arXiv subjects

Diego del-Castillo-Negrete

Publications and source records attributed to Diego del-Castillo-Negrete.

17 recordsLinked to original sources

arXiv2026

An efficient probabilistic scheme for the exit time probability of $α$-stable Lévy process

The α-stable Lévy process, commonly used to describe Lévy flight, is characterized by discontinuous jumps and is widely used to model anomalous transport phenomena. In this study, we investigate the associated exit problem and propose a method to compute the exit time probability, which quantifies the likelihood that a trajectory starting from an initial condition exits a bounded region in phase space within a given time. This estimation plays a key role in understanding anomalous diffusion behavior. The proposed method approximates the α-stable process by combining a Brownian motion with a compound Poisson process. The exit time probability is then modeled using a framework based on partial integro-differential equations (PIDEs). The Feynman-Kac formula provides a probabilistic representation of the solution, involving conditional expectations over stochastic differential equations. These expectations are computed via tailored quadrature rules and interpolation techniques. The proposed method achieves first-order convergence in time and offers significant computational advantages over standard Monte Carlo and deterministic approaches. In particular, it avoids assembling and solving large dense linear systems, resulting in improved efficiency. We demonstrate the method's accuracy and performance through two numerical examples, highlighting its applicability to physical transport problems.

math.NA↗

arXiv2025

Generative AI Models for Learning Flow Maps of Stochastic Dynamical Systems in Bounded Domains

Simulating stochastic differential equations (SDEs) in bounded domains, presents significant computational challenges due to particle exit phenomena, which requires accurate modeling of interior stochastic dynamics and boundary interactions. Despite the success of machine learning-based methods in learning SDEs, existing learning methods are not applicable to SDEs in bounded domains because they cannot accurately capture the particle exit dynamics. We present a unified hybrid data-driven approach that combines a conditional diffusion model with an exit prediction neural network to capture both interior stochastic dynamics and boundary exit phenomena. Our ML model consists of two major components: a neural network that learns exit probabilities using binary cross-entropy loss with rigorous convergence guarantees, and a training-free diffusion model that generates state transitions for non-exiting particles using closed-form score functions. The two components are integrated through a probabilistic sampling algorithm that determines particle exit at each time step and generates appropriate state transitions. The performance of the proposed approach is demonstrated via three test cases: a one-dimensional simplified problem for theoretical verification, a two-dimensional advection-diffusion problem in a bounded domain, and a three-dimensional problem of interest to magnetically confined fusion plasmas.

stat.ML↗

arXiv2023

A pseudo-reversible normalizing flow for stochastic dynamical systems with various initial distributions

We present a pseudo-reversible normalizing flow method for efficiently generating samples of the state of a stochastic differential equation (SDE) with different initial distributions. The primary objective is to construct an accurate and efficient sampler that can be used as a surrogate model for computationally expensive numerical integration of SDE, such as those employed in particle simulation. After training, the normalizing flow model can directly generate samples of the SDE's final state without simulating trajectories. Existing normalizing flows for SDEs depend on the initial distribution, meaning the model needs to be re-trained when the initial distribution changes. The main novelty of our normalizing flow model is that it can learn the conditional distribution of the state, i.e., the distribution of the final state conditional on any initial state, such that the model only needs to be trained once and the trained model can be used to handle various initial distributions. This feature can provide a significant computational saving in studies of how the final state varies with the initial distribution. We provide a rigorous convergence analysis of the pseudo-reversible normalizing flow model to the target probability density function in the Kullback-Leibler divergence metric. Numerical experiments are provided to demonstrate the effectiveness of the proposed normalizing flow model.

math.NA↗

arXiv2022

A divergence-free constrained magnetic field interpolation method for scattered data

An interpolation method to evaluate magnetic fields given unstructured, scattered magnetic data is presented. The method is based on the reconstruction of the global magnetic field using a superposition of orthogonal functions. The coefficients of the expansion are obtained by minimizing a cost function defined as the L^2 norm of the difference between the ground truth and the reconstructed magnetic field evaluated on the training data. The divergence-free condition is incorporated as a constrain in the cost function allowing the method to achieve arbitrarily small errors in the magnetic field divergence. An exponential decay of the approximation error is observed and compared with the less favorable algebraic decay of local splines. Compared to local methods involving computationally expensive search algorithms, the proposed method exhibits a significant reduction of the computational complexity of the field evaluation, while maintaining a small error in the divergence even in the presence of magnetic islands and stochasticity. Applications to the computation of Poincaré sections using data obtained from numerical solutions of the magnetohydrodynamic equations in toroidal geometry are presented and compared with local methods currently in use.

