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.

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.

Minimax optimization problems are an important class of optimization problems arising from modern machine learning and traditional research areas. While there have been many numerical algorithms for solving smooth convex-concave minimax problems, numerical algorithms for nonsmooth convex-concave minimax problems are rare. This paper aims to develop an efficient numerical algorithm for a structured nonsmooth convex-concave minimax problem. A semi-proximal point method (SPP) is proposed, in which a quadratic convex-concave function is adopted for approximating the smooth part of the objective function and semi-proximal terms are added in each subproblem. This construction enables the subproblems at each iteration are solvable and even easily solved when the semiproximal terms are cleverly chosen. We prove the global convergence of our algorithm under mild assumptions, without requiring strong convexity-concavity condition. Under the locally metrical subregularity of the solution mapping, we prove that our algorithm has the linear rate of convergence. Preliminary numerical results are reported to verify the efficiency of our algorithm.

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.

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.

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.

An essential feature of the subdiffusion equations with the α-order time fractional derivative is the weak singularity at the initial time. The weak regularity of the solution is usually characterized by a regularity parameter σ ∈ (0, 1) ∪ (1, 2). Under this general regularity assumption, we present a rigorous analysis for the truncation errors and develop a new tool to obtain the stability results, i.e., a refined discrete fractional-type Grönwall inequality (DFGI). After that, we obtain the pointwise-in-time error estimate of the widely used L1 scheme for nonlinear subdiffusion equations. The present results fill the gap on some interesting convergence results of L1 scheme on σ ∈ (0, α) ∪ (α, 1) ∪ (1, 2]. Numerical experiments are provided to demonstrate the effectiveness of our theoretical analysis.

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.

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 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 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.

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.

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.

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.

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 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.

This paper deals with numerical methods for solving one-dimensional (1D) and twodimensional (2D) initial-boundary value problems (IBVPs) of space-fractional sine-Gordon equations (SGEs) with distributed delay. For 1D problems, we construct a kind of oneparameter finite difference (OPFD) method. It is shown that, under a suitable condition, the proposed method is convergent with second order accuracy both in time and space. In implementation, the preconditioned conjugate gradient (PCG) method with the Strang circulant preconditioner is carried out to improve the computational efficiency of the OPFD method. For 2D problems, we develop another kind of OPFD method. For such a method, two classes of accelerated schemes are suggested, one is alternative direction implicit (ADI) scheme and the other is ADI-PCG scheme. In particular, we prove that ADI scheme can arrive at second-order accuracy in time and space. With some numerical experiments, the computational effectiveness and accuracy of the methods are further verified. Moreover, for the suggested methods, a numerical comparison in computational efficiency is presented.

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.

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.

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.

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.

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.

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.

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.

A Riemannian gradient descent algorithm and a truncated variant are presented to solve systems of phaseless equations |Ax|² = y. The algorithms are developed by exploiting the inherent low rank structure of the problem based on the embedded manifold of rank-1 positive semidefinite matrices. Theoretical recovery guarantee has been established for the truncated variant, showing that the algorithm is able to achieve successful recovery when the number of equations is proportional to the number of unknowns. Two key ingredients in the analysis are the restricted well conditioned property and the restricted weak correlation property of the associated truncated linear operator. Empirical evaluations show that our algorithms are competitive with other state-of-the-art first order nonconvex approaches with provable guarantees.

In this paper, we offer a new sparse recovery strategy based on the generalized error function. The introduced penalty function involves both the shape and the scale parameters, making it extremely flexible. For both constrained and unconstrained models, the theoretical analysis results in terms of the null space property, the spherical section property and the restricted invertibility factor are established. The practical algorithms via both the iteratively reweighted $\ell_1$ and the difference of convex functions algorithms are presented. Numerical experiments are carried out to demonstrate the benefits of the suggested approach in a variety of circumstances. Its practical application in magnetic resonance imaging (MRI) reconstruction is also investigated.

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.

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.

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 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.

The mixed-integer quadratically constrained quadratic fractional programming (MIQCQFP) problem often appears in various fields such as engineering practice, management science and network communication. However, most of the solutions to such problems are often designed for their unique circumstances. This paper puts forward a new global optimization algorithm for solving the problem MIQCQFP. We first convert the MIQCQFP into an equivalent generalized bilinear fractional programming (EIGBFP) problem with integer variables. Secondly, we linearly underestimate and linearly overestimate the quadratic functions in the numerator and the denominator respectively, and then give a linear fractional relaxation technique for EIGBFP on the basis of non-negative numerator. After that, combining rectangular adjustment-segmentation technique and midpointsampling strategy with the branch-and-bound procedure, an efficient algorithm for solving MIQCQFP globally is proposed. Finally, a series of test problems are given to illustrate the effectiveness, feasibility and other performance of this algorithm.

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.

