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In this article, we study the semi discrete and fully discrete formulations for a Kirchhoff type quasilinear integro-differential equation involving time-fractional derivative of order $\alpha \in (0,1) $. For the semi discrete formulation of the equation under consideration, we discretize the space domain using a conforming FEM and keep the time variable continuous. We modify the standard Ritz-Volterra projection operator to carry out error analysis for the semi discrete formulation of the considered equation. In general, solutions of the time-fractional partial differential equations (PDEs) have a weak singularity near time $t=0$. Taking this singularity into account, we develop a new linearized fully discrete numerical scheme for the considered equation on a graded mesh in time. We derive a priori bounds on the solution of this fully discrete numerical scheme using a new weighted $H^{1}(\Omega)$ norm. We prove that the developed numerical scheme has an accuracy rate of $O(P^{-1}+N^{-(2-\alpha)})$ in $L^{\infty}(0,T;L^{2}(\Omega))$ as well as in $L^{\infty}(0,T;H^{1}_{0}(\Omega))$, where $P$ and $N$ are degrees of freedom in the space and time directions respectively. The robustness and efficiency of the proposed numerical scheme are demonstrated by some numerical examples.

An infinite set is orbit-finite if, up to permutations of the underlying structure of atoms, it has only finitely many elements. We study a generalisation of linear programming where constraints are expressed by an orbit-finite system of linear inequalities. As our principal contribution we provide a decision procedure for checking if such a system has a real solution, and for computing the minimal/maximal value of a linear objective function over the solution set. We also show undecidability of these problems in case when only integer solutions are considered. Therefore orbit-finite linear programming is decidable, while orbit-finite integer linear programming is not.

This paper presents an accelerated proximal gradient method for multiobjective optimization, in which each objective function is the sum of a continuously differentiable, convex function and a closed, proper, convex function. Extending first-order methods for multiobjective problems without scalarization has been widely studied, but providing accelerated methods with accurate proofs of convergence rates remains an open problem. Our proposed method is a multiobjective generalization of the accelerated proximal gradient method, also known as the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA), for scalar optimization. The key to this successful extension is solving a subproblem with terms exclusive to the multiobjective case. This approach allows us to demonstrate the global convergence rate of the proposed method ($O(1 / k^2)$), using a merit function to measure the complexity. Furthermore, we present an efficient way to solve the subproblem via its dual representation, and we confirm the validity of the proposed method through some numerical experiments.

Wikipedia is one of the most successful collaborative projects in history. It is the largest encyclopedia ever created, with millions of users worldwide relying on it as the first source of information as well as for fact-checking and in-depth research. As Wikipedia relies solely on the efforts of its volunteer-editors, its success might be particularly affected by toxic speech. In this paper, we analyze all 57 million comments made on user talk pages of 8.5 million editors across the six most active language editions of Wikipedia to study the potential impact of toxicity on editors' behaviour. We find that toxic comments consistently reduce the activity of editors, leading to an estimated loss of 0.5-2 active days per user in the short term. This amounts to multiple human-years of lost productivity when considering the number of active contributors to Wikipedia. The effects of toxic comments are even greater in the long term, as they significantly increase the risk of editors leaving the project altogether. Using an agent-based model, we demonstrate that toxicity attacks on Wikipedia have the potential to impede the progress of the entire project. Our results underscore the importance of mitigating toxic speech on collaborative platforms such as Wikipedia to ensure their continued success.

We consider the computations of the action ground state for a rotating nonlinear Schr\"odinger equation. It reads as a minimization of the action functional under the Nehari constraint. In the focusing case, we identify an equivalent formulation of the problem which simplifies the constraint. Based on it, we propose a normalized gradient flow method with asymptotic Lagrange multiplier and establish the energy-decaying property. Popular optimization methods are also applied to gain more efficiency. In the defocusing case, we prove that the ground state can be obtained by the unconstrained minimization. Then the direct gradient flow method and unconstrained optimization methods are applied. Numerical experiments show the convergence and accuracy of the proposed methods in both cases, and comparisons on the efficiency are discussed. Finally, the relation between the action and the energy ground states are numerically investigated.

