Long-time Convergence Research Articles

Self-interacting CBO: Existence, uniqueness, and long-time convergence

A self-interacting dynamics that mimics the standard Consensus-Based Optimization (CBO) model is introduced. This single-particle dynamics is shown to converge to a unique invariant measure that approximates the global minimum of a given function. As an application, its connection to CBO with Personal Best introduced by C. Totzeck and M.-T. Wolfram (Math. Biosci. Eng., 2020) has been established.

Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds

This note is concerned with two families of operators related to the fractional Laplacian, the first arising from the Caffarelli-Silvestre extension problem and the second from the fractional heat equation. They both include the Poisson semigroup. We show that on a complete, connected, and non-compact Riemannian manifold of non-negative Ricci curvature, in both cases, the solution with L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} initial data behaves asymptotically as the mass times the fundamental solution. Similar long-time convergence results remain valid on more general manifolds satisfying the Li-Yau two-sided estimate of the heat kernel. The situation changes drastically on hyperbolic space, and more generally on rank one non-compact symmetric spaces: we show that for the Poisson semigroup, the convergence to the Poisson kernel fails -but remains true under the additional assumption of radial initial data.

Long-time behavior of stochastic Hamilton-Jacobi equations

The long-time behavior of stochastic Hamilton-Jacobi equations is analyzed, including the stochastic mean curvature flow as a special case. In a variety of settings, new and sharpened results are obtained. Among them are (i) a regularization by noise phenomenon for the mean curvature flow with homogeneous noise which establishes that the inclusion of noise speeds up the decay of solutions, and (ii) the long-time convergence of solutions to spatially inhomogeneous stochastic Hamilton-Jacobi equations. A number of motivating examples about nonlinear stochastic partial differential equations are presented in the appendix.

Confined run-and-tumble model with boundary aggregation: Long-time behavior and convergence to the confined Fokker–Planck model

The motile micro-organisms such as Escherichia coli, sperm, or some seaweed are usually modeled by self-propelled particles that move with the run-and-tumble process. Individual-based stochastic models are usually employed to model the aggregation phenomenon at the boundary, which is an active research field that has attracted a lot of biologists and biophysicists. Self-propelled particles at the microscale have complex behaviors, while characteristics at the population level are more important for practical applications but rely on individual behaviors. Kinetic PDE models that describe the time evolution of the probability density distribution of the motile micro-organisms are widely used. However, how to impose the appropriate boundary conditions that take into account the boundary aggregation phenomena is rarely studied. In this paper, we propose the boundary conditions for a 2D confined run-and-tumble model (CRTM) for self-propelled particle populations moving between two parallel plates with a run-and-tumble process. The proposed model satisfies the relative entropy inequality and thus long-time convergence. We establish the relation between CRTM and the confined Fokker–Planck model (CFPM) studied in [J. Fu, B. Perthame and M. Tang, Fokker–Plank system for movement of micro-organism population in confined environment, J. Statist. Phys. 184 (2021) 1–25]. We prove theoretically that when the tumble is highly forward peaked and frequent enough, CRTM converges asymptotically to the CFPM. A numerical comparison of the CRTM with aggregation and CFPM is given. The time evolution of both the deterministic PDE model and individual-based stochastic simulations are displayed, which match each other well.

Long-time convergence of a nonlocal Burgers’ equation towards the local N-wave

We study the long-time behaviour of the unique weak solution of a nonlocal regularisation of the (inviscid) Burgers equation where the velocity is approximated by a one-sided convolution with an exponential kernel. The initial datum is assumed to be positive, bounded, and integrable. The asymptotic profile is given by the ‘N-wave’ entropy solution of the Burgers equation. The key ingredients of the proof are a suitable scaling argument and a nonlocal Oleinik-type estimate.

Recent advances in the long-time analysis of killed degenerate processes and their particle approximation

We review some recent results of quantitative long-time convergence for the law of a killed Markov process conditioned to survival toward a quasi-stationary distribution, and on the analogous question for the particle systems used in practice to sample these distributions. With respect to the existing literature, one of the novelties of these works is the degeneracy of the underlying process with respect to classical elliptic diffusion, namely it can be a non-elliptic hypoelliptic diffusion, a piecewise deterministic Markov process or an Euler numerical scheme.

