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In this contribution we apply an adaptive model hierarchy, consisting of a full-order model, a reduced basis reduced order model, and a machine learning surrogate, to parametrized linear-quadratic optimal control problems. The involved reduced order models are constructed adaptively and are called in such a way that the model hierarchy returns an approximate solution of given accuracy for every parameter value. At the same time, the fastest model of the hierarchy is evaluated whenever possible and slower models are only queried if the faster ones are not sufficiently accurate. The performance of the model hierarchy is studied for a parametrized heat equation example with boundary value control.

The current study investigates the asymptotic spectral properties of a finite difference approximation of nonlocal Helmholtz equations with a Caputo fractional Laplacian and a variable coefficient wave number $\mu$, as it occurs when considering a wave propagation in complex media, characterized by nonlocal interactions and spatially varying wave speeds. More specifically, by using tools from Toeplitz and generalized locally Toeplitz theory, the present research delves into the spectral analysis of nonpreconditioned and preconditioned matrix-sequences. We report numerical evidences supporting the theoretical findings. Finally, open problems and potential extensions in various directions are presented and briefly discussed.

By interpreting planar polynomial curves as complex-valued functions of a real parameter, an inner product, norm, metric function, and the notion of orthogonality may be defined for such curves. This approach is applied to the complex pre-image polynomials that generate planar Pythagorean-hodograph (PH) curves, to facilitate the implementation of bounded modifications of them that preserve their PH nature. The problems of bounded modifications under the constraint of fixed curve end points and end tangent directions, and of increasing the arc length of a PH curve by a prescribed amount, are also addressed.

The broad class of multivariate unified skew-normal (SUN) distributions has been recently shown to possess fundamental 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 such a 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 substantially 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.

The incidence of vertebral fragility fracture is increased by the presence of preexisting pathologies such as metastatic disease. Computational tools could support the fracture prediction and consequently the decision of the best medical treatment. Anyway, validation is required to use these tools in clinical practice. To address this necessity, in this study subject-specific homogenized finite element models of single vertebrae were generated from micro CT images for both healthy and metastatic vertebrae and validated against experimental data. More in detail, spine segments were tested under compression and imaged with micro CT. The displacements field could be extracted for each vertebra singularly using the digital volume correlation full-field technique. Homogenized finite element models of each vertebra could hence be built from the micro CT images, applying boundary conditions consistent with the experimental displacements at the endplates. Numerical and experimental displacements and strains fields were eventually compared. In addition, the outcomes of a micro CT based homogenized model were compared to the ones of a clinical-CT based model. Good agreement between experimental and computational displacement fields, both for healthy and metastatic vertebrae, was found. Comparison between micro CT based and clinical-CT based outcomes showed strong correlations. Furthermore, models were able to qualitatively identify the regions which experimentally showed the highest strain concentration. In conclusion, the combination of experimental full-field technique and the in-silico modelling allowed the development of a promising pipeline for validation of fracture risk predictors, although further improvements in both fields are needed to better analyse quantitatively the post-yield behaviour of the vertebra.

Sine-skewed circular distributions are identifiable and have easily-computable trigonometric moments and a simple random number generation algorithm, whereas they are known to have relatively low levels of asymmetry. This study proposes a new family of circular distributions that can be skewed more significantly than that of existing models. It is shown that a subfamily of the proposed distributions is identifiable with respect to parameters and all distributions in the subfamily have explicit trigonometric moments and a simple random number generation algorithm. The maximum likelihood estimation for model parameters is considered and its finite sample performances are investigated by numerical simulations. Some real data applications are illustrated for practical purposes.

We introduce a lower bounding technique for the min max correlation clustering problem and, based on this technique, a combinatorial 4-approximation algorithm for complete graphs. This improves upon the previous best known approximation guarantees of 5, using a linear program formulation (Kalhan et al., 2019), and 40, for a combinatorial algorithm (Davies et al., 2023a). We extend this algorithm by a greedy joining heuristic and show empirically that it improves the state of the art in solution quality and runtime on several benchmark datasets.

We consider a model selection problem for structural equation modeling (SEM) with latent variables for diffusion processes based on high-frequency data. First, we propose the quasi-Akaike information criterion of the SEM and study the asymptotic properties. Next, we consider the situation where the set of competing models includes some misspecified parametric models. It is shown that the probability of choosing the misspecified models converges to zero. Furthermore, examples and simulation results are given.

Mesh-based Graph Neural Networks (GNNs) have recently shown capabilities to simulate complex multiphysics problems with accelerated performance times. However, mesh-based GNNs require a large number of message-passing (MP) steps and suffer from over-smoothing for problems involving very fine mesh. In this work, we develop a multiscale mesh-based GNN framework mimicking a conventional iterative multigrid solver, coupled with adaptive mesh refinement (AMR), to mitigate challenges with conventional mesh-based GNNs. We use the framework to accelerate phase field (PF) fracture problems involving coupled partial differential equations with a near-singular operator due to near-zero modulus inside the crack. We define the initial graph representation using all mesh resolution levels. We perform a series of downsampling steps using Transformer MP GNNs to reach the coarsest graph followed by upsampling steps to reach the original graph. We use skip connectors from the generated embedding during coarsening to prevent over-smoothing. We use Transfer Learning (TL) to significantly reduce the size of training datasets needed to simulate different crack configurations and loading conditions. The trained framework showed accelerated simulation times, while maintaining high accuracy for all cases compared to physics-based PF fracture model. Finally, this work provides a new approach to accelerate a variety of mesh-based engineering multiphysics problems

Inner products of neural network feature maps arises in a wide variety of machine learning frameworks as a method of modeling relations between inputs. This work studies the approximation properties of inner products of neural networks. It is shown that the inner product of a multi-layer perceptron with itself is a universal approximator for symmetric positive-definite relation functions. In the case of asymmetric relation functions, it is shown that the inner product of two different multi-layer perceptrons is a universal approximator. In both cases, a bound is obtained on the number of neurons required to achieve a given accuracy of approximation. In the symmetric case, the function class can be identified with kernels of reproducing kernel Hilbert spaces, whereas in the asymmetric case the function class can be identified with kernels of reproducing kernel Banach spaces. Finally, these approximation results are applied to analyzing the attention mechanism underlying Transformers, showing that any retrieval mechanism defined by an abstract preorder can be approximated by attention through its inner product relations. This result uses the Debreu representation theorem in economics to represent preference relations in terms of utility functions.