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This paper introduces a formulation of the variable density incompressible Navier-Stokes equations by modifying the nonlinear terms in a consistent way. For Galerkin discretizations, the formulation leads to full discrete conservation of mass, squared density, momentum, angular momentum and kinetic energy without the divergence-free constraint being strongly enforced. In addition to favorable conservation properties, the formulation is shown to make the density field invariant to global shifts. The effect of viscous regularizations on conservation properties is also investigated. Numerical tests validate the theory developed in this work. The new formulation shows superior performance compared to other formulations from the literature, both in terms of accuracy for smooth problems and in terms of robustness.

Inversion for parameters of a physical process typically requires taking expensive measurements, and the task of finding an optimal set of measurements is known as the optimal design problem. Surprisingly, measurement locations in optimal designs are sometimes extremely clustered, and researchers often avoid measurement clusterization by modifying the optimal design problem. We consider a certain flavor of the optimal design problem, based on the Bayesian D-optimality criterion, and suggest an analytically tractable model for D-optimal designs in Bayesian linear inverse problems over Hilbert spaces. We demonstrate that measurement clusterization is a generic property of D-optimal designs, and prove that correlated noise between measurements mitigates clusterization. We also give a full characterization of D-optimal designs under our model: We prove that D-optimal designs uniformly reduce uncertainty in a select subset of prior covariance eigenvectors. Finally, we show how measurement clusterization is a consequence of the characterization mentioned above and the pigeonhole principle.

On the basis of the recent group classification of the one-dimensional magnetohydrodynamics (MHD) equations in cylindrical geometry, the construction of symmetry-preserving finite-difference schemes with conservation laws is carried out. New schemes are constructed starting from the classical completely conservative Samarsky-Popov schemes. In the case of finite conductivity, schemes are derived that admit all the symmetries and possess all the conservation laws of the original differential model, including previously unknown conservation laws. In the case of a frozen-in magnetic field (when the conductivity is infinite), various schemes are constructed that possess conservation laws, including those preserving entropy along trajectories of motion. The peculiarities of constructing schemes with an extended set of conservation laws for specific forms of entropy and magnetic fluxes are discussed.

Atmospheric systems incorporating thermal dynamics must be stable with respect to both energy and entropy. While energy conservation can be enforced via the preservation of the skew-symmetric structure of the Hamiltonian form of the equations of motion, entropy conservation is typically derived as an additional invariant of the Hamiltonian system, and satisfied via the exact preservation of the chain rule. This is particularly challenging since the function spaces used to represent the thermodynamic variables in compatible finite element discretisations are typically discontinuous at element boundaries. In the present work we negate this problem by constructing our equations of motion via weighted averages of skew-symmetric formulations using both flux form and material form advection of thermodynamic variables, which allow for the necessary cancellations required to conserve entropy without the chain rule. We show that such formulations allow for the stable simulation of both the thermal shallow water and 3D compressible Euler equations on the sphere using mixed compatible finite elements without entropy damping.

The electric vehicle sharing problem (EVSP) arises from the planning and operation of one-way electric car-sharing systems. It aims to maximize the total rental time of a fleet of electric vehicles while ensuring that all the demands of the customer are fulfilled. In this paper, we expand the knowledge on the complexity of the EVSP by showing that it is NP-hard to approximate it to within a factor of $n^{1-\epsilon}$ in polynomial time, for any $\epsilon > 0$, where $n$ denotes the number of customers, unless P = NP. In addition, we also show that the problem does not have a monotone structure, which can be detrimental to the development of heuristics employing constructive strategies. Moreover, we propose a novel approach for the modeling of the EVSP based on energy flows in the network. Based on the new model, we propose a relax-and-fix strategy and an exact algorithm that uses a warm-start solution obtained from our heuristic approach. We report computational results comparing our formulation with the best-performing formulation in the literature. The results show that our formulation outperforms the previous one concerning the number of optimal solutions obtained, optimality gaps, and computational times. Previously, $32.7\%$ of the instances remained unsolved (within a time limit of one hour) by the best-performing formulation in the literature, while our formulation obtained optimal solutions for all instances. To stress our approaches, two more challenging new sets of instances were generated, for which we were able to solve $49.5\%$ of the instances, with an average optimality gap of $2.91\%$ for those not solved optimally.

Recently, several studies consider the stochastic optimization problem but in a heavy-tailed noise regime, i.e., the difference between the stochastic gradient and the true gradient is assumed to have a finite $p$-th moment (say being upper bounded by $\sigma^{p}$ for some $\sigma\geq0$) where $p\in(1,2]$, which not only generalizes the traditional finite variance assumption ($p=2$) but also has been observed in practice for several different tasks. Under this challenging assumption, lots of new progress has been made for either convex or nonconvex problems, however, most of which only consider smooth objectives. In contrast, people have not fully explored and well understood this problem when functions are nonsmooth. This paper aims to fill this crucial gap by providing a comprehensive analysis of stochastic nonsmooth convex optimization with heavy-tailed noises. We revisit a simple clipping-based algorithm, whereas, which is only proved to converge in expectation but under the additional strong convexity assumption. Under appropriate choices of parameters, for both convex and strongly convex functions, we not only establish the first high-probability rates but also give refined in-expectation bounds compared with existing works. Remarkably, all of our results are optimal (or nearly optimal up to logarithmic factors) with respect to the time horizon $T$ even when $T$ is unknown in advance. Additionally, we show how to make the algorithm parameter-free with respect to $\sigma$, in other words, the algorithm can still guarantee convergence without any prior knowledge of $\sigma$. Furthermore, an initial distance adaptive convergence rate is provided if $\sigma$ is assumed to be known.

