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This work proposes a model-reduction approach for the material point method on nonlinear manifolds. Our technique approximates the $\textit{kinematics}$ by approximating the deformation map using an implicit neural representation that restricts deformation trajectories to reside on a low-dimensional manifold. By explicitly approximating the deformation map, its spatiotemporal gradients -- in particular the deformation gradient and the velocity -- can be computed via analytical differentiation. In contrast to typical model-reduction techniques that construct a linear or nonlinear manifold to approximate the (finite number of) degrees of freedom characterizing a given spatial discretization, the use of an implicit neural representation enables the proposed method to approximate the $\textit{continuous}$ deformation map. This allows the kinematic approximation to remain agnostic to the discretization. Consequently, the technique supports dynamic discretizations -- including resolution changes -- during the course of the online reduced-order-model simulation. To generate $\textit{dynamics}$ for the generalized coordinates, we propose a family of projection techniques. At each time step, these techniques: (1) Calculate full-space kinematics at quadrature points, (2) Calculate the full-space dynamics for a subset of `sample' material points, and (3) Calculate the reduced-space dynamics by projecting the updated full-space position and velocity onto the low-dimensional manifold and tangent space, respectively. We achieve significant computational speedup via hyper-reduction that ensures all three steps execute on only a small subset of the problem's spatial domain. Large-scale numerical examples with millions of material points illustrate the method's ability to gain an order of magnitude computational-cost saving -- indeed $\textit{real-time simulations}$ -- with negligible errors.

This paper introduces a new simulation-based inference procedure to model and sample from multi-dimensional probability distributions given access to i.i.d. samples, circumventing the usual approaches of explicitly modeling the density function or designing Markov chain Monte Carlo. Motivated by the seminal work on distance and isomorphism between metric measure spaces, we propose a new notion called the Reversible Gromov-Monge (RGM) distance and study how RGM can be used to design new transform samplers to perform simulation-based inference. Our RGM sampler can also estimate optimal alignments between two heterogeneous metric measure spaces $(\mathcal{X}, \mu, c_{\mathcal{X}})$ and $(\mathcal{Y}, \nu, c_{\mathcal{Y}})$ from empirical data sets, with estimated maps that approximately push forward one measure $\mu$ to the other $\nu$, and vice versa. Analytic properties of the RGM distance are derived; statistical rate of convergence, representation, and optimization questions regarding the induced sampler are studied. Synthetic and real-world examples showcasing the effectiveness of the RGM sampler are also demonstrated.

In this work, we introduce a novel approach to formulating an artificial viscosity for shock capturing in nonlinear hyperbolic systems by utilizing the property that the solutions of hyperbolic conservation laws are not reversible in time in the vicinity of shocks. The proposed approach does not require any additional governing equations or a priori knowledge of the hyperbolic system in question, is independent of the mesh and approximation order, and requires the use of only one tunable parameter. The primary novelty is that the resulting artificial viscosity is unique for each component of the conservation law which is advantageous for systems in which some components exhibit discontinuities while others do not. The efficacy of the method is shown in numerical experiments of multi-dimensional hyperbolic conservation laws such as nonlinear transport, Euler equations, and ideal magnetohydrodynamics using a high-order discontinuous spectral element method on unstructured grids.

We provide a decision theoretic analysis of bandit experiments. The setting corresponds to a dynamic programming problem, but solving this directly is typically infeasible. Working within the framework of diffusion asymptotics, we define suitable notions of asymptotic Bayes and minimax risk for bandit experiments. For normally distributed rewards, the minimal Bayes risk can be characterized as the solution to a nonlinear second-order partial differential equation (PDE). Using a limit of experiments approach, we show that this PDE characterization also holds asymptotically under both parametric and non-parametric distribution of the rewards. The approach further describes the state variables it is asymptotically sufficient to restrict attention to, and therefore suggests a practical strategy for dimension reduction. The upshot is that we can approximate the dynamic programming problem defining the bandit experiment with a PDE which can be efficiently solved using sparse matrix routines. We derive the optimal Bayes and minimax policies from the numerical solutions to these equations. The proposed policies substantially dominate existing methods such as Thompson sampling. The framework also allows for substantial generalizations to the bandit problem such as time discounting and pure exploration motives.

