arXiv subjects

Thanh Tran

Publications and source records attributed to Thanh Tran.

60 records · Page 4Linked to original sources

arXiv2014

A mixed discontinuous Galerkin method for the time harmonic elasticity problem with reduced symmetry

The aim of this paper is to analyze a mixed discontinuous Galerkin discretization of the time-harmonic elasticity problem. The symmetry of the Cauchy stress tensor is imposed weakly, as in the traditional dual-mixed setting. We show that the discontinuous Galerkin scheme is well-posed and uniformly stable with respect to the mesh parameter $h$ and the Lamé coefficient $λ$. We also derive optimal a-priori error bounds in the energy norm. Several numerical tests are presented in order to illustrate the performance of the method and confirm the theoretical results.

math.NA↗

arXiv2014

A shape calculus based method for a transmission problem with random interface

The present work is devoted to approximation of the statistical moments of the unknown solution of a class of elliptic transmission problems in $\mathbb R^3$ with randomly perturbed interfaces. Within this model, the diffusion coefficient has a jump discontinuity across the random transmission interface which models linear diffusion in two different media separated by an uncertain surface. We apply the shape calculus approach to approximate solution's perturbation by the so-called shape derivative, correspondingly statistical moments of the solution's perturbation are approximated by the moments of the shape derivative. We characterize the shape derivative as a solution of a related homogeneous transmission problem with nonzero jump conditions which can be solved with the aid of boundary integral equations. We develop a rigorous theoretical framework for this method, particularly i) extending the method to the case of unbounded domains and ii) closing the gaps and clarifying and adapting results in the existing literature. The theoretical findings are supported by and illustrated in two particular examples.

math.NA↗

arXiv2013

On a decoupled linear FEM integrator for Eddy-current-LLG

We propose a numerical integrator for the coupled system of the eddy-current equation with the nonlinear Landau-Lifshitz-Gilbert equation. The considered effective field contains a general field contribution, and we particularly cover exchange, anisotropy, applied field, and magnetic field (stemming from the eddy-current equation). Even though the considered problem is nonlinear, our scheme requires only the solution of two linear systems per time-step. Moreover, our algorithm decouples both equations so that in each time-step, one linear system is solved for the magnetization, and afterwards one linear system is solved for the magnetic field. Unconditional convergence -- at least of a subsequence -- towards a weak solution is proved, and our analysis even provides existence of such weak solutions. Numerical experiments with a micromagnetic benchmark problem underline the performance of the proposed algorithm.

math.NA↗

arXiv2013

A mixed method for Dirichlet problems with radial basis functions

We present a simple discretization by radial basis functions for the Poisson equation with Dirichlet boundary condition. A Lagrangian multiplier using piecewise polynomials is used to accommodate the boundary condition. This simplifies previous attempts to use radial basis functions in the interior domain to approximate the solution and on the boundary to approximate the multiplier, which technically requires that the mesh norm in the interior domain is significantly smaller than that on the boundary. Numerical experiments confirm theoretical results.

math.NA↗

arXiv2012

Radial basis functions for the solution of hypersingular operators on open surfaces

We analyze the approximation by radial basis functions of a hypersingular integral equation on an open surface. In order to accommodate the homogeneous essential boundary condition along the surface boundary, scaled radial basis functions on an extended surface and Lagrangian multipliers on the extension are used. We prove that our method converges quasi-optimally. Approximation results for scaled radial basis functions indicate that, for highly regular radial basis functions, the achieved convergence rates are close to the one of low-order conforming boundary element schemes. Numerical experiments confirm our conclusions.

math.NA↗

arXiv2009

Capturing Data Uncertainty in High-Volume Stream Processing

We present the design and development of a data stream system that captures data uncertainty from data collection to query processing to final result generation. Our system focuses on data that is naturally modeled as continuous random variables. For such data, our system employs an approach grounded in probability and statistical theory to capture data uncertainty and integrates this approach into high-volume stream processing. The first component of our system captures uncertainty of raw data streams from sensing devices. Since such raw streams can be highly noisy and may not carry sufficient information for query processing, our system employs probabilistic models of the data generation process and stream-speed inference to transform raw data into a desired format with an uncertainty metric. The second component captures uncertainty as data propagates through query operators. To efficiently quantify result uncertainty of a query operator, we explore a variety of techniques based on probability and statistical theory to compute the result distribution at stream speed. We are currently working with a group of scientists to evaluate our system using traces collected from the domains of (and eventually in the real systems for) hazardous weather monitoring and object tracking and monitoring.

cs.DB↗