Francesco Ballarin

This paper focuses on the efficient numerical algorithms of a three-field Biot’s consolidation model. The approach begins with the introduction of innovative monolithic and global-in-time iterative decoupled algorithms, which incorporate the backward differentiation formulas for time discretization. In each iteration, these algorithms involve solving a diffusion subproblem over the entire temporal domain, followed by solving a generalized Stokes subproblem over the same time interval. To accelerate the global-in-time iterative process, we present a reduced order modeling approach based on proper orthogonal decomposition, aimed at reducing the primary computational cost from the generalized Stokes subproblem. The effectiveness of this novel method is validated both theoretically and through numerical experiments.

Hierarchical Model (HiMod) reduction is an effective Reduced Order Modelling technique for problems defined on elongated, pipe-like domains. It is particularly suitable when a dominant dynamics is aligned with the longitudinal direction, while transverse effects are locally significant but spatially limited. When applied to two-field problems such as the Stokes equations, a main challenge is to ensure the stability of the reduced formulation, particularly the inf-sup condition for pressure discretization. In this work, we provide a rigorous analysis showing that the inf-sup condition holds whenever the number of velocity modes is at least equal to the number of pressure modes, thereby extending previous heuristic approaches. The proof exploits the separation of variables in HiMod and is valid for pipe-like domains under some geometric assumptions. Numerical assessment confirms the theoretical findings, providing a solid foundation for stable and efficient HiMod reduction in incompressible flow problems.

Parametric PDE-constrained optimal control with pointwise state constraints requires solving a sequence of restricted Schur-complement systems over parameter-dependent inactive sets. Under a primal active-set strategy, each inactive-set solve is symmetric positive definite, but the active set may change non-smoothly with the parameter, causing the restricted operator to vary in dimension, sparsity pattern, and spectrum. This volatility limits the direct reuse of sparse factorizations, multigrid hierarchies, and instance-to-instance Krylov recycling. We propose a reusable spectral-deflation strategy that anchors the coarse space to a single full-domain reference Schur complement. Low reference eigenmodes are computed once, restricted online to each inactive set, and used as an A-DEF2 deflation basis for Jacobi-preconditioned conjugate gradient, with optional POD enrichment, Rayleigh–Ritz reselection, coarse-grid or analytical reference modes, and conditioning safeguards. Given the active set, the method preserves the high-fidelity inactive-set system and solves it to the prescribed CG tolerance; it accelerates the linear algebra rather than replacing the optimal-control solve with a surrogate. We interpret its effectiveness through a spectral-coherence perspective, motivated by interlacing and perturbation arguments and assessed empirically with principal-angle diagnostics. Across diffusion, convection–diffusion, nonlinear thermal, and conjugate-heat-transfer benchmarks, the proposed deflation reduces CG iterations by roughly 55--98$\%$, providing a hardware-independent measure of acceleration. In GPU deployments, the reusable basis also yields substantial wall-time gains over CPU sparse-direct and algebraic-multigrid baselines, reflecting both accelerator throughput and the fact that the reference basis is built once while competing solver structures are rebuilt per instance. With coarse-grid or analytical reference modes, the offline cost amortizes within a single parameter sweep; with a fine-grid eigensolve, the per-instance gains are partially precompute-limited. All reported timings isolate the inactive-set linear-solve kernel; reducing the surrounding active-set outer loop is outside the scope of this work.

The approximate deconvolution Leray reduced order model (ADL‐ROM) uses spatial filtering to increase the ROM stability, and approximate deconvolution to increase the ROM accuracy. In the under‐resolved numerical simulation of convection‐dominated flows, ADL‐ROM was shown to be significantly more stable than the standard ROM and more accurate than the Leray ROM. In this paper, we equip ADL‐ROM with a new van Cittert AD operator and prove a priori error bounds for both the AD operator and the ADL‐ROM. To our knowledge, these are the first numerical analysis results for approximate deconvolution in a ROM context. We illustrate these numerical analysis results in the numerical simulation of convection‐dominated flows.

Galerkin reduced order models (ROMs), e.g., based on proper orthogonal decomposition (POD) or reduced basis methods, have achieved significant success in the numerical simulation of fluid flows. The ROM numerical analysis, however, is still being developed. In this paper, we take a step in this direction and present a survey of a priori error bounds, with a particular focus on POD-based ROMs. Specifically, we outline the main components of ROM a priori error bounds, emphasize their practical importance, and discuss significant contributions to a priori error bounds for ROMs for fluids.