In this study, we integrate the established obstacle problem formulation from ice sheet modeling [1, 2] with cutting-edge deep learning methodologies to enhance ice thickness predictions, specifically targeting the Greenland ice sheet. By harmonizing the mathematical structure with an energy minimization framework tailored for neural network approximations, our method’s efficacy is confirmed through both 1D and 2D numerical simulations. Utilizing the NSIDC-0092 dataset for Greenland [22] and incorporating bedrock topography for model pre-training, we register notable advances in prediction accuracy. Our research underscores the potent combination of traditional mathematical models and advanced computational techniques in delivering precise ice thickness estimations.

Image smoothing has been extensively used in various fields, e.g., edge extraction, image abstraction, and image detail enhancement. Many existing optimization-based image smoothing methods have been proposed in recent years. The downside of these methods is that the results often have unclear edges and missing structures. To obtain satisfactory smoothing results, we design a novel optimization model by introducing an anisotropic $L_0$ gradient intensity. Specifically, a weighted matrix $T$ is imposed to control further the sparsity of the gradient measured by $L_0$-norm. Since the proposed model is non-convex and non-smooth, we apply the half quadratic splitting (HQS) algorithm to solve it effectively. In addition, to obtain a more suitable regularization parameter $λ,$ we utilize an adaptive parameter selection method based on Morozovs discrepancy principle. Finally, we conduct numerical experiments to illustrate the superiority of our method over some state-of-the-art methods.

In this paper, we develop a linear and decoupled fully discrete mixed finite element scheme for the active fluid model. The scheme employs an auxiliary variable to reformulate the fourth-order derivative term, an implicit-explicit treatment to deal with the nonlinear terms and the second-order pressure-projection method to split the velocity and pressure. Through rigorous theoretical analysis, the unique solvability, unconditional stability and error estimates of the numerical scheme are obtained. Then, several numerical experiments are presented to verify the efficiency and accuracy of the proposed scheme. Finally, the comparison of simulation results with laboratory results, including the motion direction of active fluid changes from disorder to order and reversal in 2D and 3D, demonstrate that the scheme can accurately capture and handle the complex dynamics of active fluid motion.

We introduce two global-in-time domain decomposition methods, namely the Steklov-Poincare method and the Robin method, for solving a fluid-structure interaction system. These methods allow us to formulate the coupled system as a space-time interface problem and apply iterative algorithms directly to the evolutionary problem. Each time-dependent subdomain problem is solved independently, which enables the use of different time discretization schemes and time step sizes in the subsystems. This leads to an efficient way of simulating time-dependent phenomena. We present numerical tests for both non-physical and physical problems, with various mesh sizes and time step sizes to demonstrate the accuracy and efficiency of the proposed methods.

In this paper, we present and analysis a difference virtual element method (DVEM) for the three dimensional (3D) elliptic equation on general cylindrical domains. This method combines the dimension splitting method and operator splitting technique to transform the virtual element solution of 3D elliptic equation into a series of virtual element solution of 2D elliptic equation based on $(x, y)$ plane, where the central difference discretization is adopted in the $z$-direction. This allows us to solve partial differential equations on cylindrical domains at the low cost in mesh generation compared with 3D virtual element method. The $H^1$-norm error estimation of the DVEM is analysed in this paper. Finally, some numerical examples are performed to verify the theoretical predictions and showcase the efficiency of the proposed method.

Markowitz’s criterion aims to balance expected return and risk when optimizing the portfolio. The expected return level is usually fixed according to the risk appetite of an investor, then the risk is minimized at this fixed return level. However, the investor may not know which return level is suitable for her/him and the current financial circumstance. It motivates us to find a novel approach that adaptively optimizes this return level and the portfolio at the same time. It not only relieves the trouble of deciding the return level during an investment but also gets more adaptive to the ever-changing financial market than a subjective return level. In order to solve the new model, we propose an exact, convergent, and efficient Krasnoselskii-Mann Proximity Algorithm based on the proximity operator and Krasnoselskii-Mann momentum technique. Extensive experiments show that the proposed method achieves significant improvements over state-of-the-art methods in portfolio optimization. This finding may contribute a new perspective on the relationship between return and risk in portfolio optimization.