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We study the numerical solution of a Cahn-Hilliard/Allen-Cahn system with strong coupling through state and gradient dependent non-diagonal mobility matrices. A fully discrete approximation scheme in space and time is proposed which preserves the underlying gradient flow structure and leads to dissipation of the free-energy on the discrete level. Existence and uniqueness of the discrete solution is established and relative energy estimates are used to prove optimal convergence rates in space and time under minimal smoothness assumptions. Numerical tests are presented for illustration of the theoretical results and to demonstrate the viability of the proposed methods.

The paper addresses an error analysis of an Eulerian finite element method used for solving a linearized Navier--Stokes problem in a time-dependent domain. In this study, the domain's evolution is assumed to be known and independent of the solution to the problem at hand. The numerical method employed in the study combines a standard Backward Differentiation Formula (BDF)-type time-stepping procedure with a geometrically unfitted finite element discretization technique. Additionally, Nitsche's method is utilized to enforce the boundary conditions. The paper presents a convergence estimate for several velocity--pressure elements that are inf-sup stable. The estimate demonstrates optimal order convergence in the energy norm for the velocity component and a scaled $L^2(H^1)$-type norm for the pressure component.

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.

In this paper, we discuss the Dutch power market, which is comprised of a day-ahead market and an intraday balancing market that operates like an auction. Due to fluctuations in power supply and demand, there is often an imbalance that leads to different prices in the two markets, providing an opportunity for arbitrage. To address this issue, we restructure the problem and propose a collaborative dual-agent reinforcement learning approach for this bi-level simulation and optimization of European power arbitrage trading. We also introduce two new implementations designed to incorporate domain-specific knowledge by imitating the trading behaviours of power traders. By utilizing reward engineering to imitate domain expertise, we are able to reform the reward system for the RL agent, which improves convergence during training and enhances overall performance. Additionally, the tranching of orders increases bidding success rates and significantly boosts profit and loss (P&L). Our study demonstrates that by leveraging domain expertise in a general learning problem, the performance can be improved substantially, and the final integrated approach leads to a three-fold improvement in cumulative P&L compared to the original agent. Furthermore, our methodology outperforms the highest benchmark policy by around 50% while maintaining efficient computational performance.

Multiscale Finite Element Methods (MsFEMs) are now well-established finite element type approaches dedicated to multiscale problems. They first compute local, oscillatory, problem-dependent basis functions that generate a suitable discretization space, and next perform a Galerkin approximation of the problem on that space. We investigate here how these approaches can be implemented in a non-intrusive way, in order to facilitate their dissemination within industrial codes or non-academic environments. We develop an abstract framework that covers a wide variety of MsFEMs for linear second-order partial differential equations. Non-intrusive MsFEM approaches are developed within the full generality of this framework, which may moreover be beneficial to steering software development and improving the theoretical understanding and analysis of MsFEMs.

In this paper, we make the first attempt to apply the boundary integrated neural networks (BINNs) for the numerical solution of two-dimensional (2D) elastostatic and piezoelectric problems. BINNs combine artificial neural networks with the well-established boundary integral equations (BIEs) to effectively solve partial differential equations (PDEs). The BIEs are utilized to map all the unknowns onto the boundary, after which these unknowns are approximated using artificial neural networks and resolved via a training process. In contrast to traditional neural network-based methods, the current BINNs offer several distinct advantages. First, by embedding BIEs into the learning procedure, BINNs only need to discretize the boundary of the solution domain, which can lead to a faster and more stable learning process (only the boundary conditions need to be fitted during the training). Second, the differential operator with respect to the PDEs is substituted by an integral operator, which effectively eliminates the need for additional differentiation of the neural networks (high-order derivatives of neural networks may lead to instability in learning). Third, the loss function of the BINNs only contains the residuals of the BIEs, as all the boundary conditions have been inherently incorporated within the formulation. Therefore, there is no necessity for employing any weighing functions, which are commonly used in traditional methods to balance the gradients among different objective functions. Moreover, BINNs possess the ability to tackle PDEs in unbounded domains since the integral representation remains valid for both bounded and unbounded domains. Extensive numerical experiments show that BINNs are much easier to train and usually give more accurate learning solutions as compared to traditional neural network-based methods.

We examine the last-iterate convergence rate of Bregman proximal methods - from mirror descent to mirror-prox and its optimistic variants - as a function of the local geometry induced by the prox-mapping defining the method. For generality, we focus on local solutions of constrained, non-monotone variational inequalities, and we show that the convergence rate of a given method depends sharply on its associated Legendre exponent, a notion that measures the growth rate of the underlying Bregman function (Euclidean, entropic, or other) near a solution. In particular, we show that boundary solutions exhibit a stark separation of regimes between methods with a zero and non-zero Legendre exponent: the former converge at a linear rate, while the latter converge, in general, sublinearly. This dichotomy becomes even more pronounced in linearly constrained problems where methods with entropic regularization achieve a linear convergence rate along sharp directions, compared to convergence in a finite number of steps under Euclidean regularization.

The goal of this short note is to discuss the relation between Kullback--Leibler divergence and total variation distance, starting with the celebrated Pinsker's inequality relating the two, before switching to a simple, yet (arguably) more useful inequality, apparently not as well known, due to Bretagnolle and Huber. We also discuss applications of this bound for minimax testing lower bounds.

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.