With modern computational resources rapidly advancing towards exascale, large-scale simulations useful for understanding natural and man-made phenomena are becoming increasingly accessible. Consequently, the role of analyzing the resulting data is also becoming more integral to advancing science. However, the numerical nature of the analysis, compounded by the large scales of data, poses serious challenges for computer scientists, e.g., managing data efficiently, developing scalable algorithms, and obtaining consistency with the theory. In this talk, I will discuss my research on addressing some of these contemporary challenges in the context of vector fields, which are an important form of scientific data, and represent a multitude of physical phenomena, such as wind flow and ocean currents. In particular, I will discuss new theories and computational frameworks to enable consistent streamline-based analysis of vector fields.
 
Analysis based on streamlines (tangent curves to the field), such as vector field topology, is well-understood for steady (time-independent) vector fields. However, since most such techniques are numerical in nature, the residing numerical errors can potentially violate consistency with the underlying theory, with serious physical consequences. I will present new combinatorial approaches that provide consistency guarantees during analysis, and uncertainty visualization of the unavoidable errors. This consistent computational framework, however, is rendered under-utilized since it is not yet applicable to unsteady (time-varying) vector fields, which represent most phenomena of practical interest. This is because although the streamline-based analysis is widely successful for steady vector fields, in general, it lacks physical meaning for the unsteady case. To this end, I will also discuss a new theory that enables analyzing unsteady vector fields as sequences of steady fields, thus obtaining consistency in their analysis. In contrast to the state-of-the-art analysis of unsteady vector fields, these techniques are more scalable and allow better data management by accessing the data on a per-time-step basis. I will conclude with further avenues that this research opens for new and widely applicable data analysis models for vector fields.