Related papers: Optimal-transport-based mesh adaptivity on the pla…
Numerical computations of stationary states of fast-rotating Bose-Einstein condensates require high spatial resolution due to the presence of a large number of quantized vortices. In this paper we propose a low-order finite element method…
We survey the (old and new) regularity theory for the Monge-Amp\`ere equation, show its connection to optimal transportation, and describe the regularity properties of a general class of Monge-Amp\`ere type equations arising in that…
A moving mesh finite element method is studied for the numerical solution of a phase-field model for brittle fracture. The moving mesh partial differential equation approach is employed to dynamically track crack propagation. Meanwhile, the…
A structure-preserving Finite Element Method (FEM) for the transport equation in one- and two-dimensional domains is presented. This Distributed Parameter System (DPS) has non-collocated boundary control and observation, and reveals a…
We model incompressible flows with an adaptive stabilized finite element method Stokes flows, which solves a discretely stable saddle-point problem to approximate the velocity-pressure pair. Additionally, this saddle-point problem delivers…
An Adaptive Mesh in Phase Space (AMPS) methodology has been developed for solving multi-dimensional kinetic equations by the discrete velocity method. A Cartesian mesh for both configuration (r) and velocity (v) spaces is produced using a…
The inverse reflector problem aims to design a freeform reflecting surface that can direct the light from a specified source to produce the desired illumination in the target area, which is significant in the field of geometrical…
We consider the asymptotic behavior of solutions to the Monge--Amp\`ere equations with slow convergence rate at infinity and fulfill previous results under faster convergence rate by Bao--Li--Zhang [Calc. Var PDE. 52(2015). pp. 39-63].…
We introduce and analyze a statistical estimator for Monge transport maps: solutions to the quadratic optimal transport problem in Euclidean space. For absolutely continuous source measures, this map is uniquely defined as the gradient of a…
We present a numerical method to solve the optimal transport problem with a quadratic cost when the source and target measures are periodic probability densities. This method is based on a numerical resolution of the corresponding…
The multimesh finite element method is a technique for solving partial differential equations on multiple non-matching meshes by enforcing interface conditions using Nitsche's method. Since the non-matching meshes can result in arbitrarily…
The multimesh finite element method enables the solution of partial differential equations on a computational mesh composed by multiple arbitrarily overlapping meshes. The discretization is based on a continuous--discontinuous function…
Chemical accuracy serves as an important metric for assessing the effectiveness of the numerical method in Kohn--Sham density functional theory. It is found that to achieve chemical accuracy, not only the Kohn--Sham wavefunctions but also…
Chaotic free surface flows are challenging problems to simulate numerically, mainly due to the significant changes in geometry and frequent topological changes. Methods that track the evolution of the fluid in a Lagrangian formulation are a…
The Monge-Amp\`ere equation is a fully nonlinear partial differential equation (PDE) of fundamental importance in analysis, geometry and in the applied sciences. In this paper we solve the Dirichlet problem associated with the…
This work describes a concise algorithm for the generation of triangular meshes with the help of standard adaptive finite element methods. We demonstrate that a generic adaptive finite element solver can be repurposed into a triangular mesh…
During recent decades, there has been a substantial development in optimal mass transport theory and methods. In this work, we consider multi-marginal problems wherein only partial information of each marginal is available, which is a setup…
A surface moving mesh method is presented for general surfaces with or without explicit parameterization. The method can be viewed as a nontrivial extension of the moving mesh partial differential equation method that has been developed for…
In this paper we present a new GPU-oriented mesh optimization method based on high-order finite elements. Our approach relies on node movement with fixed topology, through the Target-Matrix Optimization Paradigm (TMOP) and uses a global…
Optimal mass transport, also known as the earth mover's problem, is an optimization problem with important applications in various disciplines, including economics, probability theory, fluid dynamics, cosmology and geophysics to cite a few.…