Special Session 130: kinetic theory, analysis and application

Fast spectral method for the linearized Boltzmann collision operator

Zhenning Cai National University of Singapore Singapore Co-Author(s):

Abstract: We develop a fast numerical method for the Boltzmann collision operator linearized about the local Maxwellian. The algorihtm is based on the Fourier spectral method, and the computational cost is $O(N^4 \log N)$ for all variable hard sphere models. To achieve better numerical stability, we couple the Fourier method and the Burnett spectral method to handle the tail of the distribution function. In this talk, we will also discuss the potential use of our method in solving the steady-state Boltzmann equation.

An asymptotic preserving scheme for kinetic models with singular limit

Alina Chertock North Carolina State University USA Co-Author(s):    Yan Bokai and Changhui Tan

Abstract: We propose a new class of asymptotic preserving schemes to solve kinetic equations with a mono-kinetic singular limit. The main idea in dealing with singularity is to transform the equations by appropriate scalings in velocity. In particular, we study two biologically related kinetic systems. We derive the scaling factors and prove that the rescaled solution does not have a singular limit under appropriate spatial non-oscillatory assumptions, which can be verified numerically by a newly developed asymptotic preserving scheme. We conducted a few numerical experiments demonstrating the schemes` accuracy, stability, efficiency, and asymptotic preserving properties.

Model Reduction for Multiscale Dynamics on Networks

Weiqi Chu University of Massachusetts Amherst USA Co-Author(s):    Weiqi Chu, Qin Li, Alina Chertock

Abstract: Consider a complex system consisting of a large number of interacting agents, coupled through pairwise interactions with nonhomogeneous weights. Simulating the dynamics and identifying patterns of collective behavior can become computationally expensive, particularly as the system size grows, making most of the related algorithms unscalable. In this talk, I propose a model-reduction framework that transforms heterogeneous interacting particle systems into multi-community mean-field models by accounting for the network community structures for reduction. I will also introduce two structure-preserving numerical methods for solving these reduced mean-field equations.

Forward and inverse computation for radiative transfer via hp-adaptive mesh refinement

Shukai Du Syracuse University USA Co-Author(s):    Shukai Du, Samuel N. Stechmann

Abstract: The forward and inverse problems for radiative transfer are critical in many applications, such as climate modeling, optical tomography, and remote sensing. Both problems present major challenges, particularly large memory requirements and computational expense, due to the high dimensionality of the equation and the iterative nature of solving the inverse problem. To tackle these challenges, we investigate the hp-adaptive mesh refinement approach, which has proved effective in efficiently representing solutions where they are smooth with high-order approximations, while also providing the flexibility to resolve local features through adaptive refinements. For the forward problem, we demonstrate that exponential convergence with respect to degrees of freedom (DOFs) can be achieved even when the solution exhibits certain levels of sharp gradients. For the inverse problem, we introduce a goal-oriented hp-adaptive mesh refinement method that can blend the two optimization processes -- one for inversion and one for mesh adaptivity -- thereby reducing computational cost and memory requirements. Numerical tests are presented to validate the theoretical predictions.

A mean-field approach for the asymptotic tracking of continuum target clouds

Seung Yeal Ha Seoul National University Korea Co-Author(s):    Seung Yeal Ha and Hyunjin Ahn

Abstract: In this talk, we propose a new coupled kinetic system arising from the asymptotic tracking of a continuum target cloud, and study its asymptotic tracking property. For the proposed kinetic system, we present an energy functional which is monotonic and distance between particle trajectories corresponding to kinetic equations for target, and tracking ensembles tend to zero asymptotically under a suitable sufficient framework. The framework is formulated in terms of system parameters and initial data. This is a joint work with Hyunjin Ahn (Myongji Univ. Korea)

Unique identification for discretized inverse problems

Ruhui Jin University of Wisconsin-Madison USA Co-Author(s):    Qin Li, Anjali Nair, Sam Stechmann

