arXiv subjects

Eitan Tadmor

Publications and source records attributed to Eitan Tadmor.

At least 19 recordsLinked to original sources

arXiv2026

Sampling and Optimization meet Enhanced Flows

It is well known that the computational realization of Gibbs probability measures, $e^{-\mathbb{U}(\mathbf{x})}/Z$, plays a central role in sampling and optimization. In this paper, we introduce two types of dynamics that exhibit rapid convergence towards these Gibbs measures. The mechanism driving this rapid convergence is the enhanced dissipation associated with these transport-diffusion dynamics. Motivated by these enhanced dynamics, we design numerical algorithms for sampling from the target Gibbs measure. Finally, we provide the corresponding particle systems that may yield other effective numerical samplers.

math.OC↗

arXiv2026

Finite-time breakdown of the Euler-alignment system for supercritical initial data

We study finite-time breakdown of classical solutions to the Euler-alignment system through the degeneration of the associated Lagrangian flow. This approach allows us to characterize singularity formation in terms of the loss of local invertibility of the flow and the resulting concentration of density along characteristics. For the case of constant communication kernels, we derive an explicit formula for the flow and obtain an exact pointwise breakdown criterion in arbitrary dimension. In two dimensions, this criterion admits a closed-form reformulation in terms of the symmetric part of the initial velocity gradient and the initial vorticity. For general non-constant kernels, we derive sufficient conditions for finite-time degeneracy by combining a leading compressive mechanism with perturbative control of the nonlocal remainder. These conditions provide quantitative supercritical breakdown criteria in arbitrary dimension, complementing the existing subcritical global-regularity theory for multidimensional Euler-alignment systems.

math.AP↗

arXiv2026

Optimal drift optimizer for non-convex optimization

We study a finite-horizon stochastic control criterion for non-convex optimization in which Brownian exploration is balanced against a quadratic control cost. Rather than emphasizing the classical Hopf--Cole representation, we isolate the exact drift selected by the criterion and reorganize it in a form adapted to optimization. The key object is the conditional terminal law of the optimal process. We show that this law is a Gibbs measure for a proximally penalized energy, yielding three exact representations of the drift: potential, averaged-gradient, and barycentric. We then analyze two asymptotic regimes relevant for optimization. As terminal time is approached, the drift recovers a scaled gradient-descent field. In the low-temperature regime, assuming a unique global minimizer, the conditional terminal law concentrates on it even in the presence of nonglobal local minima, and the drift converges to an affine attraction field toward it. In the nondegenerate case we also derive Laplace asymptotics for the drift, the value function, and the covariance of the conditional terminal law. Finally, we record a simple gradient-free discretization suggested by the barycentric formula.

math.OC↗

arXiv2026

Variational formulation of hyperbolic conservation laws

Entropy functions played a key role in the development of mathematical theory for hyperbolic conservation laws. The notion of entropy, which is intimately connected with symmetry, is an extension \emph{imposed} on nonlinear systems of conservation laws. In this context, Friedrichs raised the question whether the assumed symmetries can also be derived. We introduce a variational formulation that addresses Friedrichs' question: an entropy function is derived as an extremal object, from which we deduce, rather than impose, the maximum entropy production principle of Dafermos.

math.AP↗

arXiv2026

Lagrangian formulation and Eulerian closure in alignment dynamics

We investigate a continuum Lagrangian $p$-alignment system given by a nonlocal mean-field system of ordinary differential equations for interacting agents with weak initial data. We first establish global well-posedness of the Lagrangian dynamics and derive quantitative flocking estimates. We next construct Eulerian variables from the possibly non-injective Lagrangian flow via pushforward and disintegration, which leads to an Euler--Reynolds--alignment system featuring a nonnegative Reynolds stress and, for $p>2$, a nonlinear defect force induced by microscopic velocity fluctuations. Assuming only heavy-tailed interaction, we then show that these defect terms vanish asymptotically, leading to asymptotic mono-kinetic closure in the long-time limit. In the linear case $p=2$, we further obtain global weak solutions to the Euler--alignment system, including a sharp one-dimensional critical-threshold characterization and a global result in higher dimensions under a large-coupling condition. Finally, we establish a uniform-in-time mean-field stability estimate for the particle Cucker--Smale system in the linear regime and deduce uniform-in-time convergence toward the mono-kinetic Eulerian limit; for general $p\ge2$, we also obtain a finite-time mean-field convergence result toward the associated kinetic/Lagrangian alignment dynamics.

