Sharp Convergence Research Articles

Vanishing shear viscosity limit for the compressible planar MHD system with boundary layer

This paper is devoted to the study of the vanishing shear viscosity limit and strong boundary layer problem for the compressible, viscous, and heat-conducting planar MHD equations. The main aim is to obtain a sharp convergence rate which is usually connected to the boundary layer thickness. However, The convergence rate would be possibly slowed down due to the presence of the strong boundary layer effect and the interactions among the magnetic field, temperature, and fluids through not only the velocity equations but also the strongly nonlinear terms in the temperature equation. Our main strategy is to construct some new functions via asymptotic matching method which can cancel some quantities decaying in a lower speed. It leads to a sharp L∞ convergence rate as the shear viscosity vanishes for global-in-time solution with arbitrarily large initial data.

A stationary set method for estimating oscillatory integrals

We propose a new method of estimating oscillatory integrals, which we call a “stationary set” method. We use it to obtain the sharp convergence exponents of Tarry’s problems in dimension two for every degree k\ge 2 . As a consequence, we obtain sharp L^{\infty} \to L^{p} Fourier extension estimates for a family of monomial surfaces.

Alternating minimization for generalized rank-1 matrix sensing: sharp predictions from a random initialization

Abstract We consider the problem of estimating the factors of a rank-$1$ matrix with i.i.d. Gaussian, rank-$1$ measurements that are nonlinearly transformed and corrupted by noise. Considering two prototypical choices for the nonlinearity, we study the convergence properties of a natural alternating update rule for this non-convex optimization problem starting from a random initialization. We show sharp convergence guarantees for a sample-split version of the algorithm by deriving a deterministic one-step recursion that is accurate even in high-dimensional problems. Notably, while the infinite-sample population update is uninformative and suggests exact recovery in a single step, the algorithm—and our deterministic one-step prediction—converges geometrically fast from a random initialization. Our sharp, non-asymptotic analysis also exposes several other fine-grained properties of this problem, including how the nonlinearity and noise level affect convergence behaviour. On a technical level, our results are enabled by showing that the empirical error recursion can be predicted by our deterministic one-step updates within fluctuations of the order $n^{-1/2}$ when each iteration is run with $n$ observations. Our technique leverages leave-one-out tools originating in the literature on high-dimensional $M$-estimation and provides an avenue for sharply analyzing complex iterative algorithms from a random initialization in other high-dimensional optimization problems with random data.

Propagation of Moments and Sharp Convergence Rate for Inhomogeneous Noncutoff Boltzmann Equation with Soft Potentials

Propagation of Moments and Sharp Convergence Rate for Inhomogeneous Noncutoff Boltzmann Equation with Soft Potentials

Sharp convergence rates for empirical optimal transport with smooth costs

Sharp convergence rates for empirical optimal transport with smooth costs

A sharp convergence theorem for the mean curvature flow in the sphere

A sharp convergence theorem for the mean curvature flow in the sphere

Sharp convergence for sequences of Schrödinger means and related generalizations

For decreasing sequences $\{t_{n}\}_{n=1}^{\infty }$ converging to zero and initial data $f\in H^s(\mathbb {R}^N)$ , $N\geq 2$ , we consider the almost everywhere convergence problem for sequences of Schrödinger means ${\rm e}^{it_{n}\Delta }f$ , which was proposed by Sjölin, and was open until recently. In this paper, we prove that if $\{t_n\}_{n=1}^{\infty }$ belongs to Lorentz space ${\ell }^{r,\infty }(\mathbb {N})$ , then the a.e. convergence results hold for $s>\min \{\frac {r}{\frac {N+1}{N}r+1},\,\frac {N}{2(N+1)}\}$ . Inspired by the work of Lucà-Rogers, we construct a counterexample to show that our a.e. convergence results are sharp (up to endpoints). Our results imply that when $0< r<\frac {N}{N+1}$ , there is a gain over the a.e. convergence result from Du-Guth-Li and Du-Zhang, but not when $r\geq \frac {N}{N+1}$ , even though we are in the discrete case. Our approach can also be applied to get the a.e. convergence results for the fractional Schrödinger means and nonelliptic Schrödinger means.

