Explicit Convergence Research Articles

Parametric copula adjusted for non- and semiparametric regression

We consider a multivariate response regression model where each coordinate is described by a location-scale non- or semiparametric regression and where the dependence structure of the “noise term” is described by a parametric copula. Our goal is to estimate the associated Euclidean copula parameter, given a sample of the response and the covariate. In the absence of the copula sample, the usual oracle ranks are no longer computable. Instead, we study the normal scores estimator for the Gaussian copula and generalized pseudo-likelihood estimation for general parametric copulas, both based on residual ranks calculated from preliminary non- or semiparametric estimators of the location and scale functions. We show that the residual-based estimators are asymptotically equivalent to their oracle counterparts and provide explicit rate of convergence. Partially to serve this objective, we also study weighted convergence of the residual empirical process under the non- or semiparametric regression model.

On explicit L2-convergence rate estimate for piecewise deterministic Markov processes in MCMC algorithms

We establish $L^2$-exponential convergence rate for three popular piecewise deterministic Markov processes for sampling: the randomized Hamiltonian Monte Carlo method, the zigzag process, and the bouncy particle sampler. Our analysis is based on a variational framework for hypocoercivity, which combines a Poincar\'{e}-type inequality in time-augmented state space and a standard $L^2$ energy estimate. Our analysis provides explicit convergence rate estimates, which are more quantitative than existing results.

A Note on Fokker–Planck Equations and Graphons

Fokker-Planck equations represent a suitable description of the finite-time behavior for a large class of particle systems as the size of the population tends to infinity. Recently, the theory of graph limits has been introduced in the classical mean-field framework to account for heterogeneous interactions among particles. In many instances, such network heterogeneity is preserved in the limit which turns from being a single Fokker-Planck equation (also known as McKean-Vlasov) to an infinite system of non-linear partial differential equations (PDE) coupled by means of a graphon. While appealing from an applied viewpoint, few rigorous results exist on the graphon particle system. This note addresses such limit system focusing on the relation between interaction network and initial conditions: if the system initial datum and the graphon degrees satisfy a suitable condition, a significantly simpler representation of the solution is available. Interesting conclusions can be drawn from this result: substantially different graphons can give rise to exactly the same particle behavior, e.g., if the initial condition is independent of the graphon, the particle behavior is indistinguishable from the well-known mean-field limit as long as the graphon has constant degree; step-kernels, a building block of graph limit theory, can be used to approximate the graphon particle system with a finite system of Fokker-Planck equations and an explicit rate of convergence.

Convergence Analysis and Numerical Studies for Linearly Elastic Peridynamics with Dirichlet-Type Boundary Conditions

The nonlocal models of peridynamics have successfully predicted fractures and deformations for a variety of materials. In contrast to local mechanics, peridynamic boundary conditions must be defined on a finite volume region outside the body. Therefore, theoretical and numerical challenges arise in order to properly formulate Dirichlet-type nonlocal boundary conditions, while connecting them to the local counterparts. While a careless imposition of local boundary conditions leads to a smaller effective material stiffness close to the boundary and an artificial softening of the material, several strategies were proposed to avoid this unphysical surface effect. In this work, we study convergence of solutions to nonlocal state-based linear elastic model to their local counterparts as the interaction horizon vanishes, under different formulations and smoothness assumptions for nonlocal Dirichlet-type boundary conditions. Our results provide explicit rates of convergence that are sensitive to the compatibility of the nonlocal boundary data and the extension of the solution for the local model. In particular, under appropriate assumptions, constant extensions yield \(\frac{1}{2}\) order convergence rates and linear extensions yield \(\frac{3}{2}\) order convergence rates. With smooth extensions, these rates are improved to quadratic convergence. We illustrate the theory for any dimension \(d\ge 2\) and numerically verify the convergence rates with a number of two-dimensional benchmarks, including linear patch tests, manufactured solutions, and domains with curvilinear surfaces. Numerical results show a first-order convergence for constant extensions and second-order convergence for linear extensions, which suggests a possible room of improvement in the future convergence analysis.

Quantitative dynamics of irreversible enzyme reaction–diffusion systems * *We would like to thank Ján Eliaš for the discussion concerning the case of non-diffusive enzyme and complex molecules.

