Langevin Stochastic Differential Equation Research Articles

Non-asymptotic convergence bounds for modified tamed unadjusted Langevin algorithm in non-convex setting

We consider the problem of sampling from a high-dimensional target distribution πβ on Rd with density proportional to θ↦e−βU(θ) using explicit numerical schemes based on discretising the Langevin stochastic differential equation (SDE). In recent literature, taming has been proposed and studied as a method for ensuring stability of Langevin-based numerical schemes in the case of super-linearly growing drift coefficients for the Langevin SDE. In particular, the Tamed Unadjusted Langevin Algorithm (TULA) was proposed in [2] to sample from such target distributions with the gradient of the potential U being super-linearly growing. However, theoretical guarantees in Wasserstein distances for Langevin-based algorithms have traditionally been derived assuming strong convexity of the potential U. In this paper, we propose a novel taming factor and derive, under a setting with possibly non-convex potential U and super-linearly growing gradient of U, non-asymptotic theoretical bounds in Wasserstein-1 and Wasserstein-2 distances between the law of our algorithm, which we name the modified Tamed Unadjusted Langevin Algorithm (mTULA), and the target distribution πβ. We obtain respective rates of convergence O(λ) and O(λ1/2) in Wasserstein-1 and Wasserstein-2 distances for the discretisation error of mTULA in step size λ. High-dimensional numerical simulations which support our theoretical findings are presented to showcase the applicability of our algorithm.

New method for the investigation of mode coupling in graded-index polymer photonic crystal fibers using the Langevin stochastic differential equation

The mode coupling in a graded-index polymer photonic crystal fiber (GI PPCF) with a solid core has been investigated using the Langevin equation. Based on the computer-simulated Langevin force, the Langevin equation is numerically integrated. The numerical solutions of the Langevin equation align with those of the time-independent power flow equation (TI PFE). We showed that by solving the Langevin equation, which is a stochastic differential equation, one can successfully treat a mode coupling in GI PPCFs, which is an intrinsically stochastic process. We demonstrated that, in terms of effectiveness, the Langevin equation is preferable compared to the TI PFE. The GI PPCF achieves the equilibrium mode distribution (EMD) at a coupling length that is even shorter than the conventional GI plastic optical fiber (POF). The application of multimode GI PCFs in communications and optical fiber sensor systems will benefit from these findings.

Liouville models of particle-laden flow

Langevin (stochastic differential) equations are routinely used to describe particle-laden flows. They predict Gaussian probability density functions (PDFs) of a particle's trajectory and velocity, even though experimentally observed dynamics might be highly non-Gaussian. Our Liouville approach overcomes this dichotomy by replacing the Wiener process in the Langevin models with a (small) set of random variables, whose distributions are tuned to match the observed statistics. This strategy gives rise to an exact (deterministic, first-order, hyperbolic) Liouville equation that describes the evolution of a joint PDF in the augmented phase-space spanned by the random variables and the particle position and velocity. Analytical PDF solutions for canonical models of particle-laden flows serve to establish a relationship between the Langevin and Liouville approaches. Finally, our framework is used to derive a new analytical PDF model for fluidized homogeneous heating systems.

Stochastic analysis of air–sea heat fluxes variability in the North Atlantic in 1979–2022 based on reanalysis data

A dynamic stochastic model based on the Langevin stochastic differential equation is introduced for the reanalysis data of the ERA5 database to model and analyze the behavior of latent and sensible air–sea heat fluxes in the North Atlantic for the period 1979–2022. The point estimates of the random coefficients (the drift vector and the diffusion matrix) of this type of equation for the entire period under consideration are presented. The numerical methods and software tools for statistical analysis of time evolution of the coefficients as well as determination their relationships and the behavior of their maxima, averages and minima at various time intervals (days, months, years), are developed. A strong seasonality for the coefficients of the equation is demonstrated. The spatiotemporal variability of the dynamic and stochastic components of the coefficients of the Langevin equation and their relationship with jet streams of different regions of the North Atlantic is analyzed. The presence of non-trivial positive trends in the drift and diffusion coefficients, especially for the latent fluxes, within the interannual variability is demonstrated. One indicates a quantitative increase in the air–sea interaction on the interannual scale. Numerical estimation was carried out using high-performance computing cluster with software implementation in the Python programming language. The tools for dynamic visualization of various quantities on geographical maps of the region under consideration are also presented.

