Monday 22nd of May 2023, 15:15-16:15 in HG03.085
Wouter van Harten
Shape uncertainty quantification with compactly supported basis functions.
In this talk, we consider a linear, elliptic, partial differential equation (PDE) on a domain \(D(y)\) where the domain is parametrized by \(y \in [-1, 1]^d\), \(d\) possibly countably infinite. Polynomial expansions of parametric linear elliptic PDEs recently got a lot of attention when first, it was shown that these expansions converge at least with a rate \(1/p - 1\), when \((\|ψ_j\|_∞)_{j ∈ ℕ} \in l^p\) and therefore, the convergence does not suffer from the curse of dimensionality. Later, for affine parametric diffusion coefficient \(a(y)\), faster convergence rates were observed and proven for wavelet-type expansions by utilizing a pointwise bound on \((ψ_j)_{j ∈ ℕ}\) instead of the previously exploited \(L^∞\)-bound. In this work, we expand these results to the linear elliptic equation with parametrized boundary by employing a mapping approach to handle the parametrized domain. In particular, we show theoretically and illustrate numerically that locality in the support of the functions allows us to achieve higher convergence rates than globally supported basis functions.