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This paper studies the convergence of a spatial semidiscretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. For non-smooth initial values, the regularity of the mild solution is investigated, and an error estimate is derived with the spatial $ L^2 $-norm. For smooth initial values, two error estimates with the general spatial $ L^q $-norms are established.

This paper aims first to perform robust continuous analysis of a mixed nonlinear formulation for stress-assisted diffusion of a solute that interacts with an elastic material, and second to propose and analyse a virtual element formulation of the model problem. The two-way coupling mechanisms between the Herrmann formulation for linear elasticity and the reaction-diffusion equation (written in mixed form) consist of diffusion-induced active stress and stress-dependent diffusion. The two sub-problems are analysed using the extended Babu\v{s}ka--Brezzi--Braess theory for perturbed saddle-point problems. The well-posedness of the nonlinearly coupled system is established using a Banach fixed-point strategy under the smallness assumption on data. The virtual element formulations for the uncoupled sub-problems are proven uniquely solvable by a fixed-point argument in conjunction with appropriate projection operators. We derive the a priori error estimates, and test the accuracy and performance of the proposed method through computational simulations.

We develop a novel and efficient discontinuous Galerkin spectral element method (DG-SEM) for the spherical rotating shallow water equations in vector invariant form. We prove that the DG-SEM is energy stable, and discretely conserves mass, vorticity, and linear geostrophic balance on general curvlinear meshes. These theoretical results are possible due to our novel entropy stable numerical DG fluxes for the shallow water equations in vector invariant form. We experimentally verify these results on a cubed sphere mesh. Additionally, we show that our method is robust, that is can be run stably without any dissipation. The entropy stable fluxes are sufficient to control the grid scale noise generated by geostrophic turbulence without the need for artificial stabilisation.

In this survey, we address the worst-case, average-case, and generic-case time complexity of the word problem and some other algorithmic problems in several classes of groups and show that it is often the case that the average-case complexity of the word problem is linear with respect to the length of an input word, which is as good as it gets if one considers groups given by generators and defining relations. At the same time, there are other natural algorithmic problems, for instance, the geodesic (decision) problem or Whitehead's automorphism problem, where the average-case time complexity can be sublinear, even constant.

In this article, we consider the problems of finding in $d+1$ dimensions a minimum-volume axis-parallel box, a minimum-volume arbitrarily-oriented box and a minimum-volume convex body into which a given set of $d$-dimensional unit-radius balls can be packed under translations. The computational problem is neither known to be NP-hard nor to be in NP. We give a constant-factor approximation algorithm for each of these containers based on a reduction to finding a shortest Hamiltonian path in a weighted graph, which in turn models the problem of stabbing the centers of the input balls while keeping them disjoint. We also show that for $n$ such balls, a container of volume $O(n^{\frac{d-1}{d}})$ is always sufficient and sometimes necessary. As a byproduct, this implies that for $d \geq 2$ there is no finite size $(d+1)$-dimensional convex body into which all $d$-dimensional unit-radius balls can be packed simultaneously.

In this paper, we further investigate the problem of selecting a set of design points for universal kriging, which is a widely used technique for spatial data analysis. Our goal is to select the design points in order to make simultaneous predictions of the random variable of interest at a finite number of unsampled locations with maximum precision. Specifically, we consider as response a correlated random field given by a linear model with an unknown parameter vector and a spatial error correlation structure. We propose a new design criterion that aims at simultaneously minimizing the variation of the prediction errors at various points. We also present various efficient techniques for incrementally building designs for that criterion scaling well for high dimensions. Thus the method is particularly suitable for big data applications in areas of spatial data analysis such as mining, hydrogeology, natural resource monitoring, and environmental sciences or equivalently for any computer simulation experiments. We have demonstrated the effectiveness of the proposed designs through two illustrative examples: one by simulation and another based on real data from Upper Austria.

High-frequency issues have been remarkably challenges in numerical methods for partial differential equations. In this paper, a learning based numerical method (LbNM) is proposed for Helmholtz equation with high frequency. The main novelty is using Tikhonov regularization method to stably learn the solution operator by utilizing relevant information especially the fundamental solutions. Then applying the solution operator to a new boundary input could quickly update the solution. Based on the method of fundamental solutions and the quantitative Runge approximation, we give the error estimate. This indicates interpretability and generalizability of the present method. Numerical results validates the error analysis and demonstrates the high-precision and high-efficiency features.

Open sets are central to mathematics, especially analysis and topology, in ways few notions are. In most, if not all, computational approaches to mathematics, open sets are only studied indirectly via their 'codes' or 'representations'. In this paper, we study how hard it is to compute, given an arbitrary open set of reals, the most common representation, i.e. a countable set of open intervals. We work in Kleene's higher-order computability theory, which was historically based on the S1-S9 schemes and which now has an intuitive lambda calculus formulation due to the authors. We establish many computational equivalences between on one hand the 'structure' functional that converts open sets to the aforementioned representation, and on the other hand functionals arising from mainstream mathematics, like basic properties of semi-continuous functions, the Urysohn lemma, and the Tietze extension theorem. We also compare these functionals to known operations on regulated and bounded variation functions, and the Lebesgue measure restricted to closed sets. We obtain a number of natural computational equivalences for the latter involving theorems from mainstream mathematics.

In this paper, we propose a non-parametric score to evaluate the quality of the solution to an iterative algorithm for Independent Component Analysis (ICA) with arbitrary Gaussian noise. The novelty of this score stems from the fact that it just assumes a finite second moment of the data and uses the characteristic function to evaluate the quality of the estimated mixing matrix without any knowledge of the parameters of the noise distribution. We also provide a new characteristic function-based contrast function for ICA and propose a fixed point iteration to optimize the corresponding objective function. Finally, we propose a theoretical framework to obtain sufficient conditions for the local and global optima of a family of contrast functions for ICA. This framework uses quasi-orthogonalization inherently, and our results extend the classical analysis of cumulant-based objective functions to noisy ICA. We demonstrate the efficacy of our algorithms via experimental results on simulated datasets.