In this work we investigate the convergence to equilibrium for mass action reaction–diffusion systems which model irreversible enzyme reactions. Using the standard entropy method in this situation is not feasible as the irreversibility of the system implies that the concentrations of the substrate and the complex decay to zero. The key idea we utilise in this work to circumvent this issue is to introduce a family of cut-off partial entropy-like functionals which, when combined with the dissipation of a mass like term of the substrate and the complex, yield an explicit exponential convergence to equilibrium. This method is also applicable in the case where the enzyme and complex molecules do not diffuse, corresponding to chemically relevant situation where these molecules are large in size.

Space-Time Fractional KdV–Burger–Kuramato Equation with Time Dependent Variable Coefficients: Lie Symmetry, Explicit Power Series Solution, Convergence Analysis and Conservation Laws

Space-Time Fractional KdV–Burger–Kuramato Equation with Time Dependent Variable Coefficients: Lie Symmetry, Explicit Power Series Solution, Convergence Analysis and Conservation Laws

Halting Time is Predictable for Large Models: A Universality Property and Average-Case Analysis

Average-case analysis computes the complexity of an algorithm averaged over all possible inputs. Compared to worst-case analysis, it is more representative of the typical behavior of an algorithm, but remains largely unexplored in optimization. One difficulty is that the analysis can depend on the probability distribution of the inputs to the model. However, we show that this is not the case for a class of large-scale problems trained with first-order methods including random least squares and one-hidden layer neural networks with random weights. In fact, the halting time exhibits a universality property: it is independent of the probability distribution. With this barrier for average-case analysis removed, we provide the first explicit average-case convergence rates showing a tighter complexity not captured by traditional worst-case analysis. Finally, numerical simulations suggest this universality property holds for a more general class of algorithms and problems.

CTRW modeling of quantum measurement and fractional equations of quantum stochastic filtering and control

Initially developed in the framework of quantum stochastic calculus, the main equations of quantum stochastic filtering were later on derived as the limits of Markov models of discrete measurements under appropriate scaling. In many branches of modern physics it became popular to extend random walk modeling to the continuous time random walk (CTRW) modeling, where the time between discrete events is taken to be non-exponential. In the present paper we apply the CTRW modeling to the continuous quantum measurements yielding the new fractional in time evolution equations of quantum filtering and thus new fractional equations of quantum mechanics of open systems. The related quantum control problems and games turn out to be described by the fractional Hamilton-Jacobi-Bellman (HJB) equations on Riemannian manifolds. By-passing we provide a full derivation of the standard quantum filtering equations, in a modified way as compared with existing texts, which (i) provides explicit rates of convergence (that are not available via the tightness of martingales approach developed previously) and (ii) allows for the direct applications of the basic results of CTRWs to deduce the final fractional filtering equations.

Boundedness and asymptotic behavior in a Keller-Segel(-Navier)-Stokes system with indirect signal production

This paper deals with the Keller-Segel(-Navier)-Stokes system with indirect signal production(⋆){nt+u⋅∇n=Δn−∇⋅(n∇v)+rn−μn2,vt+u⋅∇v=Δv−v+w,wt+u⋅∇w=Δw−w+n,ut+κ(u⋅∇)u=Δu−∇P+n∇Φ,∇⋅u=0 in a bounded and smooth domain Ω⊂RN(N=2,3) with no-flux boundary for n, v, w and no-slip boundary for u, where r∈R, μ≥0, κ∈{0,1} and Φ∈W2,∞(Ω). In the case without logistic source (r=μ=0), it is proved that for all suitably regular initial data, the associated initial-boundary value problem for the spatially two-dimensional Navier-Stokes system (⋆) admits a globally bounded classical solution. This result improves and extends the previously known ones. We point out that the same result to the corresponding two-dimensional Navier-Stokes system with direct signal production holds necessarily imposing some saturated chemotactic sensitivity, logistic damping or small total initial population mass. In the case coupled with logistic source (r∈R, μ>0), it is shown that for any reasonably regular initial data, the corresponding initial-boundary value problem for the spatially three-dimensional Stokes system (⋆) possesses a globally bounded classical solution, and that this solution stabilizes toward the corresponding spatially homogeneous equilibrium with the explicit convergence rates for the cases r<0, r=0 and r>0. We underline that the global boundedness of classical solution to the corresponding three-dimensional Stokes system with direct signal production was obtained only for μ≥23 (or sublinear signal production), and that the convergence result to the corresponding system with direct signal production was established only for r=0 and μ≥23. Our results rigorously confirm that the indirect signal production mechanism genuinely contributes to the global boundedness of classical solution to the Keller-Segel(-Navier)-Stokes system.

