• Abstract

In this paper, by introducing a definition of parameterized comparison matrix of a given complex square matrix, the solvability of a parameterized class of complex nonsymmetric algebraic Riccati equations (NAREs) is discussed. The existence and uniqueness of the extremal solutions of the NAREs is proved. Some classical numerical methods can be applied to compute the extremal solutions of the NAREs, mainly including the Schur method, the basic fixed-point iterative methods, Newton's method and the doubling algorithms. Furthermore, the linear convergence of the basic fixed-point iterative methods and the quadratic convergence of Newton's method and the doubling algorithms are also shown. Moreover, some concrete parameter selection strategies in complex number field for the doubling algorithms are also given. Numerical experiments demonstrate that our numerical methods are effective.

• Abstract

In this paper, we consider the generalized singular value decompositions for two tensors via the T-product. We investigate and discuss in detail the structures of the quotient singular value decomposition (T-QSVD) and product singular value decomposition (T-PSVD) for two tensors. The algorithms are presented with numerical examples illustrating the results. For applications, we consider color image watermarking processing with T-QSVD and T-PSVD. There are two advantages to T-QSVD and T-PSVD approaches on color watermark processing: two color watermarks can be processed simultaneously and only one key needs to be saved.

• Abstract

This paper presents error estimates in both an energy norm and the $L^2$- norm for the weak Galerkin (WG) finite element methods for elliptic problems with low regularity solutions. The error analysis for the continuous Galerkin finite element remains same regardless of regularity. A totally different analysis is needed for discontinuous finite element methods if the elliptic regularity is lower than H-1.5. Numerical results confirm the theoretical analysis.

• Abstract

An abstract framework of numerical method is devised for solving an elliptic hemivariational inequality with convex constraint. Convergence of the method is explored under the minimal solution regularity available from the well-posedness of the hemivariational inequality. A Céa-type inequality is derived for error estimation. As a typical example, a virtual element method is proposed to solve a frictionless unilateral contact problem and its optimal error estimates are obtained as well. Numerical results are reported to show the performance of the proposed method.

• Abstract

The variants of randomized Kaczmarz and randomized Gauss-Seidel algorithms are two effective stochastic iterative methods for solving ridge regression problems. For solving ordinary least squares regression problems, the greedy randomized Gauss-Seidel (GRGS) algorithm always performs better than the randomized Gauss-Seidel algorithm (RGS) when the system is overdetermined. In this paper, inspired by the greedy modification technique of the GRGS algorithm, we extend the variant of the randomized Gauss-Seidel algorithm, obtaining a variant of greedy randomized Gauss-Seidel (VGRGS) algorithm for solving ridge regression problems. In addition, we propose a relaxed VGRGS algorithm and the corresponding convergence theorem is established. Numerical experiments show that our algorithms outperform the VRK-type and the VRGS algorithms when $m>n$.

• Abstract

In this paper, we establish a novel fractional model arising in the chemical reaction and develop an efficient spectral method for the three-dimensional Riesz-like space fractional nonlinear coupled reaction-diffusion equations. Based on the backward difference method for time stepping and the Legendre-Galerkin spectral method for space discretization, we construct a fully discrete numerical scheme which leads to a linear algebraic system. Then a direct method based on the matrix diagonalization approach is proposed to solve the linear algebraic system, where the cost of the algorithm is of a small multiple of $N^4$ ( $N$ is the polynomial degree in each spatial coordinate) flops for each time level. In addition, the stability and convergence analysis are rigorously established. We obtain the optimal error estimate in space, and the results also show that the fully discrete scheme is unconditionally stable and convergent of order one in time. Furthermore, numerical experiments are presented to confirm the theoretical claims. As the applications of the proposed method, the fractional Gray-Scott model is solved to capture the pattern formation with an analysis of the properties of the fractional powers.

