The quadratic Wasserstein metric for inverse data matching
It addresses computational challenges in inverse data matching for researchers in applied mathematics and computational science, offering insights into metric selection for improved robustness and optimization.
This work analyzes the quadratic Wasserstein (W2) distance for solving inverse problems, showing that it smooths the inversion process to be robust against high-frequency noise but reduces resolution, and leads to better convexity in optimization compared to classical distances like L2 and H-1.
This work characterizes, analytically and numerically, two major effects of the quadratic Wasserstein ($W_2$) distance as the measure of data discrepancy in computational solutions of inverse problems. First, we show, in the infinite-dimensional setup, that the $W_2$ distance has a smoothing effect on the inversion process, making it robust against high-frequency noise in the data but leading to a reduced resolution for the reconstructed objects at a given noise level. Second, we demonstrate that for some finite-dimensional problems, the $W_2$ distance leads to optimization problems that have better convexity than the classical $L^2$ and $H^{-1}$ distances, making it a more preferred distance to use when solving such inverse matching problems.