• Abstract

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.

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Image restoration based on total variation has been widely studied owing to its edge-preservation 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.

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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 (Batch-OMP) 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 $ℓ_1$-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 $ℓ_1$-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 $ℓ_1$-regularized least squares, it is more effective than the other seven algorithms.

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This paper investigates a semilinear stochastic fractional Rayleigh-Stokes equation featuring a Riemann-Liouville fractional derivative of order $α ∈ (0, 1)$ in time and a fractional time-integral noise. The study begins with an examination of the solution’s existence, uniqueness, and regularity. The spatial discretization is then carried out using a finite element method, and the error estimate is analyzed. A convolution quadrature method generated by the backward Euler method is employed for the time discretization resulting in a fully discrete scheme. The error estimate for the fully discrete solution is considered based on the regularity of the solution, and a strong convergence rate is established. The paper concludes with numerical tests to validate the theoretical findings.

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This work focuses on the temporal average of the backward Euler-Maruyama (BEM) method, which is used to approximate the ergodic limit of stochastic ordinary differential equations (SODEs). We give the central limit theorem (CLT) of the temporal average of the BEM method, which characterizes its asymptotics in distribution. When the deviation order is smaller than the optimal strong order, we directly derive the CLT of the temporal average through that of original equations and the uniform strong order of the BEM method. For the case that the deviation order equals to the optimal strong order, the CLT is established via the Poisson equation associated with the generator of original equations. Numerical experiments are performed to illustrate the theoretical results. The main contribution of this work is to generalize the existing CLT of the temporal average of numerical methods to that for SODEs with super-linearly growing drift coefficients.

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In this paper, we consider numerical solutions of the fractional diffusion equation with the $α$ order time fractional derivative defined in the Caputo-Hadamard sense. A high order time-stepping scheme is constructed, analyzed, and numerically validated. The contribution of the paper is twofold: 1) regularity of the solution to the underlying equation is investigated, 2) a rigorous stability and convergence analysis for the proposed scheme is performed, which shows that the proposed scheme is $3 + α$ order accurate. Several numerical examples are provided to verify the theoretical statement.

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This paper presents three regularized models for the logarithmic Klein-Gordon equation. By using a modified Crank-Nicolson method in time and the Galerkin finite element method (FEM) in space, a fully implicit energy-conservative numerical scheme is constructed for the local energy regularized model that is regarded as the best one among the three regularized models. Then, the cut-off function technique and the time-space error splitting technique are innovatively combined to rigorously analyze the unconditionally optimal and high-accuracy convergence results of the numerical scheme without any coupling condition between the temporal step size and the spatial mesh width. The theoretical framework is uniform for the other two regularized models. Finally, numerical experiments are provided to verify our theoretical results. The analytical techniques in this work are not limited in the FEM, and can be directly extended into other numerical methods. More importantly, this work closes the gap for the unconditional error/stability analysis of the numerical methods for the logarithmic systems in higher dimensional spaces.

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We design and analyze an iterative two-grid algorithm for the finite element discretizations of strongly nonlinear elliptic boundary value problems in this paper. We propose an iterative two-grid algorithm, in which a nonlinear problem is first solved on the coarse space, and then a symmetric positive definite problem is solved on the fine space. The main contribution in this paper is to establish a first convergence analysis, which requires dealing with four coupled error estimates, for the iterative two-grid methods. We also present some numerical experiments to confirm the efficiency of the proposed algorithm.

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This paper is concerned with the numerical solution of Volterra integro-differential equations with noncompact operators. The focus is on the problems with weakly singular solutions. To handle the initial weak singularity of the solution, a fractional collocation method is applied. A rigorous $hp$-version error analysis of the numerical method under a weighted $H^1$-norm is carried out. The result shows that the method can achieve high order convergence for such equations. Numerical experiments are also presented to confirm the effectiveness of the proposed method.