Abstract: Theoretical inverse problems are often studied in an ideal infinite-dimensional setting. The well-posedness theory provides a unique reconstruction of the parameter function, when an infinite amount of data is given. Through the lens of PDE-constrained optimization, this means one attains the zero-loss property of the mismatch function in this setting. This is no longer true in computations when we are limited to finite amount of measurements due to experimental or economical reasons. Consequently, one must compromise the goal, from inferring a function, to a discrete approximation. What is the reconstruction power of a fixed number of data observations? How many parameters can one reconstruct? Here we describe a probabilistic approach, and spell out the interplay of the observation size $(r)$ and the number of parameters to be uniquely identified $(m)$. The technical pillar here is the random sketching strategy, in which the matrix concentration inequality and sampling theory are largely employed. By analyzing a randomly subsampled Hessian matrix, we attain a well-conditioned reconstruction problem with high probability. Our main theory is validated in numerical experiments, using an elliptic inverse problem as an example.

On the dynamical low-rank numerical method for kinetic equations

Christian Klingenberg Wuerzburg University Germany Co-Author(s):    Lena Baumann (Wuerzburg, Germany), Lukas Einkemmer (Innsbruck, Austria) and Jonas Kusch (As, Norway)

Abstract: The numerical solution of kinetic equations often requires a high computational effort and memory cost due to the potentially six-dimensional phase space. One approach to overcome this difficulty is the reduced order method dynamical low-rank approximation. It has recently gained an increasing interest as it has been shown to provide accurate numerical solutions of kinetic PDEs in various applications while reducing the computational time significantly. This talk will focus on a research project that has the goal to devise an efficient numerical method for solving a BGK-type kinetic equation $$ \partial_t f+ v \cdot \nabla f= \sigma (M - f) . $$ Our approach has the potential to bring about large savings in computational time. We build on the low-rank approximation technique used by Einkemmer, Jingwei Hu, Ying, SIAM J. Sci. Comput. (2021). We show how we have made progress in reducing the numerical effort even further by proving stability estimates for a related system of kinetic equations, see Baumann, Einkemmer, Klingenberg, Kusch, SIAM J. Sci. Comput. (2024). This is joint work with Lena Baumann (W\urzburg, Germany), Lukas Einkemmer (Innsbruck,Austria) and Jonas Kusch (As, Norway).

A Hybrid Finite-Difference-Particle Method for Chemotaxis Models

Alexander Kurganov Southern University of Science and Technology Peoples Rep of China Co-Author(s):    Alina Chertock, Shumo Cui, Chenxi Wang

Abstract: Chemotaxis systems play a crucial role in modeling the dynamics of bacterial and cellular behaviors, including propagation, aggregation, and pattern formation, all under the influence of chemical signals. One notable characteristic of these systems is their ability to simulate concentration phenomena, where cell density undergoes rapid growth near specific concentration points or along certain curves. Such growth can result in singular, spiky structures and lead to finite-time blowups. Our investigation focuses on the dynamics of the Patlak-Keller-Segel chemotaxis system and its two-species extensions. In the latter case, different species may exhibit distinct chemotactic sensitivities, giving rise to very different rates of cell density growth. Such a situation may be extremely challenging for numerical methods as they may fail to accurately capture the blowup of the slower-growing species mainly due to excessive numerical dissipation. We propose a hybrid finite-difference-particle (FDP) method, in which a sticky particle method is used to solve the chemotaxis equation(s), while finite-difference schemes are employed to solve the chemoattractant equation. Thanks to the low-dissipation nature of the particle method, the proposed hybrid scheme is particularly adept at capturing the blowup behaviors in both one- and two-species cases. The proposed hybrid FDP methods are tested on a series of challenging examples, and the obtained numerical results demonstrate that our hybrid method can provide sharp resolution of the singular structures even with a relatively small number of particles. Moreover, in the two-species case, our method adeptly captures the blowing-up solution for the component with lower chemotactic sensitivity, a feature not observed in other works.

