Joint-sparse recovery for high-dimensional parametric PDEs

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 5, 2018 - 1:55pm for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Nick Dexter – University of Tennessee – ndexter@utk.eduhttps://bredesencenter.utk.edu/people/nick-dexter/
Organizer
Wenjing Liao
We present and analyze a novel sparse polynomial approximation method for the solution of PDEs with stochastic and parametric inputs. Our approach treats the parameterized problem as a problem of joint-sparse signal reconstruction, i.e., the simultaneous reconstruction of a set of signals sharing a common sparsity pattern from a countable, possibly infinite, set of measurements. Combined with the standard measurement scheme developed for compressed sensing-based polynomial approximation, this approach allows for global approximations of the solution over both physical and parametric domains. In addition, we are able to show that, with minimal sample complexity, error estimates comparable to the best s-term approximation, in energy norms, are achievable, while requiring only a priori bounds on polynomial truncation error. We perform extensive numerical experiments on several high-dimensional parameterized elliptic PDE models to demonstrate the superior recovery properties of the proposed approach.