Explicit Convergence Research Articles

An application of the multiplicative Sewing Lemma to the high order weak approximation of stochastic differential equations

We introduce a variant of the multiplicative Sewing Lemma in [Gerasimovičs, Hocquet, Nilssen; J. Funct. Anal. 281 (2021)] which yields arbitrary high order weak approximations to stochastic differential equations, extending the cubature approximation on Wiener space introduced by Lyons and Victoir. Our analysis allows to derive stability estimates and explicit weak convergence rates. As a particular example, a cubature approximation for stochastic differential equations driven by continuous Gaussian martingales is given.

Rigorous data‐driven computation of spectral properties of Koopman operators for dynamical systems

AbstractKoopman operators are infinite‐dimensional operators that globally linearize nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. However, Koopman operators can have continuous spectra and infinite‐dimensional invariant subspaces, making computing their spectral information a considerable challenge. This paper describes data‐driven algorithms with rigorous convergence guarantees for computing spectral information of Koopman operators from trajectory data. We introduce residual dynamic mode decomposition (ResDMD), which provides the first scheme for computing the spectra and pseudospectra of general Koopman operators from snapshot data without spectral pollution. Using the resolvent operator and ResDMD, we compute smoothed approximations of spectral measures associated with general measure‐preserving dynamical systems. We prove explicit convergence theorems for our algorithms (including for general systems that are not measure‐preserving), which can achieve high‐order convergence even for chaotic systems when computing the density of the continuous spectrum and the discrete spectrum. Since our algorithms have error control, ResDMD allows aposteri verification of spectral quantities, Koopman mode decompositions, and learned dictionaries. We demonstrate our algorithms on the tent map, circle rotations, Gauss iterated map, nonlinear pendulum, double pendulum, and Lorenz system. Finally, we provide kernelized variants of our algorithms for dynamical systems with a high‐dimensional state space. This allows us to compute the spectral measure associated with the dynamics of a protein molecule with a 20,046‐dimensional state space and compute nonlinear Koopman modes with error bounds for turbulent flow past aerofoils with Reynolds number >105 that has a 295,122‐dimensional state space.

Spectral analysis of implicit 2 stage block Runge-Kutta preconditioners

We analyze the recently introduced family of preconditioners in [15] for the stage equations of implicit Runge-Kutta methods for two stage methods. We give explicit formulas for the eigenvalues and eigenvectors of the preconditioned systems for a general method and use these to give explicit convergence estimates of preconditioned GMRES for some common choices of the implicit Runge-Kutta methods. This analysis also allows us to qualitatively predict and explain the main observed features of the GMRES convergence behavior, not only bound it. We illustrate our analysis with numerical experiments. We also consider the direction of numerical optimization for improving the preconditioners performance, as suggested in [15]. We consider two different ways – both distinct to the one introduced in [15] – and numerically optimize these, using the explicit bounds obtained beforehand.

Explicit Convergence Rates for the M/G/1 Queue under Perturbation

This paper establishes convergence rates for discrete-time Markov chains on a countable state space that are stochastically ordered starting from a stationary distribution under perturbation. We investigate the explicit criteria to obtain the ordinary ergodicity, geometric ergodicity and polynomial ergodicity for the embedded M/G/1 queue under perturbation. The explicit geometric convergence rates for the original system and the system under perturbation are caculated. Our bounds in the geometric case and polynomial case are closely connected to the first hitting times. Two examples are provided to illustrate our result.

Compressible Euler–Maxwell limit for global smooth solutions to the Vlasov–Maxwell–Boltzmann system

Two fundamental models in plasma physics are given by the Vlasov–Maxwell–Boltzmann system and the compressible Euler–Maxwell system which both capture the complex dynamics of plasmas under the self-consistent electromagnetic interactions at the kinetic and fluid levels, respectively. It has remained a long-standing open problem to rigorously justify the hydrodynamic limit from the former to the latter as the Knudsen number [Formula: see text] tends to zero. In this paper, we give an affirmative answer to the problem for smooth solutions to both systems near constant equilibrium in the whole space in case when only the dynamics of electrons is taken into account. The explicit rate of convergence in [Formula: see text] over an almost global time interval is also obtained for well-prepared data. For the proof, one of main difficulties occurs to the cubic growth of large velocities due to the action of the classical transport operator on local Maxwellians and we develop new [Formula: see text]-dependent energy estimates basing on the macro–micro decomposition to characterize the asymptotic limit in the compressible setting.

Rate of Convergence in the Smoluchowski-Kramers Approximation for Mean-field Stochastic Differential Equations

In this paper we study a second-order mean-field stochastic differential systems describing the movement of a particle under the influence of a time-dependent force, a friction, a mean-field interaction and a space and time-dependent stochastic noise. Using techniques from Malliavin calculus, we establish explicit rates of convergence in the zero-mass limit (Smoluchowski-Kramers approximation) in the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-distances and in the total variation distance for the position process, the velocity process and a re-scaled velocity process to their corresponding limiting processes.

