Mathematics of Computation最新文献

This paper analyses conforming and nonconforming virtual element formulations of arbitrary polynomial degrees on general polygonal meshes for the coupling of solid and fluid phases in deformable porous plates. The governing equations consist of one fourth-order equation for the transverse displacement of the middle surface coupled with a second-order equation for the pressure head relative to the solid with mixed boundary conditions. We propose novel enrichment operators that connect nonconforming virtual element spaces of general degree to continuous Sobolev spaces. These operators satisfy additional orthogonal and best-approximation properties (referred to as conforming companion operators in the context of finite element methods), which play an important role in the nonconforming methods. This paper proves a priori error estimates in the best-approximation form, and derives residual–based reliable and efficient a posteriori error estimates in appropriate norms, and shows that these error bounds are robust with respect to the main model parameters. The computational examples illustrate the numerical behaviour of the suggested virtual element discretisations and confirm the theoretical findings on different polygonal meshes with mixed boundary conditions.

Quantiles are used as a measure of risk in many stochastic systems. We study the estimation of quantiles with the Hilbert space-filling curve (HSFC) sampling scheme that transforms specifically chosen one-dimensional points into high dimensional stratified samples while still remaining the extensibility. We study the convergence and asymptotic normality for the estimate based on HSFC. By a generalized Dvoretzky–Kiefer–Wolfowitz inequality for independent but not identically distributed samples, we establish the strong consistency for such an estimator. We find that under certain conditions, the distribution of the quantile estimator based on HSFC is asymptotically normal. The asymptotic variance is of O ( n − 1 − 1 / d ) O(n^{-1-1/d}) when using n n HSFC-based quadrature points in dimension d d , which is more efficient than the Monte Carlo sampling and the Latin hypercube sampling. Since the asymptotic variance does not admit an explicit form, we establish an asymptotically valid confidence interval by the batching method. We also prove a Bahadur representation for the quantile estimator based on HSFC. Numerical experiments show that the quantile estimator is asymptotically normal with a comparable mean squared error rate of randomized quasi-Monte Carlo (RQMC) sampling. Moreover, the coverage of the confidence intervals constructed with HSFC is better than that with RQMC.

The active set method aims at finding the correct active set of the optimal solution and it is a powerful method for solving strictly convex quadratic problems with bound constraints. To guarantee the finite step convergence, existing active set methods all need strict conditions or some additional strategies, which can significantly impact the efficiency of the algorithm. In this paper, we propose a random active set method that introduces randomness in the active set’s update process. We prove that the algorithm can converge in a finite number of iterations with probability one, without any extra conditions on the problem or any supplementary strategies. At last, numerical experiments show that the algorithm can obtain the correct active set within a few iterations, and it has better efficiency and robustness than the existing methods.

We establish a uniform-in-scaling error estimate for the asymptotic preserving (AP) scheme proposed by Xu and Wang [Commun. Math. Sci. 21 (2023), pp. 1–23] for the Lévy-Fokker-Planck (LFP) equation. The main difficulties stem not only from the interplay between the scaling and numerical parameters but also the slow decay of the tail of the equilibrium state. We tackle these problems by separating the parameter domain according to the relative size of the scaling parameter ε varepsilon : in the regime where ε varepsilon is large, we design a weighted norm to mitigate the issue caused by the fat tail, while in the regime where ε varepsilon is small, we prove a strong convergence of LFP towards its fractional diffusion limit with an explicit convergence rate. This method extends the traditional AP estimates to cases where uniform bounds are unavailable. Our result applies to any dimension and to the whole span of the fractional power.