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We provide a novel dimension-free uniform concentration bound for the empirical risk function of constrained logistic regression. Our bound yields a milder sufficient condition for a uniform law of large numbers than conditions derived by the Rademacher complexity argument and McDiarmid's inequality. The derivation is based on the PAC-Bayes approach with second-order expansion and Rademacher-complexity-based bounds for the residual term of the expansion.

We show that Pfaffians or contiguity relations of hypergeometric functions of several variables give a direct sampling algorithm from toric models in statistics, which is a Markov chain on a lattice generated by a matrix $A$. A correspondence among graphical models and $A$-hypergeometric system is discussed and we give a sum formula of special values of $A$-hypergeometric polynomials. Some hypergeometric series which are interesting in view of statistics are presented.

This paper proposes a new parameterized enhanced shift-splitting (PESS) preconditioner to solve the three-by-three block saddle point problem (SPP). Additionally, we introduce a local PESS (LPESS) preconditioner by relaxing the PESS preconditioner. Necessary and sufficient criteria are established for the convergence of the proposed PESS iterative process for any initial guess. Furthermore, we meticulously investigate the spectral bounds of the PESS and LPESS preconditioned matrices. Moreover, empirical investigations have been performed for the sensitivity analysis of the proposed PESS preconditioner, which unveils its robustness. Numerical experiments are carried out to demonstrate the enhanced efficiency and robustness of the proposed PESS and LPESS preconditioners compared to the existing state-of-the-art preconditioners.

We present the numerical analysis of a finite element method (FEM) for one-dimensional Dirichlet problems involving the logarithmic Laplacian (the pseudo-differential operator that appears as a first-order expansion of the fractional Laplacian as the exponent $s\to 0^+$). Our analysis exhibits new phenomena in this setting; in particular, using recently obtained regularity results, we prove rigorous error estimates and provide a logarithmic order of convergence in the energy norm using suitable $\log$-weighted spaces. Numerical evidence suggests that this type of rate cannot be improved. Moreover, we show that the stiffness matrix of logarithmic problems can be obtained as the derivative of the fractional stiffness matrix evaluated at $s=0$. Lastly, we investigate the relationship between the discrete eigenvalue problem and its convergence to the continuous one.

This paper presents a scalable physics-based block preconditioner for mixed-dimensional models in beam-solid interaction and their application in engineering. In particular, it studies the linear systems arising from a regularized mortar-type approach for embedding geometrically exact beams into solid continua. Due to the lack of block diagonal dominance of the arising 2 x 2 block system, an approximate block factorization preconditioner is used. It exploits the sparsity structure of the beam sub-block to construct a sparse approximate inverse, which is then not only used to explicitly form an approximation of the Schur complement, but also acts as a smoother within the prediction step of the arising SIMPLE-type preconditioner. The correction step utilizes an algebraic multigrid method. Although, for now, the beam sub-block is tackled by a one-level method only, the multi-level nature of the computationally demanding correction step delivers a scalable preconditioner in practice. In numerical test cases, the influence of different algorithmic parameters on the quality of the sparse approximate inverse is studied and the weak scaling behavior of the proposed preconditioner on up to 1000 MPI ranks is demonstrated, before the proposed preconditioner is finally applied for the analysis of steel-reinforced concrete structures in civil engineering.

We present a proof showing that the weak error of a system of $n$ interacting stochastic particles approximating the solution of the McKean-Vlasov equation is $\mathcal O(n^{-1})$. Our proof is based on the Kolmogorov backward equation for the particle system and bounds on the derivatives of its solution, which we derive more generally using the variations of the stochastic particle system. The convergence rate is verified by numerical experiments, which also indicate that the assumptions made here and in the literature can be relaxed.