Greedy PSB methods with explicit superlinear convergence

Greedy PSB methods with explicit superlinear convergence

Kolmogorov continuity and stability of sample paths of entropy solutions of stochastic conservation laws

This paper is concerned with sample paths and path-based properties of the entropy solution to conservation laws with stochastic forcing. We derive a series of uniform maximal-type estimates for the viscous perturbation and establish the existence of stochastic entropy solution that has Hölder continuous sample paths. This information is then carefully choreographed with Kružkov’s technique to obtain stronger continuous dependence estimates, based on the nonlinearities, for the sample paths of the solutions. Finally, convergence of sample paths is established for vanishing viscosity approximation along with an explicit rate of convergence.

Temporal Semi-discretizations of a Backward Semilinear Stochastic Evolution Equation

This paper studies the convergence of three temporal semi-discretizations for a backward semilinear stochastic evolution equation. For general terminal value and general coefficient with Lipschitz continuity, the convergence of the first two temporal semi-discretizations is established, and an explicit convergence rate is derived for the third temporal semi-discretization. The third temporal semi-discretization is applied to a general stochastic linear quadratic control problem, and the convergence of a temporally semi-discrete approximation to the optimal control is established.

Stochastic inviscid Leray-[formula omitted] model with transport noise: Convergence rates and CLT

We consider the stochastic inviscid Leray-α model on the torus driven by transport noise. Under a suitable scaling of the noise, we prove that the weak solutions converge, in some negative Sobolev spaces, to the unique solution of the deterministic viscous Leray-α model. This implies that transport noise regularizes the inviscid Leray-α model so that it enjoys approximate weak uniqueness. Interpreting such limit result as a law of large numbers, we study the underlying central limit theorem and provide an explicit convergence rate.

The dependency of spectral gaps on the convergence of the inverse iteration for a nonlinear eigenvector problem

In this paper, we consider the generalized inverse iteration for computing ground states of the Gross–Pitaevskii eigenvector (GPE) problem. For that we prove explicit linear convergence rates that depend on the maximum eigenvalue in magnitude of a weighted linear eigenvalue problem. Furthermore, we show that this eigenvalue can be bounded by the first spectral gap of a linearized Gross–Pitaevskii operator, recovering the same rates as for linear eigenvector problems. With this we establish the first local convergence result for the basic inverse iteration for the GPE without damping. We also show how our findings directly generalize to extended inverse iterations, such as the Gradient Flow Discrete Normalized (GFDN) proposed in [W. Bao and Q. Du, Computing the ground state solution of Bose–Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput. 25 (2004) 1674–1697] or the damped inverse iteration suggested in [P. Henning and D. Peterseim, Sobolev gradient flow for the Gross–Pitaevskii eigenvalue problem: Global convergence and computational efficiency, SIAM J. Numer. Anal. 58 (2020) 1744–1772]. Our analysis also reveals why the inverse iteration for the GPE does not react favorably to spectral shifts. This empirical observation can now be explained with a blow-up of a weighting function that crucially contributes to the convergence rates. Our findings are illustrated by numerical experiments.

On the effectiveness of the Moulinec–Suquet discretization for composite materials

AbstractMoulinec and Suquet introduced a method for computational homogenization based on the fast Fourier transform which turned out to be rather computationally efficient. The underlying discretization scheme was subsequently identified as an approach based on trigonometric polynomials, coupled to the trapezoidal rule to substitute full integration. For problems with smooth solutions, the power of spectral methods is well‐known. However, for heterogeneous microstructures, there are jumps in the coefficients, and the solution fields are not smooth enough due to discontinuities across material interfaces. Previous convergence results only provided convergence of the discretization per se, that is, without explicit rates, and could not explain the effectiveness of the discretization observed in practice. In this work, we provide such explicit convergence rates for the local strain as well as the stress field and the effective stresses based on more refined techniques. More precisely, we consider a class of industrially relevant, discontinuous elastic moduli separated by sufficiently smooth interfaces and show rates which are known to be sharp from numerical experiments. The applied techniques are of independent interest, that is, we employ a local smoothing strategy, utilize Féjer means as well as Bernstein estimates and rely upon recently established superconvergence results for the effective elastic energy in the Galerkin setting. The presented results shed theoretical light on the effectiveness of the Moulinec–Suquet discretization in practice. Indeed, the obtained convergence rates coincide with those obtained for voxel finite element methods, which typically require higher computational effort.

