Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-diffusion equations

Computational costs of numerically solving multidimensional partial differential equations (PDEs) increase significantly when the spatial dimensions of the PDEs are high, due to large number of spatial grid points. For multidimensional reaction-diffusion equations, stiffness of the system provides additional challenges for achieving efficient numerical simulations. In this paper, we propose a class of Krylov implicit integration factor (IIF) discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion equations on high spatial dimensions. The key ingredient of spatial DG discretization is the multiwavelet bases on nested sparse grids, which can significantly reduce the numbers of degrees of freedom. To deal with the stiffness of the DG spatial operator in discretizing reaction-diffusion equations, we apply the efficient IIF time discretization methods, which are a class of exponential integrators. Krylov subspace approximations are used to evaluate the large size matrix exponentials resulting from IIF schemes for solving PDEs on high spatial dimensions. Stability and error analysis for the semi-discrete scheme are performed. Numerical examples of both scalar equations and systems in two and three spatial dimensions are provided to demonstrate the accuracy and efficiency of the methods. The stiffness of the reaction-diffusion equations is resolved well and large time step size computations are obtained

A Personalised Ranking Framework with Multiple Sampling Criteria for Venue Recommendation

Recommending a ranked list of interesting venues to users based on their preferences has become a key functionality in Location-Based Social Networks (LBSNs) such as Yelp and Gowalla. Bayesian Personalised Ranking (BPR) is a popular pairwise recommendation technique that is used to generate the ranked list of venues of interest to a user, by leveraging the user's implicit feedback such as their check-ins as instances of positive feedback, while randomly sampling other venues as negative instances. To alleviate the sparsity that affects the usefulness of recommendations by BPR for users with few check-ins, various approaches have been proposed in the literature to incorporate additional sources of information such as the social links between users, the textual content of comments, as well as the geographical location of the venues. However, such approaches can only readily leverage one source of additional information for negative sampling. Instead, we propose a novel Personalised Ranking Framework with Multiple sampling Criteria (PRFMC) that leverages both geographical influence and social correlation to enhance the effectiveness of BPR. In particular, we apply a multi-centre Gaussian model and a power-law distribution method, to capture geographical influence and social correlation when sampling negative venues, respectively. Finally, we conduct comprehensive experiments using three large-scale datasets from the Yelp, Gowalla and Brightkite LBSNs. The experimental results demonstrate the effectiveness of fusing both geographical influence and social correlation in our proposed PRFMC framework and its superiority in comparison to BPR-based and other similar ranking approaches. Indeed, our PRFMC approach attains a 37% improvement in MRR over a recently proposed approach that identifies negative venues only from social links