Chris Bailey-Kellogg and Naren Ramakrishnan
Qualitative assessment of scientific computations is an emerging application area that applies a data-driven approach to characterize, at a high level, phenomena including conditioning of matrices, sensitivity to various types of error propagation, and algorithmic convergence behavior. This paper develops a spatial aggregation approach that formalizes such analysis in terms of model selection utilizing spatial structures extracted from matrix perturbation datasets. We focus in particular on the characterization of matrix eigenstructure, both analyzing sensitivity of computations with spectral portraits and determining eigenvalue multiplicity with Jordan portraits. Our approach employs spatial reasoning to overcome noise and sparsity by detecting mutually reinforcing interpretations, and to guide subsequent data sampling. It enables quantitative evaluation of properties of a scientific computation 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 methodology more rigorous than the common approach of visual inspection, but it also is often substantially more efficient, due to well-defined stopping criteria. Results show that the mechanism efficiently samples perturbation space and successfully uncovers high-level properties of matrices.