• Abstract

To model a non-stationary signal as a superposition of amplitude and frequency-modulated Fourier-like oscillatory modes is important to extract information, such as the underlying dynamics, hidden in the signal. Recently, the synchrosqueezed wavelet transform (SST) and its variants have been developed to estimate instantaneous frequencies and separate the components of non-stationary multicomponent signals. The short-time Fourier transform-based SST (FSST for short) reassigns the frequency variable to sharpen the time-frequency representation and to separate the components of a multicomponent non-stationary signal. However FSST works well only with multicomponent signals having slowly changing frequencies. To deal with multicomponent signals having fast-changing frequencies, the secondorder FSST (FSST2 for short) was proposed. The key point for FSST2 is to construct a phase transformation of a signal which is the instantaneous frequency when the signal is a linear chirp. In this paper we consider a phase transformation for FSST2 which has a simple expression than that used in the literature. In the study the theoretical analysis of FSST2 with this phase transformation, we observe that the proof for the error bounds for the instantaneous frequency estimation and component recovery is simpler than that with the conventional phase transformation. We also provide some experimental results which show that this FSST2 performs well in non-stationary multicomponent signal separation.

• Abstract

We present a simple and fast explicit hybrid numerical scheme for the motion by mean curvature on curved surfaces in three-dimensional (3D) space. We numerically solve the Allen-Cahn (AC) and conservative Allen-Cahn (CAC) equations on a triangular surface mesh. We use the operator splitting method and an explicit hybrid numerical method. For the AC equation, we solve the diffusion term using a discrete Laplace-Beltrami operator on the triangular surface mesh and solve the reaction term using the closed-form solution, which is obtained using the separation of variables. Next, for the CAC equation, we additionally solve the time-space dependent Lagrange multiplier using an explicit scheme. Our numerical scheme is computationally fast and efficient because we use an explicit hybrid numerical scheme. We perform various numerical experiments to demonstrate the robustness and efficiency of the proposed scheme.

• Abstract

A new fast algorithm based on the augmented immersed interface method and a fast Poisson solver is proposed to solve three dimensional elliptic interface problems with a piecewise constant but discontinuous coefficient. In the new approach, an augmented variable along the interface, often the jump in the normal derivative along the interface is introduced so that a fast Poisson solver can be utilized. Thus, the solution of the Poisson equation depends on the augmented variable which should be chosen such that the original flux jump condition is satisfied. The discretization of the flux jump condition is done by a weighted least squares interpolation using the solution at the grid points, the jump conditions, and the governing PDEs in a neighborhood of control points on the interface. The interpolation scheme is the key to the success of the augmented IIM particularly. In this paper, the key new idea is to select interpolation points along the normal direction in line with the flux jump condition. Numerical experiments show that the method maintains second order accuracy of the solution and can reduce the CPU time by 20-50%. The number of the GMRES iterations is independent of the mesh size.

• Abstract

We present a new algorithm for generating layer-adapted meshes for the finite element solution of singularly perturbed problems based on mesh partial differential equations (MPDEs). The ultimate goal is to design meshes that are similar to the well-known Bakhvalov meshes, but can be used in more general settings: specifically two-dimensional problems for which the optimal mesh is not tensorproduct in nature. Our focus is on the efficient implementation of these algorithms, and numerical verification of their properties in a variety of settings. The MPDE is a nonlinear problem, and the efficiency with which it can be solved depends adversely on the magnitude of the perturbation parameter and the number of mesh intervals. We resolve this by proposing a scheme based on $h$-refinement. We present fully working FEniCS codes [Alnaes $et$ $al$., Arch. Numer. Softw., 3 (100) (2015)] that implement these methods, facilitating their extension to other problems and settings.

• Abstract

The p-step backward difference formula (BDF) for solving systems of ODEs can be formulated as all-at-once linear systems that are solved by parallelin-time preconditioned Krylov subspace solvers (see McDonald $et$ $al$. [36] and Lin and Ng [32]). However, when the BDF$p$ ($2 ≤ p ≤ 6$) method is used to solve timedependent PDEs, the generalization of these studies is not straightforward as $p$-step BDF is not selfstarting for $p ≥ 2$. In this note, we focus on the 2-step BDF which is often superior to the trapezoidal rule for solving the Riesz fractional diffusion equations, and show that it results into an all-at-once discretized system that is a low-rank perturbation of a block triangular Toeplitz system. We first give an estimation of the condition number of the all-at-once systems and then, capitalizing on previous work, we propose two block circulant (BC) preconditioners. Both the invertibility of these two BC preconditioners and the eigenvalue distributions of preconditioned matrices are discussed in details. An efficient implementation of these BC preconditioners is also presented, including the fast computation of dense structured Jacobi matrices. Finally, numerical experiments involving both the one- and two-dimensional Riesz fractional diffusion equations are reported to support our theoretical findings.