Iterative solvers for partial differential equations with dissipative structure: Operator preconditioning and optimal control

TL;DR

This paper addresses iterative solution of large-scale PDE-derived systems with a dissipative port-Hamiltonian structure, modeled as $\mathcal{A}=\mathcal{H}+\mathcal{S}$ where $\mathcal{H}$ is symmetric and nonnegative and $\mathcal{S}$ is skew-symmetric. By leveraging operator preconditioning with the symmetric part, the authors analyze when the preconditioned operator $I+\mathcal{H}^{-1}\mathcal{S}$ remains bounded and, consequently, when the spectrum—and thus convergence—can be mesh-size independent for elliptic and parabolic problems. They derive boundedness results for several PDE classes (advection-diffusion, Stokes, Oseen, wave with damping, Kelvin-Voigt beam) and, in the optimal-control setting, present two strategies: direct splitting and condensed formulations, both solved via tailored Krylov methods (GMRES, Rapoport, Widlund) with AMG or incomplete Cholesky preconditioners. Numerical experiments illustrate robust, scalable performance, particularly when using algebraic multigrid preconditioning for the symmetric part and, in projected CG, a constraint preconditioner to confine iterations to the feasible subspace. Overall, the work provides a rigorous and practical framework for fast, structure-aware solvers in dissipative PDEs and PDE-constrained optimization, with clear routes to extension to nonlinear and DAEs.

Abstract

This work considers the iterative solution of large-scale problems subject to non-symmetric matrices or operators arising in discretizations of (port-)Hamiltonian partial differential equations. We consider problems governed by an operator $\mathcal{A}=\mathcal{H}+\mathcal{S}$ with symmetric part $\mathcal{H}$ that is positive (semi-)definite and skew-symmetric part $\mathcal{S}$. Prior work has shown that the structure and sparsity of the associated linear system enables Krylov subspace solvers such as the generalized minimal residual method (GMRES) or short recurrence variants such as Widlund's or Rapoport's method using the symmetric part $\mathcal{H}$, or an approximation of it, as preconditioner. In this work, we analyze the resulting condition numbers, which are crucial for fast convergence of these methods, for various partial differential equations (PDEs) arising in diffusion phenomena, fluid dynamics, and elasticity. We show that preconditioning with the symmetric part leads to a condition number uniform in the mesh size in case of elliptic and parabolic PDEs where $\mathcal{H}^{-1}\mathcal{S}$ is a bounded operator. Further, we employ the tailored Krylov subspace methods in optimal control by means of a condensing approach and a constraint preconditioner for the optimality system. We illustrate the results by various large-scale numerical examples and discuss efficient evaluations of the preconditioner, such as incomplete Cholesky factorization or the algebraic multigrid method.

Figures (9)

• Figure 1: Condition numbers for Stokes equation with $L^2$ pressure regularization $(s_1,s_2) = (1,0)$ (left) and $H^1$ pressure regularization $(s_1,s_2) = (1,1)$ (right).
• Figure 2: Condition numbers for Oseen equation for the original system (left) and the Schur complement (right).
• Figure 3: Condition numbers for wave equation with momentum damping in first-order formulation.
• Figure 4: Condition numbers for beam equation with Kelvin-Voigt damping.
• Figure 5: Advection-Diffusion-Reaction equation: Solution of condensed system with outer tolerance $\mathrm{cgtol}=10^{-4}$ and $\lambda \in\{10^{-1},10^{-2},10^{-3}\}$ (top to bottom).

Theorems & Definitions (17)

• Lemma 1
• proof
• Proposition 2: Preconditioning
• proof
• Remark 3
• Proposition 4
• proof
• Proposition 5
• proof
• Proposition 6