Qualitative assessment of scientific computations is an emerging application area for the use of data mining and imagistic reasoning techniques. It aims to characterize broad, explicative, features of numerical computations by analyzing datasets expressed in suitable spatial contexts. We are focusing on assessment of scientific computations by characterization of matrix eigenstructure. Recently, the spectral portrait has emerged as a popular tool for graphically visualizing eigenstructure, by illustrating how the eigenvalues of a matrix change as perturbations (e.g. due to numerical error) are introduced. For example (figure above left), moving outward from the original eigenvalues, level curves for increasing perturbation magnitudes surround increasingly large regions; these regions represent eigenvalues that are equivalent with respect to perturbations of those magnitudes. Analysis of level curves reveals information about a matrix (e.g. nonnormality and defective eigenvalues) and the effects of different algorithms and numerical approximations. A similar approach (figure above right) introduces positive and negative perturbations to a matrix that has repeated eigenvalues; the computed eigenvalues then appear as vertices of regular polygons (with degree determined by number of repeats) centered on the original eigenvalues. Analysis of a superposition of such perturbations reveals the matrix Jordan form (in particular, eigenvalue multiplicities) from symmetries among the samples.

We are developing an active data mining framework, using a spatial aggregation approach, for multi-level qualitative analysis of data from matrix computations. The highest-level spatial structures specify qualitative models of matrix properties, including sensitivity (modeled by correspondences, where smaller curves surrounding separate eigenvalues merge into a large curve surrounding multiple ones), and multiplicity (modeled by symmetries of samples under rotation around the original eigenvalue). Quantitative confidence metrics for proposed models evaluate their consistency with underlying samples, encoding the intuition that "good" models effectively overcome noise and sparsity by uncovering mutually reinforcing interpretations (e.g. correspondence and symmetry). Model comparison further allows detection of ambiguity and handling of it by selection of additional perturbation samples most suitable for refining and discriminating among the models. This approach thus permits efficient, explainable data collection and analysis. It helps overcome noise and sparsity by leveraging domain knowledge to detect mutually reinforcing interpretations of spatial data.

• Naren Ramakrishnan (computer science, Virginia Tech)