In many engineering applications governed by PDEs, the parameters of the equations (coefficients, forcing terms, boundary and initial conditions, shape of the domain) are not known exactly but rather affected by a certain degree of uncertainties, and can be described by means of random variables (or random fields).
Uncertainty Quantification (UQ) aims at estimating how the randomness of these “input” parameters affects the “outputs” of the PDE, typically its solution or functionals thereof. UQ techniques are often based on repeatedly solving the PDE at hand for different combinations of the input parameters (i.e., a sampling approach), which requires a significant computational effort.
To reduce such effort, so-called “multi-level” and “multi-index” methods have recently been proposed. These methods explore the variability of the PDE outputs using a hierarchy of suitably chosen discretization levels to balance the PDE discretization error and the sampling error, and refine the discretization of the PDE only when needed. We emphasize in particular that these methodologies are completely “black-box”, in the sense that they allow reuse of legacy PDE solvers, and are moreover
In this talk we describe in detail one such method, i.e., the so-called Multi-Index Stochastic Collocation method (MISC), which is closely related to the quite popular Sparse-Grids Stochastic Collocation method for the approximation of PDE with random data.
In particular, this method relies on PDE solvers with tensor structure. To this end, we use Isogeometric Analysis (IGA), which is a technique introduced in the early 2000 to bridge the gap between Computer Aided Design (CAD) and PDE-based engineering analysis.
The core idea of IGA is to use the basis functions used by CAD designers to describe geometries (typically Cubic Splines or Non-Uniform Rational B-Splines) as a basis for the approximation of the solution of the PDE as well; the PDE is then solved a traditional Galerkin approach.
Beside the fact that its tensorized construction makes IGA very suitable in the MISC framework, attractive features of IGA are the simplified treatment of complex geometries, and the fact that basis functions with high-order and high-degree of regularity can be easily generated, thanks to the flexibility of the splines bases.
This is a joint work with Joakim Beck and Raul Tempone (KAUST), Abdul-Lateef Haji-Ali (Oxford) and Fabio Nobile (EPFL).