In this paper, we study the low-rank matrix completion problem with Poisson observations, where only partial entries are available and the observations are in the presence of Poisson noise. We propose a novel model composed of the Kullback-Leibler (KL) divergence by using the maximum likelihood estimation of Poisson noise, and total variation (TV) and nuclear norm constraints. Here the nuclear norm and TV constraints are utilized to explore the approximate low-rankness and piecewise smoothness of the underlying matrix, respectively. The advantage of these two constraints in the proposed model is that the low-rankness and piecewise smoothness of the underlying matrix can be exploited simultaneously, and they can be regularized for many real-world image data. An upper error bound of the estimator of the proposed model is established with high probability, which is not larger than that of only TV or nuclear norm constraint. To the best of our knowledge, this is the first work to utilize both low-rank and TV constraints with theoretical error bounds for matrix completion under Poisson observations. Extensive numerical examples on both synthetic data and real-world images are reported to corroborate the superiority of the proposed approach.

In this paper, we introduce a type of tensor neural network. For the first time, we propose its numerical integration scheme and prove the computational complexity to be the polynomial scale of the dimension. Based on the tensor product structure, we develop an efficient numerical integration method by using fixed quadrature points for the functions of the tensor neural network. The corresponding machine learning method is also introduced for solving high-dimensional problems. Some numerical examples are also provided to validate the theoretical results and the numerical algorithm.

The alternating direction method of multipliers (ADMM) has been extensively investigated in the past decades for solving separable convex optimization problems, and surprisingly, it also performs efficiently for nonconvex programs. In this paper, we propose a symmetric ADMM based on acceleration techniques for a family of potentially nonsmooth and nonconvex programming problems with equality constraints, where the dual variables are updated twice with different stepsizes. Under proper assumptions instead of the socalled Kurdyka-Lojasiewicz inequality, convergence of the proposed algorithm as well as its pointwise iteration-complexity are analyzed in terms of the corresponding augmented Lagrangian function and the primal-dual residuals, respectively. Performance of our algorithm is verified by numerical examples corresponding to signal processing applications in sparse nonconvex/convex regularized minimization.

A mixed finite element method is presented for the Biot consolidation problem in poroelasticity. More precisely, the displacement is approximated by using the Crouzeix-Raviart nonconforming finite elements, while the fluid pressure is approximated by using the node conforming finite elements. The well-posedness of the fully discrete scheme is established, and a corresponding priori error estimate with optimal order in the energy norm is also derived. Numerical experiments are provided to validate the theoretical results.

Fractional Klein-Kramers equation can well describe subdiffusion in phase space. In this paper, we develop the fully discrete scheme for time-fractional Klein-Kramers equation based on the backward Euler convolution quadrature and local discontinuous Galerkin methods. Thanks to the obtained sharp regularity estimates in temporal and spatial directions after overcoming the hypocoercivity of the operator, the complete error analyses of the fully discrete scheme are built. It is worth mentioning that the convergence of the provided scheme is independent of the temporal regularity of the exact solution. Finally, numerical results are proposed to verify the correctness of the theoretical results.

Optimization problem of cardinality constrained mean-variance (CCMV) model for sparse portfolio selection is considered. To overcome the difficulties caused by cardinality constraint, an exact penalty approach is employed, then CCMV problem is transferred into a difference-of-convex-functions (DC) problem. By exploiting the DC structure of the gained problem and the superlinear convergence of semismooth Newton (ssN) method, an inexact proximal DC algorithm with sieving strategy based on a majorized ssN method (siPDCA-mssN) is proposed. For solving the inner problems of siPDCA-mssN from dual, the second-order information is wisely incorporated and an efficient mssN method is employed. The global convergence of the sequence generated by siPDCA-mssN is proved. To solve large-scale CCMV problem, a decomposed siPDCA-mssN (DsiPDCA-mssN) is introduced. To demonstrate the efficiency of proposed algorithms, siPDCA-mssN and DsiPDCA-mssN are compared with the penalty proximal alternating linearized minimization method and the CPLEX(12.9) solver by performing numerical experiments on realword market data and large-scale simulated data. The numerical results demonstrate that siPDCA-mssN and DsiPDCA-mssN outperform the other methods from computation time and optimal value. The out-of-sample experiments results display that the solutions of CCMV model are better than those of other portfolio selection models in terms of Sharp ratio and sparsity.