From Schr\{o}dinger to diffusion- speckle formation of light in random media and the Gaussian conjecture

Anjali Nair University of Chicago USA Co-Author(s):    Guillaume Bal

Abstract: A well-known conjecture in physical literature states that high frequency waves propagating over long distances through turbulence eventually become complex Gaussian distributed. The intensity of such wave fields then follows an exponential law, consistent with speckle formation observed in physical experiments. Though fairly well-accepted and intuitive, this conjecture is not entirely supported by any detailed mathematical derivation. In this talk, I will discuss some recent results demonstrating the Gaussian conjecture in a weak-coupling regime of the paraxial approximation. The paraxial approximation is a high frequency approximation of the Helmholtz equation, where backscattering is ignored. This takes the form of a Schr\{o}dinger equation with a random potential and is often used to model laser propagation through turbulence. The proof relies on the asymptotic closeness of statistical moments of the wavefield under the paraxial approximation, its white noise limit and the complex Gaussian distribution itself. I will describe two scaling regimes, one is a kinetic scaling where the second moment is given by a transport equation and a second diffusive scaling, where the second moment follows an anomalous diffusion. In both cases, the limiting complex Gaussian distribution is fully characterized by its first and second moments. An additional stochastic continuity/tightness criterion allows to show the convergence of these distributions over spaces of H\{o}lder-continuous functions. This is joint work with Guillaume Bal.

The sticky particle dynamics with alignment interactions

Changhui Tan University of South Carolina USA Co-Author(s):    Trevor Leslie

Abstract: In this talk, I will introduce the Euler-alignment system in collective dynamics, which models flocking behavior. The discussion will center on weak solutions, with the goal of isolating a unique solution through the use of an entropic selection principle. Notably, this selection principle aligns with the sticky particle rules applied in the agent-based Cucker-Smale dynamics. I will present an analytical convergence result and discuss the formation of both finite- and infinite-time clusters.

On the exponential weak flocking for the kinetic Cucker-Smale model with non-compact support

XINYU WANG Seoul National University Peoples Rep of China Co-Author(s):    Seung-Yeal Ha, Xinyu Wang,Xiaoping Xue

Abstract: We study the propagation of the second spatial-velocity moments for the kinetic Cucker-Smale model with non-compact spatial support. In contrast to compact support, non-compact support leads to a lower bound of zero for the communication weight, which makes the previous approach break down. To address this challenge, we consider two types of initial distributions: exponential decay distributions and polynomial decay distributions. Moreover, our approach uses the infinite-particle mean-field approximation as an intermediary step to analyze the kinetic Cucker-Smale model, with conservation laws of mass and momentum. When initial distributions belong to the aforementioned types of decaying classes and coupling strength exceeds a certain threshold, we show the weak flocking behavior of the kinetic Cucker-Smale model. Specifically, the second velocity moment of the solution centered around the initial average velocity converges to zero, and the second spatial moment around the position of the center of mass remains uniformly bounded in time. The emergence of weak flocking behavior illustrates that even for non-compact support, a certain degree of aggregation can be maintained for the kinetic Cucker-Smale model, as long as the initial distribution exhibits relative concentration.

Shock Profiles for the Long-Range Boltzmann Equation

Dominic L Wynter University of Texas at Austin USA Co-Author(s):

Abstract: The Boltzmann equation models gas dynamics in the low density or high Mach number regime, using a statistical description of molecular interactions. Shock wave solutions have been constructed for the Boltzmann equation with hard-sphere particle interactions, and more recently for the related Landau equation of plasma dynamics by Albritton, Bedrossian, and Novack. Along similar lines as these works, we construct traveling shock solutions for the Boltzmann equation when molecular interactions are long-range. We prove existence and uniqueness up to translation near compressible Navier-Stokes shock profiles, using stability estimates for the Boltzmann equation and the stability theory of viscous shocks.