Factor Modeling for Clustering High-Dimensional Time Series

We propose a new unsupervised learning method for clustering a large number of time series based on a latent factor structure. Each cluster is characterized by its own cluster-specific factors in addition to some common factors which impact on all the time series concerned. Our setting also offers the flexibility that some time series may not belong to any clusters. The consistency with explicit convergence rates is established for the estimation of the common factors, the cluster-specific factors, and the latent clusters. Numerical illustration with both simulated data as well as a real data example is also reported. As a spin-off, the proposed new approach also advances significantly the statistical inference for the factor model of Lam and Yao. Supplementary materials for this article are available online.

Distributed Optimization Framework for Mobile Data Collection With Energy Harvesting in Duty-Cycle WSNs

The increased dynamics and complexity of wireless sensor networks (WSNs) raises higher requirements on the flexibility and applicability of data collection algorithms. Therefore, a novel distributed optimization framework for data collection with energy harvesting in duty-cycle (DC) WSNs is developed in this article, which employs the mobile collector (called SenCar) to collect data from the selected sensors (called anchors) to circumvent the bottlenecks caused by the variations of energy distribution. In order to guarantee the optimality of mobile data collection, it is divided into two steps: Firstly, a method for calculating the optimal number of anchors is developed and then an adaptive anchor selection algorithm is proposed to determine those anchors. Secondly, the mobile data collection problem of DC WSNs is formulated as a delay optimization problem constrained by flow conversation, congestion control and energy balance. Then, we focus on designing the distributed algorithm to resolve that optimization problem, in which each sensor node only needs to communicate with its neighbors to make decisions without any global information and can locally adjust the information weight of its neighbors. Besides, the accelerated distributed algorithm with the uncoordinated step size is also proposed to improve the convergence rate. Furthermore, the explicit convergence analysis of the proposed algorithm is provided in this paper. Finally, the numerical results show that our algorithm can quickly converge to the optimal solution and has obvious superiority in adjusting link flow, storing energy, and extending network lifetime.

Sodicity stress differently influences physiological traits and anti-oxidant enzymes in pear and peach cultivars.

The growth and physiological responses to sodicity stress of pear and peach are poorly understood. Insights into how sodicity stress alters tree physiology remain vital to developing salt tolerant scion and rootstock cultivars. The effects of sodicity stress (soil pHs ~8.8) on tree growth and physiological traits of field grown trees of pear cultivars Punjab Beauty and Patharnakh, and peach cultivars Partap and Shan-e-Punjab were recorded using standard procedures. Sodicity-induced changes in oxidative stressors, proline, anti-oxidant enzymes and leaf ions were measured to draw inferences. Sodicity-induced reductions in vegetative growth were particularly marked in Patharnakh pear and Partap peach compared with other cultivars. Although sodicity stress triggered a significant increase in leaf malondialdehyde (MDA) and hydrogen peroxide (H2O2), their levels relative to controls were much higher in peach than in pear; reflecting that peach suffered from greater oxidative stress. Interestingly, MDA and H2O2 levels did not seem to be deleterious enough to trigger proline-induced osmotic adjustment in pears. The activities of anti-oxidant enzymes strongly varied with the cultivar; specifically, the sodicity-induced increases in CAT and SOD activities were much higher in Punjab Beauty pear and Shan-e-Punjab peach. Principal Component Analysis revealed an explicit convergence between CAT and SOD activities in Punjab Beauty and Shan-e-Punjab cultivars in response to sodicity-induced oxidative stress. Correlation analysis revealed that leaf Na+ strongly inhibited tree growth in peach than in pear. Leaf K+ and proline were found to be the major osmolytes in sodicity-stressed pear and peach cultivars, respectively. We have for the first time studied the effects of sodicity stress on important tree growth and physiological traits of commercially important pear and peach cultivars. Our findings revealed a marked suppressive effect of sodicity stress on tree growth in peach than in pear. The sodicity-induced upticks in leaf malondialdehyde, hydrogen peroxide and Na+ seemed to induce proline-mediated osmotic adjustment in peach but not in pear. The overall better sodicity tolerance in pear compared to peach was ascribed to increased activities of anti-oxidant enzymes catalase and superoxide dismutase enzymes together with restricted Na+ uptake and better leaf K+ levels. Further investigations are needed to elucidate the effects of sodicity stress on genetic and transcriptional changes, and on fruit yield and quality.

On quantitative hypocoercivity estimates based on Harris-type theorems

This Review concerns recent results on the quantitative study of convergence toward the stationary state for spatially inhomogeneous kinetic equations. We focus on analytical results obtained by means of certain probabilistic techniques from the ergodic theory of Markov processes. These techniques are sometimes referred to as Harris-type theorems. They provide constructive proofs for convergence results in the L1 (or total variation) setting for a large class of initial data. The convergence rates can be made explicit (for both geometric and sub-geometric rates) by tracking the constants appearing in the hypotheses. Harris-type theorems are particularly well-adapted for equations exhibiting non-explicit and non-equilibrium steady states since they do not require prior information on the existence of stationary states. This allows for significant improvements of some already-existing results by relaxing assumptions and providing explicit convergence rates. We aim to present Harris-type theorems, providing a guideline on how to apply these techniques to kinetic equations at hand. We discuss recent quantitative results obtained for kinetic equations in gas theory and mathematical biology, giving some perspectives on potential extensions to nonlinear equations.

