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We consider a Johnson-N\'ed\'elec FEM-BEM coupling, which is a direct and non-symmetric coupling of finite and boundary element methods, in order to solve interface problems for the magnetostatic Maxwell's equations with the magnetic vector potential ansatz. In the FEM-domain, equations may be non-linear, whereas they are exclusively linear in the BEM-part to guarantee the existence of a fundamental solution. First, the weak problem is formulated in quotient spaces to avoid resolving to a saddle point problem. Second, we establish in this setting well-posedness of the arising problem using the framework of Lipschitz and strongly monotone operators as well as a stability result for a special type of non-linearity, which is typically considered in magnetostatic applications. Then, the discretization is performed in the isogeometric context, i.e., the same type of basis functions that are used for geometry design are considered as ansatz functions for the discrete setting. In particular, NURBS are employed for geometry considerations, and B-Splines, which can be understood as a special type of NURBS, for analysis purposes. In this context, we derive a priori estimates w.r.t. h-refinement, and point out to an interesting behavior of BEM, which consists in an amelioration of the convergence rates, when a functional of the solution is evaluated in the exterior BEM-domain. This improvement may lead to a doubling of the convergence rate under certain assumptions. Finally, we end the paper with a numerical example to illustrate the theoretical results, along with a conclusion and an outlook.

Developing reinforcement learning algorithms that satisfy safety constraints is becoming increasingly important in real-world applications. In multi-agent reinforcement learning (MARL) settings, policy optimisation with safety awareness is particularly challenging because each individual agent has to not only meet its own safety constraints, but also consider those of others so that their joint behaviour can be guaranteed safe. Despite its importance, the problem of safe multi-agent learning has not been rigorously studied; very few solutions have been proposed, nor a sharable testing environment or benchmarks. To fill these gaps, in this work, we formulate the safe MARL problem as a constrained Markov game and solve it with policy optimisation methods. Our solutions -- Multi-Agent Constrained Policy Optimisation (MACPO) and MAPPO-Lagrangian -- leverage the theories from both constrained policy optimisation and multi-agent trust region learning. Crucially, our methods enjoy theoretical guarantees of both monotonic improvement in reward and satisfaction of safety constraints at every iteration. To examine the effectiveness of our methods, we develop the benchmark suite of Safe Multi-Agent MuJoCo that involves a variety of MARL baselines. Experimental results justify that MACPO/MAPPO-Lagrangian can consistently satisfy safety constraints, meanwhile achieving comparable performance to strong baselines.

We prove new optimality results for adaptive mesh refinement algorithms for non-symmetric, indefinite, and time-dependent problems by proposing a generalization of quasi-orthogonality which follows directly from the inf-sup stability of the underlying problem. This completely removes a central technical difficulty in modern proofs of optimal convergence of adaptive mesh refinement algorithms and leads to simple optimality proofs for the Taylor-Hood discretization of the stationary Stokes problem, a finite-element/boundary-element discretization of an unbounded transmission problem, and an adaptive time-stepping scheme for parabolic equations. The main technical tool are new stability bounds for the $LU$-factorization of matrices together with a recently established connection between quasi-orthogonality and matrix factorization.

The stable roommates problem can admit multiple different stable matchings. We have different criteria for deciding which one is optimal, but computing those is often NP-hard. We show that the problem of finding generous or rank-maximal stable matchings in an instance of the roommates problem with incomplete lists is NP-hard even when the preference lists are at most length 3. We show that just maximising the number of first choices or minimising the number of last choices is NP-hard with the short preference lists. We show that the number of $R^{th}$ choices, where $R$ is the minimum-regret of a given instance of SRI, is 2-approximable among all the stable matchings. Additionally, we show that the problem of finding a stable matching that maximises the number of first choices does not admit a constant time approximation algorithm and is W[1]-hard with respect to the number of first choices. We implement integer programming and constraint programming formulations for the optimality criteria of SRI. We find that constraint programming outperforms integer programming and an earlier answer set programming approach by Erdam et. al. (2020) for most optimality criteria. Integer programming outperforms constraint programming and answer set programming on the almost stable roommates problem.

Consider $k$ independent random samples from $p$-dimensional multivariate normal distributions. We are interested in the limiting distribution of the log-likelihood ratio test statistics for testing for the equality of $k$ covariance matrices. It is well known from classical multivariate statistics that the limit is a chi-square distribution when $k$ and $p$ are fixed integers. Jiang and Yang~\cite{JY13} and Jiang and Qi~\cite{JQ15} have obtained the central limit theorem for the log-likelihood ratio test statistics when the dimension $p$ goes to infinity with the sample sizes. In this paper, we derive the central limit theorem when either $p$ or $k$ goes to infinity. We also propose adjusted test statistics which can be well approximated by chi-squared distributions regardless of values for $p$ and $k$. Furthermore, we present numerical simulation results to evaluate the performance of our adjusted test statistics and the log-likelihood ratio statistics based on classical chi-square approximation and the normal approximation.

