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We prove several results showing that every locally finite Borel graph whose large-scale geometry is ‘tree-like’ induces a treeable equivalence relation. In particular, our hypotheses hold if each component of the original graph either has bounded tree-width or is quasi-isometric to a tree, answering a question of Tucker-Drob. In the latter case, we moreover show that there exists a Borel quasi-isometry to a Borel forest, under the additional assumption of (componentwise) bounded degree. We also extend these results on quasi-treeings to Borel proper metric spaces. In fact, our most general result shows treeability of countable Borel equivalence relations equipped with an abstract wallspace structure on each class obeying some local finiteness conditions, which we call aproper walling. The proof is based on the Stone duality between proper wallings and median graphs (i.e., CAT(0) cube complexes). Finally, we strengthen the conclusion of treeability in these results to hyperfiniteness in the case where the original graph has one (selected) end per component, generalizing the same result for trees due to Dougherty–Jackson–Kechris. 

Model predictive control (MPC) has been applied to many platforms in robotics and autonomous systems for its capability to predict a system’s future behavior while incorporating constraints that a system may have. To enhance the performance of a system with an MPC controller, one can manually tunethe MPC’s cost function. However, it can be challenging due to the possibly high dimension of the parameter space as well as the potential difference between the open-loop cost function in MPC and the overall closed-loop performance metric function. This letter presents Difffune-MPC, a novel learning method, to learn the cost function of an MPC in a closed-loop manner. The proposed framework is compatible with the scenario where the time interval for performance evaluation and MPC’s planning horizon have different lengths. We show the auxiliary problem whose solution admits the analytical gradients of MPC and discuss its variations in different MPC settings, including nonlinear MPCs that are solved using sequential quadratic programming. Simulation results demonstrate the learning capability of DiffTune-MPC and the generalization capability of the learned MPC parameters. 

Light transmission through a sea ice cover has strong implications for the heat content of the upper ocean, the magnitude of bottom and lateral ice melt, and primary productivity in the ocean. Light transmittance in the vicinity of the Multidisciplinary Drifting Observatory for the Study of Arctic Climate (MOSAiC) Central Observatory was estimated by driving a two-stream radiative transfer model with physical property observations. Data include point and transect observations of snow depth, surface scattering layer thickness, ice thickness, and pond depth. The temporal evolution of light transmittance at specific sites and the spatial variability along transect lines were computed. Ponds transmitted 4–6 times as much solar energy per unit area as bare ice. On July 25, ponds covered about 18% of the area and contributed roughly 50% of the sunlight transmitted through the ice cover. Approximating the transmittance along a transect line using average values for the physical properties will always result in lower light transmittance than finding the average light transmittance using the full distribution of points. Transmitted solar energy calculated using the standard five ice thickness categories and three surface types used in the Los Alamos sea ice model CICE, the sea ice component of many weather and climate models, was only about 1 W m−2 less than using all the points along the transect. This minor difference suggests that the important processes and resulting feedbacks relating to solar transmittance can be represented in models that use five or more categories of ice thickness distributions. 

Safe control designs for robotic systems remain challenging because of the difficulties of explicitly solving optimal control with nonlinear dynamics perturbed by stochastic noise. However, recent technological advances in computing devices enable online optimization or sampling-based methods to solve control problems. For example, Control Barrier Functions (CBFs) have been proposed to numerically solve convex optimization problems that ensure the control input to stay in the safe set. Model Predictive Path Integral (MPPI) control uses forward sampling of stochastic differential equations to solve optimal control problems online. Both control algorithms are widely used for nonlinear systems because they avoid calculating the derivatives of the nonlinear dynamic functions. In this paper, we use Stochastic Control Barrier Functions (SCBFs) constraints to limit sample regions in the samplingbased algorithm, ensuring safety in a probabilistic sense and improving sample efficiency with a stochastic differential equation. We also show that our algorithm needs fewer samples than the original MPPI algorithm does by providing a sampling complexity analysis. 

Abstract Melt ponds forming on Arctic sea ice in summer significantly reduce the surface albedo and impact the heat and mass balance of the sea ice. Therefore, their areal coverage, which can undergo rapid change, is crucial to monitor. We present a revised method to extract melt pond fraction (MPF) from Sentinel‐2 satellite imagery, which is evaluated by MPF products from higher‐resolution satellite and helicopter‐borne imagery. The analysis of melt pond evolution during the MOSAiC campaign in summer 2020, shows a split of the Central Observatory (CO) into a level ice and a highly deformed ice part, the latter of which exhibits exceptional early melt pond formation compared to the vicinity. Average CO MPFs are 17% before and 23% after the major drainage. Arctic‐wide analysis of MPF for years 2017–2021 shows a consistent seasonal cycle in all regions and years.