arXiv subjects

Q. T. Le Gia

Publications and source records attributed to Q. T. Le Gia.

11 recordsLinked to original sources

arXiv2026

Spectral Bounds for Kernel Quadrature

A bottleneck in the theory of kernel methods in machine learning is the storage requirement. To ameliorate this, a standard trick is to replace the kernel with an explicit feature map. Perhaps, the most well known example is the Gaussian kernel which can be expressed in terms of the Fourier features. Analytically, the kernel $K$ can be expressed in terms of an integral expression that involves a possibly asymmetric kernel $G$ representing the feature map. Numerically, one needs to approximate this integral by a suitable numerical integration scheme, typically Monte Carlo. In this paper, we demonstrate that the eigenvalues of $K$ are approximated much better by the eigenvalues of the kernel obtained by discretizing the integral using suitable quadrature formulas instead. We illustrate this fact in the case of the Gaussian kernel and neural tangent kernels on the unit sphere of a four-dimensional Euclidean space corresponding to the sigmoid and ReLU activation functions.

math.NA↗

arXiv2025

Bayesian inference calibration of the modulus of elasticity

This work uses the Bayesian inference technique to infer the Young modulus from the stochastic linear elasticity equation. The Young modulus is modeled by a finite Karhunen Loéve expansion, while the solution to the linear elasticity equation is approximated by the finite element method. The high-dimensional integral involving the posterior density and the quantity of interest is approximated by a higher-order quasi-Monte Carlo method.

math.NA↗

arXiv2025

A simple modification to mitigate locking in conforming FEM for nearly incompressible elasticity

Due to the divergence-instability, the accuracy of low-order conforming finite element methods (FEMs) for nearly incompressible elasticity equations deteriorates as the Lamé parameter $λ\to\infty$, or equivalently as the Poisson ratio $ν\to1/2$. This effect is known as {\itshape locking} or {\itshape non-robustness}. For the piecewise linear case, the error in the ${\bf L}^2$-norm of the standard Galerkin conforming FEM is bounded by~$Cλh^2$, resulting in poor accuracy for practical values of~$h$ if $λ$ is sufficiently large. In this short paper, we show that the locking phenomenon can be reduced by replacing $λ$ with~$λ_h=λμ/(μ+λh/L)<λ$ in the stiffness matrix, where $μ$ is the second Lamé parameter and $L$ is the diameter of the body $Ω$. We prove that with this modification, the error in the ${\bf L}^2$-norm is bounded by $Ch$ for a constant $C$ that does not depend on $λ$. Numerical experiments confirm this convergence behaviour and show that, for practical meshes, our method is more accurate than the standard method if $λ$ is larger than about $μL/h$. Our analysis also shows that the error in the ${\bf H}^1$-norm is bounded by $Cλ_h^{1/2}\,h$, which improves the $Cλ^{1/2}\,h$ estimate for the case of conforming FEM.

math.NA↗

arXiv2024

Quasi-Monte Carlo sparse grid Galerkin finite element methods for linear elasticity equations with uncertainties

We explore a linear inhomogeneous elasticity equation with random Lamé parameters. The latter are parameterized by a countably infinite number of terms in separated expansions. The main aim of this work is to estimate expected values (considered as an infinite dimensional integral on the parametric space corresponding to the random coefficients) of linear functionals acting on the solution of the elasticity equation. To achieve this, the expansions of the random parameters are truncated, a high-order quasi-Monte Carlo (QMC) is combined with a sparse grid approach to approximate the high dimensional integral, and a Galerkin finite element method (FEM) is introduced to approximate the solution of the elasticity equation over the physical domain. The error estimates from (1) truncating the infinite expansion, (2) the Galerkin FEM, and (3) the QMC sparse grid quadrature rule are all studied. For this purpose, we show certain required regularity properties of the continuous solution with respect to both the parametric and physical variables. To achieve our theoretical regularity and convergence results, some reasonable assumptions on the expansions of the random coefficients are imposed. Finally, some numerical results are delivered.

math.NA↗

arXiv2024

On approximation for time-fractional stochastic diffusion equations on the unit sphere

This paper develops a two-stage stochastic model to investigate evolution of random fields on the unit sphere $\bS^2$ in $\R^3$. The model is defined by a time-fractional stochastic diffusion equation on $\bS^2$ governed by a diffusion operator with the time-fractional derivative defined in the Riemann-Liouville sense. In the first stage, the model is characterized by a homogeneous problem with an isotropic Gaussian random field on $\bS^2$ as an initial condition. In the second stage, the model becomes an inhomogeneous problem driven by a time-delayed Brownian motion on $\bS^2$. The solution to the model is given in the form of an expansion in terms of complex spherical harmonics. An approximation to the solution is given by truncating the expansion of the solution at degree $L\geq1$. The rate of convergence of the truncation errors as a function of $L$ and the mean square errors as a function of time are also derived. It is shown that the convergence rates depend not only on the decay of the angular power spectrum of the driving noise and the initial condition, but also on the order of the fractional derivative. We study sample properties of the stochastic solution and show that the solution is an isotropic Hölder continuous random field. Numerical examples and simulations inspired by the cosmic microwave background (CMB) are given to illustrate the theoretical findings.