Fast finite-time observer-based sliding mode controller design for a class of uncertain nonlinear systems with input saturation

In this paper, a novel fast finite-time velocity observer-based terminal sliding mode controller is proposed for a class of multi-input multi-output nonlinear systems in the presence of unknown time-varying matched uncertainties and input saturation constraint. The considered nonlinear system is a chain of interdependent second-order nonlinear subsystems that can describe a multitude of real practical applications. The notion of an auxiliary system is introduced and utilized to deal with the input saturation. Due to the hardship of measuring and accessibility to all the system states, a robust output feedback sliding mode controller is designed based on nonsingular fast terminal sliding surfaces exploiting the estimated states and the auxiliary system’s states which is capable of alleviating the control signal gains and enhancing the convergence rate. By noticing that the separation principle does not hold in nonlinear systems, analyzing the finite-time stability of the closed-loop system is a challenging problem and will be assured via an innovative Lyapunov function candidate. Moreover, an explicit adjustable finite convergence time is derived. Simulation results on two practical systems consisting of a two-degree-of-freedom rigid robotic manipulator and Van der Pol oscillator illustrate the effectiveness and superiority of the suggested control scheme.

A natural regularization of the adsorption integral equation with Langmuir-kernel

Adsorption integral equations are used to compute adsorption energy distributions by measured total isotherms. Since these types of equations are Fredholm integral equations of the first kind on bounded domains, they are unstable or ill-posed, respectively. Hence, there is a need for regularization. In this work, we present general regularizations based on the fourier transform for a special kernel, the Langmuir kernel. The regularization parameters are chosen as zeros or minimizers of simple functions depending on the mean absolute error of a transformed total isotherm. In difference to many other solutions proposed, an explicit error and convergence analysis is made. Additionally, we consider adsorption energy distributions with sharp peaks or for ideal adsorbents. Here, we construct a regularization for computing averages of the adsorption energy distribution and the maximal approximation error is estimated uniformly.

Tangent-point energies and ropelength as Gamma-limits of discrete tangent-point energies on biarc curves

Using interpolation with biarc curves we prove Γ-convergence of discretized tangent-point energies to the continuous tangent-point energies in the C^{1}-topology, as well as to the ropelength functional. As a consequence, discrete almost minimizing biarc curves converge to minimizers of the continuous tangent-point energies, and to ropelength minimizers, respectively. In addition, taking point-tangent data from a given C^{1,1}-curve γ, we establish convergence of the discrete energies evaluated on biarc curves interpolating these data, to the continuous tangent-point energy of γ, together with an explicit convergence rate.

Residual dynamic mode decomposition: robust and verified Koopmanism

Dynamic mode decomposition (DMD) describes complex dynamic processes through a hierarchy of simpler coherent features. DMD is regularly used to understand the fundamental characteristics of turbulence and is closely related to Koopman operators. However, verifying the decomposition, equivalently the computed spectral features of Koopman operators, remains a significant challenge due to the infinite-dimensional nature of Koopman operators. Challenges include spurious (unphysical) modes and dealing with continuous spectra, which both occur regularly in turbulent flows. Residual dynamic mode decomposition (ResDMD), introduced by Colbrook & Townsend (Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems. 2021. arXiv:2111.14889), overcomes such challenges through the data-driven computation of residuals associated with the full infinite-dimensional Koopman operator. ResDMD computes spectra and pseudospectra of general Koopman operators with error control and computes smoothed approximations of spectral measures (including continuous spectra) with explicit high-order convergence theorems. ResDMD thus provides robust and verified Koopmanism. We implement ResDMD and demonstrate its application in various fluid dynamic situations at varying Reynolds numbers from both numerical and experimental data. Examples include vortex shedding behind a cylinder, hot-wire data acquired in a turbulent boundary layer, particle image velocimetry data focusing on a wall-jet flow and laser-induced plasma acoustic pressure signals. We present some advantages of ResDMD: the ability to resolve nonlinear and transient modes verifiably; the verification of learnt dictionaries; the verification of Koopman mode decompositions; and spectral calculations with reduced broadening effects. We also discuss how a new ordering of modes via residuals enables greater accuracy than the traditional modulus ordering (e.g. when forecasting) with a smaller dictionary. This result paves the way for more significant dynamic compression of large datasets without sacrificing accuracy.