Algorithmic stability is an important notion that has proven powerful for deriving generalization bounds for practical algorithms. The last decade has witnessed an increasing number of stability bounds for different algorithms applied on different classes of loss functions. While these bounds have illuminated various properties of optimization algorithms, the analysis of each case typically required a different proof technique with significantly different mathematical tools. In this study, we make a novel connection between learning theory and applied probability and introduce a unified guideline for proving Wasserstein stability bounds for stochastic optimization algorithms. We illustrate our approach on stochastic gradient descent (SGD) and we obtain time-uniform stability bounds (i.e., the bound does not increase with the number of iterations) for strongly convex losses and non-convex losses with additive noise, where we recover similar results to the prior art or extend them to more general cases by using a single proof technique. Our approach is flexible and can be generalizable to other popular optimizers, as it mainly requires developing Lyapunov functions, which are often readily available in the literature. It also illustrates that ergodicity is an important component for obtaining time-uniform bounds -- which might not be achieved for convex or non-convex losses unless additional noise is injected to the iterates. Finally, we slightly stretch our analysis technique and prove time-uniform bounds for SGD under convex and non-convex losses (without additional additive noise), which, to our knowledge, is novel.

In automated warehouses, teams of mobile robots fulfill the packaging process by transferring inventory pods to designated workstations while navigating narrow aisles formed by tightly packed pods. This problem is typically modeled as a Multi-Agent Pickup and Delivery (MAPD) problem, which is then solved by repeatedly planning collision-free paths for agents on a fixed graph, as in the Rolling-Horizon Collision Resolution (RHCR) algorithm. However, existing approaches make the limiting assumption that agents are only allowed to move pods that correspond to their current task, while considering the other pods as stationary obstacles (even though all pods are movable). This behavior can result in unnecessarily long paths which could otherwise be avoided by opening additional corridors via pod manipulation. To this end, we explore the implications of allowing agents the flexibility of dynamically relocating pods. We call this new problem Terraforming MAPD (tMAPD) and develop an RHCR-based approach to tackle it. As the extra flexibility of terraforming comes at a significant computational cost, we utilize this capability judiciously by identifying situations where it could make a significant impact on the solution quality. In particular, we invoke terraforming in response to disruptions that often occur in automated warehouses, e.g., when an item is dropped from a pod or when agents malfunction. Empirically, using our approach for tMAPD, where disruptions are modeled via a stochastic process, we improve throughput by over 10%, reduce the maximum service time (the difference between the drop-off time and the pickup time of a pod) by more than 50%, without drastically increasing the runtime, compared to the MAPD setting.

In this paper, we propose the Ordered Median Tree Location Problem (OMT). The OMT is a single-allocation facility location problem where p facilities must be placed on a network connected by a non-directed tree. The objective is to minimize the sum of the ordered weighted averaged allocation costs plus the sum of the costs of connecting the facilities in the tree. We present different MILP formulations for the OMT based on properties of the minimum spanning tree problem and the ordered median optimization. Given that ordered median hub location problems are rather difficult to solve we have improved the OMT solution performance by introducing covering variables in a valid reformulation plus developing two pre-processing phases to reduce the size of this formulations. In addition, we propose a Benders decomposition algorithm to approach the OMT. We establish an empirical comparison between these new formulations and we also provide enhancements that together with a proper formulation allow to solve medium size instances on general random graphs.

Graph convolution networks (GCN) are increasingly popular in many applications, yet remain notoriously hard to train over large graph datasets. They need to compute node representations recursively from their neighbors. Current GCN training algorithms suffer from either high computational costs that grow exponentially with the number of layers, or high memory usage for loading the entire graph and node embeddings. In this paper, we propose a novel efficient layer-wise training framework for GCN (L-GCN), that disentangles feature aggregation and feature transformation during training, hence greatly reducing time and memory complexities. We present theoretical analysis for L-GCN under the graph isomorphism framework, that L-GCN leads to as powerful GCNs as the more costly conventional training algorithm does, under mild conditions. We further propose L^2-GCN, which learns a controller for each layer that can automatically adjust the training epochs per layer in L-GCN. Experiments show that L-GCN is faster than state-of-the-arts by at least an order of magnitude, with a consistent of memory usage not dependent on dataset size, while maintaining comparable prediction performance. With the learned controller, L^2-GCN can further cut the training time in half. Our codes are available at //github.com/Shen-Lab/L2-GCN.