Multilevel Intrusive UQ Methods

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Physical processes such as heat transfer and fluid flows are typically modelled using partial differential equations (PDEs). If all the inputs (coefficients, boundary conditions etc) are known then standard numerical schemes such as finite element methods can be used to perform simulations and predict quantities of interest related to the model solution. In engineering problems, however, we frequently encounter scenarios where we are uncertain about one or more model inputs. The most common way to deal with this is to appeal to probability theory and represent uncertain inputs as functions of random variables. Estimating quantities of interest related to solutions of models with random inputs with a prescribed probability distribution is called forward uncertainty quantification (UQ). Although many algorithms for performing forward UQ exist, estimating statistical quantities of interest efficiently and accurately for complex PDE models remains an important scientific challenge. This project will make theoretical and computational advances in the development of so-called multilevel intrusive (MINT) algorithms for forward UQ that are computationally efficient and also provably accurate. Unlike sampling methods, intrusive schemes seek approximations which are polynomials of the random inputs. Standard intrusive methods are unpopular because they require the solution of huge linear systems of equations which quickly exhausts available computational resources. The main issue is that they use large tensor product approximation spaces which leads to wasted computations. Advances will be made by constructing lower-dimensional approximation spaces with flexible multilevel structure driven by an automated and accurate assessment of error.