Factorizations and fast diagonalization for the heat equation

TL;DR

The paper addresses solving linear evolution problems, especially the heat equation, by exploiting Kronecker-structured discretizations. It shows that naive diagonalization in time is unstable for the heat equation and develops three stable alternatives—LU factorization, arrowhead factorization, and low-rank modification—that replace the time-eigendecomposition while preserving efficiency and enabling space-time parallelism. The methods are analyzed for computational cost and stability, and numerical experiments demonstrate robust performance as both direct solvers and preconditioners within spline-based (isogeometric) discretizations. The work highlights practical pathways for scalable, robust time factoring in tensor-product discretizations and suggests broader applicability to related PDEs and discretization frameworks. The results indicate that, with tensor-product structure, these stable factorizations achieve favorable complexity and parallelism, while remaining applicable as preconditioners in more general settings.

Abstract

This work investigates diagonalization-based methods for efficiently solving linear evolution problems, with a particular focus on the heat equation. The plain diagonalization of the differential operator, though effective for elliptic problems where fast diagonalization can be used, exhibits instability when applied to the heat equation. To address this difficulty, we examine three alternative approaches, based on LU factorization, a suitable arrowhead factorization, and a low-rank modification. These methods introduce more robust factorizations of the time derivative, ensuring both computational efficiency and stability.

Figures (5)

• Figure 1: Generalized eigenvectors for the pencil $(\mathbf{A}_t, \mathbf{M}_t )$, with associated eigenvalues for $p_t =3$ and $N_{t}=32$. The real part is in solid line, while the imaginary part is in dashed line.
• Figure 2: Real part (solid line) and imaginary part (dashed line) of splines corresponding to the columns of $\mathbf{U}_t$, as function of time $t \in [0,1]$, with associated diagonal entry in $\mathbf{\Delta}_t$. Discretization with $p_t = 3$ and $N_{t} = 32$.
• Figure 3: Setup cost for $N_s = N_t^3$.
• Figure 4: Application costs for $N_s = N_t^3$.
• Figure 5: Computational domains $\Omega_s$.