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We propose a method to numerically compute fractional derivatives (or the fractional Laplacian) on the whole real line via Riesz fractional integrals. The compactified real line is divided into a number of intervals, thus amounting to a multi-domain approach; after transformations in accordance with the underlying $Z_{q}$ curve ensuring analyticity of the respective integrands, the integrals over the different domains are computed with a Clenshaw-Curtis algorithm. As an example, we consider solitary waves for fractional Korteweg-de Vries equations and compare these to results obtained with a discrete Fourier transform.

The present work provides a comprehensive study of symmetric-conjugate operator splitting methods in the context of linear parabolic problems and demonstrates their additional benefits compared to symmetric splitting methods. Relevant applications include nonreversible systems and ground state computations for linear Schr\"odinger equations based on the imaginary time propagation. Numerical examples confirm the favourable error behaviour of higher-order symmetric-conjugate splitting methods and illustrate the usefulness of a time stepsize control, where the local error estimation relies on the computation of the imaginary parts and thus requires negligible costs.

In this paper, we are concerned about the lattice Boltzmann methods (LBMs) based on vector-kinetic models for hyperbolic partial differential equations. In addition to usual lattice Boltzmann equation (LBE) derived by explicit discretisation of vector-kinetic equation (VKE), we also consider LBE derived by semi-implicit discretisation of VKE and compare the relaxation factors of both. We study the properties such as H-inequality, total variation boundedness and positivity of both the LBEs, and infer that the LBE due to semi-implicit discretisation naturally satisfies all the properties while the LBE due to explicit discretisation requires more restrictive condition on relaxation factor compared to the usual condition obtained from Chapman-Enskog expansion. We also derive the macroscopic finite difference form of the LBEs, and utilise it to establish the consistency of LBEs with the hyperbolic system. Further, we extend this LBM framework to hyperbolic conservation laws with source terms, such that there is no spurious numerical convection due to imbalance between convection and source terms. We also present a D$2$Q$9$ model that allows upwinding even along diagonal directions in addition to the usual upwinding along coordinate directions. The different aspects of the results are validated numerically on standard benchmark problems.

The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices $A_n$ arising from numerical discretizations of differential equations. Indeed, when the mesh fineness parameter $n$ tends to infinity, these matrices $A_n$ give rise to a sequence $\{A_n\}_n$, which often turns out to be a GLT sequence. In this paper, we extend the theory of GLT sequences in several directions: we show that every GLT sequence enjoys a normal form, we identify the spectral symbol of every GLT sequence formed by normal matrices, and we prove that, for every GLT sequence $\{A_n\}_n$ formed by normal matrices and every continuous function $f:\mathbb C\to\mathbb C$, the sequence $\{f(A_n)\}_n$ is again a GLT sequence whose spectral symbol is $f(\kappa)$, where $\kappa$ is the spectral symbol of $\{A_n\}_n$. In addition, using the theory of GLT sequences, we prove a spectral distribution result for perturbed normal matrices.

One tuple of probability vectors is more informative than another tuple when there exists a single stochastic matrix transforming the probability vectors of the first tuple into the probability vectors of the other. This is called matrix majorization. Solving an open problem raised by Mu et al, we show that if certain monotones - namely multivariate extensions of R\'{e}nyi divergences - are strictly ordered between the two tuples, then for sufficiently large $n$, there exists a stochastic matrix taking the $n$-fold Kronecker power of each input distribution to the $n$-fold Kronecker power of the corresponding output distribution. The same conditions, with non-strict ordering for the monotones, are also necessary for such matrix majorization in large samples. Our result also gives conditions for the existence of a sequence of statistical maps that asymptotically (with vanishing error) convert a single copy of each input distribution to the corresponding output distribution with the help of a catalyst that is returned unchanged. Allowing for transformation with arbitrarily small error, we find conditions that are both necessary and sufficient for such catalytic matrix majorization. We derive our results by building on a general algebraic theory of preordered semirings recently developed by one of the authors. This also allows us to recover various existing results on majorization in large samples and in the catalytic regime as well as relative majorization in a unified manner.

