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The numerical solution of a linear Schr\"odinger equation in the semiclassical regime is very well understood in a torus $\mathbb{T}^d$. A raft of modern computational methods are precise and affordable, while conserving energy and resolving high oscillations very well. This, however, is far from the case with regard to its solution in $\mathbb{R}^d$, a setting more suitable for many applications. In this paper we extend the theory of splitting methods to this end. The main idea is to derive the solution using a spectral method from a combination of solutions of the free Schr\"odinger equation and of linear scalar ordinary differential equations, in a symmetric Zassenhaus splitting method. This necessitates detailed analysis of certain orthonormal spectral bases on the real line and their evolution under the free Schr\"odinger operator.

We consider the deformation of a geological structure with non-intersecting faults that can be represented by a layered system of viscoelastic bodies satisfying rate- and state-depending friction conditions along the common interfaces. We derive a mathematical model that contains classical Dieterich- and Ruina-type friction as special cases and accounts for possibly large tangential displacements. Semi-discretization in time by a Newmark scheme leads to a coupled system of non-smooth, convex minimization problems for rate and state to be solved in each time step. Additional spatial discretization by a mortar method and piecewise constant finite elements allows for the decoupling of rate and state by a fixed point iteration and efficient algebraic solution of the rate problem by truncated non-smooth Newton methods. Numerical experiments with a spring slider and a layered multiscale system illustrate the behavior of our model as well as the efficiency and reliability of the numerical solver.

Partial Differential Equations (PDEs) describe several problems relevant to many fields of applied sciences, and their discrete counterparts typically involve the solution of sparse linear systems. In this context, we focus on the analysis of the computational aspects related to the solution of large and sparse linear systems with HPC solvers, by considering the performances of direct and iterative solvers in terms of computational efficiency, scalability, and numerical accuracy. Our aim is to identify the main criteria to support application-domain specialists in the selection of the most suitable solvers, according to the application requirements and available resources. To this end, we discuss how the numerical solver is affected by the regular/irregular discretisation of the input domain, the discretisation of the input PDE with piecewise linear or polynomial basis functions, which generally result in a higher/lower sparsity of the coefficient matrix, and the choice of different initial conditions, which are associated with linear systems with multiple right-hand side terms. Finally, our analysis is independent of the characteristics of the underlying computational architectures, and provides a methodological approach that can be applied to different classes of PDEs or with approximation problems.

While many works exploiting an existing Lie group structure have been proposed for state estimation, in particular the Invariant Extended Kalman Filter (IEKF), few papers address the construction of a group structure that allows casting a given system into the IEKF framework, namely making the dynamics group affine and the observations invariant. In this paper we introduce a large class of systems encompassing most problems involving a navigating vehicle encountered in practice. For those systems we introduce a novel methodology that systematically provides a group structure for the state space, including vectors of the body frame such as biases. We use it to derive observers having properties akin to those of linear observers or filters. The proposed unifying and versatile framework encompasses all systems where IEKF has proved successful, improves state-of-the art "imperfect" IEKF for inertial navigation with sensor biases, and allows addressing novel examples, like GNSS antenna lever arm estimation.

Recommender system is one of the most important information services on today's Internet. Recently, graph neural networks have become the new state-of-the-art approach of recommender systems. In this survey, we conduct a comprehensive review of the literature in graph neural network-based recommender systems. We first introduce the background and the history of the development of both recommender systems and graph neural networks. For recommender systems, in general, there are four aspects for categorizing existing works: stage, scenario, objective, and application. For graph neural networks, the existing methods consist of two categories, spectral models and spatial ones. We then discuss the motivation of applying graph neural networks into recommender systems, mainly consisting of the high-order connectivity, the structural property of data, and the enhanced supervision signal. We then systematically analyze the challenges in graph construction, embedding propagation/aggregation, model optimization, and computation efficiency. Afterward and primarily, we provide a comprehensive overview of a multitude of existing works of graph neural network-based recommender systems, following the taxonomy above. Finally, we raise discussions on the open problems and promising future directions of this area. We summarize the representative papers along with their codes repositories in //github.com/tsinghua-fib-lab/GNN-Recommender-Systems.

