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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 vector of parameters in general probit, tobit, and multinomial probit models, these distributions yield posteriors that still belong to the SUN family. Although such a core 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 an important 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 general 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 proposes two algorithms to impose seepage boundary conditions in the context of Richards' equation for groundwater flows in unsaturated media. Seepage conditions are non-linear boundary conditions, that can be formulated as a set of unilateral constraints on both the pressure head and the water flux at the ground surface, together with a complementarity condition: these conditions in practice require switching between Neumann and Dirichlet boundary conditions on unknown portions on the boundary. Upon realizing the similarities of these conditions with unilateral contact problems in mechanics, we take inspiration from that literature to propose two approaches: the first method relies on a strongly consistent penalization term, whereas the second one is obtained by an hybridization approach, in which the value of the pressure on the surface is treated as a separate set of unknowns. The flow problem is discretized in mixed form with div-conforming elements so that the water mass is preserved. Numerical experiments show the validity of the proposed strategy in handling the seepage boundary conditions on geometries with increasing complexity.

We consider the problem of approximating an unknown function in a nonlinear model class from point evaluations. When obtaining these point evaluations is costly, minimising the required sample size becomes crucial. Recently, an increasing focus has been on employing adaptive sampling strategies to achieve this. These strategies are based on linear spaces related to the nonlinear model class, for which the optimal sampling measures are known. However, the resulting optimal sampling measures depend on an orthonormal basis of the linear space, which is known rarely. Consequently, sampling from these measures is challenging in practice. This manuscript presents a sampling strategy that iteratively refines an estimate of the optimal sampling measure by updating it based on previously drawn samples. This strategy can be performed offline and does not require evaluations of the sought function. We establish convergence and illustrate the practical performance through numerical experiments. Comparing the presented approach with standard Monte Carlo sampling demonstrates a significant reduction in the number of samples required to achieve a good estimation of an orthonormal basis.

Ensemble Kalman Inversion (EKI) has been proposed as an efficient method for solving inverse problems with expensive forward models. However, the method is based on the assumption that we proceed through a sequence of Gaussian measures in moving from the prior to the posterior, and that the forward model is linear. In this work, we introduce Sequential Kalman Monte Carlo (SKMC) samplers, where we exploit EKI and Flow Annealed Kalman Inversion (FAKI) within a Sequential Monte Carlo (SMC) sampling scheme to perform efficient gradient-free inference in Bayesian inverse problems. FAKI employs normalizing flows (NF) to relax the Gaussian ansatz of the target measures in EKI. NFs are able to learn invertible maps between a Gaussian latent space and the original data space, allowing us to perform EKI updates in the Gaussianized NF latent space. However, FAKI alone is not able to correct for the model linearity assumptions in EKI. Errors in the particle distribution as we move through the sequence of target measures can therefore compound to give incorrect posterior moment estimates. In this work we consider the use of EKI and FAKI to initialize the particle distribution for each target in an adaptive SMC annealing scheme, before performing t-preconditioned Crank-Nicolson (tpCN) updates to distribute particles according to the target. We demonstrate the performance of these SKMC samplers on three challenging numerical benchmarks, showing significant improvements in the rate of convergence compared to standard SMC with importance weighted resampling at each temperature level. Code implementing the SKMC samplers is available at //github.com/RichardGrumitt/KalmanMC.

Predicting quantum operator matrices such as Hamiltonian, overlap, and density matrices in the density functional theory (DFT) framework is crucial for understanding material properties. Current methods often focus on individual operators and struggle with efficiency and scalability for large systems. Here we introduce a novel deep learning model, SLEM (strictly localized equivariant message-passing) for predicting multiple quantum operators, that achieves state-of-the-art accuracy while dramatically improving computational efficiency. SLEM's key innovation is its strict locality-based design, constructing local, equivariant representations for quantum tensors while preserving physical symmetries. This enables complex many-body dependence without expanding the effective receptive field, leading to superior data efficiency and transferability. Using an innovative SO(2) convolution technique, SLEM reduces the computational complexity of high-order tensor products and is therefore capable of handling systems requiring the $f$ and $g$ orbitals in their basis sets. We demonstrate SLEM's capabilities across diverse 2D and 3D materials, achieving high accuracy even with limited training data. SLEM's design facilitates efficient parallelization, potentially extending DFT simulations to systems with device-level sizes, opening new possibilities for large-scale quantum simulations and high-throughput materials discovery.

We propose a topological mapping and localization system able to operate on real human colonoscopies, despite significant shape and illumination changes. The map is a graph where each node codes a colon location by a set of real images, while edges represent traversability between nodes. For close-in-time images, where scene changes are minor, place recognition can be successfully managed with the recent transformers-based local feature matching algorithms. However, under long-term changes -- such as different colonoscopies of the same patient -- feature-based matching fails. To address this, we train on real colonoscopies a deep global descriptor achieving high recall with significant changes in the scene. The addition of a Bayesian filter boosts the accuracy of long-term place recognition, enabling relocalization in a previously built map. Our experiments show that ColonMapper is able to autonomously build a map and localize against it in two important use cases: localization within the same colonoscopy or within different colonoscopies of the same patient. Code: //github.com/jmorlana/ColonMapper.

A new two-parameter discrete distribution, namely the PoiG distribution is derived by the convolution of a Poisson variate and an independently distributed geometric random variable. This distribution generalizes both the Poisson and geometric distributions and can be used for modelling over-dispersed as well as equi-dispersed count data. A number of important statistical properties of the proposed count model, such as the probability generating function, the moment generating function, the moments, the survival function and the hazard rate function. Monotonic properties are studied, such as the log concavity and the stochastic ordering are also investigated in detail. Method of moment and the maximum likelihood estimators of the parameters of the proposed model are presented. It is envisaged that the proposed distribution may prove to be useful for the practitioners for modelling over-dispersed count data compared to its closest competitors.

We prove the convergence of a damped Newton's method for the nonlinear system resulting from a discretization of the second boundary value problem for the Monge-Ampere equation. The boundary condition is enforced through the use of the notion of asymptotic cone. The differential operator is discretized based on a discrete analogue of the subdifferential.

In this work we propose a discretization of the second boundary condition for the Monge-Ampere equation arising in geometric optics and optimal transport. The discretization we propose is the natural generalization of the popular Oliker-Prussner method proposed in 1988. For the discretization of the differential operator, we use a discrete analogue of the subdifferential. Existence, unicity and stability of the solutions to the discrete problem are established. Convergence results to the continuous problem are given.

In the finite difference approximation of the fractional Laplacian the stiffness matrix is typically dense and needs to be approximated numerically. The effect of the accuracy in approximating the stiffness matrix on the accuracy in the whole computation is analyzed and shown to be significant. Four such approximations are discussed. While they are shown to work well with the recently developed grid-over finite difference method (GoFD) for the numerical solution of boundary value problems of the fractional Laplacian, they differ in accuracy, economics to compute, performance of preconditioning, and asymptotic decay away from the diagonal line. In addition, two preconditioners based on sparse and circulant matrices are discussed for the iterative solution of linear systems associated with the stiffness matrix. Numerical results in two and three dimensions are presented.