physics.comp-ph↗

arXiv2022

Transport, flow topology and Lagrangian conditional statistics in edge plasma turbulence

Lagrangian statistics and particle transport in edge plasma turbulence are investigated using the Hasegawa-Wakatani model and its modified version. The latter shows the emergence of pronounced zonal flows. Different values of the adiabaticity parameter are considered. The main goal is to characterize the role of coherent structures, i.e., vortices and zonal flows, and their impact on the Lagrangian statistics of particles. Computationally intensive long time simulations following ensembles of test particles over hundreds of eddy turnover times are considered in statistically stationary turbulent flows. The flow topology is characterized using the Lagrangian Okubo-Weiss criterion, and the flow can thus be split into topologically different domains. In elliptic and hyperbolic regions, the probability density functions (pdfs) of the residence time have self-similar algebraic decaying tails. However, in the intermediate regions the pdfs do exhibit exponentially decaying tails. Topologically conditioned pdfs of the Lagrangian velocity, and acceleration and density fluctuations are likewise computed. The differences between the classical Hasegawa-Wakatani system and its modified version are assessed and the role of zonal flows is highlighted. The density flux spectrum which characterizes the contributions of different length scales is studied and its inertial scaling is found to be in agreement with predictions based on dimensional arguments. Analyzing the angular change of particle tracers at different time scales, corresponding to coarse grained curvature, completes the study and the multiscale geometric statistics quantify the directional properties of the particle motion in the different flow regimes.

physics.plasm-ph↗

arXiv2021

A Feynman-Kac based numerical method for the exit time probability of a class of transport problems

The exit time probability, which gives the likelihood that an initial condition leaves a prescribed region of the phase space of a dynamical system at, or before, a given time, is arguably one of the most natural and important transport problems. Here we present an accurate and efficient numerical method for computing this probability for systems described by non-autonomous (time-dependent) stochastic differential equations (SDEs) or their equivalent Fokker-Planck partial differential equations. The method is based on the direct approximation of the Feynman-Kac formula that establishes a link between the adjoint Fokker-Planck equation and the forward SDE. The Feynman-Kac formula is approximated using the Gauss-Hermite quadrature rules and piecewise cubic Hermite interpolating polynomials, and a GPU accelerated matrix representation is used to compute the entire time evolution of the exit time probability using a single pass of the algorithm. The method is unconditionally stable, exhibits second-order convergence in space, first-order convergence in time, and is straightforward to parallelize. Applications are presented to the advection-diffusion of a passive tracer in a fluid flow exhibiting chaotic advection, and to the runaway acceleration of electrons in a plasma in the presence of an electric field, collisions, and radiation damping. Benchmarks against analytical solutions as well as comparisons with explicit and implicit finite difference standard methods for the adjoint Fokker-Planck equation are presented.

physics.comp-ph↗

arXiv2021

A new method to compute periodic orbits in general symplectic maps

The search of high-order periodic orbits has been typically restricted to problems with symmetries that help to reduce the dimension of the search space. Well-known examples include reversible maps with symmetry lines. The present work proposes a new method to compute high-order periodic orbits in twist maps without the use of symmetries. The method is a combination of the parameterization method in Fourier space and a Newton-Gauss multiple shooting scheme. The parameterization method has been successfully used in the past to compute quasi-periodic invariant circles. However, this is the first time that this method is used in the context of periodic orbits. Numerical examples are presented showing the accuracy and efficiency of the proposed method. The method is also applied to verify the renormalization prediction of the residues' convergence at criticality (extensively studied in reversible maps) in the relatively unexplored case of maps without symmetries.

math.DS↗

arXiv2020

A sparse-grid probabilistic scheme for approximation of the runaway probability of electrons in fusion tokamak simulation

Runaway electrons (RE) generated during magnetic disruptions present a major threat to the safe operation of plasma nuclear fusion reactors. A critical aspect of understanding RE dynamics is to calculate the runaway probability, i.e., the probability that an electron in the phase space will runaway on, or before, a prescribed time. Such probability can be obtained by solving the adjoint equation of the underlying Fokker-Planck equation that controls the electron dynamics. In this effort, we present a sparse-grid probabilistic scheme for computing the runaway probability. The key ingredient of our approach is to represent the solution of the adjoint equation as a conditional expectation, such that discretizing the differential operator reduces to the approximation of a set of integrals. Adaptive sparse grid interpolation is utilized to approximate the map from the phase space to the runaway probability. The main novelties of this effort are the integration of the sparse-grid method into the probabilistic numerical scheme for computing escape probability, as well as the demonstration in computing RE probabilities. Two numerical examples are given to illustrate that the proposed method can achieve $\mathcal{O}(Δt)$ convergence, as well as the adaptive refinement strategy can effectively handle the sharp transition layer between the runaway and non-runaway regions.