In this paper, we consider the strong convergence of the time-space fractional diffusion equation driven by fractional Gaussian noise with Hurst index H ∈ (1/2, 1). A sharp regularity estimate of the mild solution and the numerical scheme constructed by finite element method for integral fractional Laplacian and backward Euler convolution quadrature for Riemann-Liouville time fractional derivative are proposed. With the help of inverse Laplace transform and fractional Ritz projection, we obtain the accurate error estimates in time and space. Finally, our theoretical results are accompanied by numerical experiments.

The second-order serendipity virtual element method is studied for the semilinear pseudo-parabolic equations on curved domains in this paper. Nonhomogeneous Dirichlet boundary conditions are taken into account, the existence and uniqueness are investigated for the weak solution of the nonhomogeneous initial-boundary value problem. The Nitschebased projection method is adopted to impose the boundary conditions in a weak way. The interpolation operator is used to deal with the nonlinear term. The Crank-Nicolson scheme is employed to discretize the temporal variable. There are two main features of the proposed scheme: (i) the internal degrees of freedom are avoided no matter what type of mesh is utilized, and (ii) the Jacobian is simple to calculate when Newton’s iteration method is applied to solve the fully discrete scheme. The error estimates are established for the discrete schemes and the theoretical results are illustrated through some numerical examples.

In this paper, we analyze two classes of spectral volume (SV) methods for one-dimensional hyperbolic equations with degenerate variable coefficients. Two classes of SV methods are constructed by letting a piecewise k-th order (k ≥ 1 is an integer) polynomial to satisfy the conservation law in each control volume, which is obtained by refining spectral volumes (SV) of the underlying mesh with k Gauss-Legendre points (LSV) or Radaus points (RSV) in each SV. The L²-norm stability and optimal order convergence properties for both methods are rigorously proved for general non-uniform meshes. Surprisingly, we discover some very interesting superconvergence phenomena: At some special points, the SV flux function approximates the exact flux with (k+2)-th order and the SV solution itself approximates the exact solution with (k + 3/2)-th order, some superconvergence behaviors for element averages errors have been also discovered. Moreover, these superconvergence phenomena are rigorously proved by using the so-called correction function method. Our theoretical findings are verified by several numerical experiments.

In this paper, we investigate the theoretical and numerical analysis of the stochastic Volterra integro-differential equations (SVIDEs) driven by Lévy noise. The existence, uniqueness, boundedness and mean square exponential stability of the analytic solutions for SVIDEs driven by Lévy noise are considered. The split-step theta method of SVIDEs driven by Lévy noise is proposed. The boundedness of the numerical solution and strong convergence are proved. Moreover, its mean square exponential stability is obtained. Some numerical examples are given to support the theoretical results.

In the paper, we propose a novel linearly implicit structure-preserving algorithm, which is derived by combing the invariant energy quadratization approach with the exponential time differencing method, to construct efficient and accurate time discretization scheme for a large class of Hamiltonian partial differential equations (PDEs). The proposed scheme is a linear system, and can be solved more efficient than the original energy-preserving exponential integrator scheme which usually needs nonlinear iterations. Various experiments are performed to verify the conservation, efficiency and good performance at relatively large time step in long time computations.

In this paper, we shall prove a Wong-Zakai approximation for stochastic Volterra equations under appropriate assumptions. We may apply it to a class of stochastic differential equations with the kernel of fractional Brownian motion with Hurst parameter H ∈ (1/2, 1) and subfractional Brownian motion with Hurst parameter H ∈ (1/2, 1). As far as we know, this is the first result on stochastic Volterra equations in this topic.

