Related papers: Time fractional gradient flows: Theory and numeric…

The paper introduces a one dimensional analogy of Poincare's "section" method. Its application to numerical study of sequences of fractional parts is considered. As a result, the possibility of Cantorian structure of time flow is deduced. A…

We study a scalar integro-differential conservation law. The equation was first derived in [2] as the slow erosion limit of granular flow. Considering a set of more general erosion functions, we study the initial boundary value problem for…

We study the statistical properties of the iterates generated by gradient descent, applied to the fundamental problem of least squares regression. We take a continuous-time view, i.e., consider infinitesimal step sizes in gradient descent,…

We study the high-frequency limit of non-autonomous gradient flows in metric spaces of energy functionals comprising an explicitly time-dependent perturbation term which might oscillate in a rapid way, but fulfills a certain Lipschitz…

We consider a general optimal control problem in the setting of gradient flows. Two approximations of the problem are presented, both relying on the variational reformulation of gradient-flow dynamics via the Weighted-Energy-Dissipation…

We present a general framework for the rigorous numerical analysis of time-fractional nonlinear parabolic partial differential equations, with a fractional derivative of order $\alpha\in(0,1)$ in time. The framework relies on three…

In this contribution we present an exploratory study of several novel methods for numerical stochastic perturbation theory. For the investigation we consider observables defined through the gradient flow in the simple {\phi}^4 theory.

Over the past few decades, there has been substantial interest in evolution equations that involving a fractional-order derivative of order $\alpha\in(0,1)$ in time, due to their many successful applications in engineering, physics, biology…

Convolutional neural networks are widely used in imaging and image recognition. Learning such networks from training data leads to the minimization of a non-convex function. This makes the analysis of standard optimization methods such as…

In this paper we investigate the following fractional order in time Cauchy problem \begin{equation*} \begin{cases} \mathbb{D}_{t}^{\alpha }u(t)+Au(t)=f(u(t)), & 1<\alpha <2, u(0)=u_{0},\,\,\,u^{\prime }(0)=u_{1}. & \end{cases}%…

This paper derives physically meaningful boundary conditions for fractional diffusion equations, using a mass balance approach. Numerical solutions are presented, and theoretical properties are reviewed, including well-posedness and steady…

We initiate an investigation into whether fractional calculus, with its intrinsic long-tailed memory and nonlocal features, can provide a viable model for gravitational-wave memory effects. We consider two toy constructions: ($i$) a…

Time-fractional parabolic equations with a Caputo time derivative are considered. For such equations, we explore and further develop the new methodology of the a-posteriori error estimation and adaptive time stepping proposed in [7]. We…

Starting from the microscopic description of a normal fluid in terms of any kind of local interacting many-particle theory we present a well defined step by step procedure to derive the hydrodynamic equations for the macroscopic phenomena.…

An initial-boundary value problem with a Caputo time derivative of fractional order $\alpha\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. For this problem, we give a simple framework…

The study of systems with memory requires methods which are different from the methods used in regular dynamics. Systems with power-law memory in many cases can be described by fractional differential equations, which are…

This paper focuses on the convergence problem of the emerging fractional order gradient descent method, and proposes three solutions to overcome the problem. In fact, the general fractional gradient method cannot converge to the real…

This article overviews how gradient flows, and discretizations thereof, are useful to design and analyze optimization and sampling algorithms. The interplay between optimization, sampling, and gradient flows is an active research area; our…

Within the framework of fractional calculus with variable order the evolution of space in the adiabatic limit is investigated. Based on the Caputo definition of a fractional derivative using the fractional quantum harmonic oscillator a…

Fractional derivatives are a well-studied generalization of integer order derivatives. Naturally, for optimization, it is of interest to understand the convergence properties of gradient descent using fractional derivatives. Convergence…