As large-scale numerical simulations permeate science and industry, the resulting deluge of data brings about an urgent need for effective analysis tools to help researchers turn this information into actual insight. Scientific visualization plays an important role in this process by creating a visual interface to massive dataset sets that affords an intuitive basis for interpretation,
assessment, and decision making. However, the rapidly growing size and complexity of scientific datasets put an increasing emphasis on the ability of visualization methods to clearly convey a high-level picture of the data by characterizing its inherent structure across spatial and temporal scales. In this talk I will describe a general strategy built upon a principled mathematical framework to identify salient structures in vector and tensor fields, which are ubiquitous in practical scenarios. Our methodology combines concepts from topology and dynamical systems, differential geometry, and computer vision to extract important features from large-scale multivariate
datasets, thus producing a concise geometric signature that lends itself to automatic processing and insightful visual representations. I will illustrate this basic approach in the context of problems ranging from computational fluid dynamics and solid mechanics to fusion research and medical image analysis. I will also present our ongoing work on the application of a novel Lagrangian method to the structural analysis and interactive multi-scale exploration of transient flows.