We study the numerical solution of forward and inverse time-harmonic acoustic scattering problems by randomly shaped obstacles in three-dimensional space using a fast isogeometric boundary element method. Within the isogeometric framework, realizations of the random scatterer can efficiently be computed by simply updating the NURBS mappings which represent the scatterer. This way, we end up with a random deformation field. In particular, we show that it suffices to know the deformation field’s expectation and covariance at the scatterer’s boundary to model the surface’s Karhunen–Loève expansion. Leveraging on the isogeometric framework, we employ multilevel quadrature methods to approximate quantities of interest such as the scattered wave’s expectation and variance. By computing the wave’s Cauchy data at an artificial, fixed interface enclosing the random obstacle, we can also directly infer quantities of interest in free space. Adopting the Bayesian paradigm, we finally compute the expected shape and variance of the scatterer from noisy measurements of the scattered wave at the artificial interface. Numerical results for the forward and inverse problems validate the proposed approach.

Motivated by evolving shapes such as moving cells, we construct examples of evolving stochastic surfaces by transformation of solutions to stochastic PDEs on spheres. We focus on the stochastic heat equation and its approximation to understand the transformation and simulation methods for the surfaces.

In this talk, we focus on the numerical approximation of time-harmonic electromagnetic fields for the Maxwell lossy cavity problem on uncertain domains. To deal with the different problem geometries, a shape parametrization framework that maps physical domains to a fixed polyhedral nominal domain is adopted. We discuss multilevel Monte Carlo sampling and multilevel sparse-grid quadrature for computing the expectation of the solutions with respect to uncertain domain ensembles. In addition, we analyze sparse-grid interpolation to compute surrogates of the domain-to-solution mappings. A rigorous fully discrete error analysis is provided, and we prove that dimension-independent algebraic convergence is achieved.

We consider the solution to an elliptic partial differential equation on a domain which is subject to uncertain changes in its shape.

When representing uncertain shape variations, using localized basis functions can be appealing from a modeling point of view, as they offer more geometrical flexibility compared to globally supported basis functions. In this talk, we will see that locality of basis functions can also be convenient in terms of approximation properties with respect to the uncertain parameter. Extending ideas from [1,2], it is indeed possible to prove, using pointwise summability bounds, that sparse polynomial approximations to the parameter-to-solution map may converge faster if localized functions are used in the shape representation. We will also see that this approximability result goes beyond shape uncertainty, and it applies in fact to many other parameter-to-solution maps, as long as they are smooth and have some given sparsity properties.

[1] M. Bachmayr, A. Cohen, G. Migliorati. Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients, ESAIM: Mathematical Modelling and Numerical Analysis 51(1): 321-339, 2017.

[2] M. Bachmayr, A. Cohen, R. DeVore, G. Migliorati, Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients, ESAIM: Mathematical Modelling and Numerical Analysis 51(1): 341-363, 2017