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This paper presents a novel generic asymptotic expansion formula of expectations of multidimensional Wiener functionals through a Malliavin calculus technique. The uniform estimate of the asymptotic expansion is shown under a weaker condition on the Malliavin covariance matrix of the target Wiener functional. In particular, the method provides a tractable expansion for the expectation of an irregular functional of the solution to a multidimensional rough differential equation driven by fractional Brownian motion with Hurst index $H<1/2$, without using complicated fractional integral calculus for the singular kernel. In a numerical experiment, our expansion shows a much better approximation for a probability distribution function than its normal approximation, which demonstrates the validity of the proposed method.

This paper focuses on investigating Stein's invariant shrinkage estimators for large sample covariance matrices and precision matrices in high-dimensional settings. We consider models that have nearly arbitrary population covariance matrices, including those with potential spikes. By imposing mild technical assumptions, we establish the asymptotic limits of the shrinkers for a wide range of loss functions. A key contribution of this work, enabling the derivation of the limits of the shrinkers, is a novel result concerning the asymptotic distributions of the non-spiked eigenvectors of the sample covariance matrices, which can be of independent interest.

In this paper, we first investigate the global existence of a solution for the stochastic fractional nonlinear Schr\"odinger equation with radially symmetric initial data in a suitable energy space $H^{\alpha}$. We then show that the stochastic fractional nonlinear Schr\"odinger equation in the Stratonovich sense forms an infinite-dimensional stochastic Hamiltonian system, with its phase flow preserving symplecticity. Finally, we develop a stochastic midpoint scheme for the stochastic fractional nonlinear Schr\"odinger equation from the perspective of symplectic geometry. It is proved that the stochastic midpoint scheme satisfies the corresponding symplectic law in the discrete sense. A numerical example is conducted to validate the efficiency of the theory.

In the present contribution we propose a novel conforming Finite Element scheme for the time-dependent Navier-Stokes equation, which is proven to be both convection quasi-robust and pressure robust. The method is built combining a "divergence-free" velocity/pressure couple (such as the Scott-Vogelius element), a Discontinuous Galerkin in time approximation, and a suitable SUPG-curl stabilization. A set of numerical tests, in accordance with the theoretical results, is included.

In this work is considered a spectral problem, involving a second order term on the domain boundary: the Laplace-Beltrami operator. A variational formulation is presented, leading to a finite element discretization. For the Laplace-Beltrami operator to make sense on the boundary, the domain is smooth: consequently the computational domain (classically a polygonal domain) will not match the physical one. Thus, the physical domain is discretized using high order curved meshes so as to reduce the \textit{geometric error}. The \textit{lift operator}, which is aimed to transform a function defined on the mesh domain into a function defined on the physical one, is recalled. This \textit{lift} is a key ingredient in estimating errors on eigenvalues and eigenfunctions. A bootstrap method is used to prove the error estimates, which are expressed both in terms of \textit{finite element approximation error} and of \textit{geometric error}, respectively associated to the finite element degree $k\ge 1$ and to the mesh order~$r\ge 1$. Numerical experiments are led on various smooth domains in 2D and 3D, which allow us to validate the presented theoretical results.

In this paper, we propose a computationally valid and theoretically justified methods, the likelihood ratio scan method (LRSM), for estimating multiple change-points in a piecewise stationary generalized conditional integer-valued autoregressive process. LRSM with the usual window parameter $h$ is more satisfied to be used in long-time series with few and even change-points vs. LRSM with the multiple window parameter $h_{mix}$ performs well in short-time series with large and dense change-points. The computational complexity of LRSM can be efficiently performed with order $O((\log n)^3 n)$. Moreover, two bootstrap procedures, namely parametric and block bootstrap, are developed for constructing confidence intervals (CIs) for each of the change-points. Simulation experiments and real data analysis show that the LRSM and bootstrap procedures have excellent performance and are consistent with the theoretical analysis.

In this paper, we plan to show an eigenvalue algorithm for block Hessenberg matrices by using the idea of non-commutative integrable systems and matrix-valued orthogonal polynomials. We introduce adjacent families of matrix-valued $\theta$-deformed bi-orthogonal polynomials, and derive corresponding discrete non-commutative hungry Toda lattice from discrete spectral transformations for polynomials. It is shown that this discrete system can be used as a pre-precessing algorithm for block Hessenberg matrices. Besides, some convergence analysis and numerical examples of this algorithm are presented.

In this study, our main objective is to address the challenge of solving elliptic equations with quasiperiodic coefficients. To achieve accurate and efficient computation, we introduce the projection method, which enables the embedding of quasiperiodic systems into higher-dimensional periodic systems. To enhance the computational efficiency, we propose a compressed storage strategy for the stiffness matrix by its multi-level block circulant structure, reducing memory requirements while preserving accuracy. Furthermore, we design a diagonal preconditioner to efficiently solve the resulting high-dimensional linear system by reducing the condition number of the stiffness matrix. These techniques collectively contribute to the computational effectiveness of our proposed approach. We demonstrate the effectiveness and accuracy of our approach through a series of numerical examples. Moreover, we apply our method to achieve a highly accurate computation of the homogenized coefficients for a quasiperiodic multiscale elliptic equation.

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.

In this paper, by using $|x|=2\max\{0,x\}-x$, a class of maximum-based iteration methods is established to solve the generalized absolute value equation $Ax-B|x|=b$. Some convergence conditions of the proposed method are presented. By some numerical experiments, the effectiveness and feasibility of the proposed method are confirmed.