This paper introduces a universal accelerated fractional-order gradient method (AFOGM) and analyzes its convergence from the perspective of dynamical systems. Under an appropriate regularity condition for non-convex objective functions, we show that AFOGM attains R-linear convergence, providing an explicit description of its convergence parameter region. Numerical experiments on standard benchmark problems further demonstrate that AFOGM, including the fractional-order Heavy Ball (FOHB) method and the fractional-order Nesterov’s accelerated gradient (FONAG), possesses a larger convergence domain than classical gradient descent methods. Finally, we employ AFOGM as an optimizer for deep neural networks, where it outperforms classical momentum-based methods on the MNIST and Fashion-MNIST datasets.
This article investigates an inverse source problem for a subdiffusion equation with multiple fractional derivatives. The goal is to recover a temporally varying source term with low regularity from distributed measurements. The main challenges, posed by the variable coefficients and the source’s weak regularity, are addressed by establishing new coercivity properties of the fractional derivatives. This key innovation allows us to frame the analysis similarly to elliptic equations, proving the well-posedness of the forward prob- lem via the Lax-Milgram theorem and deriving stability estimates for the inverse problem under low regularity assumptions. A stability estimate is provided, and the effectiveness of the approach is supported by numerical experiments.
The objective of the paper is to develop the quadratic discontinuous finite volume methods (DFVM) for solving elliptic equations with variable coefficients. The proposed numerical schemes are composed of the discontinuous Galerkin (DG) method with different interior penalty formulations (IIPG, NIPG, SIPG) and the finite volume element method which the trial function is discontinuous quadratic element function. Subsequently, with the specialized projection techniques, we built up a bridge between the bilinear form of DFVM and that of DG method, which simplifies the analysis and proves the optimal error estimate of the schemes in the broken H1 norm. Finally, we provide numerical simulations to validate the theoretical findings.
In this paper, we introduce novel high-dimensional delayed stochastic Navier-Stokes models (HD-DSNSMs) characterized by multi-fractional Gaussian noise (mGn) and then establish sufficient conditions for the existence and uniqueness of solutions. Since obtaining exact solutions for these models is mostly impossible, this work aims to develop an efficient approach by applying an approximation of multi-fractional Brownian motion (mBm) and then using physics-informed deep neural networks to predict solutions for the HD-DSNSMs characterized by mGn. The deep neural network parameters are optimized by minimizing a mean-squared error loss function, which is derived from the discretization of the HD-DSNSMs characterized by mGn via the standard Euler-Maruyama scheme. Optimization is carried out with the Adam algorithm, and the model employs the activation function sin. Four test problems are conducted to show the efficiency and accuracy of the method. The results demonstrate the ability of the method to extract solution patterns even when pressure and delay increase. This approach proves adaptable and remarkably efficient in solving the discussed models, as evidenced by the research outcome. Our results illustrate that the HD-DSNSMs characterized by mGn improve the classical DSNSMs.
We propose a two-step Gauss-Newton method (TS-GNM) for solving nonsmooth equations. At each iteration, the TS-GNM solves both a Gauss-Newton equation and an approximate Gauss-Newton equation. A second-order derivative-free line search strategy is designed to ensure the global convergence of TS-GNM. Under the nonsingularity condition and the strong semismoothness of the underlying function, we prove that the TS-GNM converges quadratically. Furthermore, we demonstrate that the TS-GNM achieves a cubic convergence rate when the generalized Jacobian is locally Lipschitz continuous at the solutions. Finally, we pay particular attention to the absolute value equation and present some numerical results.