Unified analysis of finite-size error for periodic Hartree-Fock and second order Møller-Plesset perturbation theory

Despite decades of practice, finite-size errors in many widely used electronic structure theories for periodic systems remain poorly understood. For periodic systems using a general Monkhorst-Pack grid, there has been no comprehensive and rigorous analysis of the finite-size error in the Hartree-Fock theory (HF) and the second order Møller-Plesset perturbation theory (MP2), which are the simplest wavefunction based method, and the simplest post-Hartree-Fock method, respectively. Such calculations can be viewed as a multi-dimensional integral discretized with certain trapezoidal rules. Due to the Coulomb singularity, the integrand has many points of discontinuity in general, and standard error analysis based on the Euler-Maclaurin formula gives overly pessimistic results. The lack of analytic understanding of finite-size errors also impedes the development of effective finite-size correction schemes. We propose a unified analysis to obtain sharp convergence rates of finite-size errors for the periodic HF and MP2 theories. Our main technical advancement is a generalization of the result of Lyness [Math. Comp. 30 (1976), pp. 1–23] for obtaining sharp convergence rates of the trapezoidal rule for a class of non-smooth integrands. Our result is applicable to three-dimensional bulk systems as well as low dimensional systems (such as nanowires and 2D materials). Our unified analysis also allows us to prove the effectiveness of the Madelung-constant correction to the Fock exchange energy, and the effectiveness of a recently proposed staggered mesh method for periodic MP2 calculations (see X. Xing, X. Li, and L. Lin [J. Chem. Theory Comput. 17 (2021), pp. 4733–4745]). Our analysis connects the effectiveness of the staggered mesh method with integrands with removable singularities, and suggests a new staggered mesh method for reducing finite-size errors of periodic HF calculations.

Case study of a special event of low‐level windshear and turbulence at the Hong Kong International Airport

AbstractA case study of a special event of low‐level windshear and turbulence at the Hong Kong International Airport is performed. The mountainous Lantau Island to the south of the Airport may lead to terrain‐disrupted airflow disturbances in the Airport area in certain wind directions, resulting in low‐level windshear and turbulence to the aircraft. There are a number of novel features in the flow field in the airport region. First, over the seas to the west of the airport, there is sharp convergence between northerly flow and southerly flow within a short distance. Second, in the temperature measurements over this region, there are sharp increases and decreases in temperature. The performance of a high‐resolution numerical weather prediction model in predicting these features is studied. It is shown that, though the model is quite successful in reproducing some features, it may not be able to capture some novel features, such as a sharp rise of temperature and sharp convergence of the airflow. The case study may be a useful reference for aviation weather forecasters and mountain meteorologists.

An augmented interface approach in fictitious domain methods

We introduce a new type of fictitious domain method to solve elliptic problems defined over curved complicated domains. In fictitious domain methods with a penalty approach, given domain Ω is embedded into a rectangular domain R and a penalty is applied over the whole fictitious part R ﹨ Ω . We apply the L 2 penalty over a small strip Ω δ around ∂Ω in R ﹨ Ω and present a priori estimates with the sharp convergence of the penalized problem's solution to the original problem's solution. We use the linear finite element method to solve the penalized problem and attain the desired error estimates. Numerical experiments ensure the theoretical estimations and forecast the optimal convergence irrespective of the shape and convexity of the domain.

Optimal and Sharp Convergence Rate of Solutions for a Semilinear Heat Equation with a Critical Exponent and Exponentially Approaching Initial Data

Optimal and Sharp Convergence Rate of Solutions for a Semilinear Heat Equation with a Critical Exponent and Exponentially Approaching Initial Data

Convergence rates for the Allen–Cahn equation with boundary contact energy: the non-perturbative regime

