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Adv Comput Math 2021 Jun 2;47(3). Epub 2021 May 2.

Applied Mathematics and Computational Science, 4700 King Abdullah University of Science and Technology, Thuwal, 23955-6900, Saudi Arabia.

Consider a linear elliptic PDE defined over a stochastic stochastic geometry a function of random variables. In many application, quantify the uncertainty propagated to a Quantity of Interest (QoI) is an important problem. The random domain is split into large and small variations contributions. Read More

Adv Comput Math 2021 26;47(4):51. Epub 2021 Jun 26.

Dipartimento di Matematica DIMA, Università di Genova, Genoa, Italy.

We investigate the use of the so-called variably scaled kernels (VSKs) for learning tasks, with a particular focus on support vector machine (SVM) classifiers and kernel regression networks (KRNs). Concerning the kernels used to train the models, under appropriate assumptions, the VSKs turn out to be and than the standard ones. Numerical experiments and applications to breast cancer and coronavirus disease 2019 (COVID-19) data support our claims. Read More

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 2018 Oct 19;44(5):1651-1672. Epub 2018 Mar 19.

Facebook Artificial Intelligence Research, 1 Facebook Way, Menlo Park, CA 94025.

Randomized algorithms provide solutions to two ubiquitous problems: (1) the distributed calculation of a principal component analysis or singular value decomposition of a highly rectangular matrix, and (2) the distributed calculation of a low-rank approximation (in the form of a singular value decomposition) to an arbitrary matrix. Carefully honed algorithms yield results that are uniformly superior to those of the stock, deterministic implementations in Spark (the popular platform for distributed computation); in particular, whereas the stock software will without warning return left singular vectors that are far from numerically orthonormal, a significantly burnished randomized implementation generates left singular vectors that are numerically orthonormal to nearly the machine precision. 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