• Abstract

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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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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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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In this paper, we introduce the weak Galerkin (WG) method for solving the coupled Stokes and Darcy-Forchheimer flows problem with the Beavers-Joseph-Saffman interface condition in bounded domains. We define the WG spaces in the polygonal meshes and construct corresponding discrete schemes. We prove the existence and uniqueness of the WG scheme by the discrete inf-sup condition and monotone operator theory. Then, we derive the optimal error estimates for the velocity and pressure. Numerical experiments are presented to verify the efficiency of the WG method.

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Superconvergence of differential structure on discretized surfaces is studied in this paper. The newly introduced geometric supercloseness provides us with a fundamental tool to prove the superconvergence of gradient recovery on deviated surfaces. An algorithmic framework for gradient recovery without exact geometric information is introduced. Several numerical examples are documented to validate the theoretical results.

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This paper is concerned with the new error analysis of a Hodge-decomposition based finite element method for the time-dependent Ginzburg-Landau equations in superconductivity. In this approach, the original equation of magnetic potential $\boldsymbol{A}$ is replaced by a new system consisting of four scalar variables. As a result, the conventional Lagrange finite element method (FEM) can be applied to problems defined on non-smooth domains. It is known that due to the low regularity of $\boldsymbol{A},$ conventional FEM, if applied to the original Ginzburg-Landau system directly, may converge to the unphysical solution. The main purpose of this paper is to establish an optimal error estimate for the order parameter in spatial direction, as previous analysis only gave a sub-optimal convergence rate analysis for all three variables due to coupling of variables. The analysis is based on a nonstandard quasi-projection for ψ and the corresponding negative-norm estimate for the classical Ritz projection. Our numerical experiments confirm the optimal convergence of $ψ_h.$

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In this paper, we develop a fully discrete virtual element scheme based on the local pressure projection stabilization for a three-field poroelasticity problem with a storage coefficient $c_0 ≥ 0.$ We not only provide the well-posedness of the proposed scheme by proving a weaker form of the discrete inf-sup condition, but also show optimal error estimates for all unknowns, whose generic constants are independent of the Lamé coefficient $λ.$ Moreover, our proposed scheme avoids pressure oscillation and applies to general polygonal elements, including hanging-node elements. Finally, we numerically validate the good performance of our virtual element scheme.

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This paper constructs the first mixed finite element for the linear elasticity problem in 3D using $P_3$ polynomials for the stress and discontinuous $P_2$ polynomials for the displacement on tetrahedral meshes under some mild mesh conditions. The degrees of freedom of the stress space as well as the corresponding nodal basis are established by characterizing a space of certain piecewise constant symmetric matrices on a patch around each edge. Macro-element techniques are used to define a stable interpolation to prove the discrete inf-sup condition. Optimal convergence is obtained theoretically.

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The recently developed DOC kernels technique has been successful in the stability and convergence analysis for variable time-step BDF2 schemes. However, it may not be readily applicable to problems exhibiting an initial singularity. In the numerical simulations of solutions with initial singularity, variable time-step schemes like the graded mesh are always adopted to achieve the optimal convergence, whose first adjacent time-step ratio may become pretty large so that the acquired restriction is not satisfied. In this paper, we revisit the variable time-step implicit-explicit two-step backward differentiation formula (IMEX BDF2) scheme to solve the parabolic integro-differential equations (PIDEs) with initial singularity. We obtain the sharp error estimate under a mild restriction condition of adjacent time-step ratios $r_k := τ_k/τ_{k−1} < r_{max} = 4.8645$ $(k ≥ 3)$ and a much mild requirement on the first ratio, i.e. $r_2 > 0.$ This leads to the validation of our analysis of the variable time-step IMEX BDF2 scheme when the initial singularity is dealt by a simple strategy, i.e. the graded mesh $t_k = T (k/N)^{\gamma}.$ In this situation, the convergence order of $\mathcal{O}(N^{− min\{2,γα\}})$ is achieved, where $N$ denotes the total number of mesh points and $α$ indicates the regularity of the exact solution. This is, the optimal convergence will be achieved by taking $\gamma_{{\rm opt}} = 2/α.$ Numerical examples are provided to demonstrate our theoretical analysis.

