A priori error analysis of discrete-ordinate weak Galerkin method for radiative transfer equation

This research article discusses a numerical solution of the radiative transfer equation based on the weak Galerkin finite element method. We discretize the angular variable by means of the discrete-ordinate method. Then the resulting semi-discrete hyperbolic system is approximated using the weak Galerkin method. The stability result for the proposed numerical method is devised. A \emph{priori} error analysis is established under the suitable norm. In order to examine the theoretical results, numerical experiments are carried out

Analysis of an implicitly extended Crank-Nicolson scheme for the heat equation on a time-dependent domain

We consider a time-stepping scheme of Crank-Nicolson type for the heat equation on a moving domain in Eulerian coordinates. As the spatial domain varies between subsequent time steps, an extension of the solution from the previous time step is required. Following Lehrenfeld \& Olskanskii [ESAIM: M2AN, 53(2):\,585-614, 2019], we apply an implicit extension based on so-called ghost-penalty terms. For spatial discretisation, a cut finite element method is used. We derive a complete a priori error analysis in space and time, which shows in particular second-order convergence in time under a parabolic CFL condition. Finally, we present numerical results in two and three space dimensions that confirm the analytical estimates, even for much larger time steps

Implementing the Lean Sigma Framework in an Indian SME: a case study

Lean and Six Sigma are two widely acknowledged business process improvement strategies available to organisations today for achieving dramatic results in cost, quality and time by focusing on process performance. Lately, Lean and Six Sigma practitioners are integrating the two strategies into a more powerful and effective hybrid, addressing many of the weaknesses and retaining most of the strengths of each strategy. Lean Sigma combines the variability reduction tools and techniques from Six Sigma with the waste and non-value added elimination tools and techniques from Lean Manufacturing, to generate savings to the bottom-line of an organisation. This paper proposes a Lean Sigma framework to reduce the defect occurring in the final product (automobile accessories) manufactured by a die-casting process. The proposed framework integrates Lean tools (current state map, 5S System, and Total Productive Maintenance (TPM)) within Six Sigma DMAIC methodology to enhance the bottom-line results and win customer loyalty. Implementation of the proposed framework shows dramatic improvement in the key metrics (defect per unit (DPU), process capability index, mean and standard deviation of casting density, yield, and overall equipment effectiveness (OEE)) and a substantial financial savings is generated by the organisation

Kernel Instrumental Variable Regression

Instrumental variable (IV) regression is a strategy for learning causal relationships in observational data. If measurements of input X and output Y are confounded, the causal relationship can nonetheless be identified if an instrumental variable Z is available that influences X directly, but is conditionally independent of Y given X and the unmeasured confounder. The classic two-stage least squares algorithm (2SLS) simplifies the estimation problem by modeling all relationships as linear functions. We propose kernel instrumental variable regression (KIV), a nonparametric generalization of 2SLS, modeling relations among X, Y, and Z as nonlinear functions in reproducing kernel Hilbert spaces (RKHSs). We prove the consistency of KIV under mild assumptions, and derive conditions under which convergence occurs at the minimax optimal rate for unconfounded, single-stage RKHS regression. In doing so, we obtain an efficient ratio between training sample sizes used in the algorithm's first and second stages. In experiments, KIV outperforms state of the art alternatives for nonparametric IV regression.Comment: 41 pages, 11 figures. Advances in Neural Information Processing Systems. 201