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Adv Comput Math 2020 Mar 4;46(3). Epub 2020 May 4.

Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston MA 02215.

In this paper we introduce concepts from uncertainty quantification (UQ) and numerical analysis for the efficient evaluation of stochastic high dimensional Newton iterates. In particular, we develop complex analytic regularity theory of the solution with respect to the random variables. This justifies the application of sparse grids for the computation of statistical measures. Read More

Adv Comput Math 2013 Aug;39(2):327-347

Institute for Computational Engineering and Sciences, University of Texas at Austin,

In a similar fashion to estimates shown for Harmonic, Wachspress, and Sibson coordinates in [Gillette et al., AiCM, to appear], we prove interpolation error estimates for the mean value coordinates on convex polygons suitable for standard finite element analysis. Our analysis is based on providing a uniform bound on the gradient of the mean value functions for all convex polygons of diameter one satisfying certain simple geometric restrictions. Read More

Adv Comput Math 2012 Oct;37(3):417-439

Department of Mathematics, University of Texas at Austin,

We prove the optimal convergence estimate for first order interpolants used in finite element methods based on three major approaches for generalizing barycentric interpolation functions to convex planar polygonal domains. The Wachspress approach explicitly constructs rational functions, the Sibson approach uses Voronoi diagrams on the vertices of the polygon to define the functions, and the Harmonic approach defines the functions as the solution of a PDE. We show that given certain conditions on the geometry of the polygon, each of these constructions can obtain the optimal convergence estimate. Read More