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.

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.

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.

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.

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.

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.

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.

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.

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.