Forcing in DNS of stationary isotropic turbulence and its effects on the closure of diffusion current in the PDF kinetic equation for high-inertia particle pairs

The relative motion of inertial, monodisperse particle pairs in stationary isotropic turbulence is investigated by evolving the Langevin stochastic differential equations governing the pair relative velocities and separations in the limit of Stokes number ≫1. The stochastic force in the Langevin equations is evaluated from the pair relative-velocity-space diffusivity that is equal to 1/τv2 multiplied by the time integral of the Eulerian two-time correlation of fluid velocity differences seen by particle pairs that are nearly stationary (τv is the particle viscous relaxation time). The Eulerian two-time correlation of the fluid velocity-difference field is computed using direct numerical simulation (DNS) of isotropic turbulence seeded with fixed particles. Since numerical forcing of the energy-containing eddies is employed to achieve stationarity in DNS, the impact of the forcing scheme on the two-time correlation is quantified as a function of separation r and the Taylor micro-scale Reynolds number Reλ. As the DNS-computed correlation is needed to evaluate the stochastic force in Langevin simulations (LS), a key objective is to quantify the effects of forcing scheme on the LS predictions of pair relative motion. Deterministic forcing (DF) and stochastic forcing (SF) are employed to achieve stationarity in the DNS runs. In SF, one needs to specify the time scale Tf of the Uhlenbeck–Ornstein processes that constitute the forcing. The Eulerian two-time correlations were computed using both DNS with DF and DNS with SF, the latter for Tf=TE/4,TE/2,TE,2TE, and 4TE, where TE is the large-eddy turnover time obtained from the corresponding DNS with DF. It is seen that the correlations obtained from DNS with DF are higher than those from DNS with SF for all five Tf’s, the differences being substantially more pronounced at larger separations and for higher Reλ. The two-time correlations computed using DNS with DF and DNS with SF were then utilized to perform Langevin simulations for particle Stokes numbers based on Kolmogorov time scale, Stη≥10. For Reλ=80, the pair relative-velocity variances predicted by the Langevin simulations based on DF were not significantly different from those predicted by the Langevin simulations based on SF. However, for Reλ=130 and Stη=10,20, the variances obtained from LS based on DF were significantly higher than the variances from LS based on SF. The overprediction of relative velocity variances by LS-DF may be attributed to the slower decay of Eulerian two-time correlations when deterministic forcing is employed. The effects of forcing scheme on the radial distribution function (RDF) computed from the Langevin simulations are less pronounced compared to its effects on the variances. Nevertheless, for Reλ=130 and separations r=L/2 and L, where L is the integral length scale, the RDF values obtained from LS based on DF were higher than those from LS based on SF.

Selfsimilar stochastic differential equations

Diffusion in a logarithmic potential (DLP) attracted significant interest in physics recently. The dynamics of DLP are governed by a Langevin stochastic differential equation (SDE) whose underpinning potential is logarithmic, and that is driven by Brownian motion. The SDE that governs DLP is a particular case of a selfsimilar SDE: one that is driven by a selfsimilar motion, and that produces a selfsimilar motion. This paper establishes the pivotal role of selfsimilar SDEs via two novel universality results. I) Selfsimilar SDEs emerge universally, on the macro level, when applying scaling limits to micro-level SDEs. II) Selfsimilar SDEs emerge universally when applying the Lamperti transformation to stationary SDEs. Using the universality results, this paper further establishes: a novel statistical-analysis approach to selfsimilar Ito diffusions; and the focal importance of DLP.

Orientational dynamics of magnetotactic bacteria in Earth's magnetic field-a simulation study.

We investigate through simulations the phenomena of magnetoreception to enable an understanding of the minimum requirements of a fail-safe mechanism, operational at the cellular level, to sense a weak magnetic field at ambient temperature in a biologically active environment. To do this, we use magnetotactic bacteria (MTB) as our model system. The magnetic field sensing ability of these bacteria is due to the presence of magnetosomes, which are internal membrane-bound organelles that contain an iron-based magnetic mineral crystal. These magnetosomes are usually found arranged in a chain aligned with the long axis of the bacterial body. This arrangement yields an overall magnetic dipole moment to the bacterial cell. To simulate this orientation process, we set up a rotational Langevin stochastic differential equation and solve it repeatedly over appropriate time steps for isolated spherical shaped MTB as well as for a more realistic model of spheroidal MTB with flagella. The orientation process appears to depend on shape parameters with spheroidal MTB showing a slower response time compared to spherical MTB. Further, our simulation also reveals that the alignment to the external magnetic field is more robust for an MTB when compared to single magnetosome. For the simulation involving magnetosomes, we include an extra torque that arises from the twisting of an attachment tether and enhance the viscosity of the surrounding medium to mimic intracellular conditions in the governing Langevin equation. The response time of alignment is found to be substantially reduced when one includes a dipole interaction term with a neighboring magnetosome and the alignment becomes less robust with increase in inter dipole distance. The alignment process can thereby be said to be very sensitively dependent on the distance between magnetosomes. Simulating the process of alignment between two neighboring magnetosomes, both in the absence and presence of an ambient magnetic field, we conclude that alignment between these dipoles at the distances typical in an MTB is highly probable and it would be the locked unit that responds to changes in the external magnetic field.