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Consider the inverse scattering of time-harmonic acoustic waves by a mixed-type scatterer consisting of an inhomogeneous penetrable medium with a conductive transmission condition and various impenetrable obstacles with different kinds of boundary conditions. Based on the establishment of the well-posedness result of the direct problem, we intend to develop a modified factorization method to simultaneously reconstruct both the support of the inhomogeneous conductive medium and the shape and location of various impenetrable obstacles by means of the far-field data for all incident plane waves at a fixed wave number. Numerical examples are carried out to illustrate the feasibility and effectiveness of the proposed inversion algorithms.

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We discuss and analyze the virtual element method on general polygonal meshes for the time-dependent Poisson-Nernst-Planck (PNP) equations, which are a nonlinear coupled system widely used in semiconductors and ion channels. After presenting the semi-discrete scheme, the optimal $H^1$ norm error estimates are presented for the time-dependent PNP equations, which are based on some error estimates of a virtual element energy projection. The Gummel iteration is used to decouple and linearize the PNP equations and the error analysis is also given for the iteration of fully discrete virtual element approximation. The numerical experiment on different polygonal meshes verifies the theoretical convergence results and shows the efficiency of the virtual element method.

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A novel overlapping domain decomposition splitting algorithm based on a Crank-Nicolson method is developed for the stochastic nonlinear Schrödinger equation driven by a multiplicative noise with non-periodic boundary conditions. The proposed algorithm can significantly reduce the computational cost while maintaining the similar conservation laws. Numerical experiments are dedicated to illustrating the capability of the algorithm for different spatial dimensions, as well as the various initial conditions. In particular, we compare the performance of the overlapping domain decomposition splitting algorithm with the stochastic multi-symplectic method in [S. Jiang et al., Commun. Comput. Phys., 14 (2013), 393–411] and the finite difference splitting scheme in [J. Cui et al., J. Differ. Equ., 266 (2019), 5625–5663]. We observe that our proposed algorithm has excellent computational efficiency and is highly competitive. It provides a useful tool for solving stochastic partial differential equations.

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We present a decoupled, linearly implicit numerical scheme with energy stability and mass conservation for solving the coupled Cahn-Hilliard system. The time-discretization is done by leap-frog method with the scalar auxiliary variable (SAV) approach. It only needs to solve three linear equations at each time step, where each unknown variable can be solved independently. It is shown that the semi-discrete scheme has second-order accuracy in the temporal direction. Such convergence results are proved by a rigorous analysis of the boundedness of the numerical solution and the error estimates at different time-level. Numerical examples are presented to further confirm the validity of the methods.

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In this paper, a radial basis function method combined with the stochastic Galerkin method is considered to approximate elliptic optimal control problem with random coefficients. This radial basis function method is a meshfree approach for solving high dimensional random problem. Firstly, the optimality system of the model problem is derived and represented as a set of deterministic equations in high-dimensional parameter space by finite-dimensional noise assumption. Secondly, the approximation scheme is established by using finite element method for the physical space, and compactly supported radial basis functions for the parameter space. The radial basis functions lead to the sparsity of the stiff matrix with lower condition number. A residual type a posteriori error estimates with lower and upper bounds are derived for the state, co-state and control variables. An adaptive algorithm is developed to deal with the physical and parameter space, respectively. Numerical examples are presented to illustrate the theoretical results.

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In this paper, we introduce a new class of explicit numerical methods called the tamed stochastic Runge-Kutta-Chebyshev (t-SRKC) methods, which apply the idea of taming to the stochastic Runge-Kutta-Chebyshev (SRKC) methods. The key advantage of our explicit methods is that they can be suitable for stochastic differential equations with non-globally Lipschitz coefficients and stiffness. Under certain non-globally Lipschitz conditions, we study the strong convergence of our methods and prove that the order of strong convergence is 1/2. To show the advantages of our methods, we compare them with some existing explicit methods (including the Euler-Maruyama method, balanced Euler-Maruyama method and two types of SRKC methods) through several numerical examples. The numerical results show that our t-SRKC methods are efficient, especially for stiff stochastic differential equations.

