Modern numerical simulation using models based on partial differential equations is characterized by high dimensionality together with parameter dependence. High dimensionality is required to achieve accuracy in discrete approximations, especially for three-dimensional or multi-component models. Parameter dependence stems from a variety of sources. Parameters may correspond to time or to specific terms such as Reynolds numbers in fluids, for which the properties of solutions are wanted at a variety of parameter choices. Alternatively, they may be used to specify components of a model such as material properties, geometry, or boundary conditions, for which it is desired to perform multiple simulations. These same sets of parameters may instead be uncertain and treated as random variables, for which simulations are used to identify statistical properties of solutions. All these scenarios require the computation of many discrete solutions, which may be prohibitively expensive when the discrete models are large in scale. Our aim in this project is to explore, develop and refine computational algorithms to reduce these costs. Our emphasis will be on effective use of reduced basis methods, which project high-dimensional models into subspaces of significantly smaller dimension with the aim of quickly and efficiently constructing accurate approximate solutions over a wide range of parameters.
The potential impact of this approach lies in its use in wide varieties of engineering and scientific simulations. These include models of groundwater flows and other environmental phenomena, where uncertain properties such as boundary conditions or permeabilities of media in which fluids are found are treated as parameters; aerodynamic simulations, where material properties of structures or qualities of combustible material are parameters that must be analyzed for their effects on efficiency and safety; and in models of biological processes, for example blood flows or chemical reactions in cells, which depend on parameters such as fluid viscosity or reaction rates. The development of efficient algorithmic strategies based on reduction of problem size will significantly enhance the prospects of performing such simulations quickly, enabling engineers and scientists to use the results of simulations in the field and for real-time decision making.
This project concerned the development and mathematical analysis of new computational algorithms for the solution of partial differential equations used in the mathematical modeling of physical processes. The intellectual merit of this work comes from the fundamental aim of constructing efficient computational algorithms for important and classical mathematical models. We have developed new algorithms for accurately computing models of fluid flow and electromagnatic flows and shown how to efficiently analyze statistical properties of the solutions when they are affected by problem-dependent parameters. The broader impact of the work lies in the use of such computational strategies in wide varieties of engineering and scientific simulations. Mathematical models represent an important tool to help develop an understanding of features of physical phenomena when it is difficult or impossible to obtain such understanding from purely physical considerations. Examples of how these ideas are used include the identification of concentrations of solvents contained in fluids flowing through underground aquifers when the porosity of the media where the fluids lie is not well understood; and the impact of magnetic fields on the properties of electrically conductiong fluids, which has applications in the study of electrolytes (used to study hydration in physiology) and plasmas (for models of fusion), among many others. In order for such computations to be useful, they need to be done quickly and efficiently. The work done done in this project demonstrates that accurate solutions of such models can be computed efficiently using new computational algorithms developed in the project.