  Data mining has traditionally focused on the task of drawing inferences from
large datasets. However, many scientific and engineering domains, such as fluid
dynamics and aircraft design, are characterized by scarce data, due to the
expense and complexity of associated experiments and simulations. In such
data-scarce domains, it is advantageous to focus the data collection effort on
only those regions deemed most important to support a particular data mining
objective. This paper describes a mechanism that interleaves bottom-up data
mining, to uncover multi-level structures in spatial data, with top-down
sampling, to clarify difficult decisions in the mining process. The mechanism
exploits relevant physical properties, such as continuity, correspondence, and
locality, in a unified framework. This leads to effective mining and sampling
decisions that are explainable in terms of domain knowledge and data
characteristics. This approach is demonstrated in two diverse applications --
mining pockets in spatial data, and qualitative determination of Jordan forms
of matrices.