math.AP↗

arXiv2025

Entropy decrease and emergence of order in collective dynamics

We study the hydrodynamic description of collective dynamics driven by velocity {\it alignment}. It is known that such Euler alignment systems must flock towards a limiting ``flocking'' velocity, provided their solutions remain globally smooth. To address this question of global existence we proceed in two steps. (i) Entropy and closure. The system lacks a closure, reflecting lack of detailed energy balance in collective dynamics. We discuss the decrease of entropy and the asymptotic behavior towards a mono-kinetic closure; and (ii) Mono-kinetic closure. We prove that global regularity persists for all time for a large class of initial conditions satisfying a critical threshold condition, which is intimately linked to the decrease of entropy. The result applies in any number of spatial dimensions, thus addressing the open question of existence beyond two dimensions.

math.AP↗

arXiv2025

Large-time behavior of pressureless Euler--Poisson equations with background states

We study the large-time asymptotic behavior of solutions to the one-dimensional damped pressureless Euler-Poisson system with variable background states, subject to a neutrality condition. In the case where the background density converges asymptotically to a positive constant, we establish the convergence of global classical solutions toward the corresponding equilibrium state. The proof combines phase plane analysis with hypocoercivity-type estimates. As an application, we analyze the damped pressureless Euler--Poisson system arising in cold plasma ion dynamics, where the electron density is modeled by a Maxwell-Boltzmann relation. We show that solutions converge exponentially to the steady state under suitable a priori bounds on the density and velocity fields. Our results provide a rigorous characterization of asymptotic stability for damped Euler-Poisson systems with nontrivial background structures.

math.AP↗

arXiv2024

Constructions of bounded solutions of $div\, {\mathbf u}=f$ in critical spaces

We construct uniformly bounded solutions of the equation $div\, {\mathbf u}=f$ for arbitrary data $f$ in the critical spaces $L^d(\Omega)$, where $\Omega$ is a domain of ${\mathbb R}^d$. This question was addressed by Bourgain & Brezis, [On the equation ${\rm div}\, Y=f$ and application to control of phases, JAMS 16(2) (2003) 393-426], who proved that although the problem has a uniformly bounded solution, it is critical in the sense that there exists no linear solution operator for general $L^d$-data. We first discuss the validity of this existence result under weaker conditions than $f\in L^d(\Omega)$, and then focus our work on constructive processes for such uniformly bounded solutions. In the $d=2$ case, we present a direct one-step explicit construction, which generalizes for $d>2$ to a $(d-1)$-step construction based on induction. An explicit construction is proposed for compactly supported data in $L^{2,\infty}(\Omega)$ in the $d=2$ case. We also present constructive approaches based on optimization of a certain loss functional adapted to the problem. This approach provides a two-step construction in the $d=2$ case. This optimization is used as the building block of a hierarchical multistep process introduced in [E. Tadmor, Hierarchical construction of bounded solutions in critical regularity spaces, CPAM 69(6) (2016) 1087-1109] that converges to a solution in more general situations.

math.AP↗

arXiv2024

Swarm-based gradient descent meets simulated annealing

We introduce a novel method for non-convex optimization, called Swarm-based Simulated Annealing (SSA), which is at the interface between the swarm-based gradient-descent (SBGD) [J. Lu et. al., ArXiv:2211.17157; E.Tadmor and A. Zenginoglu, Acta Applicandae Math., 190, 2024] and Simulated Annealing (SA) [V. Cerny, J. optimization theory and appl., 45:41-51, 1985; S.Kirkpatrick et. al., Science, 220(4598):671-680, 1983; S. Geman and C.-R. Hwang, SIAM J. Control and Optimization, 24(5):1031-1043, 1986]. Similar to SBGD, we introduce a swarm of agents, each identified with a position, ${\mathbf x}$ and mass $m$, to explore the ambient space. Similar to SA, the agents proceed in the gradient descent direction, and are subject to Brownian motion. The annealing rate, however, is dictated by a decreasing function of their mass. As a consequence, instead of the SA protocol for time-decreasing temperature, we let the swarm decide how to `cool down' agents, depending on their accumulated mass over time. The dynamics of masses is coupled with the dynamics of positions: agents at higher ground transfer (part of) their mass to those at lower ground. Consequently, resulting SSA optimizer is dynamically divided between heavier, cooler agents viewed as `leaders' and lighter, warmer agents viewed as `explorers'. Mean-field convergence analysis and benchmark optimizations demonstrate the effectiveness of the swarm-based method as a multi-dimensional global optimizer.