We propose a novel finite element-based physics-informed operator learning framework that allows for predicting spatiotemporal dynamics governed by partial differential equations (PDEs). The proposed framework employs a loss function inspired by the finite element method (FEM) with the implicit Euler time integration scheme. A transient thermal conduction problem is considered to benchmark the performance. The proposed operator learning framework takes a temperature field at the current time step as input and predicts a temperature field at the next time step. The Galerkin discretized weak formulation of the heat equation is employed to incorporate physics into the loss function, which is coined finite operator learning (FOL). Upon training, the networks successfully predict the temperature evolution over time for any initial temperature field at high accuracy compared to the FEM solution. The framework is also confirmed to be applicable to a heterogeneous thermal conductivity and arbitrary geometry. The advantages of FOL can be summarized as follows: First, the training is performed in an unsupervised manner, avoiding the need for a large data set prepared from costly simulations or experiments. Instead, random temperature patterns generated by the Gaussian random process and the Fourier series, combined with constant temperature fields, are used as training data to cover possible temperature cases. Second, shape functions and backward difference approximation are exploited for the domain discretization, resulting in a purely algebraic equation. This enhances training efficiency, as one avoids time-consuming automatic differentiation when optimizing weights and biases while accepting possible discretization errors. Finally, thanks to the interpolation power of FEM, any arbitrary geometry can be handled with FOL, which is crucial to addressing various engineering application scenarios.

The broad class of multivariate unified skew-normal (SUN) distributions has been recently shown to possess important conjugacy properties. When used as priors for the coefficients vector in probit, tobit, and multinomial probit models, these distributions yield posteriors that still belong to the SUN family. Although this result has led to important advancements in Bayesian inference and computation, its applicability beyond likelihoods associated with fully-observed, discretized, or censored realizations from multivariate Gaussian models remains yet unexplored. This article covers such a gap by proving that the wider family of multivariate unified skew-elliptical (SUE) distributions, which extends SUNs to more general perturbations of elliptical densities, guarantees conjugacy for broader classes of models, beyond those relying on fully-observed, discretized or censored Gaussians. Such a result leverages the closure under linear combinations, conditioning and marginalization of SUE to prove that this family is conjugate to the likelihood induced by multivariate regression models for fully-observed, censored or dichotomized realizations from skew-elliptical distributions. This advancement enlarges the set of models that enable conjugate Bayesian inference to general formulations arising from elliptical and skew-elliptical families, including the multivariate Student's t and skew-t, among others.

This paper introduces a prognostic method called FLASH that addresses the problem of joint modelling of longitudinal data and censored durations when a large number of both longitudinal and time-independent features are available. In the literature, standard joint models are either of the shared random effect or joint latent class type. Combining ideas from both worlds and using appropriate regularisation techniques, we define a new model with the ability to automatically identify significant prognostic longitudinal features in a high-dimensional context, which is of increasing importance in many areas such as personalised medicine or churn prediction. We develop an estimation methodology based on the EM algorithm and provide an efficient implementation. The statistical performance of the method is demonstrated both in extensive Monte Carlo simulation studies and on publicly available real-world datasets. Our method significantly outperforms the state-of-the-art joint models in predicting the latent class membership probability in terms of the C-index in a so-called ``real-time'' prediction setting, with a computational speed that is orders of magnitude faster than competing methods. In addition, our model automatically identifies significant features that are relevant from a practical perspective, making it interpretable.

This paper aims first to perform robust continuous analysis of a mixed nonlinear formulation for stress-assisted diffusion of a solute that interacts with an elastic material, and second to propose and analyse a virtual element formulation of the model problem. The two-way coupling mechanisms between the Herrmann formulation for linear elasticity and the reaction-diffusion equation (written in mixed form) consist of diffusion-induced active stress and stress-dependent diffusion. The two sub-problems are analysed using the extended Babu\v{s}ka--Brezzi--Braess theory for perturbed saddle-point problems. The well-posedness of the nonlinearly coupled system is established using a Banach fixed-point strategy under the smallness assumption on data. The virtual element formulations for the uncoupled sub-problems are proven uniquely solvable by a fixed-point argument in conjunction with appropriate projection operators. We derive the a priori error estimates, and test the accuracy and performance of the proposed method through computational simulations.