In this paper, we analyze the convergence properties of a stochastic augmented Lagrangian method for solving stochastic convex programming problems with inequality constraints. Approximation models for stochastic convex programming problems are constructed from stochastic observations of real objective and constraint functions. Based on relations between solutions of the primal problem and solutions of the dual problem, it is proved that the convergence of the algorithm from the perspective of the dual problem. Without assumptions on how these random models are generated, when estimates are merely sufficiently accurate to the real objective and constraint functions with high enough, but fixed, probability, the method converges globally to the optimal solution almost surely. In addition, sufficiently accurate random models are given under different noise assumptions. We also report numerical results that show the good performance of the algorithm for different convex programming problems with several random models.

We propose a simple iterative (SI) algorithm for the maxcut problem through fully using an equivalent continuous formulation. It does not need rounding at all and has advantages that all subproblems have explicit analytic solutions, the cut values are monotonically updated and the iteration points converge to a local optima in finite steps via an appropriate subgradient selection. Numerical experiments on G-set demonstrate the performance. In particular, the ratios between the best cut values achieved by SI and those by some advanced combinatorial algorithms in [Ann. Oper. Res., 248 (2017), 365-403] are at least 0.986 and can be further improved to at least 0.997 by a preliminary attempt to break out of local optima.

We develop a stabilizer free weak Galerkin (SFWG) finite element method for Brinkman equations. The main idea is to use high order polynomials to compute the discrete weak gradient and then the stabilizing term is removed from the numerical formulation. The SFWG scheme is very simple and easy to implement on polygonal meshes. We prove the well-posedness of the scheme and derive optimal order error estimates in energy and L² norm. The error results are independent of the permeability tensor, hence the SFWG method is stable and accurate for both the Stokes and Darcy dominated problems. Finally, we present some numerical experiments to verify the efficiency and stability of the SFWG method.

In this paper two dimensional elliptic interface problem with imperfect contact is considered, which is featured by the implicit jump condition imposed on the imperfect contact interface, and the jumping quantity of the unknown is related to the flux across the interface. A finite difference method is constructed for the 2D elliptic interface problems with straight and curve interface shapes. Then, the stability and convergence analysis are given for the constructed scheme. Further, in particular case, it is proved to be monotone. Numerical examples for elliptic interface problems with straight and curve interface shapes are tested to verify the performance of the scheme. The numerical results demonstrate that it obtains approximately second-order accuracy for elliptic interface equations with implicit jump condition.

Classical quasi-Newton methods are widely used to solve nonlinear problems in which the first-order information is exact. In some practical problems, we can only obtain approximate values of the objective function and its gradient. It is necessary to design optimization algorithms that can utilize inexact first-order information. In this paper, we propose an adaptive regularized quasi-Newton method to solve such problems. Under some mild conditions, we prove the global convergence and establish the convergence rate of the adaptive regularized quasi-Newton method. Detailed implementations of our method, including the subspace technique to reduce the amount of computation, are presented. Encouraging numerical results demonstrate that the adaptive regularized quasi-Newton method is a promising method, which can utilize the inexact first-order information effectively.

The numerical integration of phase-field equations is a delicate task which needs to recover at the discrete level intrinsic properties of the solution such as energy dissipation and maximum principle. Although the theory of energy dissipation for classical phase field models is well established, the corresponding theory for time-fractional phase-field models is still incomplete. In this article, we study certain nonlocal-in-time energies using the first-order stabilized semi-implicit L1 scheme. In particular, we will establish a discrete fractional energy law and a discrete weighted energy law. The extension for a (2-α)-order L1 scalar auxiliary variable scheme will be investigated. Moreover, we demonstrate that the energy bound is preserved for the L1 schemes with nonuniform time steps. Several numerical experiments are carried to verify our theoretical analysis.