math.OC↗

arXiv2024

Critical thresholds in pressureless Euler--Poisson equations with background states

We investigate the critical threshold phenomena in a large class of one dimensional pressureless Euler--Poisson (EP) equations, with non-vanishing background states. First, we establish local-in-time well-posedness in proper regularity spaces, which are adapted for a certain \textit{neutrality condition} to hold. The neutrality condition is shown to be necessary: we construct smooth solutions that exhibit instantaneous failure of the neutrality condition, which in turn yields non-existence of solutions, even locally in time, in the classical Sobolev spaces $H^s({\mathbb R})$, $s \geq 2$. Next, we study the critical threshold phenomena in the neutrality-condition-satisfying pressureless EP systems, where we distinguish between two cases. We prove that in the case of attractive forcing, the neutrality condition can further restrict the sub-critical region into its borderline, namely -- the sub-critical region is reduced to a single line in the phase plane. We then turn to provide a rather definitive answer for the critical thresholds in the case of repulsive EP systems with variable backgrounds. As an application, we analyze the critical thresholds for the damped EP system for cold plasma ion dynamics, where the density of electrons is given by the \textit{Maxwell--Boltzmann relation}.

math.AP↗

arXiv2023

Runge-Kutta methods are stable

We prove that Runge-Kutta (RK) methods for numerical integration of arbitrarily large systems of Ordinary Differential Equations are linearly stable. Standard stability arguments -- based on spectral analysis, resolvent condition or strong stability, fail to secure the stability of arbitrarily large RK systems. We explain the failure of different approaches, offer a new stability theory and demonstrate a few examples.

math.NA↗

arXiv2023

Revisting high-resolution schemes with van-Albada slope limiter

Slope limiters play an essential role in maintaining the non-oscillatory behavior of high-resolution methods for nonlinear conservation laws. The family of minmod limiters serves as the prototype example. Here, we revisit the question of non-oscillatory behavior of high-resolution central schemes in terms of the slope limiter proposed by van Albada et. al. 1982. The van Albada (vA) limiter is smoother near extrema, and consequently, in many cases, it outperforms the results obtained using the standard minmod limiter. In particular, we prove that the vA limiter ensures 1D TVD stability and demonstrate that it yields noticeable improvement in computation of one- and two-dimensional systems.

math.NA↗

arXiv2023

Swarm-based optimization with random descent

We extend our study of the swarm-based gradient descent method for non-convex optimization, [Lu, Tadmor & Zenginoglu, arXiv:2211.17157], to allow random descent directions. We recall that the swarm-based approach consists of a swarm of agents, each identified with a position, ${\mathbf x}$, and mass, $m$. The key is the transfer of mass from high ground to low(-est) ground. The mass of an agent dictates its step size: lighter agents take larger steps. In this paper, the essential new feature is the choice of direction: rather than restricting the swarm to march in the steepest gradient descent, we let agents proceed in randomly chosen directions centered around -- but otherwise different from -- the gradient direction. The random search secures the descent property while at the same time, enabling greater exploration of ambient space. Convergence analysis and benchmark optimizations demonstrate the effectiveness of the swarm-based random descent method as a multi-dimensional global optimizer.

