In this work, we discuss elasticity equations on a two-dimensional domain with random boundaries ... more 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 R. C. Augustyn, D. Nankivil, A. Mohamed, B. Maceo, F. Pierre, and J.-M. Parel. Human ocular biometry. Exp. Eye Res. 102 (2012), pp. 70–75. doi: 10.1016/j.exer.2012.06.009. F. Ballarin, A. Manzoni, G. Rozza, and S. Salsa. Shape optimization by free-form deformation: Existence results and numerical solution for Stokes flows. J. Sci. Comput. 60.3 (2014), pp. 537–563. doi: 10.1007/s10915-013-9807-8. S. C. Brenner and L.-Y. Sung. Linear finite element methods for planar linear elasticity. Math. Comp. 59 (1992), pp. 321–338. doi: 10.2307/2153060. M. C. Delfour and J.-P. Zolesio. Shapes and geometries. Advances in Design and Control. SIAM, Philadelphia, 2001. doi: 10.1137/1.9780898719826. J. Dick. Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands. Ann. Statist. 39.3...
Bulk defects in silicon wafers are key contributors to solar cell efficiency loss. The identifica... more Bulk defects in silicon wafers are key contributors to solar cell efficiency loss. The identification and characterization of these defects are critical steps in the process of improving the efficiency and reliability of solar cells. In this study, we present the first successful application of machine learning for extraction of defect parameters from temperature- and injection-dependent lifetime spectroscopy (TIDLS). With approximately half a million simulated TIDLS curves, random forest regressors are trained to obtain the defect’s energy level and both capture cross-sections. The high correlation coefficient between predicted and simulated values highlights the usefulness of this novel approach. With no prior knowledge of the physical model at work, the regressor learns the physical limitations of that model. This work pioneers the use of machine learning for lifetime spectroscopy, bringing the newest prowess of artificial intelligence to material quality inspection. It opens a n...
Bulk defects in silicon (Si) wafers are key contributors to solar cell efficiency loss.1 The iden... more Bulk defects in silicon (Si) wafers are key contributors to solar cell efficiency loss.1 The identification and characterization of these defects are critical steps in the process of improving the efficiency and reliability of solar cells. In this study, we present an application of machine learning (ML) for extraction of defect parameters from temperatureand injection-dependent lifetime spectroscopy (TIDLS). A common technique to fit TIDLS measurements data is the defect parameter solution surface (DPSS) method developed by Rein.2 Based on the Shockley-Read-Hall (SRH) equation,3,4 the DPSS method finds the best fit among the combinations of defect parameters: the defect energy level (Et) and the electron (σn) and hole (σp) capture cross-sections. Usually, the DPSS approach results in two possible defect parameter combinations, with one in the upper half of the bandgap and one in the lower half of the bandgap. In this study, using the SRH equation, approximately half a million TIDLS...
Mathematical Models and Methods in Applied Sciences
We propose and analyze deterministic multilevel (ML) approximations for Bayesian inversion of ope... more We propose and analyze deterministic multilevel (ML) approximations for Bayesian inversion of operator equations with uncertain distributed parameters, subject to additive Gaussian measurement data. The algorithms use a ML approach based on deterministic, higher-order quasi-Monte Carlo (HoQMC) quadrature for approximating the high-dimensional expectations, which arise in the Bayesian estimators, and a Petrov–Galerkin (PG) method for approximating the solution to the underlying partial differential equation (PDE). This extends the previous single-level (SL) approach from [J. Dick, R. N. Gantner, Q. T. Le Gia and Ch. Schwab, Higher order quasi-Monte Carlo integration for Bayesian estimation, Report 2016-13, Seminar for Applied Mathematics, ETH Zürich (in review)]. Compared to the SL approach, the present convergence analysis of the ML method requires stronger assumptions on holomorphy and regularity of the countably-parametric uncertainty-to-observation maps of the forward problem. As...
We discuss a Galerkin approximation scheme for the elliptic partial differential equation −∆u + ω... more We discuss a Galerkin approximation scheme for the elliptic partial differential equation −∆u + ω 2 u = f on S n ⊂ R n+1. Here ∆ is the Laplace-Beltrami operator on S n, ω is a non-zero constant and f belongs to C 2k−2 (S n), where k ≥ n/4 + 1, k is an integer. The shifts of a spherical basis function φ with φ ∈ H τ (S n) and τ> 2k ≥ n/2 + 2 are used to construct an approximate solution. An H 1 (S n)error estimate is derived under the assumption that the exact solution u belongs to C 2k (S n). Key words: spherical basis function, Galerkin method
Abstract. In this work, we describe, analyze, and implement a pseudospec-tral quadrature method f... more Abstract. In this work, we describe, analyze, and implement a pseudospec-tral 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 so-lutions 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. 1.