This paper considers the Westervelt equation, one of the most widely used models in nonlinear acoustics, and seeks to recover two spatially-dependent parameters of physical importance from time-trace boundary measurements. Specifically, these are the nonlinearity parameter $\kappa(x)$ often referred to as $B/A$ in the acoustics literature and the wave speed $c_0(x)$. The determination of the spatial change in these quantities can be used as a means of imaging. We consider identifiability from one or two boundary measurements as relevant in these applications. For a reformulation of the problem in terms of the squared slowness $\mathfrak{s}=1/c_0^2$ and the combined coefficient $\eta=\frac{B/A+2}{\varrho_0 c_0^4}$ we devise a frozen Newton method and prove its convergence. The effectiveness (and limitations) of this iterative scheme are demonstrated by numerical examples.

Variational autoencoders (VAEs) are one of the deep generative models that have experienced enormous success over the past decades. However, in practice, they suffer from a problem called posterior collapse, which occurs when the encoder coincides, or collapses, with the prior taking no information from the latent structure of the input data into consideration. In this work, we introduce an inverse Lipschitz neural network into the decoder and, based on this architecture, provide a new method that can control in a simple and clear manner the degree of posterior collapse for a wide range of VAE models equipped with a concrete theoretical guarantee. We also illustrate the effectiveness of our method through several numerical experiments.

Gaussian process regression underpins countless academic and industrial applications of machine learning and statistics, with maximum likelihood estimation routinely used to select appropriate parameters for the covariance kernel. However, it remains an open problem to establish the circumstances in which maximum likelihood estimation is well-posed, that is, when the predictions of the regression model are insensitive to small perturbations of the data. This article identifies scenarios where the maximum likelihood estimator fails to be well-posed, in that the predictive distributions are not Lipschitz in the data with respect to the Hellinger distance. These failure cases occur in the noiseless data setting, for any Gaussian process with a stationary covariance function whose lengthscale parameter is estimated using maximum likelihood. Although the failure of maximum likelihood estimation is part of Gaussian process folklore, these rigorous theoretical results appear to be the first of their kind. The implication of these negative results is that well-posedness may need to be assessed post-hoc, on a case-by-case basis, when maximum likelihood estimation is used to train a Gaussian process model.

In a seminal paper, Kannan and Lov\'asz (1988) considered a quantity $\mu_{KL}(\Lambda,K)$ which denotes the best volume-based lower bound on the covering radius $\mu(\Lambda,K)$ of a convex body $K$ with respect to a lattice $\Lambda$. Kannan and Lov\'asz proved that $\mu(\Lambda,K) \leq n \cdot \mu_{KL}(\Lambda,K)$ and the Subspace Flatness Conjecture by Dadush (2012) claims a $O(\log n)$ factor suffices, which would match the lower bound from the work of Kannan and Lov\'asz. We settle this conjecture up to a constant in the exponent by proving that $\mu(\Lambda,K) \leq O(\log^{3}(n)) \cdot \mu_{KL} (\Lambda,K)$. Our proof is based on the Reverse Minkowski Theorem due to Regev and Stephens-Davidowitz (2017). Following the work of Dadush (2012, 2019), we obtain a $(\log n)^{O(n)}$-time randomized algorithm to solve integer programs in $n$ variables. Another implication of our main result is a near-optimal flatness constant of $O(n \log^{4}(n))$.

Optimizing static risk-averse objectives in Markov decision processes is challenging because they do not readily admit dynamic programming decompositions. Prior work has proposed to use a dynamic decomposition of risk measures that help to formulate dynamic programs on an augmented state space. This paper shows that several existing decompositions are inherently inexact, contradicting several claims in the literature. In particular, we give examples that show that popular decompositions for CVaR and EVaR risk measures are strict overestimates of the true risk values. However, an exact decomposition is possible for VaR, and we give a simple proof that illustrates the fundamental difference between VaR and CVaR dynamic programming properties.