Isospectral flows appear in a variety of applications, e.g. the Toda lattice in solid state physics or in discrete models for two-dimensional hydrodynamics, with the isospectral property often corresponding to mathematically or physically important conservation laws. Their most prominent feature, i.e. the conservation of the eigenvalues of the matrix state variable, should therefore be retained when discretizing these systems. Recently, it was shown how isospectral Runge-Kutta methods can, in the Lie-Poisson case also considered in our work, be obtained through Hamiltonian reduction of symplectic Runge-Kutta methods on the cotangent bundle of a Lie group. We provide the Lagrangian analogue and, in the case of symplectic diagonal implicit Runge-Kutta methods, derive the methods through a discrete Euler-Poincare reduction. Our derivation relies on a formulation of diagonally implicit isospectral Runge-Kutta methods in terms of the Cayley transform, generalizing earlier work that showed this for the implicit midpoint rule. Our work is also a generalization of earlier variational Lie group integrators that, interestingly, appear when these are interpreted as update equations for intermediate time points. From a practical point of view, our results allow for a simple implementation of higher order isospectral methods and we demonstrate this with numerical experiments where both the isospectral property and energy are conserved to high accuracy.
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We consider the Cauchy problem for the Helmholtz equation with a domain in R^d, d>2 with N cylindrical outlets to infinity with bounded inclusions in R^{d-1}. Cauchy data are prescribed on the boundary of the bounded domains and the aim is to find solution on the unbounded part of the boundary. In 1989, Kozlov and Maz'ya proposed an alternating iterative method for solving Cauchy problems associated with elliptic,self-adjoint and positive-definite operators in bounded domains. Different variants of this method for solving Cauchy problems associated with Helmholtz-type operators exists. We consider the variant proposed by Mpinganzima et al. for bounded domains and derive the necessary conditions for the convergence of the procedure in unbounded domains. For the numerical implementation, a finite difference method is used to solve the problem in a simple rectangular domain in R^2 that represent a truncated infinite strip. The numerical results shows that by appropriate truncation of the domain and with appropriate choice of the Robin parameters, the Robin-Dirichlet alternating iterative procedure is convergent.
We introduce and analyze various Regularized Combined Field Integral Equations (CFIER) formulations of time-harmonic Navier equations in media with piece-wise constant material properties. These formulations can be derived systematically starting from suitable coercive approximations of Dirichlet-to-Neumann operators (DtN), and we present a periodic pseudodifferential calculus framework within which the well posedness of CIER formulations can be established. We also use the DtN approximations to derive and analyze Optimized Schwarz (OS) methods for the solution of elastodynamics transmission problems. The pseudodifferential calculus we develop in this paper relies on careful singularity splittings of the kernels of Navier boundary integral operators which is also the basis of high-order Nystr\"om quadratures for their discretizations. Based on these high-order discretizations we investigate the rate of convergence of iterative solvers applied to CFIER and OS formulations of scattering and transmission problems. We present a variety of numerical results that illustrate that the CFIER methodology leads to important computational savings over the classical CFIE one, whenever iterative solvers are used for the solution of the ensuing discretized boundary integral equations. Finally, we show that the OS methods are competitive in the high-frequency high-contrast regime.
We extend the Deep Galerkin Method (DGM) introduced in Sirignano and Spiliopoulos (2018)} to solve a number of partial differential equations (PDEs) that arise in the context of optimal stochastic control and mean field games. First, we consider PDEs where the function is constrained to be positive and integrate to unity, as is the case with Fokker-Planck equations. Our approach involves reparameterizing the solution as the exponential of a neural network appropriately normalized to ensure both requirements are satisfied. This then gives rise to nonlinear a partial integro-differential equation (PIDE) where the integral appearing in the equation is handled by a novel application of importance sampling. Secondly, we tackle a number of Hamilton-Jacobi-Bellman (HJB) equations that appear in stochastic optimal control problems. The key contribution is that these equations are approached in their unsimplified primal form which includes an optimization problem as part of the equation. We extend the DGM algorithm to solve for the value function and the optimal control \simultaneously by characterizing both as deep neural networks. Training the networks is performed by taking alternating stochastic gradient descent steps for the two functions, a technique inspired by the policy improvement algorithms (PIA).
