Optimal random sampling for approximation with non-orthogonal bases

Recent developments in optimal random sampling for least squares approximations have led to the identification of a (near-)optimal sampling distribution. This distribution can easily be evaluated given an orthonormal basis for the approximation space. However, many computational problems in science and engineering naturally yield building blocks that enable accurate approximation but do not form an orthonormal basis. In the first part of the talk, we will explore how numerical rounding errors affect the approximation error and the optimal sampling distribution when approximating with non-orthogonal bases. In the second part, we will demonstrate how this distribution can be computed without the need to orthogonalize the basis. This is joint work with Daan Huybrechs and Ben Adcock.