Spatial Aggregation for Qualitative Assessment of Scientific Computations

Abstract

Qualitative assessment of scientific computations is an emerging application area that applies a data-driven approach to characterize, at a high level, phenomena including condi-tioning of matrices, sensitivity to various types of error prop-agation, and algorithmic convergence behavior. This paper develops a spatial aggregation approach that formalizes such analysis in terms of model selection utilizing spatial struc-tures extracted from matrix perturbation datasets. We fo-cus in particular on the characterization of matrix eigenstruc-ture, both analyzing sensitivity of computations with spectral portraits and determining eigenvalue multiplicity with Jordan portraits. Our approach employs spatial reasoning to over-come noise and sparsity by detecting mutually reinforcing in-terpretations, and to guide subsequent data sampling. It en-ables quantitative evaluation of properties of a scientific com-putation in terms of confidence in a model, explainable in terms of the sampled data and domain knowledge about the underlying mathematical structure. Not only is our method-ology more rigorous than the common approach of visual in-spection, but it also is often substantially more efficient, due to well-defined stopping criteria. Results show that the mech-anism efficiently samples perturbation space and successfully uncovers high-level properties of matrices.