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In this paper, we propose and analyze a first-order, semi-implicit, and unconditionally energy-stable scheme for an incompressible ferrohydrodynamics flow. We consider the constitutive equation describing the behavior of magnetic fluid provided by Shliomis, which consists of the Navier-Stokes equation, the magnetization equation, and the magnetostatics equation. By using an existing regularization method, we derive some prior estimates for the solutions. We then bring up a rigorous error analysis of the temporal semi-discretization scheme based on these prior estimates. Through a series of experiments, we verify the convergence and energy stability of the proposed scheme and simulate the behavior of ferrohydrodynamics flow in the background of practical applications.

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A weak Galerkin mixed finite element method is studied for linear elasticity problems without the requirement of symmetry. The key of numerical methods in mixed formulation is the symmetric constraint of numerical stress. In this paper, we introduce the discrete symmetric weak divergence to ensure the symmetry of numerical stress. The corresponding stabilizer is presented to guarantee the weak continuity. This method does not need extra unknowns. The optimal error estimates in discrete $H^1$ and $L^2$ norms are established. The numerical examples in 2D and 3D are presented to demonstrate the efficiency and locking-free property.

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It is one of the most challenging issues in applied mathematics to approximately solve high-dimensional partial differential equations (PDEs) and most of the numerical approximation methods for PDEs in the scientific literature suffer from the so-called curse of dimensionality in the sense that the number of computational operations employed in the corresponding approximation scheme to obtain an approximation precision $ε>0$ grows exponentially in the PDE dimension and/or the reciprocal of $ε.$ Recently, certain deep learning based methods for PDEs have been proposed and various numerical simulations for such methods suggest that deep artificial neural network (ANN) approximations might have the capacity to indeed overcome the curse of dimensionality in the sense that the number of real parameters used to describe the approximating deep ANNs grows at most polynomially in both the PDE dimension $d ∈ \mathbb{N}$ and the reciprocal of the prescribed approximation accuracy $ε > 0.$ There are now also a few rigorous mathematical results in the scientific literature which substantiate this conjecture by proving that deep ANNs overcome the curse of dimensionality in approximating solutions of PDEs. Each of these results establishes that deep ANNs overcome the curse of dimensionality in approximating suitable PDE solutions at a fixed time point $T > 0$ and on a compact cube $[a, b]^d$ in space but none of these results provides an answer to the question whether the entire PDE solution on $[0, T ] × [a, b]^d$ can be approximated by deep ANNs without the curse of dimensionality. It is precisely the subject of this article to overcome this issue. More specifically, the main result of this work in particular proves for every $a ∈\mathbb{R},$ $b ∈ (a,∞)$ that solutions of certain Kolmogorov PDEs can be approximated by deep ANNs on the space-time region $[0, T ] × [a, b]^d$ without the curse of dimensionality.

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We study a class of stochastic semilinear damped wave equations driven by additive Wiener noise. Owing to the damping term, under appropriate conditions on the nonlinearity, the solution admits a unique invariant distribution. We apply semi-discrete and fully-discrete methods in order to approximate this invariant distribution, using a spectral Galerkin method and an exponential Euler integrator for spatial and temporal discretization respectively. We prove that the considered numerical schemes also admit unique invariant distributions, and we prove error estimates between the approximate and exact invariant distributions, with identification of the orders of convergence. To the best of our knowledge this is the first result in the literature concerning numerical approximation of invariant distributions for stochastic damped wave equations.

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In this paper, we present an SQP-type proximal gradient method (SQP-PG) for composite optimization problems with equality constraints. At each iteration, SQP-PG solves a subproblem to get the search direction, and takes an exact penalty function as the merit function to determine if the trial step is accepted. The global convergence of the SQP-PG method is proved and the iteration complexity for obtaining an $\epsilon$-stationary point is analyzed. We also establish the local linear convergence result of the SQP-PG method under the second-order sufficient condition. Numerical results demonstrate that, compared to the state-of-the-art algorithms, SQP-PG is an effective method for equality constrained composite optimization problems.

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