physics.comp-ph↗

arXiv2018

A fluid-kinetic framework for self-consistent runaway-electron simulations

The problem of self-consistently coupling kinetic runaway-electron physics to the macroscopic evolution of the plasma is addressed by dividing the electron population into a bulk and a tail. A probabilistic closure is adopted to determine the coupling between the bulk and the tail populations, preserving them both as genuine, non-negative distribution functions. Macroscopic one-fluid equations and the kinetic equation for the runaway-electron population are then derived, now displaying sink and source terms due to transfer of electrons between the bulk and the tail.

physics.plasm-ph↗

arXiv2017

A backward Monte-Carlo method for time-dependent runaway electron simulations

Kinetic descriptions of runaway electrons (RE) are usually based on Fokker-Planck models that determine the probability distribution function (PDF) of RE in 2-dimensional momentum space. Despite of the simplification involved, the Fokker-Planck equation can rarely be solved analytically and direct numerical approaches (e.g., continuum and particle-based Monte Carlo (MC)) can be time consuming, especially in the computation of asymptotic-type observables including the runaway probability, the slowing-down and runaway mean times, and the energy limit probability. Here we present a novel backward MC approach to these problems based on backward stochastic differential equations (BSDEs) that describe the dynamics of the runaway probability by means of the Feynman-Kac theory. The key ingredient of the backward MC algorithm is to place all the particles in a runaway state and simulate them backward from the terminal time to the initial time. As such, our approach can provide much faster convergence than direct MC methods (by significantly reducing the number of particles required to achieve a prescribed accuracy) while at the same time maintaining the advantages of particle-based methods (compared to continuum approaches). The proposed algorithm is unconditionally stable, can be parallelized as easy as the direct MC method, and its extension to dimensions higher than two is straightforward, thus paving the way for conducting large-scale RE simulation.

physics.plasm-ph↗

arXiv2016

Charged particle dynamics in the presence of non-Gaussian Lévy electrostatic fluctuations

Full orbit dynamics of charged particles in a $3$-dimensional helical magnetic field in the presence of $α$-stable Lévy electrostatic fluctuations and linear friction modeling collisional Coulomb drag is studied via Monte Carlo numerical simulations. The Lévy fluctuations are introduced to model the effect of non-local transport due to fractional diffusion in velocity space resulting from intermittent electrostatic turbulence. The probability distribution functions of energy, particle displacements, and Larmor radii are computed and showed to exhibit a transition from exponential decay, in the case of Gaussian fluctuations, to power law decay in the case of Lévy fluctuations. The absolute value of the power law decay exponents are linearly proportional to the Lévy index $α$. The observed anomalous non-Gaussian statistics of the particles' Larmor radii (resulting from outlier transport events) indicate that, when electrostatic turbulent fluctuations exhibit non-Gaussian Lévy statistics, gyro-averaging and guiding centre approximations might face limitations and full particle orbit effects should be taken into account.

physics.plasm-ph↗

arXiv2016

Global transport in a nonautonomous standard map

A non-autonomous version of the standard map with a periodic variation of the parameter is introduced and studied. Symmetry properties in the variables and parameters of the map are found and used to find relations between rotation numbers of invariant sets of the autonomous realization of the period-two case of the map. The role of the nonautonomous dynamics on period-one orbits, stability and bifurcation is studied. The critical boundaries for the global transport and the destruction of invariant circles with fixed rotation number are studied in detail using direct computation and a continuation method. In the case of global transport, the critical boundary has a particular symmetrical horn shape. The results are contrasted with similar calculations found in the literature.

math.DS↗

arXiv2016

Multiscale statistical analysis of coronal solar activity

Multi-filter images from the solar corona are used to obtain temperature maps which are analyzed using techniques based on proper orthogonal decomposition (POD) in order to extract dynamical and structural information at various scales. Exploring active regions before and after a solar flare and comparing them with quiet regions we show that the multiscale behavior presents distinct statistical properties for each case that can be used to characterize the level of activity in a region. Information about the nature of heat transport is also be extracted from the analysis.

