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Ngoc Cuong Nguyen

Publications and source records attributed to Ngoc Cuong Nguyen.

At least 37 records · Page 2Linked to original sources
arXiv2023

Hybridizable discontinuous Galerkin methods for the Monge-Ampere equation

We introduce two hybridizable discontinuous Galerkin (HDG) methods for numerically solving the Monge-Ampere equation. The first HDG method is devised to solve the nonlinear elliptic Monge-Ampere equation by using Newton's method. The second HDG method is devised to solve a sequence of the Poisson equation until convergence to a fixed-point solution of the Monge-Ampere equation is reached. Numerical examples are presented to demonstrate the convergence and accuracy of the HDG methods. Furthermore, the HDG methods are applied to r-adaptive mesh generation by redistributing a given scalar density function via the optimal transport theory. This r-adaptivity methodology leads to the Monge-Ampere equation with a nonlinear Neumann boundary condition arising from the optimal transport of the density function to conform the resulting high-order mesh to the boundary. Hence, we extend the HDG methods to treat the nonlinear Neumann boundary condition. Numerical experiments are presented to illustrate the generation of r-adaptive high-order meshes on planar and curved domains.

math.NA↗
arXiv2023

Exploring Model Complexity in Machine Learned Potentials for Simulated Properties

Machine learning (ML) enables the development of interatomic potentials that promise the accuracy of first principles methods while retaining the low cost and parallel efficiency of empirical potentials. While ML potentials traditionally use atom-centered descriptors as inputs, different models such as linear regression and neural networks can map these descriptors to atomic energies and forces. This begs the question: what is the improvement in accuracy due to model complexity irrespective of choice of descriptors? We curate three datasets to investigate this question in terms of ab initio energy and force errors: (1) solid and liquid silicon, (2) gallium nitride, and (3) the superionic conductor LGPS. We further investigate how these errors affect simulated properties with these models and verify if the improvement in fitting errors corresponds to measurable improvement in property prediction. Since linear and nonlinear regression models have different advantages and disadvantages, the results presented herein help researchers choose models for their particular application. By assessing different models, we observe correlations between fitting quantity (e.g. atomic force) error and simulated property error with respect to ab initio values. Such observations can be repeated by other researchers to determine the level of accuracy, and hence model complexity, needed for their particular systems of interest.

cond-mat.mtrl-sci↗
arXiv2023

A high-order discontinuous Galerkin approach for physics-based thermospheric modeling

The accurate prediction of aerodynamic drag on satellites orbiting in the upper atmosphere is critical to the operational success of modern space technologies, such as satellite-based communication or navigation systems, which have become increasingly popular in the last few years due to the deployment of constellations of satellites in low-Earth orbit. As a result, physics-based models of the ionosphere and thermosphere have emerged as a necessary tool for the prediction of atmospheric outputs under highly variable space weather conditions. This paper proposes a high-fidelity approach for physics-based space weather modeling based on the solution of the Navier-Stokes equations using a high-order discontinuous Galerkin method, combined with a matrix-free strategy suitable for high-performance computing on GPU architectures. The approach consists of a thermospheric model that describes a chemically frozen neutral atmosphere in non-hydrostatic equilibrium driven by the external excitation of the Sun. A novel set of variables is considered to treat the low densities present in the upper atmosphere and to accommodate the wide range of scales present in the problem. At the same time, and unlike most existing approaches, radial and angular directions are treated in a non-segregated approach. The study presents a set of numerical examples that demonstrate the accuracy of the approximation and validate the current approach against observational data along a satellite orbit, including estimates of established empirical and physics-based models of the ionosphere-thermosphere system. Finally, a 1D radial derivation of the physics-based model is presented and utilized for conducting a parametric study of the main thermal quantities under various solar conditions.

physics.space-ph↗
arXiv2022

Weak convergence of Monge-Ampère measures on compact Hermitian manifolds

We give a sufficient condition on a sequence of uniformly bounded $ω$-plurisubharmonic functions, $ω$ being a Hermitian metric, for which the sequence of associated Monge-Ampère measures converges weakly. This criterion can be used to obtained a bounded $ω$-plurisubharmonic solution to the Monge-Ampère equation.

math.CV↗
arXiv2022

The Dirichlet problem for the Monge-Ampère equation on Hermitian manifolds with boundary

We study weak quasi-plurisubharmonic solutions to the Dirichlet problem for the complex Monge-Amère equation on a general Hermitian manifold with non-empty boundary. We prove optimal subsolution theorems: for bounded and Hölder continuous quasi-plurisubharmonic functions. The continuity of the solution is proved for measures that well dominated by capacity, for example measures with $L^p$, $p>1$ densities, or moderate measures in the sense of Dinh-Nguyen-Sibony.

