We explore how Stochastic Gradient methods with momentum perform within a time-varying framework by establishing bounds on their tracking errors, specifically focusing on quadratic cases. Notably, we find that momentum methods achieve, in high drift-to-noise regimes, i.e., when the rate of change of the dynamic optimum prevails on the variance of the gradient noise, smaller neighborhood of convergence compared to stochastic gradient descent. To the best of our knowledge, this is the first proof that a momentum method can improve upon stochastic gradient descent’s tracking error bounds in a time-varying setting. We then investigate, for a given learning rate, the optimal choice of the momentum parameter that minimizes the tracking error bound.
We are concerned with the bifurcation analysis of the following Schrödinger-Bopp-Podolsky system with double criticality: {[ -Δ u+ u+ϕ |u|^3u=λ h(x)u+|u|^4u, in ℝ^3,; -Δϕ +a^2Δ ^2ϕ =4π |u|^5, in ℝ^3, ]. where h≥ 0 is a weight function and a>0 is the Bopp-Podolsky constant. The value of the real parameter λ associated with the subcritical term can determine whether a ground state solution exists. A feature of the analysis developed in this paper is given by the combined effects of the critical nonlocal term and the critical reaction term. The main result establishes a bifurcation alternative with respect to the eigenvalues of a related linear eigenvalue problem on the entire space.
This paper is devoted to the existence of positive normalized solutions for Kirchhoff-type equations in ℝ^2 with nonlinearities exhibiting critical exponential growth. We consider two classes of potentials: coercive potentials and steep potential wells. By truncating the nonlinearity and establishing uniform a priori estimates, we prove the existence of normalized solutions using a constrained minimization method that avoids the Pohozaev manifold. Our results complement the existing theory for both Schrödinger and Kirchhoff equations, and provide a general framework for different types of potentials.
In this paper, we study Liouville type problems for the three-dimensional stationary incompressible Navier–Stokes equations. Most of the previous Liouville type results focus on integrability conditions or growth conditions of the solution. This paper is devoted to establishing Liouville type theorems under some growth conditions imposed on the gradient of the solution. Using the energy method and an iteration argument, we establish Liouville type theorems provided that L^p norm of the gradient of the solution on the annulus satisfies certain growth conditions and the solution is bounded. Furthermore, by establishing new energy estimates and developing some tricky differential inequality techniques, we relax the growth conditions by logarithmic factors for 1≤ p<6/5 and hence obtain a logarithmic improvement version of Liouville type theorem.