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In this paper, we propose a novel algorithm called neuron-wise parallel subspace correction method for the finite neuron method that approximates numerical solutions of partial differential equations (PDEs) using neural network functions. Despite extremely extensive research activities in applying neural networks for numerical PDEs, there is still a serious lack of effective training algorithms that can achieve adequate accuracy, even for one-dimensional problems. Based on recent results on the spectral properties of linear layers and analysis for single neuron problems, we develop a special type of subspace correction method that optimizes the linear layer and each neuron in the nonlinear layer separately. An optimal preconditioner that resolves the ill-conditioning of the linear layer is presented for one-dimensional problems, so that the linear layer is trained in a uniform number of iterations with respect to the number of neurons. In each single neuron problem, a local minimum is found by a superlinearly convergent algorithm. Numerical experiments on function approximation problems and PDEs demonstrate better performance of the proposed method than other gradient-based methods.

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The aim of this paper is to present an improvement of the incremental Newton method proposed by Iannazzo [SIAM J. Matrix Anal. Appl., 28:2 (2006), 503–523] for approximating the principal $p$-th root of a matrix. We construct and analyze an incremental Chebyshev method with better numerical behavior. We present a convergence and numerical analysis of the method, where we compare it with the corresponding incremental Newton method. The new method has order of convergence three and is stable and more efficient than the incremental Newton method.

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In this work, an unconditionally stable, decoupled, variable time step scheme is presented for the incompressible Navier-Stokes equations. Based on a scalar auxiliary variable in exponential function, this fully discrete scheme combines the backward Euler scheme for temporal discretization with variable time step and a mixed finite element method for spatial discretization, where the nonlinear term is treated explicitly. Moreover, without any restriction on the time step, stability of the proposed scheme is discussed. Besides, error estimate is provided. Finally, some numerical results are presented to illustrate the performances of the considered numerical scheme.

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This paper is concerned with the superconvergence error estimates of a classical mixed finite element method for a nonlinear parabolic/elliptic coupled thermistor equations. The method is based on a popular combination of the lowest-order rectangular Raviart-Thomas mixed approximation for the electric potential/field $(\phi, \mathbf{θ})$ and the bilinear Lagrange approximation for temperature $u.$ In terms of the special properties of these elements above, the superclose error estimates with order $\mathcal{O}(h^2)$ are obtained firstly for all three components in such a strongly coupled system. Subsequently, the global superconvergence error estimates with order $\mathcal{O}(h^2)$ are derived through a simple and effective interpolation post-processing technique. As by a product, optimal error estimates are acquired for potential/field and temperature in the order of $\mathcal{O}(h)$ and $\mathcal{O}(h^2 ),$ respectively. Finally, some numerical results are provided to confirm the theoretical analysis.

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We study decentralized smooth optimization problems over compact submanifolds. Recasting it as a composite optimization problem, we propose a decentralized Douglas-Rachford splitting algorithm (DDRS). When the proximal operator of the local loss function does not have a closed-form solution, an inexact version of DDRS (iDDRS), is also presented. Both algorithms rely on careful integration of the nonconvex Douglas-Rachford splitting algorithm with gradient tracking and manifold optimization. We show that our DDRS and iDDRS achieve the convergence rate of $\mathcal{O}(1/k).$ The main challenge in the proof is how to handle the nonconvexity of the manifold constraint. To address this issue, we utilize the concept of proximal smoothness for compact submanifolds. This ensures that the projection onto the submanifold exhibits convexity-like properties, which allows us to control the consensus error across agents. Numerical experiments on the principal component analysis are conducted to demonstrate the effectiveness of our decentralized DRS compared with the state-of-the-art ones.

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