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1. Estimating Normalizing Constants for Log-Concave Distributions: Algorithms and Lower Bounds

   Ge, Rong; Lee, Holden; Lu, Jianfeng (June 2020, STOC 2020)

   Estimating the normalizing constant of an unnormalized probability distribution has important applications in computer science, statistical physics, machine learning, and statistics. In this work, we consider the problem of estimating the normalizing constant to within a multiplication factor of 1 ± ε for a μ-strongly convex and L-smooth function f, given query access to f(x) and ∇f(x). We give both algorithms and lowerbounds for this problem. Using an annealing algorithm combined with a multilevel Monte Carlo method based on underdamped Langevin dynamics, we show that O(d^{4/3}/\eps^2) queries to ∇f are sufficient. Moreover, we provide an information theoretic lowerbound, showing that at least d^{1-o(1)}/\eps^{2-o(1)} queries are necessary. This provides a first nontrivial lowerbound for the problem. 

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

   https://doi.org/10.1088/1361-6544/acf988  

   Lu, Yulong; Slepčev, Dejan; Wang, Lihan (September 2023, Nonlinearity)

   Abstract Motivated by the challenge of sampling Gibbs measures with nonconvex potentials, we study a continuum birth–death dynamics. We improve results in previous works (Liuet al2023Appl. Math. Optim.8748; Luet al2019 arXiv:1905.09863) and provide weaker hypotheses under which the probability density of the birth–death governed by Kullback–Leibler divergence or byχ²divergence 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. 

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3. The ⁴¹ Ar(n,y) ⁴² Ar reaction

   https://doi.org/10.1051/epjconf/202328401037  

   Sahoo, R. N.; Paul, M.; Köster, U.; Scott, R.; Tessler, M.; Zylstra, A.; Avila, M. L.; Dickerson, C.; Jayatissa, H.; Kohen, M.S.; et al (January 2023, EPJ Web of Conferences)

   Mattoon, C.M.; Vogt, R.; Escher, J.; Thompson, I. (Ed.)

   The cross-section of the thermal neutron capture⁴¹Ar(n,γ)⁴²Ar(t_1/2=32.9 y) reaction was measured by irradiating a⁴⁰Ar sample at the high-flux reactor of Institut Laue-Langevin (ILL) Grenoble, France. The signature of the two-neutron capture has been observed by measuring the growth curve and identifying the 1524.6 keV γ-lines of the shorter-lived⁴²K(12.4 h) β⁻daughter of⁴²Ar. Our preliminary value of the⁴¹Ar(n,γ)⁴²Ar thermal cross section is 240(80) mb at 25.3 meV. For the first time, direct counting of⁴²Ar was performed using the ultra-high sensitivity technique of noble gas accelerator mass spectrometry (NOGAMS) at Argonne National Laboratory, USA. 

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4. Light fluctuations are key in modulating plankton trophic dynamics and their impact on primary production

   https://doi.org/10.1002/lol2.10156  

   Morison, Françoise; Franzè, Gayantonia; Harvey, Elizabeth; Menden‐Deuer, Susanne (April 2020, Limnology and Oceanography Letters)

   Abstract Surface‐ocean mixing creates dynamic light environments with predictable effects on phytoplankton growth but unknown consequences for predation. We investigated how variations in average mixed‐layer (ML) irradiance shaped plankton trophic dynamics by incubating a Northwest‐Atlantic plankton community for 4 days at high (H) and low (L) light, followed by exposure to either sustained or reversed light intensities. In deep‐ML (sustained L), phytoplankton biomass declined (μ= −0.2 ± 0.08 d⁻¹) and grazing was absent. In shallow‐ML (sustained H), growth exceeded grazing (μ= 0.46 ± 0.07 d⁻¹;g= 0.32 ± 0.04 d⁻¹). In rapidly changing ML‐conditions simulated by switching light‐availability, growth and grazing responded on different timescales. During rapid ML‐shoaling (L to H),μimmediately increased (0.23 ± 0.01 d⁻¹) with no change in grazing. During rapid ML‐deepening (H to L),μimmediately decreased (0.02 ± 0.09 d⁻¹), whereas grazing remained high (g= 0.38 ± 0.05 d⁻¹). Predictable rate responses of phytoplankton growth (rapid) vs. grazing (delayed) to measurable light variability can provide insights into predator‐prey processes and their effects on spatio‐temporal dynamics of phytoplankton biomass. 

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5. Marcinkiewicz Averages of Smooth Orthogonal Projections on Sphere

   https://doi.org/10.1007/s00041-022-09966-y  

   Bownik, Marcin; Dziedziul, Karol; Kamont, Anna (October 2022, Journal of Fourier Analysis and Applications)

   Abstract We construct a single smooth orthogonal projection with desired localization whose average under a group action yields the decomposition of the identity operator. For any full rank lattice $$\Gamma \subset \mathbb {R}^d$$ Γ ⊂ R d , a smooth projection is localized in a neighborhood of an arbitrary precompact fundamental domain $$\mathbb {R}^d/\Gamma $$ R d / Γ . We also show the existence of a highly localized smooth orthogonal projection, whose Marcinkiewicz average under the action of SO ( d ), is a multiple of the identity on $$L^2(\mathbb {S}^{d-1})$$ L 2 ( S d - 1 ) . As an application we construct highly localized continuous Parseval frames on the sphere. 

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