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In this paper, we consider stochastic versions of three classical growth models given by ordinary differential equations (ODEs). Indeed we use stochastic versions of Von Bertalanffy, Gompertz, and Logistic differential equations as models. We assume that each stochastic differential equation (SDE) has some crucial parameters in the drift to be estimated and we use the Maximum Likelihood Estimator (MLE) to estimate them. For estimating the diffusion parameter, we use the MLE for two cases and the quadratic variation of the data for one of the SDEs. We apply the Akaike information criterion (AIC) to choose the best model for the simulated data. We consider that the AIC is a function of the drift parameter. We present a simulation study to validate our selection method. The proposed methodology could be applied to datasets with continuous and discrete observations, but also with highly sparse data. Indeed, we can use this method even in the extreme case where we have observed only one point for each path, under the condition that we observed a sufficient number of trajectories. For the last two cases, the data can be viewed as incomplete observations of a model with a tractable likelihood function; then, we propose a version of the Expectation Maximization (EM) algorithm to estimate these parameters. This type of datasets typically appears in fishery, for instance.

Recent discussions on the future of metropolitan cities underscore the pivotal role of (social) equity, driven by demographic and economic trends. More equal policies can foster and contribute to a city's economic success and social stability. In this work, we focus on identifying metropolitan areas with distinct economic and social levels in the greater Los Angeles area, one of the most diverse yet unequal areas in the United States. Utilizing American Community Survey data, we propose a Bayesian model for boundary detection based on income distributions. The model identifies areas with significant income disparities, offering actionable insights for policymakers to address social and economic inequalities. Our approach formalized as a Bayesian structural learning framework, models areal densities through finite mixture models. Efficient posterior computation is facilitated by a transdimensional Markov Chain Monte Carlo sampler. The methodology is validated via extensive simulations and applied to the income distributions in the greater Los Angeles area. We identify several boundaries in the income distributions which can be explained in light of other social dynamics such as crime rates and healthcare, showing the usefulness of such an analysis to policymakers.

Rational best approximations (in a Chebyshev sense) to real functions are characterized by an equioscillating approximation error. Similar results do not hold true for rational best approximations to complex functions in general. In the present work, we consider unitary rational approximations to the exponential function on the imaginary axis, which map the imaginary axis to the unit circle. In the class of unitary rational functions, best approximations are shown to exist, to be uniquely characterized by equioscillation of a phase error, and to possess a super-linear convergence rate. Furthermore, the best approximations have full degree (i.e., non-degenerate), achieve their maximum approximation error at points of equioscillation, and interpolate at intermediate points. Asymptotic properties of poles, interpolation nodes, and equioscillation points of these approximants are studied. Three algorithms, which are found very effective to compute unitary rational approximations including candidates for best approximations, are sketched briefly. Some consequences to numerical time-integration are discussed. In particular, time propagators based on unitary best approximants are unitary, symmetric and A-stable.

Any experiment with climate models relies on a potentially large set of spatio-temporal boundary conditions. These can represent both the initial state of the system and/or forcings driving the model output throughout the experiment. Whilst these boundary conditions are typically fixed using available reconstructions in climate modelling studies, they are highly uncertain, that uncertainty is unquantified, and the effect on the output of the experiment can be considerable. We develop efficient quantification of these uncertainties that combines relevant data from multiple models and observations. Starting from the coexchangeability model, we develop a coexchangable process model to capture multiple correlated spatio-temporal fields of variables. We demonstrate that further exchangeability judgements over the parameters within this representation lead to a Bayes linear analogy of a hierarchical model. We use the framework to provide a joint reconstruction of sea-surface temperature and sea-ice concentration boundary conditions at the last glacial maximum (19-23 ka) and use it to force an ensemble of ice-sheet simulations using the FAMOUS-Ice coupled atmosphere and ice-sheet model. We demonstrate that existing boundary conditions typically used in these experiments are implausible given our uncertainties and demonstrate the impact of using more plausible boundary conditions on ice-sheet simulation.

Functions with singularities are notoriously difficult to approximate with conventional approximation schemes. In computational applications they are often resolved with low-order piecewise polynomials, multilevel schemes or other types of grading strategies. Rational functions are an exception to this rule: for univariate functions with point singularities, such as branch points, rational approximations exist with root-exponential convergence in the rational degree. This is typically enabled by the clustering of poles near the singularity. Both the theory and computational practice of rational functions for function approximation have focused on the univariate case, with extensions to two dimensions via identification with the complex plane. Multivariate rational functions, i.e., quotients of polynomials of several variables, are relatively unexplored in comparison. Yet, apart from a steep increase in theoretical complexity, they also offer a wealth of opportunities. A first observation is that singularities of multivariate rational functions may be continuous curves of poles, rather than isolated ones. By generalizing the clustering of poles from points to curves, we explore constructions of multivariate rational approximations to functions with curves of singularities.

