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Bayesian nonparametric mixture models are common for modeling complex data. While these models are well-suited for density estimation, their application for clustering has some limitations. Miller and Harrison (2014) proved posterior inconsistency in the number of clusters when the true number of clusters is finite for Dirichlet process and Pitman--Yor process mixture models. In this work, we extend this result to additional Bayesian nonparametric priors such as Gibbs-type processes and finite-dimensional representations of them. The latter include the Dirichlet multinomial process and the recently proposed Pitman--Yor and normalized generalized gamma multinomial processes. We show that mixture models based on these processes are also inconsistent in the number of clusters and discuss possible solutions. Notably, we show that a post-processing algorithm introduced by Guha et al. (2021) for the Dirichlet process extends to more general models and provides a consistent method to estimate the number of components.

Model Predictive Control (MPC) is a well-established approach to solve infinite horizon optimal control problems. Since optimization over an infinite time horizon is generally infeasible, MPC determines a suboptimal feedback control by repeatedly solving finite time optimal control problems. Although MPC has been successfully used in many applications, applying MPC to large-scale systems -- arising, e.g., through discretization of partial differential equations -- requires the solution of high-dimensional optimal control problems and thus poses immense computational effort. We consider systems governed by parametrized parabolic partial differential equations and employ the reduced basis (RB) method as a low-dimensional surrogate model for the finite time optimal control problem. The reduced order optimal control serves as feedback control for the original large-scale system. We analyze the proposed RB-MPC approach by first developing a posteriori error bounds for the errors in the optimal control and associated cost functional. These bounds can be evaluated efficiently in an offline-online computational procedure and allow us to guarantee asymptotic stability of the closed-loop system using the RB-MPC approach in several practical scenarios. We also propose an adaptive strategy to choose the prediction horizon of the finite time optimal control problem. Numerical results are presented to illustrate the theoretical properties of our approach.

The utilization of renewable energy technologies, particularly hydrogen, has seen a boom in interest and has spread throughout the world. Ethanol steam reformation is one of the primary methods capable of producing hydrogen efficiently and reliably. This paper provides an in-depth study of the reformulated system both theoretically and numerically, as well as a plan to explore the possibility of converting the system into its conservation form. Lastly, we offer an overview of several numerical approaches for solving the general first-order quasi-linear hyperbolic equation to the particular model for ethanol steam reforming (ESR). We conclude by presenting some results that would enable the usage of these ODE/PDE solvers to be used in non-linear model predictive control (NMPC) algorithms and discuss the limitations of our approach and directions for future work.

Many applications in computational sciences and statistical inference require the computation of expectations with respect to complex high-dimensional distributions with unknown normalization constants, as well as the estimation of these constants. Here we develop a method to perform these calculations based on generating samples from a simple base distribution, transporting them by the flow generated by a velocity field, and performing averages along these flowlines. This non-equilibrium importance sampling (NEIS) strategy is straightforward to implement and can be used for calculations with arbitrary target distributions. On the theory side, we discuss how to tailor the velocity field to the target and establish general conditions under which the proposed estimator is a perfect estimator with zero-variance. We also draw connections between NEIS and approaches based on mapping a base distribution onto a target via a transport map. On the computational side, we show how to use deep learning to represent the velocity field by a neural network and train it towards the zero variance optimum. These results are illustrated numerically on benchmark examples (with dimension up to $10$), where after training the velocity field, the variance of the NEIS estimator is reduced by up to $6$ orders of magnitude than that of a vanilla estimator. We also compare the performances of NEIS with those of Neal's annealed importance sampling (AIS).

We develop a hybrid spatial discretization for the wave equation in second order form, based on high-order accurate finite difference methods and discontinuous Galerkin methods. The hybridization combines computational efficiency of finite difference methods on Cartesian grids and geometrical flexibility of discontinuous Galerkin methods on unstructured meshes. The two spatial discretizations are coupled by a penalty technique at the interface such that the overall semidiscretization satisfies a discrete energy estimate to ensure stability. In addition, optimal convergence is obtained in the sense that when combining a fourth order finite difference method with a discontinuous Galerkin method using third order local polynomials, the overall convergence rate is fourth order. Furthermore, we use a novel approach to derive an error estimate for the semidiscretization by combining the energy method and the normal mode analysis for a corresponding one dimensional model problem. The stability and accuracy analysis are verified in numerical experiments.

