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1. A Primer on Stochastic Partial Differential Equations with Spatially Correlated Noise

   https://doi.org/10.1146/annurev-conmatphys-042624-033003  

   Newhall, Katherine A (November 2024, Annual Review of Condensed Matter Physics)

   With the growing number of microscale devices from computer memory to microelectromechanical systems, such as lab-on-a-chip biosensors, and the increased ability to experimentally measure at the micro- and nanoscale, modeling systems with stochastic processes is a growing need across science. In particular, stochastic partial differential equations (SPDEs) naturally arise from continuum models—for example, a pillar magnet's magnetization or an elastic membrane's mechanical deflection. In this review, I seek to acquaint the reader with SPDEs from the point of view of numerically simulating their finite-difference approximations, without the rigorous mathematical details of assigning probability measures to the random field solutions. I stress that these simulations with spatially uncorrelated noise may not converge as the grid size goes to zero in the way that one expects from deterministic convergence of numerical schemes in two or more spatial dimensions. I then present some models with spatially correlated noise that maintain sampling of the physically relevant equilibrium distribution. Numerical simulations are presented to demonstrate the dynamics; the code is publicly available on GitHub. 

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2. Soil Seed Banks, Alternative Stable State Theory, and Ecosystem Resilience

   https://doi.org/10.1093/biosci/biab011  

   Ma, Miaojun; Collins, Scott L; Ratajczak, Zak; Du, Guozhen (March 2021, BioScience)

   null (Ed.)

   Abstract In restoration ecology, the transition from desired to degraded state is based solely on the composition of the aboveground plant community, whereas belowground propagules are often neglected. We developed a conceptual framework integrating seed bank dynamics into alternative stable state theory, highlighting the important relationship between aboveground and belowground composition. This integration emphasizes the role of resilience in systems that appear to have shifted to an “undesirable” state. Belowground propagules, especially soil seed and bud banks, provide buffering capacity and may serve as valuable indicators of potential resistance to state transition based on the degree of similarity between belowground and aboveground vegetation composition. Ecosystem states may have multiple components that differ in their rate of change, as well as in their capacity to promote resilience. We recommend that the application of alternative stable state theory from a management perspective should incorporate components of both above- and belowground vegetation. 

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3. Leader–follower equilibria to examine investment decisions in grid resilience‐enhancing measures

   https://doi.org/10.1111/risa.13921  

   Lo Prete, Chiara; Radhakrishnan, Ashish (February 2023, Risk Analysis)

   Large‐area, long‐duration power outages are increasingly common in the United States, and cost the economy billions of dollars each year. Building a strategy to enhance grid resilience requires an understanding of the optimal mix of preventive and corrective actions, the inefficiencies that arise when self‐interested parties make resilience investment decisions, and the conditions under which regulators may facilitate the realization of efficient market outcomes. We develop a bi‐level model to examine the mix of preventive and corrective measures that enhances grid resilience to a severe storm. The model represents a Stackelberg game between a regulated utility (leader) that may harden distribution feeders before a long‐duration outage and/or deploy restoration crews after the disruption, and utility customers with varying preferences for reliable power (followers) who may invest in backup generators. We show that the regulator's denial of cost recovery for the utility's preventive expenditures, coupled with the misalignment between private objectives and social welfare maximization, yields significant inefficiencies in the resilience investment mix. Allowing cost recovery for a higher share of the utility's capital expenditures in preventive measures, extending the time horizon associated with damage cost recovery, and adopting a storm restoration compensation mechanism shift the realized market outcome toward the efficient solution. If about one‐fifth of preventive resilience investments is approved by regulators, requiring utilities to pay a compensation of $365 per customer for a 3‐day outage (about seven times the level of compensation currently offered by US utilities) provides significant incentives toward more efficient preventive resilience investments. 

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4. A Second look at Exponential and Cosine Step Sizes: Simplicity, Adaptivity, and Performance

   Li, Xiaoyu; Zhuang, Zhenxun; Orabona, Francesco (July 2021, Proceedings of Machine Learning Research)

   Meila, Marina; Zhang, Tong (Ed.)

   Stochastic Gradient Descent (SGD) is a popular tool in training large-scale machine learning models. Its performance, however, is highly variable, depending crucially on the choice of the step sizes. Accordingly, a variety of strategies for tuning the step sizes have been proposed, ranging from coordinate-wise approaches (a.k.a. “adaptive” step sizes) to sophisticated heuristics to change the step size in each iteration. In this paper, we study two step size schedules whose power has been repeatedly confirmed in practice: the exponential and the cosine step sizes. For the first time, we provide theoretical support for them proving convergence rates for smooth non-convex functions, with and without the Polyak-Łojasiewicz (PL) condition. Moreover, we show the surprising property that these two strategies are adaptive to the noise level in the stochastic gradients of PL functions. That is, contrary to polynomial step sizes, they achieve almost optimal performance without needing to know the noise level nor tuning their hyperparameters based on it. Finally, we conduct a fair and comprehensive empirical evaluation of real-world datasets with deep learning architectures. Results show that, even if only requiring at most two hyperparameters to tune, these two strategies best or match the performance of various finely-tuned state-of-the-art strategies. 

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5. Introducing desirable patches to initiate ecosystem transitions and accelerate ecosystem restoration

   https://doi.org/10.1002/eap.2910  

   Eppinga, Maarten B.; Michaels, Theo K.; Santos, Maria J.; Bever, James D. (December 2023, Ecological Applications)

   Abstract Meeting restoration targets may require active strategies to accelerate natural regeneration rates or overcome the resilience associated with degraded ecosystem states. Introducing desired ecosystem patches in degraded landscapes constitutes a promising active restoration strategy, with various mechanisms potentially causing these patches to become foci from which desired species can re‐establish throughout the landscape. This study considers three mechanisms previously identified as potential drivers of introduced patch dynamics: autocatalytic nucleation, directed dispersal, and resource concentration. These mechanisms reflect qualitatively different positive feedbacks. We developed an ecological model framework that compared how the occurrence of each mechanism was reflected in spatio‐temporal patch dynamics. We then analyzed the implications of these relationships for optimal restoration design. We found that patch expansion accelerated over time when driven by the autocatalytic nucleation mechanism, while patch expansion driven by the directed dispersal or resource concentration mechanisms decelerated over time. Additionally, when driven by autocatalytic nucleation, patch expansion was independent of patch position in the landscape. However, the proximity of other patches affected patch expansion either positively or negatively when driven by directed dispersal or resource concentration. For autocatalytic nucleation, introducing many small patches was a favorable strategy, provided that each individual patch exceeded a critical patch size. Introducing a single patch or a few large patches was the most effective restoration strategy to initiate the directed dispersal mechanism. Introducing many small patches was the most effective strategy for reaching restored ecosystem states driven by a resource concentration mechanism. Our model results suggest that introducing desirable patches can substantially accelerate ecosystem restoration, or even induce a critical transition from an otherwise stable degraded state toward a desired ecosystem state. However, the potential of this type of restoration strategy for a particular ecosystem may strongly depend on the mechanism driving patch dynamics. In turn, which mechanism drives patch dynamics may affect the optimal spatial design of an active restoration strategy. Each of the three mechanisms considered reflects distinct spatio‐temporal patch dynamics, providing novel opportunities for empirically identifying key mechanisms, and restoration designs that introduce desired patches in degraded landscapes according to these patch dynamics. 

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