Analysis of continuous data assimilation scheme for the Navier–Stokes equations using variational multiscale method

In this study, we analyse a continuous data assimilation (CDA) scheme which enables us to combine an observable data with a numerical method to obtain better solutions in which these solutions are also closely similar to the current state of the system. The scheme is applied on a Navier–Stokes system which is discretized with two-step Backward Differentiation Formula (BDF2) in time and finite element in space. In order to improve the accuracy and prevent some non-physical oscillations due to the effect of small viscosity and the dominance of convection, a projection based variational multiscale method (VMS) has also been applied to the system. We present the long-time stability and long-time convergence analyses of the scheme in details and several numerical tests in order to support theoretical findings and demonstrate the promise of the method.

Characteristics of Transient Eddies During Ural Blocking Events With Different Lifetime in Winter

Ural blocking high (hereafter UB) is one of the most important weather systems influencing the weather and climate of China, with its onset and development closely connected with transient eddies. Herein, we classified the wintertime UB events during 1979–2015 into three types according to their lifetime, i.e., short, medium, and long lifetime UB events and further analyzed the characteristics of momentum and heat fluxes transported by transient eddies during the three types of UB events. The results show that the eddy momentum and heat transport by transient eddies over the Ural regions is strong, while the westerly is weak, favorable for the establishment and maintenance of the UB. Before the onset of all the UB events there are enhanced lower-level temperature disturbances, decelerated westerly and convergence of transient momentum and heat fluxes. After the onset of the UB, the upper-level jet stream is accelerated and eddy flux convergence is enhanced. Furthermore, it is found that the duration and position of convergence could be one of important factors determining the lifetime of UB event. The long-time strong convergence of transient eddies favors the maintenance of UB. During long UB events, the polar jet and the convergence of eddy momentum fluxes have the strongest intensity and northward shifted position. The convergence is further strengthened until Day +2 and weakened since Day +7, favorable for the long-time maintenance of UB. There is convergence on Day -3 of both short and medium UB events, which disappears on Day +4 during short events, while strengthened after the onset of medium events, leading to a long-time maintenance of UB. Furthermore, among the three types of UB events, the upward propagation of wave activity and E-P flux divergence are the strongest and most long-lasting during long UB events, while the weakest and shortest during short UB events, which can also verify strong atmospheric baroclinicity and long-lasting strong convergence of transient eddies are favorable for long duration of the UB.

Quasi-stationary distribution for the Langevin process in cylindrical domains, Part I: Existence, uniqueness and long-time convergence

Consider the Langevin process which models the evolution of positions (in Rd) and associated momenta (in Rd) of interacting particles. Let O be a C2 open bounded and connected set of Rd. We prove the compactness of the semigroup of the Langevin process absorbed at the boundary of the bounded-in-position domain D≔O×Rd. We then obtain the existence of a unique quasi-stationary distribution (QSD) for the Langevin process on D. We provide a spectral interpretation of this QSD and obtain an exponential convergence of the Langevin process conditioned on non-absorption towards the QSD. We also give an explicit formula for the first exit point distribution from D, starting from the QSD.

Convergence of metadynamics: Discussion of the adiabatic hypothesis

By drawing a parallel between metadynamics and self interacting models for polymers, we study the longtime convergence of the original metadynamics algorithm in the adiabatic setting, namely when the dynamics along the collective variables decouples from the dynamics along the other degrees of freedom. We also discuss the bias which is introduced when the adiabatic assumption does not hold.

Quantum digital cooling

We introduce a method for digital preparation of ground states of simulated Hamiltonians, inspired by cooling in nature and adapted to leverage the capabilities of digital quantum hardware. The cold bath is simulated by a single ancillary qubit, which is reset periodically and coupled to the system non-perturbatively. Studying this cooling method on a 1-qubit system toy model, we optimize two cooling protocols based on weak-coupling and strong-coupling approaches. Extending the insight from the 1-qubit system model, we develop two scalable protocols for larger systems. The LogSweep protocol extends the weak-coupling approach by sweeping energies to resonantly match any targeted transition. We test LogSweep on the 1D tranverse-field Ising model, demonstrating approximate ground state preparation with an error that can be made polynomially small in the computation time for all three phases of the system. The BangBang protocol extends the strong-coupling approach, and exploits a heuristics for local Hamiltonians to maximise the probability of de-exciting system transitions in the shortest possible time. Although this protocol does not promise long-time convergence, it allows for a rapid cooling to an approximation of the ground state, making this protocol appealing for near-term demonstrations.

