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Learning high-dimensional systems from incomplete data by optimal nonlinear approximations

Dr. Clayton Webster

Department of Mathematics
The University of Tennessee
Friday, February 8, 2019
2:30-3:30pm    JDT 410

Abstract: This talk will provide a brief summary of some recent theoretical and computational breakthroughs towards conquering the challenge of learning high-dimensional systems, having a certain set of constraints, from a limited amount of noisy data. We will present several innovative sparse recovery techniques and optimal nonlinear approximations, including:

  • weighted l1 minimization procedure for compressed sensing, with a precise choice of weights, for overcoming the curse of dimensionality;
  • mixed-normbasedregularizationthatsimultaneouslyreconstructsparameterizedPDEssolutionsover both physical and parametric domains; and
  • the first sharp estimates of the complexity of an artificial neural network required to recovery the best approximation in high dimensions.

Such approaches will enable the reconstruction of the entire high-dimensional solution map, with accuracy comparable to the best approximation, while utilizing an optimal number of samples.



Clayton Webster is a Distinguished Professor in the Department of Mathematics at The University of Tennessee and a Distinguished Scientist and Group Leader in the Computational and Applied Mathematics Group at Oak Ridge National Laboratory. Previously, Dr. Webster was the Director of Quantitative Trading at NextEra Energy Resources, Power Trading LLC.  Before that, he was awarded the John von Neumann Fellowship by the Department of Energy at Sandia National Laboratories. He received his Ph.D. under the supervision of Prof.Max Gunzburger, in Mathematics from Florida State University in 2007. He also earned Master’s and Bachelor’s  degrees in Pure Mathematics from McMaster University in 2003 and 2001 respectively.

His worked has earned him numerous accolades, and most recently he was awarded the DOE Career Award, and was appointed as a Frontiers of Science Fellow, by the National Academy of Sciences.

He is currently the President of the SIAM Southeastern Atlantic Section and serves as an editor for the:

SIAM Journal on Numerical, SIAM Journal for Uncertainty Quantification, Numerishe Mathematik, Results in Applied Mathematics, International Journal for Uncertainty Quantification, International Journal for Computer Mathematics, and the SIAM Book Series.