math.DG↗
arXiv2022

Proper Orthogonal Descriptors for Efficient and Accurate Interatomic Potentials

We present the proper orthogonal descriptors for efficient and accuracy representation of the potential energy surface. The potential energy surface is represented as a many-body expansion of parametrized potentials in which the potentials are functions of atom positions and parameters. The Karhunen-Loève (KL) expansion is employed to decompose the parametrized potentials into a set of proper orthogonal descriptors (PODs). Because of the rapid convergence of the KL expansion, relevant snapshots can be sampled exhaustively to represent the atomic neighborhood environment accurately with a small number of descriptors. The proper orthogonal descriptors are used to develop interatomic potentials by using a linear expansion of the descriptors and determining the expansion coefficients from a weighted least-squares regression against a density functional theory (DFT) training set. We present a comprehensive evaluation of the POD potentials on previously published DFT data sets comprising Li, Mo, Cu, Ni, Si, Ge, and Ta elements. The data sets represent a diverse pool of metals, transition metals, and semiconductors. The accuracy of the POD potentials are comparable to that of state-of-the-art machine learning potentials such as the spectral neighbor analysis potential (SNAP) and the atomic cluster expansion (ACE).

cond-mat.mtrl-sci↗
arXiv2022

The complex Sobolev Space and Hölder continuous solutions to Monge-Ampère equations

Let $X$ be a compact Kähler manifold of dimension $n$ and $ω$ a Kähler form on $X$. We consider the complex Monge-Ampère equation $(dd^c u+ω)^n=μ$, where $μ$ is a given positive measure on $X$ of suitable mass and $u$ is an $ω$-plurisubharmonic function. We show that the equation admits a Hölder continuous solution {\it if and only if} the measure $μ$, seen as a functional on a complex Sobolev space $W^*(X)$, is Hölder continuous. A similar result is also obtained for the complex Monge-Ampère equations on domains of $\mathbb{C}^n$.

math.CV↗
arXiv2020

Continuous solutions to Monge-Ampère equations on Hermitian manifolds for measures dominated by capacity

We prove the existence of a continuous quasi-plurisubharmonic solution to the Monge-Ampère equation on a compact Hermitian manifold for a very general measre on the right hand side. We admit measures dominated by capacity in a certain manner, in particular, moderate measures studied by Dinh-Nguyen-Sibony. As a consequence, we give a characterization of measures admitting Hölder continuous quasi-plurisubharmonic potential, inspired by the work of Dinh-Nguyen.

math.CV↗
arXiv2019

Stability and regularity of solutions of the Monge-Ampère equation on Hermitian manifolds

We prove stability of solutions of the complex Monge-Ampère equation on compact Hermitian manifolds, when the right hand side varies in a bounded set in $L^p, p>1$ and it is bounded away from zero. Such solutions are shown to be Hölder continuous. As an application we extend a recent result of Székelyhidi and Tosatti on Kähler-Einstein equation from Kähler to Hermitian manifolds.

math.DG↗
arXiv2018

On the Hölder continuous subsolution problem for the complex Monge-Ampère equation, II

We solve the Dirichlet problem for the complex Monge-Ampère equation on a strictly pseudoconvex with the right hand side being a positive Borel measure which is dominated by the Monge-Ampère measure of a Hölder continuous plurisubharmonic function. If the boundary data is continuous, then the solution is continuous. If the boundary is Hölder continuous, then the solution is also Hölder continuous. In particular, the answer to a question of A. Zeriahi is always affirmative.

math.CV↗
arXiv2017

On the Hölder continuous subsolution problem for the complex Monge-Ampère equation

We give a necessary and sufficient condition for positive Borel measures such that the Dirichlet problem, with zero boundary data, for the complex Monge-Ampère equation admits Hölder continuous plurisubharmonic solutions. In particular, when the subsolution has finite Monge-Ampère total mass, we obtain an affirmative answer to a question of Zeriahi.

math.CV↗
arXiv2016

Weak solutions of complex Hessian equations on compact Hermitian manifolds

We prove the existence of weak solutions of complex $m-$Hessian equations on compact Hermitian manifolds for the nonnegative right hand side belonging to $L^p, p>n/m$ ($n$ is the dimension of the manifold). For smooth, positive data the equation has been recently solved by Szekelyhidi and Zhang. We also give a stability result for such solutions.

math.CV↗
arXiv2015

The complex Monge-Ampère type equation on compact Hermitian manifolds and Applications

We prove the existence and uniqueness of continuous solutions to the complex Monge-Ampère type equation with the right hand side in $L^p$, $p>1$, on compact Hermitian manifolds. Next, we generalise results of Eyssidieux, Guedj and Zeriahi \cite{EGZ09, EGZ11} to compact Hermitian manifolds which {\em a priori} are not in the Fujiki class. These generalisations lead to a number of applications: we obtain partial results on a conjecture of Tosatti and Weinkove \cite{TW12a} and on a weak form of a conjecture of Demailly and Paun \cite{DP04}.

math.DG↗