When and why can a neural network be successfully trained? This article provides an overview of optimization algorithms and theory for training neural networks. First, we discuss the issue of gradient explosion/vanishing and the more general issue of undesirable spectrum, and then discuss practical solutions including careful initialization and normalization methods. Second, we review generic optimization methods used in training neural networks, such as SGD, adaptive gradient methods and distributed methods, and theoretical results for these algorithms. Third, we review existing research on the global issues of neural network training, including results on bad local minima, mode connectivity, lottery ticket hypothesis and infinite-width analysis.

The spatial convolution layer which is widely used in the Graph Neural Networks (GNNs) aggregates the feature vector of each node with the feature vectors of its neighboring nodes. The GNN is not aware of the locations of the nodes in the global structure of the graph and when the local structures corresponding to different nodes are similar to each other, the convolution layer maps all those nodes to similar or same feature vectors in the continuous feature space. Therefore, the GNN cannot distinguish two graphs if their difference is not in their local structures. In addition, when the nodes are not labeled/attributed the convolution layers can fail to distinguish even different local structures. In this paper, we propose an effective solution to address this problem of the GNNs. The proposed approach leverages a spatial representation of the graph which makes the neural network aware of the differences between the nodes and also their locations in the graph. The spatial representation which is equivalent to a point-cloud representation of the graph is obtained by a graph embedding method. Using the proposed approach, the local feature extractor of the GNN distinguishes similar local structures in different locations of the graph and the GNN infers the topological structure of the graph from the spatial distribution of the locally extracted feature vectors. Moreover, the spatial representation is utilized to simplify the graph down-sampling problem. A new graph pooling method is proposed and it is shown that the proposed pooling method achieves competitive or better results in comparison with the state-of-the-art methods.

Nowadays, recommender systems are present in many daily activities such as online shopping, browsing social networks, etc. Given the rising demand for reinvigoration of the tourist industry through information technology, recommenders have been included into tourism websites such as Expedia, Booking or Tripadvisor, among others. Furthermore, the amount of scientific papers related to recommender systems for tourism is on solid and continuous growth since 2004. Much of this growth is due to social networks that, besides to offer researchers the possibility of using a great mass of available and constantly updated data, they also enable the recommendation systems to become more personalised, effective and natural. This paper reviews and analyses many research publications focusing on tourism recommender systems that use social networks in their projects. We detail their main characteristics, like which social networks are exploited, which data is extracted, the applied recommendation techniques, the methods of evaluation, etc. Through a comprehensive literature review, we aim to collaborate with the future recommender systems, by giving some clear classifications and descriptions of the current tourism recommender systems.

Model-based methods for recommender systems have been studied extensively in recent years. In systems with large corpus, however, the calculation cost for the learnt model to predict all user-item preferences is tremendous, which makes full corpus retrieval extremely difficult. To overcome the calculation barriers, models such as matrix factorization resort to inner product form (i.e., model user-item preference as the inner product of user, item latent factors) and indexes to facilitate efficient approximate k-nearest neighbor searches. However, it still remains challenging to incorporate more expressive interaction forms between user and item features, e.g., interactions through deep neural networks, because of the calculation cost. In this paper, we focus on the problem of introducing arbitrary advanced models to recommender systems with large corpus. We propose a novel tree-based method which can provide logarithmic complexity w.r.t. corpus size even with more expressive models such as deep neural networks. Our main idea is to predict user interests from coarse to fine by traversing tree nodes in a top-down fashion and making decisions for each user-node pair. We also show that the tree structure can be jointly learnt towards better compatibility with users' interest distribution and hence facilitate both training and prediction. Experimental evaluations with two large-scale real-world datasets show that the proposed method significantly outperforms traditional methods. Online A/B test results in Taobao display advertising platform also demonstrate the effectiveness of the proposed method in production environments.

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.