Reconstructing Piezoelectric Responses over a Lattice: Adaptive Sampling of Low Dimensional Time Series Representations Based on Relative Isolation and Gradient Size

Abstract

We consider a $d-$dimensional lattice of points where, at each lattice point, a time series can be measured. We are interested in reconstructing the time series at all points of the lattice, to a desired error tolerance, by adaptively sampling only a subset of lattice points, each over a potentially short time interval. The method we develop is tailored to atomic force microscopy (AFM) data where time series are well-represented by constant functions at each lattice point. Through a convex weighting of a point’s relative isolation and relative gradient size, we assign a sampling priority. Our method adaptively samples the time series and then reconstructs the time series over the entire lattice. Our adaptive sampling scheme performs significantly better than sampling points homogeneously. We find that for the data provided, we can capture piezoelectric relaxation dynamics and achieve a relative $\ell _2$ reconstruction error of less than $10\%$ with a $47\%$ reduction in measurements, and less than $5\%$ with a $25\%$ reduction in measurements.

M. R. Lindstrom and W. J. Swartworth—Equal contributions.