Speaker: Michael Hecht
Multivariate Interpolation in Non-Tensorial Nodes May Lift the Curse of Dimensionality for Trefethen Functions
We extend Newton and Lagrange interpolation to arbitrary dimensions while maintaining their numerical stability and computational efficiency.
Our generalization relies on a proper choice of non-tensorial unisolvent interpolation nodes, whose number scales sub-exponentially with the space dimension.
We prove the resulting interpolation scheme to approximate all functions of the largest Hilbert space of Sobolev functions that is contained in the space of continuous functions, in all dimensions.
We empirically demonstrate that the resulting polynomial interpolant empirically reaches the optimal exponential approximation rate for the Runge function, conjecturing its optimality for a class of analytic functions we term Trefethen functions.
Combining sub-exponential node counts with exponential approximation rates, the proposed non-tensorial choice of unisolvent nodes may lift the curse of dimensionality for interpolation problems involving Trefethen functions.
In the talk we will further sketch how this interpolation approach impacts computational challenges occurring for: variational spectral PDE solvers, numerical differential geometry and regression of strongly-varying multivariate functions.
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