Computational Methods for Parameter-Dependent Partial Differential Equations

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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.