Abstract The Landau–Lifshitz–Bloch equation (LLBE) describes the evolution of the magnetic spin field in ferromagnets at high temperatures. In this paper we study the numerical approximation of the LLBE in the regime above the Curie temperature on bounded polytopal domains in $\mathbb{R}^{d}$, with $d\in \{1,2,3\}$, allowing for nonconvex domains when $d=2$. We first establish the existence and uniqueness of strong solutions to the LLBE and propose a linear, fully discrete, conforming finite element scheme for its approximation. While this scheme is shown to converge, the obtained rate is suboptimal. To address this shortcoming we introduce a viscous (pseudo-parabolic) regularization of the LLBE, which we call the $\epsilon $-LLBE. For this regularized problem we prove the unique existence of strong solutions and establish a rate of convergence of the solution $\boldsymbol{u}^\epsilon $ of the $\epsilon $-LLBE to the solution $\boldsymbol{u}$ of the LLBE as $\epsilon \to 0^{+}$. Furthermore, we propose a linear, fully discrete, conforming finite element scheme to approximate the solution of the $\epsilon $-LLBE. Given sufficiently smooth initial data error analysis is performed to show stability and uniform-in-time convergence of the scheme. Finally, several numerical simulations are presented to corroborate our theoretical results.

Abstract The use of artificial neural networks is now becoming widespread across a wide variety of tasks. In this context of very rapid development issues related to the storage and computational performance of these models emerge, since networks are sometimes very deep and comprise up to trillions of parameters. For all these reasons the use of reduced precision is increasingly being considered, although, until now, its accuracy and robustness had been approached mostly from a practical standpoint or verified by software. The aim of this work is to provide formal tools to better understand, explain and predict the accuracy and stability of neural networks when using floating-point arithmetic. To this end, we apply a rounding error analysis based on existing tools in numerical linear algebra to obtain both forward and backward error bounds. This includes both deterministic worst-case bounds as well as probabilistic bounds that are sharper on average. Since the exact backward error is not directly computable the backward error bounds are validated experimentally through a linearized backward error. These bounds both ensure the proper functioning of neural networks once trained and provide recommendations on architectures and training methods to enhance the robustness of neural networks.

Abstract Neural network-based methods have shown great promise in stably solving ill-posed inverse problems. In this work we focus on the inverse problem of recovering point sources, an important class of applied inverse problems. Despite their potential neural network-based methods for identifying point sources remain underdeveloped, primarily due to the inherent singularity of the solution. To address this challenge we develop a novel neural algorithm for identifying point sources, utilizing the singularity enrichment technique. We employ the fundamental solution and neural networks to represent the singular and regular parts, respectively, and then minimize an empirical loss involving the intensities and locations of unknown point sources and the parameters of the neural network. Moreover, by combining the conditional stability argument of the inverse problem with the generalization error of the empirical loss we conduct a rigorous error analysis of the algorithm. We demonstrate the effectiveness of the method with several challenging experiments.

Abstract This paper presents several new properties of the intrinsic $\kappa $-projection into $\kappa $-hyperbolically convex sets of $\kappa $-hyperbolic space forms, along with closed-form formulas for the intrinsic $\kappa $-projection into specific $\kappa $-hyperbolically convex sets. It also discusses the relationship between the intrinsic $\kappa $-projection, the Euclidean orthogonal projection and the Lorentz projection. These properties lay the groundwork for analyzing the gradient projection method and hold importance in their own right. Additionally, new properties of the gradient projection method to solve constrained optimization problems in $\kappa $-hyperbolic space forms are established, considering both constant and backtracking step sizes in the analysis. It is shown that every accumulation point of the sequence generated by the method for both step sizes is a stationary point for the given problem. Additionally, an iteration complexity bound is provided that upper bounds the number of iterations needed to achieve a suitable measure of stationarity for both step sizes. Finally, the properties of the constrained Fermat–Weber problem are explored, demonstrating that the sequence generated by the gradient projection method converges to its unique solution. Numerical experiments on solving the Fermat–Weber problem are presented, illustrating the theoretical findings and demonstrating the effectiveness of the proposed methods.

Abstract Ordinary differential equation (ODE) solvers with randomly sampled timestep sizes appear in the context of chaotic dynamical systems, differential equations with low regularity and, implicitly, in stochastic optimization. In this work, we propose and study the stochastic Euler dynamics—a continuous-time Markov process that is equivalent to a linear spline interpolation of a random timestep (forward) Euler method. We understand the stochastic Euler dynamics as a path-valued ansatz for the ODE solution that shall be approximated. We first obtain qualitative insights by studying deterministic Euler dynamics, which we derive through a first-order approximation to the infinitesimal generator of the stochastic Euler dynamics. In the context of linear ODEs, these deterministic Euler dynamics describe the dynamics of the expectation of the stochastic Euler dynamics. Then we show convergence of the stochastic Euler dynamics to the ODE solution by studying the associated infinitesimal generators and by a novel local truncation error analysis. Next, we prove stability by an immediate analysis of the random timestep Euler method and by deriving Foster–Lyapunov criteria for the stochastic Euler dynamics; the latter also yield bounds on the speed of convergence to stationarity. The paper ends with a discussion of second-order stochastic Euler dynamics and a series of numerical experiments that appear to verify our analytical results.

