Accuracy controlled data assimilation for parabolic problems

TL;DR

This paper develops a regularized least squares approach within a variational framework to recover approximate solutions to parabolic PDEs from incomplete data, providing error bounds, efficient preconditioners, and stopping criteria.

Contribution

It introduces a novel continuous variational formulation for data assimilation in parabolic problems, enabling error analysis and efficient numerical solution strategies.

Findings

Error bounds for recovered solutions are derived and validated.

Preconditioners with linear-time application and uniform condition numbers are constructed.

Numerical experiments confirm the theoretical error estimates and solver performance.

Abstract

This paper is concerned with the recovery of (approximate) solutions to parabolic problems from incomplete and possibly inconsistent observational data, given on a time-space cylinder that is a strict subset of the computational domain under consideration. Unlike previous approaches to this and related problems our starting point is a regularized least squares formulation in a continuous infinite-dimensional setting that is based on stable variational time-space formulations of the parabolic PDE. This allows us to derive a priori as well as a posteriori error bounds for the recovered states with respect to a certain reference solution. In these bounds the regularization parameter is disentangled from the underlying discretization. An important ingredient for the derivation of a posteriori bounds is the construction of suitable Fortin operators which allow us to control oscillation errors stemming from the discretization of dual norms. Moreover, the variational framework allows us to contrive preconditioners for the discrete problems whose application can be performed in linear time, and for which the condition numbers of the preconditioned systems are uniformly proportional to that of the regularized continuous problem. In particular, we provide suitable stopping criteria for the iterative solvers based on the a posteriori error bounds. The presented numerical experiments quantify the theoretical findings and demonstrate the performance of the numerical scheme in relation with the underlying discretization and regularization.