Related papers: On time-splitting methods for gradient flows with …
We construct the time evolution for states of Glauber dynamics for a spatial infinite particle system in terms of generating functionals. This is carried out by an Ovsjannikov-type result in a scale of Banach spaces, leading to a local (in…
In this article we investigate driven dissipative quantum dynamics of an ensemble of two-level systems given by a Markovian master equation with collective and non-collective dissipators. Exploiting the permutation symmetry in our model, we…
We consider two-dimensional flows above topography, revisiting the selective decay (or minimum-enstrophy) hypothesis of Bretherton and Haidvogel. We derive a 'condensed branch' of solutions to the variational problem where a domain-scale…
Recent evidence has shown the existence of a so-called double-descent and even triple-descent behavior for the generalization error of deep-learning models. This important phenomenon commonly appears in implemented neural network…
We consider a one-dimensional kinetic model of granular media in the case where the interaction potential is quadratic. Taking advan- tage of a simple first integral, we can use a reformulation (equivalent to the initial kinetic model for…
In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a $\Lambda$-convex energy functional featuring random and rapidly…
In approximating solutions of nonstationary problems, various approaches are used to compute the solution at a new time level from a number of simpler (sub-)problems. Among these approaches are splitting methods. Standard splitting schemes…
In this work, we study a phase-field model for curvature-driven pattern formation in biomembranes. The model is derived as a gradient flow of an energy functional that approximates the two-phase Canham--Helfrich energy. This leads to a…
The Phase-Field Method (PFM) is employed to simulate two-phase flows with the fully-coupled Cahn-Hilliard-Navier-Stokes (CHNS) equations governing the temporal evolution. The methodology minimizes the total energy functional, accounting for…
Many large-scale and distributed optimization problems can be brought into a composite form in which the objective function is given by the sum of a smooth term and a nonsmooth regularizer. Such problems can be solved via a proximal…
In this paper we present a novel multiscale splitting approach to solve multiscale Schroedinger equation, which have large different time-scales. The energy potential is based on highly oscillating functions, which are magnitudes faster…
We study the systematic numerical approximation of a class of Allen-Cahn type problems modeling the motion of phase interfaces. The common feature of these models is an underlying gradient flow structure which gives rise to a decay of an…
Gradient descent-ascent (GDA) flows play a central role in finding saddle points of bivariate functionals, with applications in optimization, game theory, and robust control. While they are well-understood in Hilbert and Banach spaces via…
In this paper we analyze a broad class of abstract doubly nonlinear evolution equations in Banach spaces, driven by nonsmooth and nonconvex energies. We provide some general sufficient conditions, on the dissipation potential and the energy…
This paper addresses the gradient flow -- the continuous-time representation of the gradient method -- with the smooth approximation of a non-differentiable objective function and presents convergence analysis framework. Similar to the…
In this work, we investigate a stochastic gradient descent method for solving inverse problems that can be written as systems of linear or nonlinear ill-posed equations in Banach spaces. The method uses only a randomly selected equation at…
We propose a variational form of the BDF2 method as an alternative to the commonly used minimizing movement scheme for the time-discrete approximation of gradient flows in abstract metric spaces. Assuming uniform semi-convexity --- but no…
This contribution is dedicated to the exploration of exponential operator splitting methods for the time integration of evolution equations. It entails the review of previous achievements as well as the depiction of novel results. The…
We investigate the test risk of continuous-time stochastic gradient flow dynamics in learning theory. Using a path integral formulation we provide, in the regime of a small learning rate, a general formula for computing the difference…
A nonlinear diffusion equation, interpreted as a Wasserstein gradient flow, is numerically solved in one space dimension using a higher-order minimizing movement scheme based on the BDF (backward differentiation formula) discretization. In…