A parallel preconditioner for the all-at-once linear system from evolutionary PDEs with Crank-Nicolson discretization

TL;DR

This work tackles parallel-in-time solution of Crank-Nicolson AaO systems for evolutionary PDEs by introducing a diagonalizable gBC-type preconditioner based on α-circulant blocks. The preconditioned matrix has most eigenvalues equal to 1 and the remaining eigenvalues confined to a positive real annulus determined by α, ensuring mesh-independent GMRES convergence under appropriate conditions. Theoretical results are complemented by numerical experiments on Heston and SABR financial PDEs, showing that the proposed P_α preconditioner accelerates convergence and remains robust to mesh refinement, while the standard P_1 can fail when the spatial discretization matrix has zero eigenvalues. Overall, the approach offers a scalable PinT strategy for CN-based evolutionary PDEs with practical impact for financial engineering applications.

Abstract

The Crank-Nicolson (CN) method is a well-known time integrator for evolutionary partial differential equations (PDEs) arising in many real-world applications. Since the solution at any time depends on the solution at previous time steps, the CN method is inherently difficult to parallelize. In this paper, we consider a parallel method for the solution of evolutionary PDEs with the CN scheme. Using an all-at-once approach, we can solve for all time steps simultaneously using a parallelizable over time preconditioner within a standard iterative method. Due to the diagonalization of the proposed preconditioner, we can prove that most eigenvalues of preconditioned matrices are equal to 1 and the others lie in the set: $\left\{z\in\mathbb{C}: 1/(1 + α) < |z| < 1/(1 - α)~{\rm and}~\Re{\rm e}(z) > 0\right\}$, where $0 < α< 1$ is a free parameter. Besides, the efficient implementation of the proposed preconditioner is described. Given certain conditions, we prove that the preconditioned GMRES method exhibits a mesh-independent convergence rate. Finally, we will verify both theoretical findings and the efficacy of the proposed preconditioner via numerical experiments on financial option pricing PDEs.

Figures (5)

• Figure 1: Eigenvalues of the original and preconditioned matrices (i.e., $\mathcal{M}$, $P^{-1}_1\mathcal{M}$ and $P^{-1}_{\alpha}\mathcal{M}$) for the model (Set I) under the uniform spatial discretization with $N_t = N_s = 2N_v = 36$ and $\alpha = 10^{-3}$.
• Figure 2: Eigenvalues of the original and preconditioned matrices (i.e., $\mathcal{M}$, $P^{-1}_1\mathcal{M}$ and $P^{-1}_{\alpha}\mathcal{M}$) for the model (Set II) under the nonuniform spatial discretization with $N_t = N_s = 2N_v = 36$ and $\alpha = 10^{-3}$.
• Figure 3: Eigenvalues of the original and preconditioned matrices (i.e., $\mathcal{M}$ and $P^{-1}_{\alpha}\mathcal{M}$) for the model (Sets III and IV) with $N_t = N_s = 2N_v = 36$ and $\alpha = 10^{-3}$
• Figure 4: Eigenvalues of the space discrete matrix $\tilde{A}$ for the model (Sets III and IV) with $N_t = N_s = 2N_v = 36$.
• Figure 5: Eigenvalues of the original and preconditioned matrices (i.e., $\mathcal{M}$, $P^{-1}_1\mathcal{M}$ and $P^{-1}_{\alpha}\mathcal{M}$) for the model (Set V) with $N_t = N_s = 2N_v = 36$ and $\alpha = 10^{-3}$.

Theorems & Definitions (21)

• Lemma 2.1
• Theorem 2.2
• Proof 1
• Theorem 2.3
• Proof 2
• Remark 3.2
• Remark 3.3
• Lemma 3.4
• Proof 3
• Lemma 3.5