Limiting behaviors of high dimensional stochastic spin ensembles

Lattice spin models in statistical physics are used to understand magnetism. Their Hamiltonians are a discrete form of a version of a Dirichlet energy, signifying a relationship to the Harmonic map heat flow equation. The Gibbs distribution, defined with this Hamiltonian, is used in the Metropolis-Hastings (M-H) algorithm to generate dynamics tending towards an equilibrium state. In the limiting situation when the inverse temperature is large, we establish the relationship between the discrete M-H dynamics and the continuous Harmonic map heat flow associated with the Hamiltonian. We show the convergence of the M-H dynamics to the Harmonic map heat flow equation in two steps: First, with fixed lattice size and proper choice of proposal size in one M-H step, the M-H dynamics acts as gradient descent and will be shown to converge to a system of Langevin stochastic differential equations (SDE). Second, with proper scaling of the inverse temperature in the Gibbs distribution and taking the lattice size to infinity, it will be shown that this SDE system converges to the deterministic Harmonic map heat flow equation. Our results are not unexpected, but show remarkable connections between the M-H steps and the SDE Stratonovich formulation, as well as reveal trajectory-wise out of equilibrium dynamics to be related to a canonical PDE system with geometric constraints.

On stochastic gradient Langevin dynamics with dependent data streams in the logconcave case

We study the problem of sampling from a probability distribution $\pi $ on $\mathbb{R}^{d}$ which has a density w.r.t. the Lebesgue measure known up to a normalization factor $x\mapsto \mathrm{e}^{-U(x)}/\int _{\mathbb{R}^{d}}\mathrm{e}^{-U(y)}\,\mathrm{d}y$. We analyze a sampling method based on the Euler discretization of the Langevin stochastic differential equations under the assumptions that the potential $U$ is continuously differentiable, $\nabla U$ is Lipschitz, and $U$ is strongly concave. We focus on the case where the gradient of the log-density cannot be directly computed but unbiased estimates of the gradient from possibly dependent observations are available. This setting can be seen as a combination of a stochastic approximation (here stochastic gradient) type algorithms with discretized Langevin dynamics. We obtain an upper bound of the Wasserstein-2 distance between the law of the iterates of this algorithm and the target distribution $\pi $ with constants depending explicitly on the Lipschitz and strong convexity constants of the potential and the dimension of the space. Finally, under weaker assumptions on $U$ and its gradient but in the presence of independent observations, we obtain analogous results in Wasserstein-2 distance.

Simulation of Gating Currents of the Shaker K Channel Using a Brownian Model of the Voltage Sensor

The physical mechanism underlying the voltage-dependent gating of K channels is usually addressed theoretically using molecular dynamics simulations. However, besides being computationally very expensive, this approach is presently unable to fully predict the behavior of fundamental variables of channel gating such as the macroscopic gating current, and hence, it is presently unable to validate the model. To fill this gap, here we propose a voltage-gating model that treats the S4 segment as a Brownian particle moving through a gating channel pore and adjacent internal and external vestibules. In our model, charges on the S4 segment are screened by charged residues localized on neighboring segments of the channel protein and by ions present in the vestibules, whose dynamics are assessed using a flux conservation equation. The electrostatic voltage spatial profile is consistently assessed by applying the Poisson equation to all the charges present in the system. The treatment of the S4 segment as a Brownian particle allows description of the dynamics of a single S4 segment using the Langevin stochastic differential equation or the behavior of a population of S4 segments—useful for assessing the macroscopic gating current—using the Fokker-Planck equation. The proposed model confirms the gating charge transfer hypothesis with the movement of the S4 segment among five different stable positions where the gating charges interact in succession with the negatively charged residues on the channel protein. This behavior produces macroscopic gating currents quite similar to those experimentally found.