In this work, we analyze the three-step backward differentiation formula (BDF3) method for solving the Allen-Cahn equation on variable grids. For BDF2 method, the discrete orthogonal convolution (DOC) kernels are positive, the stability and convergence analysis are well established in [Liao and Zhang, Math. Comp., 90 (2021), 1207–1226] and [Chen, Yu, and Zhang, arXiv:2108.02910, 2021]. However, the numerical analysis for BDF3 method with variable steps seems to be highly nontrivial due to the additional degrees of freedom and the non-positivity of DOC kernels. By developing a novel spectral norm inequality, the unconditional stability and convergence are rigorously proved under the updated step ratio restriction r_k:= τ_k/τ_k-1 ≤ 1.405 for BDF3 method. Finally, numerical experiments are performed to illustrate the theoretical results. To the best of our knowledge, this is the first theoretical analysis of variable steps BDF3 method for the Allen-Cahn equation.

Generalized Bézier surfaces are a multi-sided generalization of classical tensor product Bézier surfaces with a simple control structure and inherit most of the appealing properties from Bézier surfaces. However, the original degree elevation changes the geometry of generalized Bézier surfaces such that it is undesirable in many applications, e.g. isogeometric analysis. In this paper, we propose an improved degree elevation algorithm for generalized Bézier surfaces preserving not only geometric consistency but also parametric consistency. Based on the knot insertion of B-splines, a novel knot insertion algorithm for generalized Bézier surfaces is also proposed. Then the proposed algorithms are employed to increase degrees of freedom for multi-sided computational domains parameterized by generalized Bézier surfaces in isogeometric analysis, corresponding to the traditional p-, h-, and k-refinements. Numerical examples demonstrate the effectiveness and superiority of our method.

We propose a new method for smoothly interpolating a given set of data points on Grassmann and Stiefel manifolds using a generalization of the De Casteljau algorithm. To that end, we reduce interpolation problem to the classical Euclidean setting, allowing us to directly leverage the extensive toolbox of spline interpolation. The interpolated curve enjoy a number of nice properties: The solution exists and is optimal in many common situations. For applications, the structures with respect to chosen Riemannian metrics are detailed resulting in additional computational advantages.

An efficient spectral-Galerkin method for eigenvalue problems of the integral fractional Laplacian on a unit ball of any dimension is proposed in this paper. The symmetric positive definite linear system is retained explicitly which plays an important role in the numerical analysis. And a sharp estimate on the algebraic system’s condition number is established which behaves as N^4s with respect to the polynomial degree N, where 2s is the fractional derivative order. The regularity estimate of solutions to source problems of the fractional Laplacian in arbitrary dimensions is firstly investigated in weighted Sobolev spaces. Then the regularity of eigenfunctions of the fractional Laplacian eigenvalue problem is readily derived. Meanwhile, rigorous error estimates of the eigenvalues and eigenvectors are obtained. Numerical experiments are presented to demonstrate the accuracy and efficiency and to validate the theoretical results.

A two-grid finite element method with L1 scheme is presented for solving two-dimensional time-fractional nonlinear Schrödinger equation. The finite element solution in the L^∞-norm are proved bounded without any time-step size conditions (dependent on spatialstep size). The classical L1 scheme is considered in the time direction, and the two-grid finite element method is applied in spatial direction. The optimal order error estimations of the two-grid solution in the L^p-norm is proved without any time-step size conditions. It is shown, both theoretically and numerically, that the coarse space can be extremely coarse, with no loss in the order of accuracy.

A fundamental problem in some applications including group testing and communications is to acquire the support of a K-sparse signal x, whose nonzero elements are 1, from an underdetermined noisy linear model. This paper first designs an algorithm called binary least squares (BLS) to reconstruct x and analyzes its complexity. Then, we establish two sufficient conditions for the exact reconstruction of x’s support with K iterations of BLS based on the mutual coherence and restricted isometry property of the measurement matrix, respectively. Finally, extensive numerical tests are performed to compare the efficiency and effectiveness of BLS with those of batch orthogonal matching pursuit (BatchOMP) which to our best knowledge is the fastest implementation of OMP, orthogonal least squares (OLS), compressive sampling matching pursuit (CoSaMP), hard thresholding pursuit (HTP), Newton-step-based iterative hard thresholding (NSIHT), Newton-step-based hard thresholding pursuit (NSHTP), binary matching pursuit (BMP) and $\ell_1$-regularized least squares. Test results show that: (1) BLS can be 10-200 times more efficient than Batch-OMP, OLS, CoSaMP, HTP, NSIHT and NSHTP with higher probability of support reconstruction, and the improvement can be 20%-80%; (2) BLS has more than 25% improvement on the support reconstruction probability than the explicit BMP algorithm with a little higher computational complexity; (3) BLS is around 100 times faster than $\ell_1$-regularized least squares with lower support reconstruction probability for small K and higher support reconstruction probability for large K. Numerical tests on the generalized space shift keying (GSSK) detection indicate that although BLS is a little slower than BMP, it is more efficient than the other seven tested sparse recovery algorithms, and although it is less effective than $\ell_1$-regularized least squares, it is more effective than the other seven algorithms.