This study investigates the misclassification excess risk bound in the context of 1-bit matrix completion, a significant problem in machine learning involving the recovery of an unknown matrix from a limited subset of its entries. Matrix completion has garnered considerable attention in the last two decades due to its diverse applications across various fields. Unlike conventional approaches that deal with real-valued samples, 1-bit matrix completion is concerned with binary observations. While prior research has predominantly focused on the estimation error of proposed estimators, our study shifts attention to the prediction error. This paper offers theoretical analysis regarding the prediction errors of two previous works utilizing the logistic regression model: one employing a max-norm constrained minimization and the other employing nuclear-norm penalization. Significantly, our findings demonstrate that the latter achieves the minimax-optimal rate without the need for an additional logarithmic term. These novel results contribute to a deeper understanding of 1-bit matrix completion by shedding light on the predictive performance of specific methodologies.

This paper proposes well-conditioned boundary integral equations based on the Burton-Miller method for solving transmission problems. The Burton-Miller method is widely accepted as a highly accurate numerical method based on the boundary integral equation for solving exterior wave problems. While this method can also be applied to solve the transmission problems, a straightforward formulation may yield ill-conditioned integral equations. Consequently, the resulting linear algebraic equations may involve a coefficient matrix with a huge condition number, leading to poor convergence of Krylov-based linear solvers. To address this challenge, our study enhances Burton-Miller-type boundary integral equations tailored for transmission problems by exploiting the Calderon formula. In cases where a single material exists in an unbounded host medium, we demonstrate the formulation of the boundary integral equation such that the underlying integral operator ${\cal A}$ is spectrally well-conditioned. Specifically, ${\cal A}$ can be designed in a simple manner that ensures ${\cal A}^2$ is bounded and has only a single eigenvalue accumulation point. Furthermore, we extend our analysis to the multi-material case, proving that the square of the proposed operator has only a few eigenvalues except for a compact perturbation. Through numerical examples of several benchmark problems, we illustrate that our formulation reduces the iteration number required by iterative linear solvers, even in the presence of material junction points; locations where three or more sub-domains meet on the boundary.

This paper studies the infinite-time stability of the numerical scheme for stochastic McKean-Vlasov equations (SMVEs) via stochastic particle method. The long-time propagation of chaos in mean-square sense is obtained, with which the almost sure propagation in infinite horizon is proved by exploiting the Chebyshev inequality and the Borel-Cantelli lemma. Then the mean-square and almost sure exponential stabilities of the Euler-Maruyama scheme associated with the corresponding interacting particle system are shown through an ingenious manipulation of empirical measure. Combining the assertions enables the numerical solutions to reproduce the stabilities of the original SMVEs. The examples are demonstrated to reveal the importance of this study.

This paper introduces general methodologies for constructing closed-form solutions to linear constant-coefficient partial differential equations (PDEs) with polynomial right-hand sides in two and three spatial dimensions. Polynomial solutions have recently regained significance in the development of numerical techniques for evaluating volume integral operators and also have potential applications in certain kinds of Trefftz finite element methods. The equations covered in this work include the isotropic and anisotropic Poisson, Helmholtz, Stokes, linearized Navier-Stokes, stationary advection-diffusion, elastostatic equations, as well as the time-harmonic elastodynamic and Maxwell equations. Several solutions to complex PDE systems are obtained by a potential representation and rely on the Helmholtz or Poisson solvers. Some of the cases addressed, namely Stokes flow, Maxwell's equations and linearized Navier-Stokes equations, naturally incorporate divergence constraints on the solution. This article provides a generic pattern whereby solutions are constructed by leveraging solutions of the lowest-order part of the partial differential operator (PDO). With the exception of anisotropic material tensors, no matrix inversion or linear system solution is required to compute the solutions. This work is accompanied by a freely-available Julia library, \texttt{ElementaryPDESolutions.jl}, which implements the proposed methodology in an efficient and user-friendly format.

We derive information-theoretic generalization bounds for supervised learning algorithms based on the information contained in predictions rather than in the output of the training algorithm. These bounds improve over the existing information-theoretic bounds, are applicable to a wider range of algorithms, and solve two key challenges: (a) they give meaningful results for deterministic algorithms and (b) they are significantly easier to estimate. We show experimentally that the proposed bounds closely follow the generalization gap in practical scenarios for deep learning.