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We present a new high-order accurate spectral element solution to the two-dimensional scalar Poisson equation subject to a general Robin boundary condition. The solution is based on a simplified version of the shifted boundary method employing a continuous arbitrary order $hp$-Galerkin spectral element method as the numerical discretization procedure. The simplification relies on a polynomial correction to avoid explicitly evaluating high-order partial derivatives from the Taylor series expansion, which traditionally have been used within the shifted boundary method. In this setting, we apply an extrapolation and novel interpolation approach to project the basis functions from the true domain onto the approximate surrogate domain. The resulting solution provides a method that naturally incorporates curved geometrical features of the domain, overcomes complex and cumbersome mesh generation, and avoids problems with small-cut-cells. Dirichlet, Neumann, and general Robin boundary conditions are enforced weakly through: i) a generalized Nitsche's method and ii) a generalized Aubin's method. For this, a consistent asymptotic preserving formulation of the embedded Robin formulations is presented. We present several numerical experiments and analysis of the algorithmic properties of the different weak formulations. With this, we include convergence studies under polynomial, $p$, increase of the basis functions, mesh, $h$, refinement, and matrix conditioning to highlight the spectral and algebraic convergence features, respectively. This is done to assess the influence of errors across variational formulations, polynomial order, mesh size, and mappings between the true and surrogate boundaries.

For problems of time-harmonic scattering by rational polygonal obstacles, embedding formulae express the far-field pattern induced by any incident plane wave in terms of the far-field patterns for a relatively small (frequency-independent) set of canonical incident angles. Although these remarkable formulae are exact in theory, here we demonstrate that: (i) they are highly sensitive to numerical errors in practice, and; (ii) direct calculation of the coefficients in these formulae may be impossible for particular sets of canonical incident angles, even in exact arithmetic. Only by overcoming these practical issues can embedding formulae provide a highly efficient approach to computing the far-field pattern induced by a large number of incident angles. Here we propose solutions for problems (i) and (ii), backed up by theory and numerical experiments. Problem (i) is solved using techniques from computational complex analysis: we reformulate the embedding formula as a complex contour integral and prove that this is much less sensitive to numerical errors. In practice, this contour integral can be efficiently evaluated by residue calculus. Problem (ii) is addressed using techniques from numerical linear algebra: we oversample, considering more canonical incident angles than are necessary, thus expanding the space of valid coefficients vectors. The coefficients vectors can then be selected using either a least squares approach or column subset selection.

Generative diffusion models have achieved spectacular performance in many areas of generative modeling. While the fundamental ideas behind these models come from non-equilibrium physics, in this paper we show that many aspects of these models can be understood using the tools of equilibrium statistical mechanics. Using this reformulation, we show that generative diffusion models undergo second-order phase transitions corresponding to symmetry breaking phenomena. We argue that this lead to a form of instability that lies at the heart of their generative capabilities and that can be described by a set of mean field critical exponents. We conclude by analyzing recent work connecting diffusion models and associative memory networks in view of the thermodynamic formulations.

In recent years, research interest in personalised treatments has been growing. However, treatment effect heterogeneity and possibly time-varying treatment effects are still often overlooked in clinical studies. Statistical tools are needed for the identification of treatment response patterns, taking into account that treatment response is not constant over time. We aim to provide an innovative method to obtain dynamic treatment effect phenotypes on a time-to-event outcome, conditioned on a set of relevant effect modifiers. The proposed method does not require the assumption of proportional hazards for the treatment effect, which is rarely realistic. We propose a spline-based survival neural network, inspired by the Royston-Parmar survival model, to estimate time-varying conditional treatment effects. We then exploit the functional nature of the resulting estimates to apply a functional clustering of the treatment effect curves in order to identify different patterns of treatment effects. The application that motivated this work is the discontinuation of treatment with Mineralocorticoid receptor Antagonists (MRAs) in patients with heart failure, where there is no clear evidence as to which patients it is the safest choice to discontinue treatment and, conversely, when it leads to a higher risk of adverse events. The data come from an electronic health record database. A simulation study was performed to assess the performance of the spline-based neural network and the stability of the treatment response phenotyping procedure. In light of the results, the suggested approach has the potential to support personalized medical choices by assessing unique treatment responses in various medical contexts over a period of time.

