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We present a semi-Lagrangian characteristic mapping method for the incompressible Euler equations on a rotating sphere. The numerical method uses a spatio-temporal discretization of the inverse flow map generated by the Eulerian velocity as a composition of sub-interval flows formed by $C^1$ spherical spline interpolants. This approximation technique has the capacity of resolving sub-grid scales generated over time without increasing the spatial resolution of the computational grid. The numerical method is analyzed and validated using standard test cases yielding third-order accuracy in the supremum norm. Numerical experiments illustrating the unique resolution properties of the method are performed and demonstrate the ability to reproduce the forward energy cascade at sub-grid scales by upsampling the numerical solution.

High-dimensional central limit theorems have been intensively studied with most focus being on the case where the data is sub-Gaussian or sub-exponential. However, heavier tails are omnipresent in practice. In this article, we study the critical growth rates of dimension $d$ below which Gaussian approximations are asymptotically valid but beyond which they are not. We are particularly interested in how these thresholds depend on the number of moments $m$ that the observations possess. For every $m\in(2,\infty)$, we construct i.i.d. random vectors $\textbf{X}_1,...,\textbf{X}_n$ in $\mathbb{R}^d$, the entries of which are independent and have a common distribution (independent of $n$ and $d$) with finite $m$th absolute moment, and such that the following holds: if there exists an $\varepsilon\in(0,\infty)$ such that $d/n^{m/2-1+\varepsilon}\not\to 0$, then the Gaussian approximation error (GAE) satisfies $$ \limsup_{n\to\infty}\sup_{t\in\mathbb{R}}\left[\mathbb{P}\left(\max_{1\leq j\leq d}\frac{1}{\sqrt{n}}\sum_{i=1}^n\textbf{X}_{ij}\leq t\right)-\mathbb{P}\left(\max_{1\leq j\leq d}\textbf{Z}_j\leq t\right)\right]=1,$$ where $\textbf{Z} \sim \mathsf{N}_d(\textbf{0}_d,\mathbf{I}_d)$. On the other hand, a result in Chernozhukov et al. (2023a) implies that the left-hand side above is zero if just $d/n^{m/2-1-\varepsilon}\to 0$ for some $\varepsilon\in(0,\infty)$. In this sense, there is a moment-dependent phase transition at the threshold $d=n^{m/2-1}$ above which the limiting GAE jumps from zero to one.

Rational function approximations provide a simple but flexible alternative to polynomial approximation, allowing one to capture complex non-linearities without oscillatory artifacts. However, there have been few attempts to use rational functions on noisy data due to the likelihood of creating spurious singularities. To avoid the creation of singularities, we use Bernstein polynomials and appropriate conditions on their coefficients to force the denominator to be strictly positive. While this reduces the range of rational polynomials that can be expressed, it keeps all the benefits of rational functions while maintaining the robustness of polynomial approximation in noisy data scenarios. Our numerical experiments on noisy data show that existing rational approximation methods continually produce spurious poles inside the approximation domain. This contrasts our method, which cannot create poles in the approximation domain and provides better fits than a polynomial approximation and even penalized splines on functions with multiple variables. Moreover, guaranteeing pole-free in an interval is critical for estimating non-constant coefficients when numerically solving differential equations using spectral methods. This provides a compact representation of the original differential equation, allowing numeric solvers to achieve high accuracy quickly, as seen in our experiments.

The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of pointwise bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop a scalable, mesh-independent algorithm for optimal design problems with pointwise bound constraints. This paper also provides a derivation of the latent variable proximal point (LVPP) algorithm, an unconditionally stable alternative to the interior point method. LVPP is an infinite-dimensional optimization algorithm that may be viewed as having an adaptive barrier function that is updated with a new informative prior at each (outer loop) optimization iteration. One of its main benefits is witnessed when analyzing the classical obstacle problem. Therein, we find that the original variational inequality can be replaced by a sequence of partial differential equations (PDEs) that are readily discretized and solved with, e.g., high-order finite elements. Throughout this work, we arrive at several unexpected contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field density-based topology optimization. The complete latent variable proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis.

In this article, we focus on the error that is committed when computing the matrix logarithm using the Gauss--Legendre quadrature rules. These formulas can be interpreted as Pad\'e approximants of a suitable Gauss hypergeometric function. Empirical observation tells us that the convergence of these quadratures becomes slow when the matrix is not close to the identity matrix, thus suggesting the usage of an inverse scaling and squaring approach for obtaining a matrix with this property. The novelty of this work is the introduction of error estimates that can be used to select a priori both the number of Legendre points needed to obtain a given accuracy and the number of inverse scaling and squaring to be performed. We include some numerical experiments to show the reliability of the estimates introduced.

