Quoc Thong Le Gia
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Quoc Thong Le Gia has a total of 69 co-authors. Here are the top co-authors by number of shared publications:
- Ian H. Sloan
School of Mathematics and Statistics, UNSW Sydney, Sydney, Australia
19 shared publications - Josef Dick
School of Mathematics and Statistics, University of New South Wales, Sydney, Australia
11 shared publications - Thanh Tran
School of Mathematics and Statistics, The University of New South Wales, Sydney, Australia
8 shared publications - Holger Wendland
Chair of Applied and Numerical Analysis, Department of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany
7 shared publications Christoph Schwab
6 shared publications- Simon Hubbert
Birkbeck, University of London
6 shared publications
Latest Publications
- Open Access
Vector-Valued Gaussian Processes for Approximating Divergence- or Rotation-free Vector Fields
In this paper, we discuss vector-valued Gaussian processes for the approximation of divergence- or rotation-free functions. We establish the theory for such Gaussian processes, then link the theory to multivariate approximation theory, and finally give error estimates for the predictive mean in various situations.
Open AccessVector-Valued Gaussian Processes for Approximating Divergence- or Rotation-free Vector Fields
In this paper, we discuss vector-valued Gaussian processes for the approximation of divergence- or rotation-free functions. We establish the theory for such Gaussian processes, then link the theory to multivariate approximation theory, and finally give error estimates for the predictive mean in various situations.
- Open Access
- 6 November 2025
- Published by Elsevier in Applied Numerical Mathematics
- Vol. 221, 18-29
A simple modification to mitigate locking in conforming FEM for nearly incompressible elasticity
- 6 November 2025
- Published by Elsevier in Applied Numerical Mathematics
- Vol. 221, 18-29
Open AccessA simple modification to mitigate locking in conforming FEM for nearly incompressible elasticity
- 5 September 2024
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Imaging Sciences
- Vol. 17 (3) , 1820-1843
Removing the Mask—Reconstructing a Real-Valued Field on the Sphere from a Masked Field by Spherical Fourier Analysis
The paper analyzes a spectral approach to reconstructing a scalar field on the sphere, given only information about a masked version of the field, together with precise information about the (smooth) mask. The theory is developed for a general mask and later specialized to the case of an axially symmetric mask. Numerical experiments are given for the case of an axial mask motivated by the cosmic microwave background, assuming that the underlying field is a realization of a Gaussian random field with an artificial angular power spectrum of moderate degree ([Formula: see text]). The recovery is highly satisfactory in the absence of noise and even in the presence of moderate noise.
- 5 September 2024
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Imaging Sciences
- Vol. 17 (3) , 1820-1843
Removing the Mask—Reconstructing a Real-Valued Field on the Sphere from a Masked Field by Spherical Fourier Analysis
The paper analyzes a spectral approach to reconstructing a scalar field on the sphere, given only information about a masked version of the field, together with precise information about the (smooth) mask. The theory is developed for a general mask and later specialized to the case of an axially symmetric mask. Numerical experiments are given for the case of an axial mask motivated by the cosmic microwave background, assuming that the underlying field is a realization of a Gaussian random field with an artificial angular power spectrum of moderate degree ([Formula: see text]). The recovery is highly satisfactory in the absence of noise and even in the presence of moderate noise.
- Open Access
- 6 March 2024
- Published by Elsevier in Journal of Computational and Applied Mathematics
- Vol. 446
On approximation for time-fractional stochastic diffusion equations on the unit sphere
- 6 March 2024
- Published by Elsevier in Journal of Computational and Applied Mathematics
- Vol. 446
Open AccessOn approximation for time-fractional stochastic diffusion equations on the unit sphere
- Open Access
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 H\"{o}lder continuous random field. Numerical examples and simulations inspired by the cosmic microwave background (CMB) are given to illustrate the theoretical findings.Open AccessOn 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 H\"{o}lder continuous random field. Numerical examples and simulations inspired by the cosmic microwave background (CMB) are given to illustrate the theoretical findings. - 28 October 2023
- Published by Australian Mathematical Publishing Association, Inc. in The ANZIAM Journal
- Vol. 64, C32–C44
Numerical solutions to an inverse problem for a non-linear Helmholtz equation
In this work, we develop numerical methods to solve forward and inverse wave problems for a nonlinear Helmholtz equation defined in a spherical shell between two concentric spheres centred at the origin. A spectral method is developed to solve the forward problem while a combination of a finite difference approximation and the least squares method are derived for the inverse problem. Numerical examples are given to verify the method. References R. Askey. Orthogonal polynomials and special functions. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, 1975. doi: 10.1137/1.9781611970470 G. Baruch, G. Fibich, and S. Tsynkov. High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension. Nonlinear Photonics. Optica Publishing Group, 2007. doi: 10.1364/np.2007.ntha6 G. Fibich and S. Tsynkov. High-Order Two-Way Artificial Boundary Conditions for Nonlinear Wave Propagation with Backscattering. J. Comput. Phys. 171 (2001), pp. 632–677. doi: 10.1006/jcph.2001.6800 G. Fibich and S. Tsynkov. Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions. J. Comput. Phys. 210 (2005), pp. 183–224. doi: 10.1016/j.jcp.2005.04.015 P. M. Morse and K. U. Ingard. Theoretical Acoustics. International Series in Pure and Applied Physics. McGraw-Hill Book Company, 1968 G. N. Watson. A treatise on the theory of Bessel functions. International Series in Pure and Applied Physics. Cambridge Mathematical Library, 1996. url: https://www.cambridge.org/au/universitypress/subjects/mathematics/real-and-complex-analysis/treatise-theory-bessel-functions-2nd-edition-1?format=PB&isbn=9780521483919
