of Friesecke and Pego (Nonlinearity 12:1601–1627, 1999) to a periodic setting and the spectral theory of the periodic Schrodinger operator with KdV cnoidal wave potential.

Theory of weakly nonlinear self-sustained detonations

Abstract: We propose a theory of weakly nonlinear multi-dimensional self sustained detonations based on asymptotic analysis of the reactive compressible Navier-Stokes equations. We show that these equations can be reduced to a model consisting of a forced, unsteady, small disturbance, transonic equation and a rate equation for the heat release. In one spatial dimension, the model simplifies to a forced Burgers equation. Through analysis, numerical calculations and comparison with the reactive Euler equations, the model is demonstrated to capture such essential dynamical characteristics of detonations as the steady-state structure, the linear stability spectrum, the period-doubling sequence of bifurcations and chaos in one-dimensional detonations and cellular structures in multidimensional detonations.

Asymptotics of the near-crack-tip stress field of a growing fatigue crack in damaged materials: Numerical experiment and analytical solution

Abstract: In this paper, an asymptotic analysis of growing fatigue near-crack-tip fields in a damaged material is performed. The integrity parameter describing the damage accumulation process in the vicinity of the crack tip is incorporated into the constitutive law of an isotropic linear elastic material. An asymptotic solution based on the eigenfunction expansion method is obtained. It is shown that the problem is reduced to a nonlinear eigenvalue problem. An analytical solution of the nonlinear eigenvalue problem is found by the artificial small parameter method. The perturbation theory approach allows us to derive the analytical presentation of the stress and integrity distributions near the crack tip. The technique proposed permits us to find higher-order terms of the asymptotic expansions of the stress components and the integrity parameter.

A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions

Abstract: In this survey paper we review several aspects related to Navier-Stokes models when some hereditary characteristics (constant, distributed or variable delay, memory, etc) appear in the formulation. First some results concerning existence and/or uniqueness of solutions are established. Next the local stability analysis of steady-state solutions is studied by using the theory of Lyapunov functions, the Razumikhin-Lyapunov technique and also by constructing appropriate Lyapunov functionals. A Gronwall-like lemma for delay equations is also exploited to provide some stability results. In the end we also include some comments concerning the global asymptotic analysis of the model, as well as some open questions and future lines for research.

An Asymptotic Analysis of a 2-D Model of Dynamically Active Compartments Coupled by Bulk Diffusion

Abstract: A class of coupled cell-bulk ODE-PDE models is formulated and analyzed in a two-dimensional domain, which is relevant to studying quorum sensing behavior on thin substrates. In this model, spatially segregated dynamically active signaling cells of a common small radius $\epsilon\ll 1$ are coupled through a passive bulk diffusion field. The method of matched asymptotic expansions is used to construct steady-state solutions and to formulate a spectral problem that characterizes the linear stability properties of these solutions, with the aim of predicting whether temporal oscillations can be triggered by the cell-bulk coupling. Phase diagrams in parameter space where such collective oscillations can occur are illustrated for two specific choices of the intracellular kinetics. In the limit of very large bulk diffusion, it is shown that the ODE-PDE system can be approximated by a finite-dimensional dynamical system, which is studied both analytically and numerically. For one illustrative example of the theory it is shown that when the number of cells exceeds some critical number, the bulk diffusion field can trigger oscillations that would otherwise not occur without the coupling. Moreover, for two specific models for the intracellular dynamics, we show that there are rather wide regions in parameter space where these triggered oscillations are synchronous in nature. Unless the bulk diffusivity is asymptotically large, it is shown that a clustered spatial configuration of cells inside the domain leads to larger regions in parameter space where synchronous collective oscillations between the small cells can occur. Finally, the linear stability analysis for these cell-bulk models is shown to be qualitatively rather similar to that of localized spot patterns for activator-inhibitor reaction-diffusion systems in the limit of long-range inhibition and short-range activation.