Despite the continuous development of the different operational ensemble prediction systems over the past decades, ensemble forecasts still might suffer from lack of calibration and/or display systematic bias, thus require some post-processing to improve their forecast skill. Here we focus on visibility, which quantity plays a crucial role e.g. in aviation and road safety or in ship navigation, and propose a parametric model where the predictive distribution is a mixture of a gamma and a truncated normal distribution, both right censored at the maximal reported visibility value. The new model is evaluated in two case studies based on visibility ensemble forecasts of the European Centre for Medium-Range Weather Forecasts covering two distinct domains in Central and Western Europe and two different time periods. The results of the case studies indicate that climatology is substantially superior to the raw ensemble; nevertheless, the forecast skill can be further improved by post-processing, at least for short lead times. Moreover, the proposed mixture model consistently outperforms the Bayesian model averaging approach used as reference post-processing technique.

We study the multiplicative hazards model with intermittently observed longitudinal covariates and time-varying coefficients. For such models, the existing {\it ad hoc} approach, such as the last value carried forward, is biased. We propose a kernel weighting approach to get an unbiased estimation of the non-parametric coefficient function and establish asymptotic normality for any fixed time point. Furthermore, we construct the simultaneous confidence band to examine the overall magnitude of the variation. Simulation studies support our theoretical predictions and show favorable performance of the proposed method. A data set from cerebral infarction is used to illustrate our methodology.

Classical inequality curves and inequality measures are defined for distributions with finite mean value. Moreover, their empirical counterparts are not resistant to outliers. For these reasons, quantile versions of known inequality curves such as the Lorenz, Bonferroni, Zenga and $D$ curves, and quantile versions of inequality measures such as the Gini, Bonferroni, Zenga and $D$ indices have been proposed in the literature. We propose various nonparametric estimators of quantile versions of inequality curves and inequality measures, prove their consistency, and compare their accuracy in a~simulation study. We also give examples of the use of quantile versions of inequality measures in real data analysis.

For solving two-dimensional incompressible flow in the vorticity form by the fourth-order compact finite difference scheme and explicit strong stability preserving (SSP) temporal discretizations, we show that the simple bound-preserving limiter in [Li H., Xie S., Zhang X., SIAM J. Numer. Anal., 56 (2018)]. can enforce the strict bounds of the vorticity, if the velocity field satisfies a discrete divergence free constraint. For reducing oscillations, a modified TVB limiter adapted from [Cockburn B., Shu CW., SIAM J. Numer. Anal., 31 (1994)] is constructed without affecting the bound-preserving property. This bound-preserving finite difference method can be used for any passive convection equation with a divergence free velocity field.

Boundary value problems involving elliptic PDEs such as the Laplace and the Helmholtz equations are ubiquitous in physics and engineering. Many such problems have alternative formulations as integral equations that are mathematically more tractable than their PDE counterparts. However, the integral equation formulation poses a challenge in solving the dense linear systems that arise upon discretization. In cases where iterative methods converge rapidly, existing methods that draw on fast summation schemes such as the Fast Multipole Method are highly efficient and well established. More recently, linear complexity direct solvers that sidestep convergence issues by directly computing an invertible factorization have been developed. However, storage and compute costs are high, which limits their ability to solve large-scale problems in practice. In this work, we introduce a distributed-memory parallel algorithm based on an existing direct solver named ``strong recursive skeletonization factorization.'' The analysis of its parallel scalability applies generally to a class of existing methods that exploit the so-called strong admissibility. Specifically, we apply low-rank compression to certain off-diagonal matrix blocks in a way that minimizes data movement. Given a compression tolerance, our method constructs an approximate factorization of a discretized integral operator (dense matrix), which can be used to solve linear systems efficiently in parallel. Compared to iterative algorithms, our method is particularly suitable for problems involving ill-conditioned matrices or multiple right-hand sides. Large-scale numerical experiments are presented to demonstrate the performance of our implementation using the Julia language.

The monotonicity of discrete Laplacian implies discrete maximum principle, which in general does not hold for high order schemes. The $Q^2$ spectral element method has been proven monotone on a uniform rectangular mesh. In this paper we prove the monotonicity of the $Q^2$ spectral element method on quasi-uniform rectangular meshes under certain mesh constraints. In particular, we propose a relaxed Lorenz's condition for proving monotonicity.