In this paper, we consider the numerical solution of decoupled mean-field forward backward stochastic differential equations with jumps (MFBSDEJs). By using finite difference approximations and the Gaussian quadrature rule, and the weak order 2.0 Itô-Taylor scheme to solve the forward mean-field SDEs with jumps, we propose a new second order scheme for MFBSDEJs. The proposed scheme allows an easy implementation. Some numerical experiments are carried out to demonstrate the stability, the effectiveness and the second order accuracy of the scheme.

This paper considers the finite element approximation to parabolic optimal control problems with measure data in a nonconvex polygonal domain. Such problems usually possess low regularity in the state variable due to the presence of measure data and the nonconvex nature of the domain. The low regularity of the solution allows the finite element approximations to converge at lower orders. We prove the existence, uniqueness and regularity results for the solution to the control problem satisfying the first order optimality condition. For our error analysis we have used piecewise linear elements for the approximation of the state and co-state variables, whereas piecewise constant functions are employed to approximate the control variable. The temporal discretization is based on the implicit Euler scheme. We derive both a priori and a posteriori error bounds for the state, control and co-state variables. Numerical experiments are performed to validate the theoretical rates of convergence.

In this paper, we present a novel penalty model called ExPen for optimization over the Stiefel manifold. Different from existing penalty functions for orthogonality constraints, ExPen adopts a smooth penalty function without using any first-order derivative of the objective function. We show that all the first-order stationary points of ExPen with a sufficiently large penalty parameter are either feasible, namely, are the first-order stationary points of the original optimization problem, or far from the Stiefel manifold. Besides, the original problem and ExPen share the same second-order stationary points. Remarkably, the exact gradient and Hessian of ExPen are easy to compute. As a consequence, abundant algorithm resources in unconstrained optimization can be applied straightforwardly to solve ExPen.

Recently, the authors proposed a low-cost approach, named Optimization Approach for Linear Systems (OPALS) for solving any kind of a consistent linear system regarding the structure, characteristics, and dimension of the coefficient matrix A. The results obtained by this approach for matrices with no structure and with indefinite symmetric part were encouraging when compare with other recent and well-known techniques. In this work, we proposed to extend the OPALS approach for solving the Linear Least-Squares Problem (LLSP) and the Minimum Norm Linear System Problem (MNLSP) using any iterative lowcost gradient-type method, avoiding the construction of the matrices A^TA or AA^T, and taking full advantage of the structure and form of the gradient of the proposed nonlinear objective function in the gradient direction. The combination of those conditions together with the choice of the initial iterate allow us to produce a novel and efficient low-cost numerical scheme for solving both problems. Moreover, the scheme presented in this work can also be used and extended for the weighted minimum norm linear systems and minimum norm linear least-squares problems. We include encouraging numerical results to illustrate the practical behavior of the proposed schemes.

In this paper, we define arbitrarily high-order energy-conserving methods for Hamiltonian systems with quadratic holonomic constraints. The derivation of the methods is made within the so-called line integral framework. Numerical tests to illustrate the theoretical findings are presented.