We investigate the randomized decision tree complexity of a specific class of read-once threshold functions. A read-once threshold formula can be defined by a rooted tree, every internal node of which is labeled by a threshold function $T_k^n$ (with output 1 only when at least $k$ out of $n$ input bits are 1) and each leaf by a distinct variable. Such a tree defines a Boolean function in a natural way. We focus on the randomized decision tree complexity of such functions, when the underlying tree is a uniform tree with all its internal nodes labeled by the same threshold function. We prove lower bounds of the form $c(k,n)^d$, where $d$ is the depth of the tree. We also treat trees with alternating levels of AND and OR gates separately and show asymptotically optimal bounds, extending the known bounds for the binary case.

The proliferation of data generation has spurred advancements in functional data analysis. With the ability to analyze multiple variables simultaneously, the demand for working with multivariate functional data has increased. This study proposes a novel formulation of the epigraph and hypograph indexes, as well as their generalized expressions, specifically tailored for the multivariate functional context. These definitions take into account the interrelations between components. Furthermore, the proposed indexes are employed to cluster multivariate functional data. In the clustering process, the indexes are applied to both the data and their first and second derivatives. This generates a reduced-dimension dataset from the original multivariate functional data, enabling the application of well-established multivariate clustering techniques which have been extensively studied in the literature. This methodology has been tested through simulated and real datasets, performing comparative analyses against state-of-the-art to assess its performance.

The Graded of Membership (GoM) model is a powerful tool for inferring latent classes in categorical data, which enables subjects to belong to multiple latent classes. However, its application is limited to categorical data with nonnegative integer responses, making it inappropriate for datasets with continuous or negative responses. To address this limitation, this paper proposes a novel model named the Weighted Grade of Membership (WGoM) model. Compared with GoM, our WGoM relaxes GoM's distribution constraint on the generation of a response matrix and it is more general than GoM. We then propose an algorithm to estimate the latent mixed memberships and the other WGoM parameters. We derive the error bounds of the estimated parameters and show that the algorithm is statistically consistent. The algorithmic performance is validated in both synthetic and real-world datasets. The results demonstrate that our algorithm is accurate and efficient, indicating its high potential for practical applications. This paper makes a valuable contribution to the literature by introducing a novel model that extends the applicability of the GoM model and provides a more flexible framework for analyzing categorical data with weighted responses.

Over the last decade, approximating functions in infinite dimensions from samples has gained increasing attention in computational science and engineering, especially in computational uncertainty quantification. This is primarily due to the relevance of functions that are solutions to parametric differential equations in various fields, e.g. chemistry, economics, engineering, and physics. While acquiring accurate and reliable approximations of such functions is inherently difficult, current benchmark methods exploit the fact that such functions often belong to certain classes of holomorphic functions to get algebraic convergence rates in infinite dimensions with respect to the number of (potentially adaptive) samples $m$. Our work focuses on providing theoretical approximation guarantees for the class of $(\boldsymbol{b},\varepsilon)$-holomorphic functions, demonstrating that these algebraic rates are the best possible for Banach-valued functions in infinite dimensions. We establish lower bounds using a reduction to a discrete problem in combination with the theory of $m$-widths, Gelfand widths and Kolmogorov widths. We study two cases, known and unknown anisotropy, in which the relative importance of the variables is known and unknown, respectively. A key conclusion of our paper is that in the latter setting, approximation from finite samples is impossible without some inherent ordering of the variables, even if the samples are chosen adaptively. Finally, in both cases, we demonstrate near-optimal, non-adaptive (random) sampling and recovery strategies which achieve close to same rates as the lower bounds.

The Koopman operator presents an attractive approach to achieve global linearization of nonlinear systems, making it a valuable method for simplifying the understanding of complex dynamics. While data-driven methodologies have exhibited promise in approximating finite Koopman operators, they grapple with various challenges, such as the judicious selection of observables, dimensionality reduction, and the ability to predict complex system behaviours accurately. This study presents a novel approach termed Mori-Zwanzig autoencoder (MZ-AE) to robustly approximate the Koopman operator in low-dimensional spaces. The proposed method leverages a nonlinear autoencoder to extract key observables for approximating a finite invariant Koopman subspace and integrates a non-Markovian correction mechanism using the Mori-Zwanzig formalism. Consequently, this approach yields a closed representation of dynamics within the latent manifold of the nonlinear autoencoder, thereby enhancing the precision and stability of the Koopman operator approximation. Demonstrations showcase the technique's ability to capture regime transitions in the flow around a circular cylinder. It also provided a low dimensional approximation for chaotic Kuramoto-Sivashinsky with promising short-term predictability and robust long-term statistical performance. By bridging the gap between data-driven techniques and the mathematical foundations of Koopman theory, MZ-AE offers a promising avenue for improved understanding and prediction of complex nonlinear dynamics.