Linear mixed models (LMMs) are instrumental for regression analysis with structured dependence, such as grouped, clustered, or multilevel data. However, selection among the covariates--while accounting for this structured dependence--remains a challenge. We introduce a Bayesian decision analysis for subset selection with LMMs. Using a Mahalanobis loss function that incorporates the structured dependence, we derive optimal linear coefficients for (i) any given subset of variables and (ii) all subsets of variables that satisfy a cardinality constraint. Crucially, these estimates inherit shrinkage or regularization and uncertainty quantification from the underlying Bayesian model, and apply for any well-specified Bayesian LMM. More broadly, our decision analysis strategy deemphasizes the role of a single "best" subset, which is often unstable and limited in its information content, and instead favors a collection of near-optimal subsets. This collection is summarized by key member subsets and variable-specific importance metrics. Customized subset search and out-of-sample approximation algorithms are provided for more scalable computing. These tools are applied to simulated data and a longitudinal physical activity dataset, and demonstrate excellent prediction, estimation, and selection ability.
We investigate the feature compression of high-dimensional ridge regression using the optimal subsampling technique. Specifically, based on the basic framework of random sampling algorithm on feature for ridge regression and the A-optimal design criterion, we first obtain a set of optimal subsampling probabilities. Considering that the obtained probabilities are uneconomical, we then propose the nearly optimal ones. With these probabilities, a two step iterative algorithm is established which has lower computational cost and higher accuracy. We provide theoretical analysis and numerical experiments to support the proposed methods. Numerical results demonstrate the decent performance of our methods.
We introduce Universal Solution Manifold Network (USM-Net), a novel surrogate model, based on Artificial Neural Networks (ANNs), which applies to differential problems whose solution depends on physical and geometrical parameters. Our method employs a mesh-less architecture, thus overcoming the limitations associated with image segmentation and mesh generation required by traditional discretization methods. Indeed, we encode geometrical variability through scalar landmarks, such as coordinates of points of interest. In biomedical applications, these landmarks can be inexpensively processed from clinical images. Our approach is non-intrusive and modular, as we select a data-driven loss function. The latter can also be modified by considering additional constraints, thus leveraging available physical knowledge. Our approach can also accommodate a universal coordinate system, which supports the USM-Net in learning the correspondence between points belonging to different geometries, boosting prediction accuracy on unobserved geometries. Finally, we present two numerical test cases in computational fluid dynamics involving variable Reynolds numbers as well as computational domains of variable shape. The results show that our method allows for inexpensive but accurate approximations of velocity and pressure, avoiding computationally expensive image segmentation, mesh generation, or re-training for every new instance of physical parameters and shape of the domain.
In this paper, a third order gas kinetic scheme is developed on the three dimensional hybrid unstructured meshes for the numerical simulation of compressible inviscid and viscous flows. A third-order WENO reconstruction is developed on the hybrid unstructured meshes, including tetrahedron, pyramid, prism and hexahedron. A simple strategy is adopted for the selection of big stencil and sub-stencils. Incorporate with the two-stage fourth-order temporal discretization and lower-upper symmetric Gauss-Seidel methods, both explicit and implicit high-order gas-kinetic schemes are developed. A variety of numerical examples, from the subsonic to supersonic flows, are presented to validate the accuracy and robustness for both inviscid and viscous flows.
We prove linear convergence of gradient descent to a global minimum for the training of deep residual networks with constant layer width and smooth activation function. We further show that the trained weights, as a function of the layer index, admits a scaling limit which is H\"older continuous as the depth of the network tends to infinity. The proofs are based on non-asymptotic estimates of the loss function and of norms of the network weights along the gradient descent path. We illustrate the relevance of our theoretical results to practical settings using detailed numerical experiments on supervised learning problems.
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.
As soon as abstract mathematical computations were adapted to computation on digital computers, the problem of efficient representation, manipulation, and communication of the numerical values in those computations arose. Strongly related to the problem of numerical representation is the problem of quantization: in what manner should a set of continuous real-valued numbers be distributed over a fixed discrete set of numbers to minimize the number of bits required and also to maximize the accuracy of the attendant computations? This perennial problem of quantization is particularly relevant whenever memory and/or computational resources are severely restricted, and it has come to the forefront in recent years due to the remarkable performance of Neural Network models in computer vision, natural language processing, and related areas. Moving from floating-point representations to low-precision fixed integer values represented in four bits or less holds the potential to reduce the memory footprint and latency by a factor of 16x; and, in fact, reductions of 4x to 8x are often realized in practice in these applications. Thus, it is not surprising that quantization has emerged recently as an important and very active sub-area of research in the efficient implementation of computations associated with Neural Networks. In this article, we survey approaches to the problem of quantizing the numerical values in deep Neural Network computations, covering the advantages/disadvantages of current methods. With this survey and its organization, we hope to have presented a useful snapshot of the current research in quantization for Neural Networks and to have given an intelligent organization to ease the evaluation of future research in this area.
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