Given the measurement matrix A and the observation signal y, the central purpose of compressed sensing is to find the most sparse solution of the underdetermined linear system y = Ax + z, where x is the s-sparse signal to be recovered and z is the noise vector. Zhou and Yu [Front. Appl. Math. Stat., 5 (2019), Article 14] recently proposed a novel non-convex weighted $\ell$_r - $\ell$₁ minimization method for effective sparse recovery. In this paper, under newly coherence-based conditions, we study the non-convex weighted $\ell$_r - $\ell$₁ minimization in reconstructing sparse signals that are contaminated by different noises.Concretely, the results reveal that if the coherence $\mu$ of measurement matrix $A$ fulfills $$ \mu<\kappa(s ; r, \alpha, N), \quad s>1, \quad \alpha^{\frac{1}{r}} N^{\frac{1}{2}}<1, $$ then any $s$-sparse signals in the noisy scenarios could be ensured to be reconstructed robustly by solving weighted $\ell$_r - $\ell$₁ minimization non-convex optimization problem. Furthermore, some central remarks are presented to clear that the reconstruction assurance is much weaker than the existing ones. To the best of our knowledge, this is the first mutual coherence-based sufficient condition for such approach.

In this paper, we consider the electromagnetic wave scattering problem from a periodic chiral structure. The scattering problem is simplified to a two-dimensional problem, and is discretized by a finite volume method combined with the perfectly matched layer (PML) technique. A residual-type a posteriori error estimate of the PML finite volume method is analyzed and the upper and lower bounds on the error are established in the H¹-norm. The crucial part of the a posteriori error analysis is to derive the error representation formula and use a L²-orthogonality property of the residual which plays a similar role as the Galerkin orthogonality. An adaptive PML finite volume method is proposed to solve the scattering problem. The PML parameters such as the thickness of the layer and the medium property are determined through sharp a posteriori error estimate. Finally, numerical experiments are presented to illustrate the efficiency of the proposed method.

In this paper, a robust residual-based a posteriori estimate is discussed for the Streamline Upwind/Petrov Galerkin (SUPG) virtual element method (VEM) discretization of convection dominated diffusion equation. A global upper bound and a local lower bound for the a posteriori error estimates are derived in the natural SUPG norm, where the global upper estimate relies on some hypotheses about the interpolation errors and SUPG virtual element discretization errors. Based on the Dörfler’s marking strategy, adaptive VEM algorithm drived by the error estimators is used to solve the problem on general polygonal meshes. Numerical experiments show the robustness of the a posteriori error estimates.

For solving the stochastic differential equations driven by fractional Brownian motion, we present the modified split-step theta method by combining truncated Euler-Maruyama method with split-step theta method. For the problem under a locally Lipschitz condition and a linear growth condition, we analyze the strong convergence and the exponential stability of the proposed method. Moreover, for the stochastic delay differential equations with locally Lipschitz drift condition and globally Lipschitz diffusion condition, we give the order of convergence. Finally, numerical experiments are done to confirm the theoretical conclusions.

Low-dose computed tomography (LDCT) contains the mixed noise of Poisson and Gaussian, which makes the image reconstruction a challenging task. In order to describe the statistical characteristics of the mixed noise, we adopt the sinogram preprocessing as a standard maximum a posteriori (MAP). Based on the fact that the sinogram of LDCT has nonlocal self-similarity property, it exhibits low-rank characteristics. The conventional way of solving the low-rank problem is implemented in matrix forms, and ignores the correlations among similar patch groups. To avoid this issue, we make use of a nonlocal KroneckerBasis-Representation (KBR) method to depict the low-rank problem. A new denoising model, which consists of the sinogram preprocessing for data fidelity and the nonlocal KBR term, is developed in this work. The proposed denoising model can better illustrate the generative mechanism of the mixed noise and the prior knowledge of the LDCT. Numerical results show that the proposed denoising model outperforms the state-of-the-art algorithms in terms of peak-signal-to-noise ratio (PSNR), feature similarity (FSIM), and normalized mean square error (NMSE).