Performance evaluation of GIMs released by different IGS ionosphere associate analysis centers in ionospheric constrained single-frequency precise point positioning

Single-frequency precise point positioning (PPP) is widely used with the advantage of cost-efficient, while its further development and application are limited by the difficulty of ionospheric delay correction and the long-time convergence, especially for real-time positioning. And the development of high-accuracy ionospheric models plays an important role for the correction of ionospheric delay in single-frequency PPP. In this paper, the final and rapid global ionospheric map (GIM) products released by Chinese Academy of Science (CAS), Center for Orbit Determination in Europe (CODE), European Space Agency (ESA), Jet Propulsion Laboratory (JPL), Universitat Politècnica de Catalunya (UPC), Wuhan University (WHU) and International GNSS Service (IGS) were used as external constraints in static, simulation kinematic and kinematic single-frequency PPP respectively. The performance and practicality of each product were analyzed by using IGS station observation data (September 2017 and January 2018, with different ionospheric activity levels) and a kinematic experiment. The results indicate that GIM products may improve convergence time in most areas of middle and high latitudes of the globe. However, the improvement of convergence time of some regions in low latitudes or near the equator with dense and uniform distributed IGS stations show even better performance than the regions in middle and high latitudes. Comparing with standard PPP (S-PPP), the convergence time of static PPP is reduced about 14–42% and 13–29% with GIM final and rapid products respectively. In simulation kinematic PPP, the convergence time improves 10–43% and 12–25% for final and rapid products respectively. In the kinematic positioning, the addition of external ionospheric constraints may speed up the convergence in the early stage of positioning and improve the accuracy to a certain extent. Among all GIM products, the final products show better performance than that of the rapid products obviously. The GIM products from CAS and CODE and UQRG from UPC express the best performance are recommended to process ionospheric delay in the high-accuracy positioning.

On the Law of Large Numbers for the Empirical Measure Process of Generalized Dyson Brownian Motion

We study the generalized Dyson Brownian motion (GDBM) of an interacting N-particle system with logarithmic Coulomb interaction and general potential V. Under reasonable condition on V, we prove the existence and uniqueness of strong solution to SDE for GDBM. We then prove that the family of the empirical measures of GDBM is tight on $$\mathcal {C}([0,T],\mathscr {P}(\mathbb {R}))$$ and all the large N limits satisfy a nonlinear McKean–Vlasov equation. Inspired by previous works due to Biane and Speicher, Carrillo, McCann and Villani, and Blower, we prove that the McKean–Vlasov equation is indeed the gradient flow of the Voiculescu free entropy on the Wasserstein space of probability measures over $$\mathbb {R}$$ . Using the optimal transportation theory, we prove that if $$V''\ge K$$ for some constant $$K\in \mathbb {R}$$ , the McKean–Vlasov equation has a unique weak solution in the space of probability measures $$\mathscr {P}(\mathbb {R})$$ . This establishes the Law of Large Numbers and the propagation of chaos for the empirical measures of GDBM with non-quadratic external potentials which are not necessarily convex. Finally, we prove the longtime convergence of the McKean–Vlasov equation for $$C^2$$ -convex potentials V.

Spacelike Mean Curvature Flow

We prove long-time existence and convergence results for spacelike solutions to mean curvature flow in the pseudo-Euclidean space mathbb {R}^{n,m}, which are entire or defined on bounded domains and satisfying Neumann or Dirichlet boundary conditions. As an application, we prove long-time existence and convergence of the {{,mathrm{G},}}_2-Laplacian flow in cases related to coassociative fibrations.

