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Offline estimators are often inadequate for real-time applications. Nevertheless, many online estimators are found by sacrificing some statistical efficiency. This paper presents a general framework to understand and construct efficient nonparametric estimators for online problems. Statistically, we choose long-run variance as an exemplary estimand and derive the first set of sufficient conditions for O(1)-time or O(1)-space update, which allows methodological generation of estimators. Our asymptotic theory shows that the generated estimators dominate existing alternatives. Computationally, we introduce mini-batch estimation to accelerate online estimators for real-time applications. Implementation issues such as automatic optimal parameters selection are discussed. Practically, we demonstrate how to use our framework with recent development in change point detection, causal inference, and stochastic approximation. We also illustrate the strength of our estimators in some classical problems such as Markov chain Monte Carlo convergence diagnosis and confidence interval construction.

In this work, we study the convergence and performance of nonlinear solvers for the Bidomain equations after decoupling the ordinary and partial differential equations of the cardiac system. We first rigorously prove that Quasi-Newton methods such as BFGS and nonlinear Conjugate-Gradient such as Fletcher-Reeves methods are globally convergent, by studying an auxiliary variational problem under physically reasonable hypotheses. Then, we compare several nonlinear solvers in terms of execution time, robustness with respect to the data and parallel scalability. Our results suggest that Quasi-Newton methods are the best choice for this type of problem, being faster than standard Newton-Krylov methods without hindering their robustness or scalability. In addition, first order methods are also competitive, and represent a better alternative for matrix-free implementations, which are suitable for GPU computing.

We develop an optimization-based algorithm for parametric model order reduction (PMOR) of linear time-invariant dynamical systems. Our method aims at minimizing the $\mathcal{H}_\infty \otimes \mathcal{L}_\infty$ approximation error in the frequency and parameter domain by an optimization of the reduced order model (ROM) matrices. State-of-the-art PMOR methods often compute several nonparametric ROMs for different parameter samples, which are then combined to a single parametric ROM. However, these parametric ROMs can have a low accuracy between the utilized sample points. In contrast, our optimization-based PMOR method minimizes the approximation error across the entire parameter domain. Moreover, due to our flexible approach of optimizing the system matrices directly, we can enforce favorable features such as a port-Hamiltonian structure in our ROMs across the entire parameter domain. Our method is an extension of the recently developed SOBMOR-algorithm to parametric systems. We extend both the ROM parameterization and the adaptive sampling procedure to the parametric case. Several numerical examples demonstrate the effectiveness and high accuracy of our method in a comparison with other PMOR methods.

The widespread adoption of nonlinear Receding Horizon Control (RHC) strategies by industry has led to more than 30 years of intense research efforts to provide stability guarantees for these methods. However, current theoretical guarantees require that each (generally nonconvex) planning problem can be solved to (approximate) global optimality, which is an unrealistic requirement for the derivative-based local optimization methods generally used in practical implementations of RHC. This paper takes the first step towards understanding stability guarantees for nonlinear RHC when the inner planning problem is solved to first-order stationary points, but not necessarily global optima. Special attention is given to feedback linearizable systems, and a mixture of positive and negative results are provided. We establish that, under certain strong conditions, first-order solutions to RHC exponentially stabilize linearizable systems. Crucially, this guarantee requires that state costs applied to the planning problems are in a certain sense `compatible' with the global geometry of the system, and a simple counter-example demonstrates the necessity of this condition. These results highlight the need to rethink the role of global geometry in the context of optimization-based control.

Point cloud compression (PCC) is a key enabler for various 3-D applications, owing to the universality of the point cloud format. Ideally, 3D point clouds endeavor to depict object/scene surfaces that are continuous. Practically, as a set of discrete samples, point clouds are locally disconnected and sparsely distributed. This sparse nature is hindering the discovery of local correlation among points for compression. Motivated by an analysis with fractal dimension, we propose a heterogeneous approach with deep learning for lossy point cloud geometry compression. On top of a base layer compressing a coarse representation of the input, an enhancement layer is designed to cope with the challenging geometric residual/details. Specifically, a point-based network is applied to convert the erratic local details to latent features residing on the coarse point cloud. Then a sparse convolutional neural network operating on the coarse point cloud is launched. It utilizes the continuity/smoothness of the coarse geometry to compress the latent features as an enhancement bit-stream that greatly benefits the reconstruction quality. When this bit-stream is unavailable, e.g., due to packet loss, we support a skip mode with the same architecture which generates geometric details from the coarse point cloud directly. Experimentation on both dense and sparse point clouds demonstrate the state-of-the-art compression performance achieved by our proposal. Our code is available at //github.com/InterDigitalInc/GRASP-Net.

