# On near-optimal time samplings for initial data best approximation

## TL;DR

This paper extends the methodology of De Vore and Zuazua for near-optimal sampling in linear PDEs, showing parameter choices depend only on operator order and adapting the algorithm to time-dependent heat equations.

## Contribution

It demonstrates that relevant sampling parameters are independent of the PDE spectrum and extends the algorithm to non-autonomous heat equations with time-dependent diffusivity.

## Key findings

- Parameter choices depend only on operator order
- Sampling sequence construction is spectrum-independent
- Algorithm extended to non-autonomous heat equations

## Abstract

Leveraging on the work of De Vore and Zuazua, we further explore their methodology and deal with two open questions presented in their paper. We show that for a class of linear evolutionary PDEs the admissible choice of relevant parameters used to construct the near-optimal sampling sequence is not influenced by the spectrum of of the operator controlling the spatial part of the PDE, but only by its order. Furthermore, we show that it is possible to extend their algorithm to a simple version of a non-autonomous heat equation in which the heat diffusivity coefficient depends explicitly on time.