Asymptotic theory for the circuit envelope analysis is developed in this paper. A typical feature of circuit envelope analysis is the existence of two significantly distinct timescales: one is the fast timescale of carrier wave, and the other is the slow timescale of modulation signal. We first perform pro forma asymptotic analysis for both the driven and autonomous systems. Then resorting to the Floquet theory of periodic operators, we make a rigorous justification for first-order asymptotic approximations. It turns out that these asymptotic results are valid at least on the slow timescale. To speed up the computation of asymptotic approximations, we propose a periodization technique, which renders the possibility of utilizing the NUFFT algorithm. Numerical experiments are presented, and the results validate the theoretical findings.

Image restoration based on total variation has been widely studied owing to its edgepreservation properties. In this study, we consider the total variation infimal convolution (TV-IC) image restoration model for eliminating mixed Poisson-Gaussian noise. Based on the alternating direction method of multipliers (ADMM), we propose a complete splitting proximal bilinear constraint ADMM algorithm to solve the TV-IC model. We prove the convergence of the proposed algorithm under mild conditions. In contrast with other algorithms used for solving the TV-IC model, the proposed algorithm does not involve any inner iterations, and each subproblem has a closed-form solution. Finally, numerical experimental results demonstrate the efficiency and effectiveness of the proposed algorithm.

In this paper, we present a nonlinear correction technique to modify the nine-point scheme proposed in [SIAM J. Sci. Comput., 30:3 (2008), 1341-1361] such that the resulted scheme preserves the positivity. We first express the flux by the cell-centered unknowns and edge unknowns based on the stencil of the nine-point scheme. Then, we use a nonlinear combination technique to get a monotone scheme. In order to obtain a cell-centered finite volume scheme, we need to use the cell-centered unknowns to locally approximate the auxiliary unknowns. We present a new method to approximate the auxiliary unknowns by using the idea of an improved multi-points flux approximation. The numerical results show that the new proposed scheme is robust, can handle some distorted grids that some existing finite volume schemes could not handle, and has higher numerical accuracy than some existing positivity-preserving finite volume schemes.

In this paper, we study a posteriori error estimates of the L1 scheme for time discretizations of time fractional parabolic differential equations, whose solutions have generally the initial singularity. To derive optimal order a posteriori error estimates, the quadratic reconstruction for the L1 method and the necessary fractional integral reconstruction for the first-step integration are introduced. By using these continuous, piecewise time reconstructions, the upper and lower error bounds depending only on the discretization parameters and the data of the problems are derived. Various numerical experiments for the one-dimensional linear fractional parabolic equations with smooth or nonsmooth exact solution are used to verify and complement our theoretical results, with the convergence of α order for the nonsmooth case on a uniform mesh. To recover the optimal convergence order 2-α on a nonuniform mesh, we further develop a time adaptive algorithm by means of barrier function recently introduced. The numerical implementations are performed on nonsmooth case again and verify that the true error and a posteriori error can achieve the optimal convergence order in adaptive mesh.

The truncated singular value decomposition has been widely used in many areas of science including engineering, and statistics, etc. In this paper, the original truncated complex singular value decomposition problem is formulated as a Riemannian optimization problem on a product of two complex Stiefel manifolds, a practical algorithm based on the generic Riemannian trust-region method of Absil et al. is presented to solve the underlying problem, which enjoys the global convergence and local superlinear convergence rate. Numerical experiments are provided to illustrate the efficiency of the proposed method. Comparisons with some classical Riemannian gradient-type methods, the existing Riemannian version of limited-memory BFGS algorithms in the MATLAB toolbox Manopt and the Riemannian manifold optimization library ROPTLIB, and some latest infeasible methods for solving manifold optimization problems, are also provided to show the merits of the proposed approach.

In this paper, we propose two families of nonconforming finite elements on n-rectangle meshes of any dimension to solve the sixth-order elliptic equations. The unisolvent property and the approximation ability of the new finite element spaces are established. A new mechanism, called the exchange of sub-rectangles, for investigating the weak continuities of the proposed elements is discovered. With the help of some conforming relatives for the H³ problems, we establish the quasi-optimal error estimate for the triharmonic equation in the broken H³ norm of any dimension. The theoretical results are validated further by the numerical tests in both 2D and 3D situations.