Adhesive joints are increasingly used in industry for a wide variety of applications because of their favorable characteristics such as high strength-to-weight ratio, design flexibility, limited stress concentrations, planar force transfer, good damage tolerance and fatigue resistance. Finding the optimal process parameters for an adhesive bonding process is challenging: the optimization is inherently multi-objective (aiming to maximize break strength while minimizing cost) and constrained (the process should not result in any visual damage to the materials, and stress tests should not result in failures that are adhesion-related). Real life physical experiments in the lab are expensive to perform; traditional evolutionary approaches (such as genetic algorithms) are then ill-suited to solve the problem, due to the prohibitive amount of experiments required for evaluation. In this research, we successfully applied specific machine learning techniques (Gaussian Process Regression and Logistic Regression) to emulate the objective and constraint functions based on a \emph{limited} amount of experimental data. The techniques are embedded in a Bayesian optimization algorithm, which succeeds in detecting Pareto-optimal process settings in a highly efficient way (i.e., requiring a limited number of extra experiments).

A power series being given as the solution of a linear differential equation with appropriate initial conditions, minimization consists in finding a non-trivial linear differential equation of minimal order having this power series as a solution. This problem exists in both homogeneous and inhomogeneous variants; it is distinct from, but related to, the classical problem of factorization of differential operators. Recently, minimization has found applications in Transcendental Number Theory, more specifically in the computation of non-zero algebraic points where Siegel's $E$-functions take algebraic values. We present algorithms for these questions and discuss implementation and experiments.

It is well-known that the parameterized family of functions representable by fully-connected feedforward neural networks with ReLU activation function is precisely the class of piecewise linear functions with finitely many pieces. It is less well-known that for every fixed architecture of ReLU neural network, the parameter space admits positive-dimensional spaces of symmetries, and hence the local functional dimension near any given parameter is lower than the parametric dimension. In this work we carefully define the notion of functional dimension, show that it is inhomogeneous across the parameter space of ReLU neural network functions, and continue an investigation - initiated in [14] and [5] - into when the functional dimension achieves its theoretical maximum. We also study the quotient space and fibers of the realization map from parameter space to function space, supplying examples of fibers that are disconnected, fibers upon which functional dimension is non-constant, and fibers upon which the symmetry group acts non-transitively.

We present a Lagrangian-Eulerian scheme to solve the shallow water equations in the case of spatially variable bottom geometry. Using a local curvilinear reference system anchored on the bottom surface, we develop an effective first-order and high-resolution space-time discretization of the no-flow surfaces and solve a Lagrangian initial value problem that describes the evolution of the balance laws governing the geometrically intrinsic shallow water equations. The evolved solution set is then projected back to the original surface grid to complete the proposed Lagrangian-Eulerian formulation. The resulting scheme maintains monotonicity and captures shocks without providing excessive numerical dissipation also in the presence of non-autonomous fluxes such as those arising from the geometrically intrinsic shallow water equation on variable topographies. We provide a representative set of numerical examples to illustrate the accuracy and robustness of the proposed Lagrangian-Eulerian formulation for two-dimensional surfaces with general curvatures and discontinuous initial conditions.

Although there is an extensive literature on the eigenvalues of high-dimensional sample covariance matrices, much of it is specialized to Mar\v{c}enko-Pastur (MP) models -- in which observations are represented as linear transformations of random vectors with independent entries. By contrast, less is known in the context of elliptical models, which violate the independence structure of MP models and exhibit quite different statistical phenomena. In particular, very little is known about the scope of bootstrap methods for doing inference with spectral statistics in high-dimensional elliptical models. To fill this gap, we show how a bootstrap approach developed previously for MP models can be extended to handle the different properties of elliptical models. Within this setting, our main theoretical result guarantees that the proposed method consistently approximates the distributions of linear spectral statistics, which play a fundamental role in multivariate analysis. Lastly, we provide empirical results showing that the proposed method also performs well for a variety of nonlinear spectral statistics.