Abstract. Boundary value problems on the unit sphere arise naturally in geophysics and oceanograp... more Abstract. Boundary value problems on the unit sphere arise naturally in geophysics and oceanography when scientists model a physical quantity on large scales. Robust numerical methods play an important role in solving these problems. In this article, we construct numerical solutions to a boundary value problem defined on a spherical sub-domain (with a sufficiently smooth boundary) using radial basis functions (RBF). The error analysis between the exact solution and the approxi-mation is provided. Numerical experiments are presented to confirm theoretical estimates. 1. Introduction. Boundary
this paper we investigate the approximation of a class of parabolic partial differential equation... more this paper we investigate the approximation of a class of parabolic partial differential equations on the unit spheres Sn ⊂ Rn+1 using spherical basis functions. Error estimates in the Sobolev norm are derived.
In this paper, we study a mesh-free method using the Galerkin method with radial basis functions ... more In this paper, we study a mesh-free method using the Galerkin method with radial basis functions (RBFs) for the exterior Neumann problem of the Laplacian with boundary condition on an oblate spheroid. This problem is reformulated as a pseudo-differential equation on the spheroid by using the Dirichlet-to-Neumann map. We show convergence of the Galerkin scheme. Our approach is particularly suitable for handling scattered data. We also propose a fast solution technique based on a domain decomposition method (obtained by the additive Schwarz operator) to precondition the illconditioned matrices arising from the Galerkin scheme. We estimate the condition number of the preconditioned system. Numerical results supporting the theoretical results are presented.
arXiv: Cosmology and Nongalactic Astrophysics, 2019
We introduce a new mathematical tool (a direction-dependent probe) to analyse the randomness of p... more 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 ...
We propose two new approaches for efficiently compressing unstructured data defined on the unit s... more We propose two new approaches for efficiently compressing unstructured data defined on the unit sphere. Both approaches are based upon a meshfree multiscale representation of functions on the unit sphere. This multiscale representation employs compactly supported radial basis functions of different scales. The first approach is based on a simple thresholding strategy after the multiscale representation is computed. The second approach employs a dynamical discarding strategy, where small coefficients are already discarded during the computation of the approximate multiscale representation. We analyse the (additional) error which comes with either compression and provide numerical experiments using topographical data of the earth.
In this work, we discuss elasticity equations on a two-dimensional domain with random boundaries ... more 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 R. C. Augustyn, D. Nankivil, A. Mohamed, B. Maceo, F. Pierre, and J.-M. Parel. Human ocular biometry. Exp. Eye Res. 102 (2012), pp. 70–75. doi: 10.1016/j.exer.2012.06.009. F. Ballarin, A. Manzoni, G. Rozza, and S. Salsa. Shape optimization by free-form deformation: Existence results and numerical solution for Stokes flows. J. Sci. Comput. 60.3 (2014), pp. 537–563. doi: 10.1007/s10915-013-9807-8. S. C. Brenner and L.-Y. Sung. Linear finite element methods for planar linear elasticity. Math. Comp. 59 (1992), pp. 321–338. doi: 10.2307/2153060. M. C. Delfour and J.-P. Zolesio. Shapes and geometries. Advances in Design and Control. SIAM, Philadelphia, 2001. doi: 10.1137/1.9780898719826. J. Dick. Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands. Ann. Statist. 39.3...
Bulk defects in silicon wafers are key contributors to solar cell efficiency loss. The identifica... more Bulk defects in silicon wafers are key contributors to solar cell efficiency loss. The identification and characterization of these defects are critical steps in the process of improving the efficiency and reliability of solar cells. In this study, we present the first successful application of machine learning for extraction of defect parameters from temperature- and injection-dependent lifetime spectroscopy (TIDLS). With approximately half a million simulated TIDLS curves, random forest regressors are trained to obtain the defect’s energy level and both capture cross-sections. The high correlation coefficient between predicted and simulated values highlights the usefulness of this novel approach. With no prior knowledge of the physical model at work, the regressor learns the physical limitations of that model. This work pioneers the use of machine learning for lifetime spectroscopy, bringing the newest prowess of artificial intelligence to material quality inspection. It opens a n...
Bulk defects in silicon (Si) wafers are key contributors to solar cell efficiency loss.1 The iden... more Bulk defects in silicon (Si) wafers are key contributors to solar cell efficiency loss.1 The identification and characterization of these defects are critical steps in the process of improving the efficiency and reliability of solar cells. In this study, we present an application of machine learning (ML) for extraction of defect parameters from temperatureand injection-dependent lifetime spectroscopy (TIDLS). A common technique to fit TIDLS measurements data is the defect parameter solution surface (DPSS) method developed by Rein.2 Based on the Shockley-Read-Hall (SRH) equation,3,4 the DPSS method finds the best fit among the combinations of defect parameters: the defect energy level (Et) and the electron (σn) and hole (σp) capture cross-sections. Usually, the DPSS approach results in two possible defect parameter combinations, with one in the upper half of the bandgap and one in the lower half of the bandgap. In this study, using the SRH equation, approximately half a million TIDLS...
