Rigorous Development of an Efficient Reduced Collocation Approach for High-Dimensional Parametric Partial Differential Equations

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Many physical phenomena depend on a range of parameters, and to understand the phenomena many repeated simulations for different parameter values are required. Such parameters may describe properties of a material, wave frequencies, uncertainties in measured data, physical states at the boundaries, or domain geometry, among others. These massively repeated simulations require significant computer time, and are frequently computationally prohibitive. Reduced basis methods were developed to resolve this issue by providing efficient and accurate surrogate solutions for the large space of parameter values, based on a relatively small number of carefully selected parameters and their related pre-computed highly accurate "snapshot" solutions. Once these "snapshot" solutions are pre-computed, computing the surrogate solutions for any parameter values is quick and efficient. Moreover, their accuracy is certified by a mathematically rigorous error bound. A variant of the reduced basis method, called the reduced collocation method, was recently introduced by the PIs that is more efficient for nonlinear problems. The goals of this project are to develop this new method to be more efficient, both in precise identification of the "snapshot" solutions and in computing the surrogate solutions once the "snapshot" solutions are found, to integrate reduced basis and reduced collocation methods with uncertainly quantification techniques to efficiently handle complex problems, and to ensure that certain strong stability properties satisfied by the underlying snapshot solutions are preserved by the reduced collocation methods. These goals will enable an efficient and powerful reduced collocation method that can be applied to a wide range of parameter-dependent phenomena. Reduced basis methods were originally developed for use with a Galerkin formulation of a partial differential equations, and recently extended by the PIs for collocation formulations, which are frequently preferred for nonlinear problems. In fact, this new reduced collocation method (RCM) is more efficient than the typical reduced basis method (RBM) for a large class of partial differential equations (PDEs). The goal of this project is to rigorously develop the new RCM in order to: (1) improve the offline stage by introducing novel approaches to building a more optimal reduced space faster; (2) make the online stage of the RCM more robust and efficient for nonlinear problems through a variety of mathematical approaches to selecting the collocation points, developing pre-conditioners, and creating a co-prime multi-grid approach; (3) integrate the RBM/RCM with uncertainty quantification approaches to efficiently handle problems with high dimensional random spaces; and (4) enhance the RBM/RCM approaches to guarantee that the surrogate solutions preserve the strong stability properties satisfied by the underlying snapshot solutions. Rigorous mathematical analysis and innovative and efficient algorithm design will be combined to improve the reduced basis approaches thereby making them more efficient and robust for large classes of problems. This project will transform the RCM to make it efficient and robust, for linear as well as nonlinear problems. Furthermore, this project will combine knowledge from the field of reduced order modeling for linear and nonlinear PDEs and the field of uncertainty quantification to create powerful new methods, hybrids of the generalized polynomial chaos method and reduced basis methods. The resulting algorithms will impact the field of uncertainty quantification by improving the efficiency and robust handling of high parameter dimensions. Finally, the RBM/RCM will be improved to guarantee the preservation of nonlinear properties such as positivity. This novel development will resolve a major concern about the quality of the surrogate solution.