We extend the recent rigorous convergence result of Abels and Moser (SIAM J Math Anal 54(1):114–172, 2022. https://doi.org/10.1137/21M1424925) concerning convergence rates for solutions of the Allen–Cahn equation with a nonlinear Robin boundary condition towards evolution by mean curvature flow with constant contact angle. More precisely, in the present work we manage to remove the perturbative assumption on the contact angle being close to 90^circ . We establish under usual double-well type assumptions on the potential and for a certain class of boundary energy densities the sub-optimal convergence rate of order {varepsilon ^{frac{1}{2}}} for general contact angles alpha in (0,pi ). For a very specific form of the boundary energy density, we even obtain from our methods a sharp convergence rate of order varepsilon ; again for general contact angles alpha in (0,pi ). Our proof deviates from the popular strategy based on rigorous asymptotic expansions and stability estimates for the linearized Allen–Cahn operator. Instead, we follow the recent approach by Fischer et al. (SIAM J Math Anal 52(6):6222–6233, 2020. https://doi.org/10.1137/20M1322182), thus relying on a relative entropy technique. We develop a careful adaptation of their approach in order to encode the constant contact angle condition. In fact, we perform this task at the level of the notion of gradient flow calibrations. This concept was recently introduced in the context of weak-strong uniqueness for multiphase mean curvature flow by Fischer et al. (arXiv:2003.05478v2).

Couplings for Andersen dynamics

Andersen dynamics is a standard method for molecular simulations, and a precursor of the Hamiltonian Monte Carlo algorithm used in MCMC inference. The stochastic process corresponding to Andersen dynamics is a PDMP (piecewise deterministic Markov process) that iterates between Hamiltonian flows and velocity randomizations of randomly selected particles. Both from the viewpoint of molecular dynamics and MCMC inference, a basic question is to understand the convergence to equilibrium of this PDMP particularly in high dimension. Here we present couplings to obtain sharp convergence bounds in the Wasserstein sense that do not require global convexity of the underlying potential energy.

Eigenvalue Fluctuations for Random Elliptic Operators in Homogenization Regime

This work is devoted to the asymptotic behavior of eigenvalues of an elliptic operator with rapidly oscillating random coefficients on a bounded domain with Dirichlet boundary conditions. A sharp convergence rate is obtained for isolated eigenvalues towards eigenvalues of the homogenized problem, as well as a quantitative two-scale expansion result for eigenfunctions. Next, a quantitative central limit theorem is established for eigenvalue fluctuations; more precisely, a pathwise characterization of eigenvalue fluctuations is obtained in terms of the so-called homogenization commutator, in parallel with the recent fluctuation theory for the solution operator.

Mechanism and Inducing Factors of Rockburst Events of Roadways Under Ultrathick Strata

The overlying strata of the Yima coalfield are ultrathick conglomerate. Aiming at the problem of frequent occurrence of rockburst events in the central Yima coalfield during 2006–2015, the characteristics of rockburst events, microseismic (MS) monitoring, and rockburst event-inducing factors were analyzed through data mining and field investigation methods. The results showed that the rockburst events in roadways mainly occurred during mining of the working face, and they occurred at a large buried depth and were within the influence of mining stress, accompanied by an abrupt energy release. The occurrence of rockburst in roadways was accompanied by a sudden release of energy. The ultrathick strata and the fault nearby were the key influence factors of rockburst events. The stress field of roadway surrounding rocks was changed because of the mining disturbance, roadway repair and maintenance, and blasting, which would change the regional stress fields in the surrounding rocks and induce roadway rockburst events. The characteristics of rockburst events were floor heave, sharp convergence of two side walls, severe damage of the supporting body, and even closure of the roadway. The occurrence of rockburst can be prevented by reducing the mining speed and injecting water into coal seam.

A sharp error estimate of Euler‐Maruyama method for stochastic Volterra integral equations

In this paper, the Euler‐Maruyama method is used to solve a class of nonlinear stochastic Volterra integral equations (SVIEs), which can be derived from stochastic fractional substantial integro‐differential equations (SFSIDEs). The existence and uniqueness of the solution are proved for the nonlinear SVIEs under the non‐Lipschitz condition. Moreover, the sharp convergence estimate with if if and O(h) if , respectively, is established for the nonlinear SVIEs, which can be revised as supplemented theory results of our recent work (J Comput Appl Math 356 (2019) 377–390). In particular, it also closes a gap that was left on the convergence analysis in (J Comput Appl Math 317 (2017) 447–457). Finally, numerical experiments are given to illustrate the effectiveness of the presented method.

