Numerical Methods for Parametric Partial Differential Equations
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One of the most significant scientific challenges of this century is the accurate description and computation of complex processes such as climate change, contaminant flow, genomics, and even social media and finance. While one can create a mathematical model for these processes, the large number of parameters in the model inhibits the use of traditional computational tools for fast and reliable predictions. In addition, there is the question of the efficacy of the mathematical model. The proposed research puts forward new mathematical ideas, based primarily on model reduction, to determine the importance of the various parameters and derive simpler models that still faithfully describe the underlying process. This, in turn, leads to more accurate and less costly computational models that can be implemented within today''s existing computing resources. The project also investigates how to quantify uncertainty in both the model and the parameters from data observations of the process.This project investigates three demanding computational tasks in parametric partial differential equations (PDEs). The first of these, called the forward problem, seeks the creation of fast and accurate online solvers for the PDE when given a parameter query. Such online solvers are used in a myriad of applications that seek to optimize performance through parameter selection. The second seeks optimal methods to compute the state of the PDE from observational data. Related to this is the third problem of estimating the parameters of the PDE from observational data. Because of the large number of parameters, traditional numerical methods for such high dimensional problems face the so-called "curse of dimensionality", i.e., they cannot obtain the desired accuracy of computation in a reasonable computational time. The proposed research circumvents this difficulty by developing novel methods of model reduction based on sparsity ad highly nonlinear approximation such as n-term dictionary approximation. Foundational results will also be established for inverse parameter estimation that prove Lipschitz smoothness for the forward and inverse maps under minimal smoothness conditions on the parameters. These foundational results are then coupled with reduced modeling to create numerical methods for parameter estimation and model verification.This award reflects NSF''s statutory mission and has been deemed worthy of support through evaluation using the Foundation''s intellectual merit and broader impacts review criteria.