相关论文: Gradient Flows for the $p$-Laplacian Arising from …
We present a structure-preserving Eulerian algorithm for solving $L^2$-gradient flows and a structure-preserving Lagrangian algorithm for solving generalized diffusions. Both algorithms employ neural networks as tools for spatial…
We study the effects of relaxational dynamics on congestion pressure in scale free networks by analyzing the properties of the corresponding gradient networks (Z. Toroczkai, K. E. Bassler, Nature {\bf 428}, 716 (2004)). Using the Family…
We develop robust and scalable fully implicit nonlinear finite element solvers for the simulations of biological transportation networks driven by the gradient flow minimization of a non-convex energy cost functional. Our approach employs a…
We introduce a class of unconditionally energy stable, high order accurate schemes for gradient flows in a very general setting. The new schemes are a high order analogue of the minimizing movements approach for generating a time discrete…
We present new analytical and numerical results for the elliptic-parabolic system of partial differential equations proposed by Hu and Cai, which models the formation of biological transport networks. The model describes the pressure field…
By using the Onsager principle as an approximation tool, we give a novel derivation for the moving finite element method for gradient flow equations. We show that the discretized problem has the same energy dissipation structure as the…
We study self-regulating processes modeling biological transportation networks. Firstly, we write the formal $L^2$-gradient flow for the symmetric tensor valued diffusivity $D$ of a broad class of entropy dissipations associated with a…
We introduce an innovative numerical technique based on convex optimization to solve a range of infinite dimensional variational problems arising from the application of the background method to fluid flows. In contrast to most existing…
An extended Maxwell viscoelastic model with a relaxation parameter is studied from mathematical and numerical points of view. It is shown that the model has a gradient flow property with respect to a viscoelastic energy. Based on the…
A finite element discretization is developed for the Cai-Hu model, describing the formation of biological networks. The model consists of a non linear elliptic equation for the pressure $p$ and a non linear reaction-diffusion equation for…
In this paper, we develop an energy dissipative numerical scheme for gradient flows of planar curves, such as the curvature flow and the elastic flow. Our study presents a general framework for solving such equations. To discretize time, we…
Contrary to many recent models of growing networks, we present a model with fixed number of nodes and links, where it is introduced a dynamics favoring the formation of links between nodes with degree of connectivity as different as…
Accelerated gradient methods are the cornerstones of large-scale, data-driven optimization problems that arise naturally in machine learning and other fields concerning data analysis. We introduce a gradient-based optimization framework for…
Much of the structure of macroscopic evolution equations for relaxation to equilibrium can be derived from symmetries in the dynamical fluctuations around the most typical trajectory. For example, detailed balance as expressed in terms of…
We prove the well-posedness of entropy solutions for a wide class of nonlocal transport equations with nonlinear mobility in one spatial dimension. The solution is obtained as the limit of approximations constructed via a deterministic…
A new class of integro-partial differential equation models is derived for the prediction of granular flow dynamics. These models are obtained using a novel limiting averaging method (inspired by techniques employed in the derivation of…
We review the theory of Gradient Flows in the framework of convex and lower semicontinuous functionals on ${\sf CAT}(\kappa)$-spaces and prove that they can be characterized by the same differential inclusion $y_t'\in-\partial^-{\sf…
If gradient systems depend on a microstructure, we want to derive a macroscopic gradient structure describing the effective behavior of the microscopic effects. We introduce a notion of evolutionary Gamma-convergence that relates the…
Networks with a prescribed power-law scaling in the spectrum of the graph Laplacian can be generated by evolutionary optimization. The Laplacian spectrum encodes the dynamical behavior of many important processes. Here, the networks are…
In this paper we make a study of a partial integral differential equation with $p$-Laplacian using a mixed finite element method. Two stable and convergent fixed point schemes are proposed to solve the nonlinear algebraic system. Using the…