A robust and time-parallel preconditioner for parabolic reconstruction problems using Isogeometric Analysis

TL;DR

This paper introduces a robust, time-parallel preconditioner for solving parabolic PDE-constrained optimization problems using Isogeometric Analysis, enabling efficient and scalable solutions through a novel reformulation and diagonalization approach.

Contribution

It proposes a new preconditioner that is robust across parameters and leverages Isogeometric Analysis with a time-parallel diagonalization technique for efficient PDE-constrained optimization.

Findings

Preconditioner is robust in regularization and diffusion parameters.

The approach enables efficient time-parallel solution of elliptic problems.

Numerical experiments confirm the theoretical efficiency and robustness.

Abstract

We consider a PDE-constrained optimization problem of tracking type with parabolic state equation. The solution to the problem is characterized by the Karush-Kuhn-Tucker (KKT) system, which we formulate using a strong variational formulation of the state equation and a super weak formulation of the adjoined state equation. This allows us to propose a preconditioner that is robust both in the regularization and the diffusion parameter. In order to discretize the problem, we use Isogeometric Analysis since it allows the construction of sufficiently smooth basis functions effortlessly. To realize the preconditioner, one has to solve a problem over the whole space time cylinder that is elliptic with respect to certain non-standard norms. Using a fast diagonalization approach in time, we reformulate the problem as a collection of elliptic problems in space only. These problems are not only smaller, but our approach also allows to solve them in a time-parallel way. We show the efficiency of the preconditioner by rigorous analysis and illustrate it with numerical experiments.