Existing inferential methods for small area data involve a trade-off between maintaining area-level frequentist coverage rates and improving inferential precision via the incorporation of indirect information. In this article, we propose a method to obtain an area-level prediction region for a future observation which mitigates this trade-off. The proposed method takes a conformal prediction approach in which the conformity measure is the posterior predictive density of a working model that incorporates indirect information. The resulting prediction region has guaranteed frequentist coverage regardless of the working model, and, if the working model assumptions are accurate, the region has minimum expected volume compared to other regions with the same coverage rate. When constructed under a normal working model, we prove such a prediction region is an interval and construct an efficient algorithm to obtain the exact interval. We illustrate the performance of our method through simulation studies and an application to EPA radon survey data.

The metriplectic formalism is useful for describing complete dynamical systems which conserve energy and produce entropy. This creates challenges for model reduction, as the elimination of high-frequency information will generally not preserve the metriplectic structure which governs long-term stability of the system. Based on proper orthogonal decomposition, a provably convergent metriplectic reduced-order model is formulated which is guaranteed to maintain the algebraic structure necessary for energy conservation and entropy formation. Numerical results on benchmark problems show that the proposed method is remarkably stable, leading to improved accuracy over long time scales at a moderate increase in cost over naive methods.

The geometric high-order regularization methods such as mean curvature and Gaussian curvature, have been intensively studied during the last decades due to their abilities in preserving geometric properties including image edges, corners, and image contrast. However, the dilemma between restoration quality and computational efficiency is an essential roadblock for high-order methods. In this paper, we propose fast multi-grid algorithms for minimizing both mean curvature and Gaussian curvature energy functionals without sacrificing the accuracy for efficiency. Unlike the existing approaches based on operator splitting and the Augmented Lagrangian method (ALM), no artificial parameters are introduced in our formulation, which guarantees the robustness of the proposed algorithm. Meanwhile, we adopt the domain decomposition method to promote parallel computing and use the fine-to-coarse structure to accelerate the convergence. Numerical experiments are presented on both image denoising and CT reconstruction problem to demonstrate the ability to recover image texture and the efficiency of the proposed method.

The minimum energy path (MEP) describes the mechanism of reaction, and the energy barrier along the path can be used to calculate the reaction rate in thermal systems. The nudged elastic band (NEB) method is one of the most commonly used schemes to compute MEPs numerically. It approximates an MEP by a discrete set of configuration images, where the discretization size determines both computational cost and accuracy of the simulations. In this paper, we consider a discrete MEP to be a stationary state of the NEB method and prove an optimal convergence rate of the discrete MEP with respect to the number of images. Numerical simulations for the transitions of some several proto-typical model systems are performed to support the theory.

Most existing works of polar codes focus on the analysis of block error probability. However, in many scenarios, bit error probability is also important for evaluating the performance of channel codes. In this paper, we establish a new framework to analyze the bit error probability of polar codes. Specifically, by revisiting the error event of bit-channel, we first introduce the conditional bit error probability as a metric to evaluate the reliability of bit-channel for both systematic and non-systematic polar codes. Guided by the concept of polar subcode, we then derive an upper bound on the conditional bit error probability of each bit-channel, and accordingly, an upper bound on the bit error probability of polar codes. Based on these, two types of construction metrics aiming at minimizing the bit error probability of polar codes are proposed, which are of linear computational complexity and explicit forms. Simulation results show that the polar codes constructed by the proposed methods can outperform those constructed by the conventional methods.

Clinical Named Entity Recognition (CNER) aims to identify and classify clinical terms such as diseases, symptoms, treatments, exams, and body parts in electronic health records, which is a fundamental and crucial task for clinical and translational research. In recent years, deep neural networks have achieved significant success in named entity recognition and many other Natural Language Processing (NLP) tasks. Most of these algorithms are trained end to end, and can automatically learn features from large scale labeled datasets. However, these data-driven methods typically lack the capability of processing rare or unseen entities. Previous statistical methods and feature engineering practice have demonstrated that human knowledge can provide valuable information for handling rare and unseen cases. In this paper, we address the problem by incorporating dictionaries into deep neural networks for the Chinese CNER task. Two different architectures that extend the Bi-directional Long Short-Term Memory (Bi-LSTM) neural network and five different feature representation schemes are proposed to handle the task. Computational results on the CCKS-2017 Task 2 benchmark dataset show that the proposed method achieves the highly competitive performance compared with the state-of-the-art deep learning methods.