Two-sample spiked model is an important issue in multivariate statistical inference. This paper focuses on testing the number of spikes in a high-dimensional generalized two-sample spiked model, which is free of Gaussian population assumption and the diagonal or block-wise diagonal restriction of population covariance matrix, and the spiked eigenvalues are not necessary required to be bounded. In order to determine the number of spikes, we first propose a general test, which relies on the partial linear spectral statistics. We establish its asymptotic normality under the null hypothesis. Then we apply the conclusion to two statistical problem, variable selection in large-dimensional linear regression and change point detection when change points and additive outliers exist simultaneously. Simulations and empirical analysis are conducted to illustrate the good performance of our methods.

It is known that standard stochastic Galerkin methods encounter challenges when solving partial differential equations with high-dimensional random inputs, which are typically caused by the large number of stochastic basis functions required. It becomes crucial to properly choose effective basis functions, such that the dimension of the stochastic approximation space can be reduced. In this work, we focus on the stochastic Galerkin approximation associated with generalized polynomial chaos (gPC), and explore the gPC expansion based on the analysis of variance (ANOVA) decomposition. A concise form of the gPC expansion is presented for each component function of the ANOVA expansion, and an adaptive ANOVA procedure is proposed to construct the overall stochastic Galerkin system. Numerical results demonstrate the efficiency of our proposed adaptive ANOVA stochastic Galerkin method for both diffusion and Helmholtz problems.

We establish optimal error bounds on time-splitting methods for the nonlinear Schr\"odinger equation with low regularity potential and typical power-type nonlinearity $ f(\rho) = \rho^\sigma $, where $ \rho:=|\psi|^2 $ is the density with $ \psi $ the wave function and $ \sigma > 0 $ the exponent of the nonlinearity. For the first-order Lie-Trotter time-splitting method, optimal $ L^2 $-norm error bound is proved for $L^\infty$-potential and $ \sigma > 0 $, and optimal $H^1$-norm error bound is obtained for $ W^{1, 4} $-potential and $ \sigma \geq 1/2 $. For the second-order Strang time-splitting method, optimal $ L^2 $-norm error bound is established for $H^2$-potential and $ \sigma \geq 1 $, and optimal $H^1$-norm error bound is proved for $H^3$-potential and $ \sigma \geq 3/2 $ (or $\sigma = 1$). Compared to those error estimates of time-splitting methods in the literature, our optimal error bounds either improve the convergence rates under the same regularity assumptions or significantly relax the regularity requirements on potential and nonlinearity for optimal convergence orders. A key ingredient in our proof is to adopt a new technique called \textit{regularity compensation oscillation} (RCO), where low frequency modes are analyzed by phase cancellation, and high frequency modes are estimated by regularity of the solution. Extensive numerical results are reported to confirm our error estimates and to demonstrate that they are sharp.

In the present paper, we propose a block variant of the extended Hessenberg process for computing approximations of matrix functions and other problems producing large-scale matrices. Applications to the computation of a matrix function such as f(A)V, where A is an nxn large sparse matrix, V is an nxp block with p<<n, and f is a function are presented. Solving shifted linear systems with multiple right hand sides are also given. Computing approximations of these matrix problems appear in many scientific and engineering applications. Different numerical experiments are provided to show the effectiveness of the proposed method for these problems.

We introduce two iterative methods, GPBiLQ and GPQMR, for solving unsymmetric partitioned linear systems. The basic mechanism underlying GPBiLQ and GPQMR is a novel simultaneous tridiagonalization via biorthogonality that allows for short-recurrence iterative schemes. Similar to the biconjugate gradient method, it is possible to develop another method, GPBiCG, whose iterate (if it exists) can be obtained inexpensively from the GPBiLQ iterate. Whereas the iterate of GPBiCG may not exist, the iterates of GPBiLQ and GPQMR are always well defined as long as the biorthogonal tridiagonal reduction process does not break down. We discuss connections between the proposed methods and some existing methods, and give numerical experiments to illustrate the performance of the proposed methods.