We present an empirical study of a range of evolutionary algorithms applied to various noisy combinatorial optimisation problems. There are three sets of experiments. The first looks at several toy problems, such as OneMax and other linear problems. We find that UMDA and the Paired-Crossover Evolutionary Algorithm (PCEA) are the only ones able to cope robustly with noise, within a reasonable fixed time budget. In the second stage, UMDA and PCEA are then tested on more complex noisy problems: SubsetSum, Knapsack and SetCover. Both perform well under increasing levels of noise, with UMDA being the better of the two. In the third stage, we consider two noisy multi-objective problems (CountingOnesCountingZeros and a multi-objective formulation of SetCover). We compare several adaptations of UMDA for multi-objective problems with the Simple Evolutionary Multi-objective Optimiser (SEMO) and NSGA-II. We conclude that UMDA, and its variants, can be highly effective on a variety of noisy combinatorial optimisation, outperforming many other evolutionary algorithms.

Siamese tracking has achieved groundbreaking performance in recent years, where the essence is the efficient matching operator cross-correlation and its variants. Besides the remarkable success, it is important to note that the heuristic matching network design relies heavily on expert experience. Moreover, we experimentally find that one sole matching operator is difficult to guarantee stable tracking in all challenging environments. Thus, in this work, we introduce six novel matching operators from the perspective of feature fusion instead of explicit similarity learning, namely Concatenation, Pointwise-Addition, Pairwise-Relation, FiLM, Simple-Transformer and Transductive-Guidance, to explore more feasibility on matching operator selection. The analyses reveal these operators' selective adaptability on different environment degradation types, which inspires us to combine them to explore complementary features. To this end, we propose binary channel manipulation (BCM) to search for the optimal combination of these operators. BCM determines to retrain or discard one operator by learning its contribution to other tracking steps. By inserting the learned matching networks to a strong baseline tracker Ocean, our model achieves favorable gains by $67.2 \rightarrow 71.4$, $52.6 \rightarrow 58.3$, $70.3 \rightarrow 76.0$ success on OTB100, LaSOT, and TrackingNet, respectively. Notably, Our tracker, dubbed AutoMatch, uses less than half of training data/time than the baseline tracker, and runs at 50 FPS using PyTorch. Code and model will be released at //github.com/JudasDie/SOTS.

The per-pixel cross-entropy loss (CEL) has been widely used in structured output prediction tasks as a spatial extension of generic image classification. However, its i.i.d. assumption neglects the structural regularity present in natural images. Various attempts have been made to incorporate structural reasoning mostly through structure priors in a cooperative way where co-occuring patterns are encouraged. We, on the other hand, approach this problem from an opposing angle and propose a new framework for training such structured prediction networks via an adversarial process, in which we train a structure analyzer that provides the supervisory signals, the adversarial structure matching loss (ASML). The structure analyzer is trained to maximize ASML, or to exaggerate recurring structural mistakes usually among co-occurring patterns. On the contrary, the structured output prediction network is trained to reduce those mistakes and is thus enabled to distinguish fine-grained structures. As a result, training structured output prediction networks using ASML reduces contextual confusion among objects and improves boundary localization. We demonstrate that ASML outperforms its counterpart CEL especially in context and boundary aspects on figure-ground segmentation and semantic segmentation tasks with various base architectures, such as FCN, U-Net, DeepLab, and PSPNet.

Interest point descriptors have fueled progress on almost every problem in computer vision. Recent advances in deep neural networks have enabled task-specific learned descriptors that outperform hand-crafted descriptors on many problems. We demonstrate that commonly used metric learning approaches do not optimally leverage the feature hierarchies learned in a Convolutional Neural Network (CNN), especially when applied to the task of geometric feature matching. While a metric loss applied to the deepest layer of a CNN, is often expected to yield ideal features irrespective of the task, in fact the growing receptive field as well as striding effects cause shallower features to be better at high precision matching tasks. We leverage this insight together with explicit supervision at multiple levels of the feature hierarchy for better regularization, to learn more effective descriptors in the context of geometric matching tasks. Further, we propose to use activation maps at different layers of a CNN, as an effective and principled replacement for the multi-resolution image pyramids often used for matching tasks. We propose concrete CNN architectures employing these ideas, and evaluate them on multiple datasets for 2D and 3D geometric matching as well as optical flow, demonstrating state-of-the-art results and generalization across datasets.

Superpixel segmentation has become an important research problem in image processing. In this paper, we propose an Iterative Spanning Forest (ISF) framework, based on sequences of Image Foresting Transforms, where one can choose i) a seed sampling strategy, ii) a connectivity function, iii) an adjacency relation, and iv) a seed pixel recomputation procedure to generate improved sets of connected superpixels (supervoxels in 3D) per iteration. The superpixels in ISF structurally correspond to spanning trees rooted at those seeds. We present five ISF methods to illustrate different choices of its components. These methods are compared with approaches from the state-of-the-art in effectiveness and efficiency. The experiments involve 2D and 3D datasets with distinct characteristics, and a high level application, named sky image segmentation. The theoretical properties of ISF are demonstrated in the supplementary material and the results show that some of its methods are competitive with or superior to the best baselines in effectiveness and efficiency.