Well-posedness and long time behavior of singular Langevin stochastic differential equations

In this paper, we study damped Langevin stochastic differential equations with singular velocity fields. We prove the strong well-posedness of such equations. Moreover, by combining the technique of Lyapunov functions with Krylov’s estimate, we also establish exponential ergodicity for the unique strong solution.

The tamed unadjusted Langevin algorithm

In this article, we consider the problem of sampling from a probability measure π having a density on Rd proportional to x↦e−U(x). The Euler discretization of the Langevin stochastic differential equation (SDE) is known to be unstable, when the potential U is superlinear. Based on previous works on the taming of superlinear drift coefficients for SDEs, we introduce the Tamed Unadjusted Langevin Algorithm (TULA) and obtain non-asymptotic bounds in V-total variation norm and Wasserstein distance of order 2 between the iterates of TULA and π, as well as weak error bounds. Numerical experiments are presented which support our findings.

Thermodynamics of Superdiffusion Generated by Lévy-Wiener Fluctuating Forces.

Scale free Lévy motion is a generalized analogue of the Wiener process. Its time derivative extends the notion of “white noise” to non-Gaussian noise sources, and as such, it has been widely used to model natural signal variations described by an overdamped Langevin stochastic differential equation. Here, we consider the dynamics of an archetypal model: a Brownian-like particle is driven by external forces, and noise is represented by uncorrelated Lévy fluctuations. An unperturbed system of that form eventually attains a steady state which is uniquely determined by the set of parameter values. We show that the analyzed Markov process with the stability index violates the detailed balance, i.e., its stationary state is quantified by a stationary probability density and nonvanishing current. We discuss consequences of the non-Gibbsian character of the stationary state of the system and its impact on the general form of the fluctuation–dissipation theorem derived for weak external forcing.

Moran-evolution of cooperation: From well-mixed to heterogeneous complex networks

Configurational arrangement of network architecture and interaction character of individuals are two most influential factors on the mechanisms underlying the evolutionary outcome of cooperation, which is explained by the well-established framework of evolutionary game theory. In the current study, not only qualitatively but also quantitatively, we measure Moran-evolution of cooperation to support an analytical agreement based on the consequences of the replicator equation in a finite population. The validity of the measurement has been double-checked in the well-mixed network by the Langevin stochastic differential equation and the Gillespie-algorithmic version of Moran-evolution, while in a structured network, the measurement of accuracy is verified by the standard numerical simulation. Considering the Birth–Death and Death–Birth updating rules through diffusion of individuals, the investigation is carried out in the wide range of game environments those relate to the various social dilemmas where we are able to draw a new rigorous mathematical track to tackle the heterogeneity of complex networks. The set of modified criteria reveals the exact fact about the emergence and maintenance of cooperation in the structured population. We find that in general, nature promotes the environment of coexistent traits.

Bond valuation for generalized Langevin processes with integrated Lévy noise

PurposeRecently, Stein et al. (2016) studied theoretical properties and parameter estimation of continuous time processes derived as solutions of a generalized Langevin equation (GLE). In this paper, the authors extend the model to a wider class of memory kernels and then propose a bond and bond option valuation model based on the extension of the generalized Langevin process of Stein et al. (2016).Design/methodology/approachBond and bond option pricing based on the proposed interest rate models presents new difficulties as the standard partial differential equation method of stochastic calculus for bond pricing cannot be used directly. The authors obtain bond and bond option prices by finding the closed form expression of the conditional characteristic function of the integrated short rate process driven by a general Lévy noise.FindingsThe authors obtain zero-coupon default-free bond and bond option prices for short rate models driven by a variety of Lévy processes, which include Vasicek model and the short rate model obtained by solving a second-order Langevin stochastic differential equation (SDE) as special cases.Originality/valueBond and bond option pricing plays an important role in capital markets and risk management. In this paper, the authors derive closed form expressions for bond and bond option prices for a wider class of interest rate models including second-order SDE models. Closed form expressions may be especially instrumental in facilitating parameter estimation in these models.