Dimension-free ergodicity of path integral molecular dynamics: a generalized Gamma calculus approach

Xuda Ye Peking University Peoples Rep of China Co-Author(s):    Zhennan Zhou

Abstract: Path integral molecular dynamics (PIMD) is a standard method for computing thermal averages in quantum canonical ensembles, with its accuracy depending on the number of beads, $D$, representing the discretization size of the Feynman path integral. Despite its widespread use in computational physics, the ergodicity of PIMD, particularly the dependence of the convergence rate on $D$, is not well understood. In this talk, I will present a rigorous analysis proving the uniform-in-$D$ ergodicity of PIMD, meaning that the convergence rate toward equilibrium is independent of the bead count $D$. This result is established for both overdamped and underdamped Langevin dynamics. Our approach relies on the generalized Gamma calculus, an advanced technique related to hypocoercivity, developed by Pierre Monmarch\`{e}, which provides deeper insight into the long-time behavior of these stochastic systems.

Random Winfree dynamics with high-order couplings

Jaeyoung Yoon Technical University of Munich Germany Co-Author(s):    Seung-Yeal Ha, Myeongju Kang, Jaeyoung Yoon and Mattia Zanella

Abstract: Recently, Winfree dynamics with high-order couplings was studied for better description of nature communications. Of particular interest in this work is the random order, which means order of coupling is regarded as a random variable. We derive the sufficient conditions under which the random Winfree dynamics converges to death-state stably and compare the theoretical results with numeric simulations.

Infinitely many solutions to the isentropic system of gas dynamics

Cheng Yu University of Florida USA Co-Author(s):

Abstract: In this talk, I will discuss the non-uniqueness of global weak solutions to the isentropic system of gas dynamics. In particular, I will show that for any initial data belonging to a dense subset of the energy space, there exists infinitely many global weak solutions to the isentropic Euler equations for any $1< \gamma\leq 1+2/n$. The proof is based on a generalization of convex integration techniques and weak vanishing viscosity limit of the Navier-Stokes equations. This talk is based on a joint work with M. Chen and A. Vasseur.

Fokker-Planck equations of neuron networks: numerical simulation and dilating the blowup solution

Zhennan Zhou Westlake University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we are concerned with the Fokker-Planck equations associated with the Nonlinear Noisy Leaky Integrate-and-Fire model for neuron networks. Due to the jump mechanism at the microscopic level, such Fokker-Planck equations are endowed with an unconventional structure: transporting the boundary flux to a specific interior point. In the first part of the talk, we present a conservative and positivity-preserving scheme for these Fokker-Planck equations, and we show that in the linear case, the semi-discrete scheme satisfies the discrete relative entropy estimate, which essentially matches the only known long-time asymptotic solution property. We also provide extensive numerical tests to verify the scheme properties, and carry out several sets of numerical experiments, including finite-time blowup, convergence to equilibrium and capturing time-period solutions of the variant models. Secondly, we introduce a new notion of genealized solutions for this model with a dynamical time rescaling, so that the extension of solutions after blowups become possible.

A PDE-based model-free algorithm for Continuous-time Reinforcement Learning

Yuhua Zhu University of California, Los Angeles USA Co-Author(s):

Abstract: This talk addresses the problem of continuous-time reinforcement learning (RL). When the underlying dynamics remain unknown and only discrete-time observations are available, how can we effectively conduct policy evaluation and policy iteration? We first highlight that while model-free RL algorithms are straightforward to implement, they are often not a reliable approximation of the true value function. On the other hand, model-based PDE approaches are more accurate, but the inverse problem is not easy to solve. To bridge this gap, we introduce a new Bellman equation, PhiBE, which integrates discrete-time information into a PDE formulation. PhiBE allows us to skip the identification of the dynamics and directly evaluate the value function using discrete-time data. Additionally, it offers a more accurate approximation of the true value function, especially in scenarios where the underlying dynamics change slowly. Moreover, we extend PhiBE to higher orders, providing increasingly accurate approximations.