Compared to on-policy policy gradient techniques, off-policy model-free deep reinforcement learning (RL) that uses previously gathered data can improve sampling efficiency. However, off-policy learning becomes challenging when the discrepancy between the distributions of the policy of interest and the policies that collected the data increases. Although the well-studied importance sampling and off-policy policy gradient techniques were proposed to compensate for this discrepancy, they usually require a collection of long trajectories that increases the computational complexity and induce additional problems such as vanishing/exploding gradients or discarding many useful experiences. Moreover, their generalization to continuous action domains is strictly limited as they require action probabilities, which is unsuitable for deterministic policies. To overcome these limitations, we introduce a novel policy similarity measure to mitigate the effects of such discrepancy. Our method offers an adequate single-step off-policy correction without any probability estimates, and theoretical results show that it can achieve a contraction mapping with a fixed unique point, which allows "safe" off-policy learning. An extensive set of empirical results indicate that our algorithm substantially improves the state-of-the-art and attains higher returns in fewer steps than the competing methods by efficiently scheduling the learning rate in Q-learning and policy optimization.

We propose an efficient way of solving optimal control problems for rigid-body systems on the basis of inverse dynamics and the multiple-shooting method. We treat all variables, including the state, acceleration, and control input torques, as optimization variables and treat the inverse dynamics as an equality constraint. We eliminate the update of the control input torques from the linear equation of Newton's method by applying condensing for inverse dynamics. The size of the resultant linear equation is the same as that of the multiple-shooting method based on forward dynamics except for the variables related to the passive joints and contacts. Compared with the conventional methods based on forward dynamics, the proposed method reduces the computational cost of the dynamics and their sensitivities by utilizing the recursive Newton-Euler algorithm (RNEA) and its partial derivatives. In addition, it increases the sparsity of the Hessian of the Karush-Kuhn-Tucker conditions, which reduces the computational cost, e.g., of Riccati recursion. Numerical experiments show that the proposed method outperforms state-of-the-art implementations of differential dynamic programming based on forward dynamics in terms of computational time and numerical robustness.

Global spectral methods offer the potential to compute solutions of partial differential equations numerically to very high accuracy. In this work, we develop a novel global spectral method for linear partial differential equations on cubes by extending ideas of Chebop2 [Townsend and Olver, J. Comput. Phys., 299 (2015)] to the three-dimensional setting utilizing expansions in tensorized polynomial bases. Solving the discretized PDE involves a linear system that can be recast as a linear tensor equation. Under suitable additional assumptions, the structure of these equations admits for an efficient solution via the blocked recursive solver [Chen and Kressner, Numer. Algorithms, 84 (2020)]. In the general case, when these assumptions are not satisfied, this solver is used as a preconditioner to speed up computations.

We study a class of nonlinear eigenvalue problems of Schr\"{o}dinger type, where the potential is singular on a set of points. Such problems are widely present in physics and chemistry, and their analysis is of both theoretical and practical interest. In particular, we study the regularity of the eigenfunctions of the operators considered, and we propose and analyze the approximation of the solution via an isotropically refined $hp$ discontinuous Galerkin (dG) method. We show that, for weighted analytic potentials and for up-to-quartic polynomial nonlinearities, the eigenfunctions belong to analytic-type non homogeneous weighted Sobolev spaces. We also prove quasi optimal a priori estimates on the error of the dG finite element method; when using an isotropically refined $hp$ space the numerical solution is shown to converge with exponential rate towards the exact eigenfunction. We conclude with a series of numerical tests to validate the theoretical results.

We study the deviation inequality for a sum of high-dimensional random matrices and operators with dependence and arbitrary heavy tails. There is an increase in the importance of the problem of estimating high-dimensional matrices, and dependence and heavy-tail properties of data are among the most critical topics currently. In this paper, we derive a dimension-free upper bound on the deviation, that is, the bound does not depend explicitly on the dimension of matrices, but depends on their effective rank. Our result is a generalization of several existing studies on the deviation of the sum of matrices. Our proof is based on two techniques: (i) a variational approximation of the dual of moment generating functions, and (ii) robustification through truncation of eigenvalues of matrices. We show that our results are applicable to several problems such as covariance matrix estimation, hidden Markov models, and overparameterized linear regression models.