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The aim of this work is to present a model reduction technique in the framework of optimal control problems for partial differential equations. We combine two approaches used for reducing the computational cost of the mathematical numerical models: domain-decomposition (DD) methods and reduced-order modelling (ROM). In particular, we consider an optimisation-based domain-decomposition algorithm for the parameter-dependent stationary incompressible Navier-Stokes equations. Firstly, the problem is described on the subdomains coupled at the interface and solved through an optimal control problem, which leads to the complete separation of the subdomain problems in the DD method. On top of that, a reduced model for the obtained optimal-control problem is built; the procedure is based on the Proper Orthogonal Decomposition technique and a further Galerkin projection. The presented methodology is tested on two fluid dynamics benchmarks: the stationary backward-facing step and lid-driven cavity flow. The numerical tests show a significant reduction of the computational costs in terms of both the problem dimensions and the number of optimisation iterations in the domain-decomposition algorithm.

In this work, we address parametric non-stationary fluid dynamics problems within a model order reduction setting based on domain decomposition. Starting from the domain decomposition approach, we derive an optimal control problem, for which we present the convergence analysis. The snapshots for the high-fidelity model are obtained with the Finite Element discretisation, and the model order reduction is then proposed both in terms of time and physical parameters, with a standard POD-Galerkin projection. We test the proposed methodology on two fluid dynamics benchmarks: the non-stationary backward-facing step and lid-driven cavity flow. Finally, also in view of future works, we compare the intrusive POD--Galerkin approach with a non--intrusive approach based on Neural Networks.

Penalties that induce smoothness are common in nonparametric regression. In many settings, the amount of smoothness in the data generating function will not be known. Simon and Shojaie (2021) derived convergence rates for nonparametric estimators under misspecified smoothness. We show that their theoretical convergence rates can be improved by working with convenient approximating functions. Properties of convolutions and higher-order kernels allow these approximation functions to match the true functions more closely than those used in Simon and Shojaie (2021). As a result, we obtain tighter convergence rates.

Algorithms for solving the linear classification problem have a long history, dating back at least to 1936 with linear discriminant analysis. For linearly separable data, many algorithms can obtain the exact solution to the corresponding 0-1 loss classification problem efficiently, but for data which is not linearly separable, it has been shown that this problem, in full generality, is NP-hard. Alternative approaches all involve approximations of some kind, including the use of surrogates for the 0-1 loss (for example, the hinge or logistic loss) or approximate combinatorial search, none of which can be guaranteed to solve the problem exactly. Finding efficient algorithms to obtain an exact i.e. globally optimal solution for the 0-1 loss linear classification problem with fixed dimension, remains an open problem. In research we report here, we detail the rigorous construction of a new algorithm, incremental cell enumeration (ICE), that can solve the 0-1 loss classification problem exactly in polynomial time. We prove correctness using concepts from the theory of hyperplane arrangements and oriented matroids. We demonstrate the effectiveness of this algorithm on synthetic and real-world datasets, showing optimal accuracy both in and out-of-sample, in practical computational time. We also empirically demonstrate how the use of approximate upper bound leads to polynomial time run-time improvements to the algorithm whilst retaining exactness. To our knowledge, this is the first, rigorously-proven polynomial time, practical algorithm for this long-standing problem.

In this paper, we investigate the impact of compression on stochastic gradient algorithms for machine learning, a technique widely used in distributed and federated learning. We underline differences in terms of convergence rates between several unbiased compression operators, that all satisfy the same condition on their variance, thus going beyond the classical worst-case analysis. To do so, we focus on the case of least-squares regression (LSR) and analyze a general stochastic approximation algorithm for minimizing quadratic functions relying on a random field. We consider weak assumptions on the random field, tailored to the analysis (specifically, expected H\"older regularity), and on the noise covariance, enabling the analysis of various randomizing mechanisms, including compression. We then extend our results to the case of federated learning. More formally, we highlight the impact on the convergence of the covariance $\mathfrak{C}_{\mathrm{ania}}$ of the additive noise induced by the algorithm. We demonstrate despite the non-regularity of the stochastic field, that the limit variance term scales with $\mathrm{Tr}(\mathfrak{C}_{\mathrm{ania}} H^{-1})/K$ (where $H$ is the Hessian of the optimization problem and $K$ the number of iterations) generalizing the rate for the vanilla LSR case where it is $\sigma^2 \mathrm{Tr}(H H^{-1}) / K = \sigma^2 d / K$ (Bach and Moulines, 2013). Then, we analyze the dependency of $\mathfrak{C}_{\mathrm{ania}}$ on the compression strategy and ultimately its impact on convergence, first in the centralized case, then in two heterogeneous FL frameworks.

