Related papers: The Geometry of r-adaptive meshes generated using …
This work proposes an $r$-adaptive finite element method (FEM) using neural networks (NNs). The method employs the Ritz energy functional as the loss function, currently limiting its applicability to symmetric and coercive problems, such as…
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…
We consider the geometric optics problem of finding a system of two reflectors that transform a spherical wavefront into a beam of parallel rays with prescribed intensity distribution. Using techniques from optimal transportation theory, it…
We propose a volumetric formulation for computing the Optimal Transport problem defined on surfaces in $\mathbb{R}^3$, found in disciplines like optics, computer graphics, and computational methodologies. Instead of directly tackling the…
Mesh adaptation for finite element approximation is a procedure used in numerous applications. The use of thin and long anisotropic triangles improves the efficiency of the procedure. When piecewise linear finite elements are used, the…
This paper deals with dynamical optimal transport metrics defined by spatial discretisation of the Benamou--Benamou formula for the Kantorovich metric $W_2$. Such metrics appear naturally in discretisations of $W_2$-gradient flow…
Monotone gradient functions play a central role in solving the Monge formulation of the optimal transport (OT) problem, which arises in modern applications ranging from fluid dynamics to robot swarm control. When the transport cost is the…
We propose a simple dynamic model of polymers under shear with an anisotropic mobility tensor. We calculate the shear viscosity, the rheo-dielectric response function, and the parallel relaxation modulus under shear flow deduced from our…
Different observations of a relation between inputs ("sources") and outputs ("targets") are often reported in terms of histograms (discretizations of the source and the target densities). Transporting these densities to each other provides…
We consider the numerical construction of minimal Lagrangian graphs, which is related to recent applications in materials science, molecular engineering, and theoretical physics. It is known that this problem can be formulated as an…
In this paper we consider Monge-Amp\`ere equations on compact Hessian manifolds, or equivalently Monge-Amp\`ere equations on certain unbounded convex domains $\Omega\subseteq \mathbb{R}^n$, with a periodicity constraint given by the action…
In many applications, Bayesian inverse problems can give rise to probability distributions which contain complexities due to the Hessian varying greatly across parameter space. This complexity often manifests itself as lower dimensional…
The question of which costs admit unique optimizers in the Monge-Kantorovich problem of optimal transportation between arbitrary probability densities is investigated. For smooth costs and densities on compact manifolds, the only known…
We demonstrate an atomic force microscopy based method for estimation of defect density by identification of threading dislocations on a non-flat surface resulting from metamorphic growth. The discussed technique can be applied as an…
The validity of two approximative linear tensor models to be used for grating based X-ray or neutron dark-field tensor tomography is investigated in a simulation study. While the dark-field contrast originating from anisotropic microscopic…
In machine learning, Optimal Transport (OT) theory is extensively utilized to compare probability distributions across various applications, such as graph data represented by node distributions and image data represented by pixel…
We investigate a new multi-marginal optimal transport problem arising from a dissociation model in the Strong Interaction Limit of Density Functional Theory. In this short note, we introduce such dissociation model, the corresponding…
The Adaptive Smoothing Method (ASM) is a data-driven approach for traffic state estimation. It interpolates unobserved traffic quantities by smoothing measurements along spatio-temporal directions defined by characteristic traffic wave…
Many numerical and learning algorithms rely on the solution of the Monge-Kantorovich problem and Wasserstein distances, which provide appropriate distributional metrics. While the natural approach is to treat the problem as an…
We study multi-marginal optimal transport problems from a probabilistic graphical model perspective. We point out an elegant connection between the two when the underlying cost for optimal transport allows a graph structure. In particular,…