Longtime convergence of the temperature-accelerated molecular dynamics method

The equations of the temperature-accelerated molecular dynamics (TAMD) method for the calculations of free energies and partition functions are analyzed. Specifically, the exponential convergence of the law of these stochastic processes is established, with a convergence rate close to the one of the limiting, effective dynamics at higher temperature obtained with infinite acceleration. It is also shown that the invariant measures of TAMD are close to a known reference measure, with an error that can be quantified precisely. Finally, a central limit theorem is proven, which allows the estimation of errors on properties calculated by ergodic time averages. These results not only demonstrate the usefulness and validity range of the TAMD equations, but they also permit in principle to adjust the parameter in these equations to optimize their efficiency.

Longtime convergence for epitaxial growth model under Dirichlet conditions

This paper continues our study on the initial-boundary value problem for a semilinear parabolic equation of fourth order which has been presented by Johnson-Orme-Hunt-Graff-Sudijono-Sauder-Orr [12] to describe the large-scale features of a growing crystal surface under molecular beam epitaxy. In the preceding paper [1], we already constructed a dynamical system generated by the problem and verified that the dynamical system has a finite-dimensional attractor (especially, every trajectory has nonempty $\omega$-limit set) and admits a Lyapunov function (of the form (3.1)). This paper is then devoted to showing longtime convergence of trajectory. We shall prove that every trajectory converges to some stationary solution as $t \to \infty$. As a matter of fact, we have obtained in [10] the similar result for the equation but under the Neumann like boundary conditions $\frac{\partial u}{\partial n}=\frac\partial{\partial n}\varDelta u=0$ on the unknown function $u$. In this paper, we want as in [1] to handle the Dirichlet boundary conditions $u=\frac{\partial u}{\partial n}=0$, maybe physically more natural conditions than before.

One-Dimensional Forward–Forward Mean-Field Games

While the general theory for the terminal-initial value problem for mean-field games (MFGs) has achieved a substantial progress, the corresponding forward–forward problem is still poorly understood—even in the one-dimensional setting. Here, we consider one-dimensional forward–forward MFGs, study the existence of solutions and their long-time convergence. First, we discuss the relation between these models and systems of conservation laws. In particular, we identify new conserved quantities and study some qualitative properties of these systems. Next, we introduce a class of wave-like equations that are equivalent to forward–forward MFGs, and we derive a novel formulation as a system of conservation laws. For first-order logarithmic forward–forward MFG, we establish the existence of a global solution. Then, we consider a class of explicit solutions and show the existence of shocks. Finally, we examine parabolic forward–forward MFGs and establish the long-time convergence of the solutions.

A new optimal transport distance on the space of finite Radon measures

We introduce a new optimal transport distance between nonnegative finite Radon measures with possibly different masses. The construction is based on non-conservative continuity equations and a corresponding modified Benamou-Brenier formula. We establish various topological and geometrical properties of the resulting metric space, derive some formal Riemannian structure, and develop differential calculus following F. Otto's approach. Finally, we apply these ideas to identify a model of animal dispersal proposed by MacCall and Cosner as a gradient flow in our formalism and obtain new long-time convergence results.

Self-similar fast-reaction limits for reaction–diffusion systems on unbounded domains

We present a unified approach to characterising fast-reaction limits of systems of either two reaction–diffusion equations, or one reaction–diffusion equation and one ordinary differential equation, on unbounded domains, motivated by models of fast chemical reactions where either one or both reactant(s) is/are mobile. For appropriate initial data, solutions of four classes of problems each converge in the fast-reaction limit k→∞ to a self-similar limit profile that has one of four forms, depending on how many components diffuse and whether the spatial domain is a half or whole line. For fixed k, long-time convergence to these same self-similar profiles is also established, thanks to a scaling argument of Kamin. Our results generalise earlier work of Hilhorst, van der Hout and Peletier to a much wider class of problems, and provide a quantitative description of the penetration of one substance into another in both the fast-reaction and long-time regimes.

Long-time convergence of numerical approximations for 2D GBBM equation

We study the long-time behavior of the finite difference solution to the generalized BBM equation in two space dimensions with dirichlet boundary conditions. The unique solvability of numerical solution is shown. It is proved that there exists a global attractor of the discrete dynamical system. Finally, we obtain the long-time stability and convergence of the difference scheme. Our results show that the difference scheme can effectively simulate the infinite dimensional dynamical systems. Numerical experiment results show that the theory is accurate and the schemes are efficient and reliable.