We consider two-phase fluid deformable surfaces as model systems for biomembranes. Such surfaces are modeled by incompressible surface Navier-Stokes-Cahn-Hilliard-like equations with bending forces. We derive this model using the Lagrange-D'Alembert principle considering various dissipation mechanisms. The highly nonlinear model is solved numerically to explore the tight interplay between surface evolution, surface phase composition, surface curvature and surface hydrodynamics. It is demonstrated that hydrodynamics can enhance bulging and furrow formation, which both can further develop to pinch-offs. The numerical approach builds on a Taylor-Hood element for the surface Navier-Stokes part, a semi-implicit approach for the Cahn-Hilliard part, higher order surface parametrizations, appropriate approximations of the geometric quantities, and mesh redistribution. We demonstrate convergence properties that are known to be optimal for simplified sub-problems.

This paper studies the identification of a linear combination of point sources from a finite number of measurements. Since the data are typically contaminated by Gaussian noise, a statistical framework for its recovery is considered. It relies on two main ingredients, first, a convex but non-smooth Tikhonov point estimator over the space of Radon measures and, second, a suitable mean-squared error based on its Hellinger-Kantorovich distance to the ground truth. Assuming standard non-degenerate source conditions as well as applying careful linearization arguments, a computable upper bound on the latter is derived. On the one hand, this allows to derive asymptotic convergence results for the mean-squared error of the estimator in the small small variance case. On the other, it paves the way for applying optimal sensor placement approaches to sparse inverse problems.

High-dimensional Partial Differential Equations (PDEs) are a popular mathematical modelling tool, with applications ranging from finance to computational chemistry. However, standard numerical techniques for solving these PDEs are typically affected by the curse of dimensionality. In this work, we tackle this challenge while focusing on stationary diffusion equations defined over a high-dimensional domain with periodic boundary conditions. Inspired by recent progress in sparse function approximation in high dimensions, we propose a new method called compressive Fourier collocation. Combining ideas from compressive sensing and spectral collocation, our method replaces the use of structured collocation grids with Monte Carlo sampling and employs sparse recovery techniques, such as orthogonal matching pursuit and $\ell^1$ minimization, to approximate the Fourier coefficients of the PDE solution. We conduct a rigorous theoretical analysis showing that the approximation error of the proposed method is comparable with the best $s$-term approximation (with respect to the Fourier basis) to the solution. Using the recently introduced framework of random sampling in bounded Riesz systems, our analysis shows that the compressive Fourier collocation method mitigates the curse of dimensionality with respect to the number of collocation points under sufficient conditions on the regularity of the diffusion coefficient. We also present numerical experiments that illustrate the accuracy and stability of the method for the approximation of sparse and compressible solutions.

In this paper we propose a geometric integrator to numerically approximate the flow of Lie systems. The highlight of this paper is to present a novel procedure that integrates the system on a Lie group intrinsically associated to the Lie system, and then generating the discrete solution of this Lie system through a given action of the Lie group on the manifold where the system evolves. One major result from the integration on the Lie group is that one is able to solve all automorphic Lie systems at the same time, and that they can be written as first-order systems of linear homogeneous ODEs in normal form. This brings a lot of advantages, since solving a linear ODE involves less numerical cost. Specifically, we use two families of numerical schemes on the Lie group, which are designed to preserve its geometrical structure: the first one based on the Magnus expansion, whereas the second is based on RKMK methods. Moreover, since the aforementioned action relates the Lie group and the manifold where the Lie system evolves, the resulting integrator preserves any geometric structure of the latter. We compare both methods for Lie systems with geometric invariants, particularly a class on Lie systems on curved spaces. As already mentioned, the milestone of this paper is to show that the method we propose preserves all the geometric invariants very faithfully, in comparison with nongeometric numerical methods.

In this paper, we develop a unified regression approach to model unconditional quantiles, M-quantiles and expectiles of multivariate dependent variables exploiting the multidimensional Huber's function. To assess the impact of changes in the covariates across the entire unconditional distribution of the responses, we extend the work of Firpo et al. (2009) by running a mean regression of the recentered influence function on the explanatory variables. We discuss the estimation procedure and establish the asymptotic properties of the derived estimators. A data-driven procedure is also presented to select the tuning constant of the Huber's function. The validity of the proposed methodology is explored with simulation studies and through an application using the Survey of Household Income and Wealth 2016 conducted by the Bank of Italy.