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This paper is a continuation of Melenk et al., "Stability analysis for electromagnetic waveguides. Part 1: acoustic and homogeneous electromagnetic waveguides" (2023), extending the stability results for homogeneous electromagnetic (EM) waveguides to the non-homogeneous case. The analysis is done using perturbation techniques for self-adjoint operators eigenproblems. We show that the non-homogeneous EM waveguide problem is well-posed with the stability constant scaling linearly with waveguide length L. The results provide a basis for proving convergence of a Discontinuous Petrov-Galerkin (DPG) discretization based on a full envelope ansatz, and the ultraweak variational formulation for the resulting modified system of Maxwell equations, see Part 1. 

Wave propagation is fundamental to applications including natural resource exploration, nuclear fusion research, and military defense, among others. However, developing accurate and efficient numerical algorithms for solving time-harmonic wave propagation problems is notoriously difficult. One difficulty is that classical discretization techniques (e.g., Galerkin finite elements, finite difference, etc.) yield indefinite discrete systems that preclude the use of many scalable solution algorithms. Significant progress has been made to develop specialized preconditioners for high-frequency wave propagation problems but robust and scalable solvers for general problems, including non-homogenous media and complex geometries, remain elusive. An alternative approach is to use minimum residual discretization methods—that yield Hermitian positive-definite discrete systems—and may be amenable to more standard preconditioners. Indeed, popularization of the first-order system least-squares methodology (FOSLS) was driven by the applicability of geometric and algebraic multigrid to otherwise indefinite problems. However, for wave propagation problems, FOSLS is known to be highly dissipative and is thus less competitive in the high-frequency regime. The discontinuous Petrov–Galerkin (DPG) method of Demkowicz and Gopalakrishnan is a minimum residual finite element method with several additional attractive properties: mesh-independent stability, a built-in error indicator, and applicability to a number of variational formulations. In the context of high-frequency wave propagation, the ultraweak DPG formulation has been observed to produce pollution error roughly commensurate to Galerkin discretizations. DPG discretizations may thus deliver accuracy typical of classical discretization techniques, but result in Hermitian positive-definite discrete systems that are often more amenable to preconditioning. A multigrid preconditioner for DPG systems, developed in the dissertation work of S. Petrides, was shown to scale efficiently in a shared-memory implementation. The primary objective of this dissertation is development of an efficient, distributed implementation of the DPG multigrid solver (DPG-MG). The distributed DPG-MG solver developed in this work will be demonstrated to be massively scalable, enabling solution of three-dimensional problems with O(10¹²) degrees of freedom on up to 460 000 CPU cores, an unprecedented scale for high-frequency wave propagation. The scalability of the DPG-MG solver will be further combined with hp-adaptivity to enable efficient solution of challenging real-world high-frequency wave propagation problems including optical fiber modeling, simulation of RF heating in tokamak devices, and seismic simulation. These applications include complex three-dimensional geometries, heterogeneous and anisotropic media, and localized features; demonstrating the robustness and versatility of the solver and tools developed in this dissertation. 

This paper presents a scalable multigrid preconditioner targeting large-scale systems arising from discontinuous Petrov-Galerkin (DPG) discretizations of high-frequency wave operators. This work is built on previously developed multigrid preconditioning techniques of Petrides and Demkowicz (Comput. Math. Appl. 87 (2021) pp. 12–26) and extends the convergence results from O(10^7) degrees of freedom (DOFs) to O(10^9) DOFs using a new scalable parallel MPI/OpenMP implementation. Novel contributions of this paper include an alternative definition of coarse-grid systems based on restriction of fine-grid operators, yielding superior convergence results. In the uniform refinement setting, a detailed convergence study is provided, demonstrating ℎand 𝑝robust convergence and linear scaling with respect to the wave frequency. The paper concludes with numerical results on hp-adaptive simulations including a large-scale seismic modeling benchmark problem with high material contrast. 

Abstract Partial differential equation (PDE)-constrained inverse problems are some of the most challenging and computationally demanding problems in computational science today. Fine meshes required to accurately compute the PDE solution introduce an enormous number of parameters and require large-scale computing resources such as more processors and more memory to solve such systems in a reasonable time. For inverse problems constrained by time-dependent PDEs, the adjoint method often employed to compute gradients and higher order derivatives efficiently requires solving a time-reversed, so-called adjoint PDE that depends on the forward PDE solution at each timestep. This necessitates the storage of a high-dimensional forward solution vector at every timestep. Such a procedure quickly exhausts the available memory resources. Several approaches that trade additional computation for reduced memory footprint have been proposed to mitigate the memory bottleneck, including checkpointing and compression strategies. In this work, we propose a close-to-ideal scalable compression approach using autoencoders to eliminate the need for checkpointing and substantial memory storage, thereby reducing the time-to-solution and memory requirements. We compare our approach with checkpointing and an off-the-shelf compression approach on an earth-scale ill-posed seismic inverse problem. The results verify the expected close-to-ideal speedup for the gradient and Hessian-vector product using the proposed autoencoder compression approach. To highlight the usefulness of the proposed approach, we combine the autoencoder compression with the data-informed active subspace (DIAS) prior showing how the DIAS method can be affordably extended to large-scale problems without the need for checkpointing and large memory.