math.PR↗

arXiv2023

Numerical solutions to an inverse problem for a non-linear Helmholtz equation

In this work, we construct numerical solutions to an inverse problem of a nonlinear Helmholtz equation defined in a spherical shell between two concentric spheres centered at the origin.Assuming that the values of the forward problem are known at sufficiently many points, we would like to determine the form of the non-linear term on the right-hand side of the equation via its Chebyshev coefficients.

math.NA↗

arXiv2020

A New Probe of Gaussianity and Isotropy applied to the CMB Maps

We introduce a new mathematical tool (a direction-dependent probe) to analyse the randomness of purported isotropic Gaussian random fields on the sphere. We apply the probe to assess the full-sky cosmic microwave background (CMB) temperature maps produced by the {\it Planck} collaboration (PR2 2015 and PR3 2018), with special attention to the inpainted maps. To study the randomness of the fields represented by each map we use the autocorrelation of the sequence of probe coefficients (which are just the full-sky Fourier coefficients $a_{\ell,0}$ if the $z$ axis is taken in the probe direction). If the field is {isotropic and Gaussian} then the probe coefficients for a given direction should be realisations of uncorrelated scalar Gaussian random variables. We introduce a particular function on the sphere (called the \emph{AC discrepancy}) that accentuates the departure from Gaussianity and isotropy. We find that for some of the maps, there are many directions for which the departures are significant, especially near the galactic plane. We also study the effect of varying the highest multipole used to calculate the AC discrepancy from the initial value of $1500$ to $2500$. In the case of Commander 2015, the AC discrepancy now exhibits antipodal "blobs" well away from the galactic plane. Finally, we look briefly at the non-inpainted Planck maps, for which the computed AC discrepancy maps have a very different character, with features that are global rather than local. For the particular case of the non-inpainted 2018 \texttt{SEVEM} map (which has visible equatorial pollution), we model with partial success the observed behaviour by an isotropic Gaussian random field added to a non-random needlet-like structure located near the galactic centre.

astro-ph.CO↗

arXiv2012

Solving parabolic equations on the unit sphere via Laplace transforms and radial basis functions

We propose a method to construct numerical solutions of parabolic equations on the unit sphere. The time discretization uses Laplace transforms and quadrature. The spatial approximation of the solution employs radial basis functions restricted to the sphere. The method allows us to construct high accuracy numerical solutions in parallel. We establish $L_2$ error estimates for smooth and nonsmooth initial data, and describe some numerical experiments.

math.NA↗

arXiv2010

Stability and preconditioning for a hybrid approximation on the sphere

This paper proposes a new preconditioning scheme for a linear system with a saddle-point structure arising from a hybrid approximation scheme on the sphere, an approximation scheme that combines (local) spherical radial basis functions and (global) spherical polynomials. Making use of a recently derived inf-sup condition [13] and the Brezzi stability and convergence theorem for this approximation scheme, we show that the linear system can be optimally preconditioned with a suitable block-diagonal preconditioner. Numerical experiments with a non-uniform distribution of data points support the theoretical conclusions.

math.NA↗

arXiv2010

A pseudospectral quadrature method for Navier-Stokes equations on rotating spheres

In this work, we describe, analyze, and implement a pseudospectral quadrature method for a global computer modeling of the incompressible surface Navier-Stokes equations on the rotating unit sphere. Our spectrally accurate numerical error analysis is based on the Gevrey regularity of the solutions of the Navier-Stokes equations on the sphere. The scheme is designed for convenient application of fast evaluation techniques such as the fast Fourier transform (FFT), and the implementation is based on a stable adaptive time discretization.

math.NA↗

arXiv2008

Localized linear polynomial operators and quadrature formulas on the sphere

The purpose of this paper is to construct universal, auto--adaptive, localized, linear, polynomial (-valued) operators based on scattered data on the (hyper--)sphere $\SS^q$ ($q\ge 2$). The approximation and localization properties of our operators are studied theoretically in deterministic as well as probabilistic settings. Numerical experiments are presented to demonstrate their superiority over traditional least squares and discrete Fourier projection polynomial approximations. An essential ingredient in our construction is the construction of quadrature formulas based on scattered data, exact for integrating spherical polynomials of (moderately) high degree. Our formulas are based on scattered sites; i.e., in contrast to such well known formulas as Driscoll--Healy formulas, we need not choose the location of the sites in any particular manner. While the previous attempts to construct such formulas have yielded formulas exact for spherical polynomials of degree at most 18, we are able to construct formulas exact for spherical polynomials of degree 178.

math.NA↗