Sharp convergence rates for Darcy’s law

This article is concerned with Darcy’s law for an incompressible viscous fluid flowing in a porous medium. We establish the sharp convergence rate in a periodically perforated and bounded domain in for where ε represents the size of solid obstacles. This is achieved by constructing two boundary layer correctors to control the boundary layers created by the incompressibility condition and the discrepancy of boundary values between the solution and the leading term in its asymptotic expansion. One of the correctors deals with the tangential boundary data, while the other handles the normal boundary data.

A sharp estimate of the discrepancy of a concatenation sequence of inversive pseudorandom numbers with consecutive primes

We consider an equidistributed concatenation sequence of pseudorandom rational numbers generated from the primes by an inversive congruential method. In particular, we determine the sharp convergence rate for the star discrepancy of said sequence. Our arguments are based on well-known discrepancy estimates for inversive congruential pseudorandom numbers together with asymptotic formulae involving prime numbers.

Application of the Ultimate Hanging Arch Length of the Layered Roof with Anchors and the Gob-Side Entry Retaining

The gob-side entry retaining plays an important role in improving working face ventilation, alleviating working face connection, and increasing mining revenue. According to the characteristics of the crossheading roof at the 2103 working face of a mine in Shanxi, a structural mechanics model of the roof was established to derive the theoretical formulae for the ultimate hanging arch length of the layered roof with anchors and the initial support resistance of the entry-side support. The influence factors of the ultimate hanging arch length were evaluated using local sensitivity analysis. Based on the theoretical study, the work proposed the collaborative support technology of the crossheading, collaborative support at the 2103 working face. The results showed that the ultimate hanging arch length was most influenced by the width of the plastic zone, followed by the width of the roadway, supporting strength, anchoring strength, layered thickness, and mining depth, while the ultimate tensile strength had little influence. The initial support resistance of the entry-side supports was closely related to the ultimate hanging arch length and the process of gob-side entry retaining. The improved entry-retaining supporting process could control the sharp surface convergence of the surrounding rocks of the entry retaining, the sinkage of the roof of the entry-retaining section was controlled below 100 mm, and that of the advanced section was controlled below 50 mm. The stability of the supports next to the entry is improved, and the needs of the site project are met.

Set structured global empirical risk minimizers are rate optimal in general dimensions

Entropy integrals are widely used as a powerful empirical process tool to obtain upper bounds for the rates of convergence of global empirical risk minimizers (ERMs), in standard settings such as density estimation and regression. The upper bound for the convergence rates thus obtained typically matches the minimax lower bound when the entropy integral converges, but admits a strict gap compared to the lower bound when it diverges. Birgé and Massart (Probab. Theory Related Fields 97 (1993) 113–150) provided a striking example showing that such a gap is real with the entropy structure alone: for a variant of the natural Hölder class with low regularity, the global ERM actually converges at the rate predicted by the entropy integral that substantially deviates from the lower bound. The counter-example has spawned a long-standing negative position on the use of global ERMs in the regime where the entropy integral diverges, as they are heuristically believed to converge at a suboptimal rate in a variety of models. The present paper demonstrates that this gap can be closed if the models admit certain degree of “set structures” in addition to the entropy structure. In other words, the global ERMs in such set structured models will indeed be rate-optimal, matching the lower bound even when the entropy integral diverges. The models with set structures we investigate include (i) image and edge estimation, (ii) binary classification, (iii) multiple isotonic regression, (iv) s-concave density estimation, all in general dimensions when the entropy integral diverges. Here, set structures are interpreted broadly in the sense that the complexity of the underlying models can be essentially captured by the size of the empirical process over certain class of measurable sets, for which matching upper and lower bounds are obtained to facilitate the derivation of sharp convergence rates for the associated global ERMs.