Related papers: A Monotone, Second Order Accurate Scheme for Curva…
We construct and analyze a projection-free linearly implicit method for the approximation of flows of harmonic maps into spheres. The proposed method is unconditionally energy stable and, under a sharp discrete regularity condition,…
We propose a new explicit pseudo-energy and momentum conserving scheme for the time integration of Hamiltonian systems. The scheme, which is formally second-order accurate, is based on two key ideas: the integration during the time-steps of…
In many contemporary optimization problems such as those arising in machine learning, it can be computationally challenging or even infeasible to evaluate an entire function or its derivatives. This motivates the use of stochastic…
We analyze the convergence rate of various momentum-based optimization algorithms from a dynamical systems point of view. Our analysis exploits fundamental topological properties, such as the continuous dependence of iterates on their…
We consider a fourth-order regularization of the curvature flow for an immersed plane curve with fixed boundary, using an elastica-type functional depending on a small positive parameter $\varepsilon$. We show that the approximating flow…
We present a general technique for the analysis of first-order methods. The technique relies on the construction of a duality gap for an appropriate approximation of the objective function, where the function approximation improves as the…
Generative modelling has seen significant advances through simulation-free paradigms such as Flow Matching, and in particular, the MeanFlow framework, which replaces instantaneous velocity fields with average velocities to enable efficient…
For solving unsteady hyperbolic conservation laws on cut cell meshes, the so called small cell problem is a big issue: one would like to use a time step that is chosen with respect to the background mesh and use the same time step on the…
A multi-scale method for the hyperbolic systems governing sediment transport in subcritical case is developed. The scale separation of this problem is due to the fact that the sediment transport is much slower than flow velocity. We first…
Two-fluid plasma flow equations describe the flow of ions and electrons with different densities, velocities, and pressures. We consider the ideal plasma flow i.e. we ignore viscous, resistive, and collision effects. The resulting system of…
We consider the motion by curvature of a network of smooth curves with multiple junctions in the plane, that is, the geometric gradient flow associated to the length functional. Such a flow represents the evolution of a two--dimensional…
We apply the stabilization technique, developed by T. Zelenyak in 1960s for parabolic equations, on curve shortening flow in $\R^3$, and derive several new monotonicity formulas. All of them share one main feature: the dependence of the…
We introduce a simple and efficient numerical method to compute mean curvature flow with obstacles. The method augments the Merrimam-Bence-Osher scheme with a pointwise update that enforces the constraint and therefore retains the…
A quasi-second order scheme is developed to obtain approximate solutions of the shallow water equationswith bathymetry. The scheme is based on a staggered finite volume scheme for the space discretization:the scalar unknowns are located in…
We consider minimization of a smooth nonconvex objective function using an iterative algorithm based on Newton's method and the linear conjugate gradient algorithm, with explicit detection and use of negative curvature directions for the…
This paper synthesizes anytime algorithms, in the form of continuous-time dynamical systems, to solve monotone variational inequalities. We introduce three algorithms that solve this problem: the projected monotone flow, the safe monotone…
The curve shortening flow is a geometric heat equation for curves and provides an accessible setting to illustrate many important concepts from nonlinear partial differential equations, including maximum principle estimates, monotonicity…
Two-fluid relativistic plasma flow equations combine the equations of relativistic hydrodynamics with Maxwell's equations for electromagnetic fields, which involve divergence constraints for the magnetic and electric fields. When developing…
An algorithm for a family of self-starting high-order implicit time integration schemes with controllable numerical dissipation is proposed for both linear and nonlinear transient problems. This work builds on the previous works of the…
Sharpness is an almost generic assumption in continuous optimization that bounds the distance from minima by objective function suboptimality. It facilitates the acceleration of first-order methods through restarts. However, sharpness…