- 28 October 2023
- Published by Australian Mathematical Publishing Association, Inc. in The ANZIAM Journal
- Vol. 64, C32–C44
Numerical solutions to an inverse problem for a non-linear Helmholtz equation
In this work, we develop numerical methods to solve forward and inverse wave problems for a nonlinear Helmholtz equation defined in a spherical shell between two concentric spheres centred at the origin. A spectral method is developed to solve the forward problem while a combination of a finite difference approximation and the least squares method are derived for the inverse problem. Numerical examples are given to verify the method. References R. Askey. Orthogonal polynomials and special functions. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, 1975. doi: 10.1137/1.9781611970470 G. Baruch, G. Fibich, and S. Tsynkov. High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension. Nonlinear Photonics. Optica Publishing Group, 2007. doi: 10.1364/np.2007.ntha6 G. Fibich and S. Tsynkov. High-Order Two-Way Artificial Boundary Conditions for Nonlinear Wave Propagation with Backscattering. J. Comput. Phys. 171 (2001), pp. 632–677. doi: 10.1006/jcph.2001.6800 G. Fibich and S. Tsynkov. Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions. J. Comput. Phys. 210 (2005), pp. 183–224. doi: 10.1016/j.jcp.2005.04.015 P. M. Morse and K. U. Ingard. Theoretical Acoustics. International Series in Pure and Applied Physics. McGraw-Hill Book Company, 1968 G. N. Watson. A treatise on the theory of Bessel functions. International Series in Pure and Applied Physics. Cambridge Mathematical Library, 1996. url: https://www.cambridge.org/au/universitypress/subjects/mathematics/real-and-complex-analysis/treatise-theory-bessel-functions-2nd-edition-1?format=PB&isbn=9780521483919
- 23 December 2022
- Published by Springer Nature in Journal of Scientific Computing
- Vol. 94 (1) , 1-30
A Multiscale RBF Method for Severely Ill-Posed Problems on Spheres
- 23 December 2022
- Published by Springer Nature in Journal of Scientific Computing
- Vol. 94 (1) , 1-30
A Multiscale RBF Method for Severely Ill-Posed Problems on Spheres
- 9 March 2022
- Published by Australian Mathematical Publishing Association, Inc. in The ANZIAM Journal
- Vol. 62, C256-C272
Elasticity equations with random domains—the shape derivative approach
In this work, we discuss elasticity equations on a two-dimensional domain with random boundaries and we apply these equations to modelling human corneas. References
- 9 March 2022
- Published by Australian Mathematical Publishing Association, Inc. in The ANZIAM Journal
- Vol. 62, C256-C272
Elasticity equations with random domains—the shape derivative approach
In this work, we discuss elasticity equations on a two-dimensional domain with random boundaries and we apply these equations to modelling human corneas. References
- Open Access
FaVeST: Fast Vector Spherical Harmonic Transforms
Vector spherical harmonics on the unit sphere of $\mathbb{R}^3$ have broad applications in geophysics, quantum mechanics and astrophysics. In the representation of a tangent vector field, one needs to evaluate the expansion and the Fourier coefficients of vector spherical harmonics. In this paper, we develop fast algorithms (FaVeST) for vector spherical harmonic transforms on these evaluations. The forward FaVeST evaluates the Fourier coefficients and has a computational cost proportional to $N\log \sqrt{N}$ for $N$ number of evaluation points. The adjoint FaVeST which evaluates a linear combination of vector spherical harmonics with a degree up to $\sqrt{M}$ for $M$ evaluation points has cost proportional to $M\log\sqrt{M}$. Numerical examples of simulated tangent fields illustrate the accuracy, efficiency and stability of FaVeST.Open AccessFaVeST: Fast Vector Spherical Harmonic Transforms
Vector spherical harmonics on the unit sphere of $\mathbb{R}^3$ have broad applications in geophysics, quantum mechanics and astrophysics. In the representation of a tangent vector field, one needs to evaluate the expansion and the Fourier coefficients of vector spherical harmonics. In this paper, we develop fast algorithms (FaVeST) for vector spherical harmonic transforms on these evaluations. The forward FaVeST evaluates the Fourier coefficients and has a computational cost proportional to $N\log \sqrt{N}$ for $N$ number of evaluation points. The adjoint FaVeST which evaluates a linear combination of vector spherical harmonics with a degree up to $\sqrt{M}$ for $M$ evaluation points has cost proportional to $M\log\sqrt{M}$. Numerical examples of simulated tangent fields illustrate the accuracy, efficiency and stability of FaVeST. - Open Access
- 11 July 2020
- Published by Springer Nature in Applied Mathematics & Optimization
- Vol. 84 (2) , 1971-2035
Stochastic Navier–Stokes Equations on a Thin Spherical Domain
Incompressible Navier–Stokes equations on a thin spherical domain $$Q_\varepsilon $$ along with free boundary conditions under a random forcing are considered. The convergence of the martingale solution of these equations to the martingale solution of the stochastic Navier–Stokes equations on a sphere $$\mathbb {S}^2$$ as the thickness converges to zero is established.- 11 July 2020
- Published by Springer Nature in Applied Mathematics & Optimization
- Vol. 84 (2) , 1971-2035
Open AccessStochastic Navier–Stokes Equations on a Thin Spherical Domain
Incompressible Navier–Stokes equations on a thin spherical domain $$Q_\varepsilon $$ along with free boundary conditions under a random forcing are considered. The convergence of the martingale solution of these equations to the martingale solution of the stochastic Navier–Stokes equations on a sphere $$\mathbb {S}^2$$ as the thickness converges to zero is established.