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

TL;DR

This work addresses PDE-constrained tracking problems with parabolic state equations by formulating the KKT system using a strong variational state form and a super-weak adjoint form. It develops a robust, operator-based preconditioner that remains stable across the regularization parameter $\alpha$, diffusion parameter $\kappa$, discretization size, and spline degree, leveraging Isogeometric Analysis for smooth space-time discretizations. A key contribution is the fast diagonalization in time, which transforms a space-time elliptic problem into a sequence of decoupled space-only elliptic problems, enabling time-parallelization of the solve. Numerical experiments confirm uniform iteration bounds and favorable condition numbers, demonstrating practical efficiency for large-scale, space-time optimized parabolic problems.

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.

Figures (3)

• Figure 1: Quarter annulus domain
• Figure 2: Initial condition $y_0$ projected into the spline space with $p=2$ and $\ell = 7$.
• Figure 3: Computed state solution in the whole space time cylinder (left) and a cut through the center circle (right).

Theorems & Definitions (18)

• Lemma 1
• Remark 1: 2--by--2 formulation
• Lemma 2
• Lemma 3
• Remark 2: Neumann and Robin conditions
• Remark 3: Boundary observation
• Remark 4: Observation on discrete points in time
• Remark 5
• Theorem 4
• Theorem 5