Numerical Methods for High Dimensional Partial Differential Equations
One of the great scientific challenges of this century is to describe accurately complex processes such as climate change, contaminant flow, genomics, and even social media and finance. The main obstacle to one''s understanding of these processes is that they depend on many parameters or variables. This not only complicates their mathematical description but also inhibits the use of modern computational tools for their accurate prediction. This research puts forward new mathematical ideas based on sparsity and model reduction to determine the importance of the various parameters and derive simpler models that still accurately describe the underlying process. This in turn leads to more accurate and less costly computational models that can be implemented with today''s computational resources. It also studies how to assimilate observational data to improve models and even suggests the most effective new data sites.The development of numerical methods for such high-dimensional problems faces the so-called curse of dimensionality, which says that traditional methods are doomed to fail. This has led to the development of myriad new techniques in analysis, computer science, and numerical computation based on ideas such as sparsity, compressed sensing, variable reduction, anisotropic smoothness, sparse grids, discrepancy theory, hashing, tensor approximation, reduced modeling, and manifold learning. This project describes the family of parametric functions to be numerically recovered as a high-dimensional manifold and then seeks to develop techniques for querying the manifold that lead to low-dimensional approximations of the manifold described either by reduced bases or by high-dimensional polynomial expansions. It also seeks to describe how high-dimensional observational data on a state of the manifold can be fused with the reduced model leading to an even more accurate description of the state. The successful completion of this program will lead to numerical methods which can be implemented on-line for executing a fast query of the manifold for any prescribed set of parameters.