math.OC↗

arXiv2023

Long time and large crowd dynamics of discrete Cucker-Smale alignment models

We provide a bird's eye view on developments in analyzing the long time, large crowd behavior of Cucker-Smale alignment dynamics. We consider a class of (fully-)discrete models, paying particular attention to general alignment protocols in which agents, with possibly time-dependent masses, are driven by a large class of heavy-tailed communication kernels. The presence of time-dependent masses allows, in particular, non-symmetric communication. While revisiting known results in the literature, we also shed new light of various aspects on the long time flocking/swarming behavior, driven by the decay of energy fluctuations and heavy-tailed connectivity. We also discuss the large crowd dynamics in terms of the hydrodynamic description of Euler alignment models.

math.DS↗

arXiv2023

The Emergence of Lines of Hierarchy in Collective Motion of Biological Systems

The emergence of large scale structures in biological systems, and in particular the formation of lines of hierarchy, is observed in many scales, from collections of cells to groups of insects to herds of animals. Motivated by phenomena in chemotaxis and phototaxis, we present a new class of alignment models which exhibit alignment into lines. The spontaneous formation of such ``fingers" can be interpreted as the emergence of leaders and followers in a system of identically interacting agents. Various numerical examples are provided, which demonstrate emergent behaviors similar to the ``fingering'' phenomenon observed in some phototaxis and chemotaxis experiments; this phenomenon is generally known as a challenging pattern to capture for existing models. The novel pairwise interactions provides a fundamental mechanism by which agents may form social hierarchy across a wide range of biological systems.

nlin.AO↗

arXiv2022

Swarm-Based Gradient Descent Method for Non-Convex Optimization

We introduce a new Swarm-Based Gradient Descent (SBGD) method for non-convex optimization. The swarm consists of agents, each is identified with a position, ${\mathbf x}$, and mass, $m$. The key to their dynamics is communication: masses are being transferred from agents at high ground to low(-est) ground. At the same time, agents change positions with step size, $h=h({\mathbf x},m)$, adjusted to their relative mass: heavier agents proceed with small time-steps in the direction of local gradient, while lighter agents take larger time-steps based on a backtracking protocol. Accordingly, the crowd of agents is dynamically divided between `heavier' leaders, expected to approach local minima, and `lighter' explorers. With their large-step protocol, explorers are expected to encounter improved position for the swarm; if they do, then they assume the role of `heavy' swarm leaders and so on. Convergence analysis and numerical simulations in one-, two-, and 20-dimensional benchmarks demonstrate the effectiveness of SBGD as a global optimizer.

math.NA↗

arXiv2022

Swarming: hydrodynamic alignment with pressure

We study the swarming behavior of hydrodynamic alignment. Alignment reflects steering towards a weighted average heading. We consider the class of so-called $p$-alignment hydrodynamics, based on $2p$-Laplacians, and weighted by a general family of symmetric communication kernels. The main new aspect here is the long time emergence behavior for a general class of pressure tensors without a closure assumption, beyond the mere requirement that they form an energy dissipative process. We refer to such pressure laws as `entropic', and prove the flocking of $p$-alignment hydrodynamics, driven by singular kernels with general class of entropic pressure tensors. These results indicate the rigidity of alignment in driving long-time flocking behavior despite the lack of thermodynamic closure.

math.AP↗

arXiv2022

Hydrodynamic alignment with pressure II. Multispecies

We study the long-time hydrodynamic behavior of systems of multi-species which arise from agent-based description of alignment dynamics. The interaction between species is governed by an array of symmetric communication kernels. We prove that the crowd of different species flock towards the mean velocity if (i) cross-interactions form a heavy-tailed connected array of kernels, while (ii) self-interactions are governed by kernels with singular heads. The main new aspect here, is that flocking behavior holds without closure assumption on the specific form of pressure tensors. Specifically, we prove the long-time flocking behavior for connected arrays of multi-species, with self-interactions governed by entropic pressure laws [E. Tadmor, Swarming: hydrodynamic alignment with pressure, ArXiv 2208.11786, (2022)] and driven by fractional $p$-alignment. In particular, it follows that such multi-species hydrodynamics approaches a mono-kinetic description. This generalizes the mono-kinetic, "pressure-less" study in [S. He and E. Tadmor, A game of alignment: collective behavior of multi-species, AIHP (c) Non Linear Anal. 38(4) (2021) 1031-1053.]

math.AP↗