Abstract We present an implicit–explicit (IMEX) scheme for semilinear wave equations with strong damping. By treating the nonlinear, nonstiff term explicitly and the linear, stiff part implicitly we obtain a method that is, not only unconditionally stable, but also highly efficient. Our main results are error bounds of the full discretization in space and time for the IMEX scheme combined with a general abstract space discretization. As an application we consider the heterogeneous multiscale method for wave equations with highly oscillating coefficients in space for which we show spatial and temporal convergence rates by using the abstract result.

Abstract We introduce a hybrid high-order method for approximating the ground state of the nonlinear Gross–Pitaevskii eigenvalue problem. Optimal convergence rates are proved for the ground state approximation, as well as for the associated eigenvalue and energy approximations. Unlike classical conforming methods, which inherently provide upper bounds on the ground state energy, the proposed approach gives rise to guaranteed and asymptotically exact lower-energy bounds. Importantly, and in contrast to previous works, they are obtained directly without the need of post-processing, leading to more accurate guaranteed lower energy bounds in practice.

Abstract Based on the proximal bundle and gradient sampling (GS) methods, we develop a robust algorithm for minimizing the locally Lipschitz function $f:\mathbb{R}^{n}\to \mathbb{R}$. As an interesting feature of the proposed method, thanks to the GS technique, we sample a set of differentiable auxiliary points from the vicinity of the current point to construct an initial piecewise linear model for the objective function. If necessary, inspired by bundle methods, we iteratively enrich the set of sampled points by using a single nonredundant auxiliary point suggested by a modified variant of Mifflin’s line search. However, we may terminate the enrichment process without achieving a descent step, which is different from classic bundle methods. Indeed, the proposed enrichment process only accepts those auxiliary points having a small gradient locality measure, which significantly improves the efficiency of the method in practice. In theory, our method keeps iterations where the objective function is differentiable, and consequently, it works only with the gradient vectors of the objective function. In contrast with existing GS methods, the radius of the sampling region is not monotone. More precisely, by proposing a nonmonotone proximity parameter based on the radius of the sampling region, we add some valuable features of the trust region philosophy to our algorithm. The convergence analysis of the proposed method is comprehensively studied using exact and inexact gradients. By means of various academic and semi-academic test problems, we demonstrate the reliability and efficiency of the proposed method in practice.1

Abstract The purpose of this paper is to introduce a new numerical method to solve multi-marginal optimal transport problems with pairwise interaction costs. The complexity of multi-marginal optimal transport generally scales exponentially in the number of marginals $m$. We introduce a one-parameter family of cost functions that interpolates between the original and a special cost function for which the problem’s complexity scales linearly in $m$. We then show that the solution to the original problem can be recovered by solving an ordinary differential equation in the parameter $\varepsilon $, whose initial condition corresponds to the solution for the special cost function mentioned above; we then present some simulations, using both explicit Euler and explicit higher order Runge–Kutta schemes to compute solutions to the ordinary differential equation, and, as a result, the multi-marginal optimal transport problem.

Abstract Vandermonde matrices are usually exponentially ill-conditioned and often result in unstable approximations. In this paper we introduce and analyse the multivariate Vandermonde with Arnoldi (V+A) method, which is based on least-squares approximation together with a Stieltjes orthogonalization process, for approximating continuous, multivariate functions on $d$-dimensional irregular domains. The V+A method addresses the ill-conditioning of the Vandermonde approximation by creating a set of discrete orthogonal bases with respect to a discrete measure. The V+A method is simple and general, relying only on the domain’s sample points. This paper analyses the sample complexity of the least-squares approximation that uses the V+A method. We show that, for a large class of domains, this approximation gives a well-conditioned and near-optimal $N$-dimensional least-squares approximation using $M={\cal O}(N^{2})$ equispaced sample points or $M={\cal O}(N^{2}\log N)$ random sample points, independently of $d$. We provide a comprehensive analysis of the error estimates and the rate of convergence of the least-squares approximation that uses the V+A method. Based on the multivariate V+A techniques we propose a new variant of the weighted V+A least-squares algorithm that uses only $M={\cal O}(N\log N)$ sample points to achieve a near-optimal approximation. Our initial numerical results validate that the V+A least-squares approximation method provides well-conditioned and near-optimal approximations for multivariate functions on (irregular) domains. Additionally, the (weighted) least-squares approximation that uses the V+A method performs competitively with state-of-the-art orthogonalization techniques and can serve as a practical tool for selecting near-optimal distributions of sample points in irregular domains.