We treat infinite horizon optimal control problems by solving the associated stationary Bellman equation numerically to compute the value function and an optimal feedback law. The dynamical systems under consideration are spatial discretizations of non linear parabolic partial differential equations (PDE), which means that the Bellman equation suffers from the curse of dimensionality. Its non linearity is handled by the Policy Iteration algorithm, where the problem is reduced to a sequence of linear equations, which remain the computational bottleneck due to their high dimensions. We reformulate the linearized Bellman equations via the Koopman operator into an operator equation, that is solved using a minimal residual method. Using the Koopman operator we identify a preconditioner for operator equation, which deems essential in our numerical tests. To overcome computational infeasability we use low rank hierarchical tensor product approximation/tree-based tensor formats, in particular tensor trains (TT tensors) and multi-polynomials, together with high-dimensional quadrature, e.g. Monte-Carlo. By controlling a destabilized version of viscous Burgers and a diffusion equation with unstable reaction term numerical evidence is given.

In this paper, by designing a normalized nonmonotone search strategy with the BarzilaiBorwein-type step-size, a novel local minimax method (LMM), which is a globally convergent iterative method, is proposed and analyzed to find multiple (unstable) saddle points of nonconvex functionals in Hilbert spaces. Compared to traditional LMMs with monotone search strategies, this approach, which does not require strict decrease of the objective functional value at each iterative step, is observed to converge faster with less computations. Firstly, based on a normalized iterative scheme coupled with a local peak selection that pulls the iterative point back onto the solution submanifold, by generalizing the Zhang-Hager (ZH) search strategy in the optimization theory to the LMM framework, a kind of normalized ZH-type nonmonotone step-size search strategy is introduced, and then a novel nonmonotone LMM is constructed. Its feasibility and global convergence results are rigorously carried out under the relaxation of the monotonicity for the functional at the iterative sequences. Secondly, in order to speed up the convergence of the nonmonotone LMM, a globally convergent Barzilai-Borwein-type LMM (GBBLMM) is presented by explicitly constructing the Barzilai-Borwein-type step-size as a trial step-size of the normalized ZH-type nonmonotone step-size search strategy in each iteration. Finally, the GBBLMM algorithm is implemented to find multiple unstable solutions of two classes of semilinear elliptic boundary value problems with variational structures: one is the semilinear elliptic equations with the homogeneous Dirichlet boundary condition and another is the linear elliptic equations with semilinear Neumann boundary conditions. Extensive numerical results indicate that our approach is very effective and speeds up the LMMs significantly.

In this paper, we introduce and analyze an augmented mixed discontinuous Galerkin (MDG) method for a class of quasi-Newtonian Stokes flows. In the mixed formulation, the unknowns are strain rate, stress and velocity, which are approximated by a discontinuous piecewise polynomial triplet $\underline{\mathcal{P}}_{k+1}^{\mathbb{S}}-\underline{\mathcal{P}}_{k+1}^{\mathbb{S}}-\mathcal{P}_k$ for k ≥ 0. Here, the discontinuous piecewise polynomial function spaces for the field of strain rate and the stress field are designed to be symmetric. In addition, the pressure is easily recovered through simple postprocessing. For the benefit of the analysis, we enrich the MDG scheme with the constitutive equation relating the stress and the strain rate, so that the well-posedness of the augmented formulation is obtained by a nonlinear functional analysis. For k ≥ 0, we get the optimal convergence order for the stress in broken $\underline{H}($ div $)$-norm and velocity in L²-norm. Furthermore, the error estimates of the strain rate and the stress in $\underline{\boldsymbol{L}}^2$-norm, and the pressure in L²-norm are optimal under certain conditions. Finally, several numerical examples are given to show the performance of the augmented MDG method and verify the theoretical results. Numerical evidence is provided to show that the orders of convergence are sharp.

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.

This paper aims to analyze the weak approximation error of a fully discrete scheme for a class of semi-linear parabolic stochastic partial differential equations (SPDEs) driven by additive fractional Brownian motions with the Hurst parameter H ∈ (1/2, 1). The spatial approximation is performed by a spectral Galerkin method and the temporal discretization by an exponential Euler method. As far as we know, the weak error analysis for approximations of fractional noise driven SPDEs is absent in the literature. A key difficulty in the analysis is caused by the lack of the associated Kolmogorov equations. In the present work, a novel and efficient approach is presented to carry out the weak error analysis for the approximations, which does not rely on the associated Kolmogorov equations but relies on the Malliavin calculus. To the best of our knowledge, the rates of weak convergence, shown to be higher than the strong convergence rates, are revealed in the fractional noise driven SPDE setting for the first time. Numerical examples corroborate the claimed weak orders of convergence.

In this paper, we investigate the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional nonlinear Korteweg-de Vries type equations. The numerical flux for the nonlinear convection term is chosen as the generalized Lax-Friedrichs flux, and the generalized alternating flux and upwind-biased flux are used for the dispersion term. The generalized Lax-Friedrichs flux with anti-dissipation property will compensate the numerical dissipation of the dispersion term, resulting in a nearly energy conservative scheme that is useful in resolving waves and is beneficial for long time simulations. To deal with the nonlinearity and different numerical flux weights, a suitable numerical initial condition is constructed, for which a modified global projection is designed. By establishing relationships between the prime variable and auxiliary variables in combination with sharp bounds for jump terms, optimal error estimates are obtained. Numerical experiments are shown to confirm the validity of theoretical results.