Mathematical Models and Methods in Applied Sciences
We propose and analyze deterministic multilevel (ML) approximations for Bayesian inversion of ope... more We propose and analyze deterministic multilevel (ML) approximations for Bayesian inversion of operator equations with uncertain distributed parameters, subject to additive Gaussian measurement data. The algorithms use a ML approach based on deterministic, higher-order quasi-Monte Carlo (HoQMC) quadrature for approximating the high-dimensional expectations, which arise in the Bayesian estimators, and a Petrov–Galerkin (PG) method for approximating the solution to the underlying partial differential equation (PDE). This extends the previous single-level (SL) approach from [J. Dick, R. N. Gantner, Q. T. Le Gia and Ch. Schwab, Higher order quasi-Monte Carlo integration for Bayesian estimation, Report 2016-13, Seminar for Applied Mathematics, ETH Zürich (in review)]. Compared to the SL approach, the present convergence analysis of the ML method requires stronger assumptions on holomorphy and regularity of the countably-parametric uncertainty-to-observation maps of the forward problem. As...
We discuss a Galerkin approximation scheme for the elliptic partial differential equation −∆u + ω... more We discuss a Galerkin approximation scheme for the elliptic partial differential equation −∆u + ω 2 u = f on S n ⊂ R n+1. Here ∆ is the Laplace-Beltrami operator on S n, ω is a non-zero constant and f belongs to C 2k−2 (S n), where k ≥ n/4 + 1, k is an integer. The shifts of a spherical basis function φ with φ ∈ H τ (S n) and τ> 2k ≥ n/2 + 2 are used to construct an approximate solution. An H 1 (S n)error estimate is derived under the assumption that the exact solution u belongs to C 2k (S n). Key words: spherical basis function, Galerkin method
Abstract. In this work, we describe, analyze, and implement a pseudospec-tral quadrature method f... more Abstract. In this work, we describe, analyze, and implement a pseudospec-tral 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 so-lutions 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. 1.
Abstract. Boundary value problems on the unit sphere arise naturally in geophysics and oceanograp... more Abstract. Boundary value problems on the unit sphere arise naturally in geophysics and oceanography when scientists model a physical quantity on large scales. Robust numerical methods play an important role in solving these problems. In this article, we construct numerical solutions to a boundary value problem defined on a spherical sub-domain (with a sufficiently smooth boundary) using radial basis functions (RBF). The error analysis between the exact solution and the approxi-mation is provided. Numerical experiments are presented to confirm theoretical estimates. 1. Introduction. Boundary
this paper we investigate the approximation of a class of parabolic partial differential equation... more this paper we investigate the approximation of a class of parabolic partial differential equations on the unit spheres Sn ⊂ Rn+1 using spherical basis functions. Error estimates in the Sobolev norm are derived.
In this paper, we study a mesh-free method using the Galerkin method with radial basis functions ... more In this paper, we study a mesh-free method using the Galerkin method with radial basis functions (RBFs) for the exterior Neumann problem of the Laplacian with boundary condition on an oblate spheroid. This problem is reformulated as a pseudo-differential equation on the spheroid by using the Dirichlet-to-Neumann map. We show convergence of the Galerkin scheme. Our approach is particularly suitable for handling scattered data. We also propose a fast solution technique based on a domain decomposition method (obtained by the additive Schwarz operator) to precondition the illconditioned matrices arising from the Galerkin scheme. We estimate the condition number of the preconditioned system. Numerical results supporting the theoretical results are presented.
arXiv: Cosmology and Nongalactic Astrophysics, 2019
We introduce a new mathematical tool (a direction-dependent probe) to analyse the randomness of p... more 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 ...
We propose two new approaches for efficiently compressing unstructured data defined on the unit s... more We propose two new approaches for efficiently compressing unstructured data defined on the unit sphere. Both approaches are based upon a meshfree multiscale representation of functions on the unit sphere. This multiscale representation employs compactly supported radial basis functions of different scales. The first approach is based on a simple thresholding strategy after the multiscale representation is computed. The second approach employs a dynamical discarding strategy, where small coefficients are already discarded during the computation of the approximate multiscale representation. We analyse the (additional) error which comes with either compression and provide numerical experiments using topographical data of the earth.
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Papers by Q. Le Gia