A blob method for inhomogeneous diffusion with applications to multi-agent control and sampling

Craig, Katy; Elamvazhuthi, Karthik; Haberland, Matt; Turanova, O. A.

doi:10.48550/arxiv.2202.12927

Cited by 3 publications

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“…Blob particle method [14] is the Wasserstein gradient flow of the regularization of KL divergence that allows for discrete measures where it becomes and interacting particle system. Such viewpoint can has been applied to sampling purposes in [18] where the authors considered the Wasserstein gradient flow for regularized χ 2 energy. A different approach to create gradient flow based interacting-particle systems for sampling is the SVGD introduced in [47].…”

Section: Introductionmentioning

confidence: 99%

Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics

Lu¹,

Slepčev²,

Wang³

2022

Preprint

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Motivated by the challenge of sampling Gibbs measures with nonconvex potentials, we study a continuum birth-death dynamics. We prove that the probability density of the birthdeath governed by Kullback-Leibler divergence or by χ 2 divergence converge exponentially fast to the Gibbs equilibrium measure with a universal rate that is independent of the potential barrier. To build a practical numerical sampler based on the pure birth-death dynamics, we consider an interacting particle system which relies on kernel-based approximations of the measure and retains the gradient-flow structure. We show on the torus that the kernelized dynamics Γ-converges, on finite time intervals, to the pure birth-death dynamics as the kernel bandwidth shrinks to zero. Moreover we provide quantitative estimates on the bias of minimizers of the energy corresponding to the kernalized dynamics. Finally we prove the long-time asymptotic results on the convergence of the asymptotic states of the kernalized dynamics towards the Gibbs measure.

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Section: Introductionmentioning

confidence: 99%

Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics

Lu¹,

Slepčev²,

Wang³

2022

Preprint

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“…Although a rigorous mathematical theory has been extensively provided over the years [57,46,17], there are particular aspects of renewed interest in view of novel applications as well as advances in mathematics. For instance, their derivation from interacting particles, with a distinction between deterministic and stochastic methods, has recently attracted attention for its implications in derivation of models in mathematical biology [22] and data science [26]. We take advantage of the gradient flow structure of nonlinear diffusions [46] to connect with nonlocal interaction equations.…”

Section: Introductionmentioning

confidence: 99%

“…In the case m = 2, in the same spirit of [10,26], the authors in [7] construct weak solutions of the quadratic porous medium equation as a localising limit (ε → 0) of a sequence of weak measure solutions of the nonlocal interaction equation (NLE), for F (x) = x 2 . The authors work directly at the level of the (nonlocal) equations by means of a time-discretisation scheme which allows to work with lack of convexity, as for instance in the case of cross-diffusion systems, or even PDEs with no purely gradient flow structure.…”

Section: Introductionmentioning

confidence: 99%

“…As in [10], finite initial log-entropy is required, thus excluding particle approximation. However, simultaneously to [7], the authors in [26] focus on a weighted (quadratic) porous medium equation which is relevant, e.g. in sampling -the weight, ρ in their notations, represents a target probability measure to be approximated from specific samples drawn from it.…”

Section: Introductionmentioning

confidence: 99%

“…This means one can achieve, so far, a qualitative result, as the initial datum needs to be approximated fast enough, c.f. [26,Theorem 1.4]. To the best of our knowledge, a quantitative result has not been achieved yet in more than one dimension.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Phase Transitions for Nonlinear Nonlocal Aggregation-Diffusion Equations

Carrillo

Gvalani

2021

Commun. Math. Phys.

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We are interested in studying the stationary solutions and phase transitions of aggregation equations with degenerate diffusion of porous medium-type, with exponent $$1< m < \infty $$ 1 < m < ∞ . We first prove the existence of possibly infinitely many bifurcations from the spatially homogeneous steady state. We then focus our attention on the associated free energy, proving existence of minimisers and even uniqueness for sufficiently weak interactions. In the absence of uniqueness, we show that the system exhibits phase transitions: we classify values of m and interaction potentials W for which these phase transitions are continuous or discontinuous. Finally, we comment on the limit $$m \rightarrow \infty $$ m → ∞ and the influence that the presence of a phase transition has on this limit.

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Birth–death dynamics for sampling: global convergence, approximations and their asymptotics

Lu,

Slepčev,

Wang

2023

Nonlinearity

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Motivated by the challenge of sampling Gibbs measures with nonconvex potentials, we study a continuum birth–death dynamics. We improve results in previous works (Liu et al 2023 Appl. Math. Optim. 87 48; Lu et al 2019 arXiv:1905.09863) and provide weaker hypotheses under which the probability density of the birth–death governed by Kullback–Leibler divergence or by χ 2 divergence converge exponentially fast to the Gibbs equilibrium measure, with a universal rate that is independent of the potential barrier. To build a practical numerical sampler based on the pure birth–death dynamics, we consider an interacting particle system, which is inspired by the gradient flow structure and the classical Fokker–Planck equation and relies on kernel-based approximations of the measure. Using the technique of Γ-convergence of gradient flows, we show that on the torus, smooth and bounded positive solutions of the kernelised dynamics converge on finite time intervals, to the pure birth–death dynamics as the kernel bandwidth shrinks to zero. Moreover we provide quantitative estimates on the bias of minimisers of the energy corresponding to the kernelised dynamics. Finally we prove the long-time asymptotic results on the convergence of the asymptotic states of the kernelised dynamics towards the Gibbs measure.

show abstract

A blob method for inhomogeneous diffusion with applications to multi-agent control and sampling

Cited by 3 publications

References 0 publications

Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics

Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics

Phase Transitions for Nonlinear Nonlocal Aggregation-Diffusion Equations

Birth–death dynamics for sampling: global convergence, approximations and their asymptotics

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