astro-ph.SR↗

arXiv2009

Wavelet-based density estimation for noise reduction in plasma simulations using particles

For given computational resources, the accuracy of plasma simulations using particles is mainly held back by the noise due to limited statistical sampling in the reconstruction of the particle distribution function. A method based on wavelet analysis is proposed and tested to reduce this noise. The method, known as wavelet based density estimation (WBDE), was previously introduced in the statistical literature to estimate probability densities given a finite number of independent measurements. Its novel application to plasma simulations can be viewed as a natural extension of the finite size particles (FSP) approach, with the advantage of estimating more accurately distribution functions that have localized sharp features. The proposed method preserves the moments of the particle distribution function to a good level of accuracy, has no constraints on the dimensionality of the system, does not require an a priori selection of a global smoothing scale, and its able to adapt locally to the smoothness of the density based on the given discrete particle data. Most importantly, the computational cost of the denoising stage is of the same order as one time step of a FSP simulation. The method is compared with a recently proposed proper orthogonal decomposition based method, and it is tested with three particle data sets that involve different levels of collisionality and interaction with external and self-consistent fields.

physics.comp-ph↗

arXiv2009

Truncation effects in superdiffusive front propagation with Lévy flights

A numerical and analytical study of the role of exponentially truncated Lévy flights in the superdiffusive propagation of fronts in reaction-diffusion systems is presented. The study is based on a variation of the Fisher-Kolmogorov equation where the diffusion operator is replaced by a $λ$-truncated fractional derivative of order $α$ where $1/λ$ is the characteristic truncation length scale. For $λ=0$ there is no truncation and fronts exhibit exponential acceleration and algebraic decaying tails. It is shown that for $λ\neq 0$ this phenomenology prevails in the intermediate asymptotic regime $(χt)^{1/α} \ll x \ll 1/λ$ where $χ$ is the diffusion constant. Outside the intermediate asymptotic regime, i.e. for $x > 1/λ$, the tail of the front exhibits the tempered decay $ϕ\sim e^{-λx}/x^{(1+α)} $, the acceleration is transient, and the front velocity, $v_L$, approaches the terminal speed $v_* = (γ- λ^αχ)/λ$ as $t\to \infty$, where it is assumed that $γ> λ^αχ$ with $γ$ denoting the growth rate of the reaction kinetics. However, the convergence of this process is algebraic, $v_L \sim v_* - α/(λt)$, which is very slow compared to the exponential convergence observed in the diffusive (Gaussian) case. An over-truncated regime in which the characteristic truncation length scale is shorter than the length scale of the decay of the initial condition, $1/ν$, is also identified. In this extreme regime, fronts exhibit exponential tails, $ϕ\sim e^{-νx}$, and move at the constant velocity, $v=(γ- λ^αχ)/ν$.

cond-mat.stat-mech↗

arXiv2003

The origin of diffusion: the case of non chaotic systems

We investigate the origin of diffusion in non-chaotic systems. As an example, we consider 1-$d$ map models whose slope is everywhere 1 (therefore the Lyapunov exponent is zero) but with random quenched discontinuities and quasi-periodic forcing. The models are constructed as non-chaotic approximations of chaotic maps showing deterministic diffusion, and represent one-dimensional versions of a Lorentz gas with polygonal obstacles (e.g., the Ehrenfest wind tree model). In particular, a simple construction shows that these maps define non-chaotic billiards in space-time. The models exhibit, in a wide range of the parameters, the same diffusive behavior of the corresponding chaotic versions. We present evidence of two sufficient ingredients for diffusive behavior in one-dimensional, non-chaotic systems: i) a finite-size, algebraic instability mechanism, and ii) a mechanism that suppresses periodic orbits.

nlin.CD↗

arXiv2002

Diffusive transport and self-consistent dynamics in coupled maps

The study of diffusion in Hamiltonian systems has been a problem of interest for a number of years. In this paper we explore the influence of self-consistency on the diffusion properties of systems described by coupled symplectic maps. Self-consistency, i.e. the back-influence of the transported quantity on the velocity field of the driving flow, despite of its critical importance, is usually overlooked in the description of realistic systems, for example in plasma physics. We propose a class of self-consistent models consisting of an ensemble of maps globally coupled through a mean field. Depending on the kind of coupling, two different general types of self-consistent maps are considered: maps coupled to the field only through the phase, and fully coupled maps, i.e. through the phase and the amplitude of the external field. The analogies and differences of the diffusion properties of these two kinds of maps are discussed in detail.

nlin.CD↗