Abstract: Modern computational science is marked by an increasingly tight integration between techniques in mathematics and computer science. This talk will present results from two problems where advances in computational tools have inspired new mathematics, and vice-versa.
Large-scale simulations increasingly utilize GPU-based computers, which has increased pressure to develop numerical methods for PDEs that parallelize with respect to time. This has sparked research into space-time finite element methods (STFEM), which naturally parallelize in the time dimension. After presenting STFEMs in general, we will describe new results on the construction of four-dimensional space-time elements, and conclude with methods for adaptively refining four-dimensional meshes. In a second example, we consider recent work on Multivariate Functional Approximation (MFA), an analysis and visualization framework for large data sets. We will discuss methods for improving MFA models with tools from Fourier analysis, and how this can guide further development of MFA.