Quantitative stability and numerical analysis of Markovian quadratic BSDEs with reflection

&lt;p style='text-indent:20px;'&gt;We study the quantitative stability of solutions to Markovian quadratic reflected backward stochastic differential equations (BSDEs) with bounded terminal data. By virtue of bounded mean oscillation martingale and change of measure techniques, we obtain stability estimates for the variation of the solutions with different underlying forward processes. In addition, we propose a truncated discrete-time numerical scheme for quadratic reflected BSDEs and obtain the explicit rate of convergence by applying the quantitative stability result.&lt;/p&gt;

Quantitative mean-field limit for interacting branching diffusions

We establish an explicit rate of convergence for some systems of mean-field interacting diffusions with logistic binary branching, towards solutions of nonlinear evolution equations with non-local self-diffusion and logistic mass growth, which were shown to describe their large population limits in [12]. The proof relies on a novel coupling argument for binary branching diffusions based on optimal transport, allowing us to sharply mimic the trajectory of the interacting binary branching population by means of a system of independent particles with suitably distributed random space-time births. We are thus able to derive an optimal convergence rate, in the dual bounded-Lipschitz distance on finite measures, for the empirical measure of the population, from the convergence rate in 2-Wasserstein distance of empirical distributions of i.i.d. samples. Our approach and results extend propagation of chaos techniques and ideas, from kinetic models to stochastic systems of interacting branching populations, and appear to be new in this setting, even in the simple case of pure binary branching diffusions.

Nonsmooth Convex Optimization to Estimate the Covid-19 Reproduction Number Space-Time Evolution With Robustness Against Low Quality Data

Daily pandemic surveillance, often achieved through the estimation of the reproduction number, constitutes a critical challenge for national health authorities to design counter-measures. In an earlier work, we proposed to formulate the estimation of the reproduction number as an optimization problem, combining data-model fidelity and space-time regularity constraints, solved by nonsmooth convex proximal minimizations. Though promising, that first formulation significantly lacks robustness against the Covid-19 data low quality (irrelevant or missing counts, pseudo-seasonalities,...) stemming from the emergency and crisis context, which significantly impairs accurate pandemic evolution assessments. The present work aims to overcome these limitations by carefully crafting a functional permitting to estimate jointly, in a single step, the reproduction number and outliers defined to model low quality data. This functional also enforces epidemiology-driven regularity properties for the reproduction number estimates, while preserving convexity, thus permitting the design of efficient minimization algorithms, based on proximity operators that are derived analytically. The explicit convergence of the proposed algorithm is proven theoretically. Its relevance is quantified on real Covid-19 data, consisting of daily new infection counts for 200+ countries and for the 96 metropolitan France counties, publicly available at Johns Hopkins University and Sant&#x00E9;-Publique-France. The procedure permits automated daily updates of these estimates, reported via animated and interactive maps. Open-source estimation procedures will be made publicly available.

Newton-Based Methods for Finding the Positive Ground State of Gross-Pitaevskii Equations

The discretization of Gross-Pitaevskii equations (GPE) leads to a nonlinear eigenvalue problem with eigenvector nonlinearity (NEPv). We use two Newton-based methods to compute the positive ground state of GPE. The first method comes from the Newton-Noda iteration for saturable nonlinear Schrödinger equations proposed by Ching-Sung Liu, which can be transferred to GPE naturally. The second method combines the idea of the root-finding methods and the idea of Newton method, in which, each subproblem involving block tridiagonal linear systems can be solved easily. We give an explicit convergence and computational complexity analysis for it. Numerical experiments are provided to support the theoretical results.

Rates of Convergence for Asymptotically Weakly Contractive Mappings in Normed Spaces

We study Krasnoselskii-Mann style iterative algorithms for approximating fixpoints of asymptotically weakly contractive mappings, with a focus on providing generalized convergence proofs along with explicit rates of convergence. More specifically, we define a new notion of being asymptotically ψ-weakly contractive with modulus, and present a series of abstract convergence theorems which both generalize and unify known results from the literature. Rates of convergence are formulated in terms of our modulus of contractivity, in conjunction with other moduli and functions which form quantitative analogues of additional assumptions that are required in each case. Our approach makes use of ideas from proof theory, in particular our emphasis on abstraction and on formulating our main results in a quantitative manner. As such, the paper can be seen as a contribution to the proof mining program.