Bayesian System Identification using auxiliary stochastic dynamical systems

Bayesian approaches to statistical inference and system identification became practical with the development of effective sampling methods like Markov Chain Monte Carlo (MCMC). However, because the size and complexity of inference problems has dramatically increased, improved MCMC methods are required. Dynamical systems based samplers are an effective extension of traditional MCMC methods. These samplers treat the posterior probability distribution as the potential energy function of a dynamical system, enabling them to better exploit the structure of the inference problem. We present an algorithm, Second-Order Langevin MCMC (SOL-MC), a stochastic dynamical system based MCMC algorithm, which uses the damped second-order Langevin stochastic differential equation (SDE) to sample a posterior distribution. We design the SDE such that the desired posterior probability distribution is its stationary distribution. Since this method is based upon an underlying dynamical system, we can utilize existing work to develop, implement, and optimize the sampler's performance. As such, we can choose parameters which speed up the convergence to the stationary distribution and reduce temporal state and energy correlations in the samples. We then apply this sampler to a system identification problem for a non-linear hysteretic structure model to investigate this method under globally identifiable and unidentifiable conditions.

High-dimensional Bayesian inference via the unadjusted Langevin algorithm

We consider in this paper the problem of sampling a high-dimensional probability distribution $\pi$ having a density w.r.t. the Lebesgue measure on $\mathbb{R}^{d}$, known up to a normalization constant $x\mapsto\pi(x)=\mathrm{e}^{-U(x)}/\int_{\mathbb{R}^{d}}\mathrm{e}^{-U(y)}\,\mathrm{d}y$. Such problem naturally occurs for example in Bayesian inference and machine learning. Under the assumption that $U$ is continuously differentiable, $\nabla U$ is globally Lipschitz and $U$ is strongly convex, we obtain non-asymptotic bounds for the convergence to stationarity in Wasserstein distance of order $2$ and total variation distance of the sampling method based on the Euler discretization of the Langevin stochastic differential equation, for both constant and decreasing step sizes. The dependence on the dimension of the state space of these bounds is explicit. The convergence of an appropriately weighted empirical measure is also investigated and bounds for the mean square error and exponential deviation inequality are reported for functions which are measurable and bounded. An illustration to Bayesian inference for binary regression is presented to support our claims.

The Foraging Brain: Evidence of Lévy Dynamics in Brain Networks.

In this research we have analyzed functional magnetic resonance imaging (fMRI) signals of different networks in the brain under resting state condition. To such end, the dynamics of signal variation, have been conceived as a stochastic motion, namely it has been modelled through a generalized Langevin stochastic differential equation, which combines a deterministic drift component with a stochastic component where the Gaussian noise source has been replaced with α-stable noise. The parameters of the deterministic and stochastic parts of the model have been fitted from fluctuating data. Results show that the deterministic part is characterized by a simple, linear decreasing trend, and, most important, the α-stable noise, at varying characteristic index α, is the source of a spectrum of activity modes across the networks, from those originated by classic Gaussian noise (α = 2), to longer tailed behaviors generated by the more general Lévy noise (1 ≤ α < 2). Lévy motion is a specific instance of scale-free behavior, it is a source of anomalous diffusion and it has been related to many aspects of human cognition, such as information foraging through memory retrieval or visual exploration. Finally, some conclusions have been drawn on the functional significance of the dynamics corresponding to different α values.

The Small-Mass Limit for Langevin Dynamics with Unbounded Coefficients and Positive Friction

A class of Langevin stochastic differential equations is shown to converge in the small-mass limit under very weak assumptions on the coefficients defining the equation. The convergence result is applied to three physically realizable examples where the coefficients defining the Langevin equation for these examples grow unboundedly either at a boundary, such as a wall, and/or at the point at infinity. This unboundedness violates the assumptions of previous limit theorems in the literature. The main result of this paper proves convergence for such examples.

Dimension-independent likelihood-informed MCMC

Many Bayesian inference problems require exploring the posterior distribution of high-dimensional parameters that represent the discretization of an underlying function. This work introduces a family of Markov chain Monte Carlo (MCMC) samplers that can adapt to the particular structure of a posterior distribution over functions. Two distinct lines of research intersect in the methods developed here. First, we introduce a general class of operator-weighted proposal distributions that are well defined on function space, such that the performance of the resulting MCMC samplers is independent of the discretization of the function. Second, by exploiting local Hessian information and any associated low-dimensional structure in the change from prior to posterior distributions, we develop an inhomogeneous discretization scheme for the Langevin stochastic differential equation that yields operator-weighted proposals adapted to the non-Gaussian structure of the posterior. The resulting dimension-independent and likelihood-informed (DILI) MCMC samplers may be useful for a large class of high-dimensional problems where the target probability measure has a density with respect to a Gaussian reference measure. Two nonlinear inverse problems are used to demonstrate the efficiency of these DILI samplers: an elliptic PDE coefficient inverse problem and path reconstruction in a conditioned diffusion.