Convergence Rates of Attractive-Repulsive MCMC Algorithms

We consider MCMC algorithms for certain particle systems which include both attractive and repulsive forces, making their convergence analysis challenging. We prove that a version of these algorithms on a bounded state space is uniformly ergodic with explicit quantitative convergence rate. We also prove that a version on an unbounded state space is still geometrically ergodic, and then use the method of shift-coupling to obtain an explicit quantitative bound on its convergence rate.

Approximation of fractional local times: Zero energy and derivatives

We consider empirical processes associated with high-frequency observations of a fractional Brownian motion (fBm) X with Hurst parameter H∈(0,1), and derive conditions under which these processes verify a (possibly uniform) law of large numbers, as well as a second order (possibly uniform) limit theorem. We devote specific emphasis to the “zero energy” case, corresponding to a kernel whose integral on the real line equals zero. Our asymptotic results are associated with explicit rates of convergence, and are expressed either in terms of the local time of X or of its derivatives: in particular, the full force of our finding applies to the “rough range” 0<H<1/3, on which the previous literature has been mostly silent. The use of the derivatives of local times for studying the fluctuations of high-frequency observations of a fBm is new, and is the main technological breakthrough of the present paper. Our results are based on the use of Malliavin calculus and Fourier analysis, and extend and complete several findings in the literature, for example, by Jeganathan (Ann. Probab. 32 (2004) 1771–1795; (2006); (2008)) and Podolskij and Rosenbaum (J. Financ. Econom. 16 (2018) 588–598).

Extreme gaps between eigenvalues of Wigner matrices

This paper proves universality of the distribution of the smallest and largest gaps between eigenvalues of generalized Wigner matrices, under some smoothness assumption for the density of the entries. The proof relies on the Erdős–Schlein–Yau dynamic approach. We exhibit a new observable that satisfies a stochastic advection equation and reduces local relaxation of the Dyson Brownian motion to a maximum principle. This observable also provides a simple and unified proof of gap universality in the bulk and the edge, which is quantitative. To illustrate this, we give the first explicit rate of convergence to the Tracy–Widom distribution for generalized Wigner matrices.

Convergence rates of support vector machines regression for functional data

Support vector machines regression (SVMR) is an important part of statistical learning theory. The main difference between SVMR and the classical least squares regression (LSR) is that SVMR uses the ϵ-insensitive loss rather than quadratic loss to measure the empirical error. In this paper, we consider SVMR method in the field of functional data analysis under the framework of reproducing kernel Hilbert spaces. The main tool used in our theoretical analysis is the concentration inequalities for suprema of some appropriate empirical processes. As a result, we establish explicit convergence rates of the prediction risk for SVMR, which coincide with the minimax lower bound obtained recently in literature for LSR.

Explicit [formula omitted]-dependent convergence regions of Newton’s method for the matrix [formula omitted]th root

For a square matrix A with no eigenvalues on the closed real axis, its principal pth root, A1/p, is uniquely defined. When all eigenvalues of A are in a suitable region in the complex plane, Newton’s method can be used to approximate A1/p, starting from the identity matrix. Such a region is called a convergence region for Newton’s method. For p=2, the convergence region is the whole region for which A1/2 is defined. For p≥3, however, the convergence region will be much smaller. In this paper, we obtain for p≥3 explicit p-dependent convergence regions that significantly expand existing explicit convergence regions.

Two-stage routing with optimized guided search and greedy algorithm on proximity graph

As interest surges in large-scale retrieval tasks, proximity graphs are now the leading paradigm. Most existing proximity graphs share the simple greedy algorithm as their routing strategy for approximate nearest neighbor search (ANNS), but this leads to two issues: low routing efficiency and local optimum; this because they ignore the special requirements of different routing stages. Generally, the routing can be divided into two stages: the stage far from the query (S1) and the stage closer to the query (S2). S1 aims to quickly route to the vicinity of the query, and the efficiency is dominant. While S2 focuses on finding the results accurately, so the accuracy is staple. We carefully design tailored routing algorithm for each stage, then combine them together to form a Two-stage routing with Optimized Guided search and Greedy algorithm (TOGG). For S1, we propose optimized guided search to quickly locate the query’s neighborhood by the guidance of the query direction. While for S2, we propose optimized greedy algorithm to comprehensively visit the vertices near the query by agilely detecting explicit and implicit convergence paths. Finally, using theoretical and experimental analysis, we demonstrate the proposed method can achieve better performance of efficiency and effectiveness than state-of-the-art work.