Abstract. We propose a novel, rigorous, and conservative invariant-domain preserving (IDP) projection technique between solution spaces of differing discretizations. The projection preserves important physical invariants of the underlying hyperbolic system, for example, positivity of material density and internal energy for the compressible Euler equations. When coupled with suitable refinement indicators, the proposed scheme enables a provably IDP adaptive numerical method for hyperbolic systems where preservation of physical properties is essential. In addition to proofs of these characteristics, we supply a detailed construction of the method in the context of a high-performance finite element code. To illustrate our proposed scheme, we study a suite of computationally challenging benchmark problems, demonstrating enhanced accuracy and efficiency properties while entirely avoiding ad hoc corrections to preserve physical invariants. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/conservation-laws/ryujin and in the supplementary materials ( ryujin-development.zip [938KB]). [Formula: see text]
Abstract. We propose a novel approach to approximate the basic reproduction number [Formula: see text] as the spectral radius of the Next-Generation Operator in time-periodic population models by characterizing the latter via evolution semigroups. Once birth/infection and transition operators are identified, we discretize them via either Fourier or Chebyshev collocation methods. Then [Formula: see text] is obtained by solving a generalized matrix eigenvalue problem. The order of convergence of the approximating reproduction numbers to the true one is shown to depend on the regularity of the model coefficients, and spectral accuracy is proved. We validate the theoretical results by discussing applications to epidemiology, viz. a large-size multigroup epidemic model with periodic contact rates, and a vector-borne disease model with seasonal vector recruitment. We illustrate how the method facilitates implementation compared to existing approaches and how it can be easily adapted to also compute type-reproduction numbers. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://gitlab.com/SIMONE.DEREGGI/R0_periodicODEs and in the supplementary materials ( R0_periodicODEs-main.zip [14.9KB]). [Formula: see text]
Abstract. Differential equations (DEs) are widely used to model the evolution of natural or engineered systems. Traditionally, the parameters in DEs are adjusted to fit data obtained through system observations. However, in fields such as politics, economics, and biology, the observed data points are often collected independently (i.e., repeated cross-sectional (RCS) data). In this paper, we first demonstrate that conventional optimization techniques struggle to accurately estimate DE parameters when RCS data exhibit various heterogeneities, leading to a significant loss of information. To address this issue, we propose a new estimation method, the emulator-informed deep-generative model (EIDGM), designed to handle RCS data. Specifically, EIDGM integrates a physics-informed neural network–based emulator that immediately generates DE solutions and a Wasserstein-generative adversarial network–based parameter generator that can effectively mimic the RCS data. We then evaluated the effectiveness of EIDGM across various models, including exponential and logistic growth models, and the Lorenz system, demonstrating its superior ability to accurately capture parameter distributions. In addition, we applied EIDGM to real-world datasets, successfully capturing diverse shapes of parameter distributions. This result highlights that EIDGM can be applied to model a wide range of systems with limited data availability, and our approach can be significantly extended to uncover the operating principles of systems based on limited data. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/CHWmath/EIDGM/tree/main and in the supplementary materials ( EIDGM-main.zip [19.5MB]). [Formula: see text]
Abstract. Shock waves in gas dynamics feature jump discontinuities that hinder numerical simulations. Viscous regularizations are prone to excessive dissipation of fine-scale structures. In this work, we propose the first inviscid regularization of the multidimensional Euler equation based on ideas from semidefinite programming, information geometry, geometric hydrodynamics, and nonlinear elasticity. The Lagrangian flow maps of Euler solutions are a dynamical system on the manifold of diffeomorphisms. We observe that shock formation arises from the manifold’s geodesic incompleteness. Our regularization embeds it into an ambient space equipped with the information geometry of the logarithmic barrier function. Thus, the diffeomorphism manifold inherits a geodesically complete geometry. The resulting regularized conservation law replaces shocks with smooth profiles without affecting oscillatory structures. One- and two-dimensional numerical experiments show its practical potential to enable higher-order methods without explicit shock capturing. While we focus on the barotropic Euler equations for concreteness and simplicity of exposition, our regularization easily extends to more general Euler and Navier–Stokes-type equations. Our approach regularizes the Wasserstein geometry of the mass density with its information geometry. The former captures the natural trajectories of physical particles and the latter that of statistical estimators. Information geometric regularization accounts for the mass density’s dual nature as a statistical/computational tool summarizing the motion of physical particles. Thus, our work is a starting point for information geometric mechanics that views solutions of continuum mechanical PDEs as parameters of statistical models for unresolved scales and uses their information geometry to evolve them in time. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/f-t-s/information_geometric_regularization_of_barotropic_euler and in the supplementary materials ( information_geometric_regularization_of_barotropic_euler-main.zip [86.2KB]). [Formula: see text]
Abstract. A wide range of applications necessitates solving large-scale ill-posed problems contaminated by noise, which are commonly formulated as linear approximation problems (possibly overdetermined or underdetermined). Krylov subspace regularization methods are particularly advantageous in this context, as they rely solely on matrix-vector multiplication. Among the most widely used techniques are least squares QR (LSQR) and conjugate gradient least squares (CGLS), both of which can be extended with flexible preconditioning to enforce solution properties such as nonnegativity or sparsity. Flexible LSQR (FLSQR) can also be combined with direct methods to create efficient hybrid approaches. The flexible Golub–Kahan process underlying FLSQR requires two long-term recurrences. Here, we introduce a novel fast flexible Golub–Kahan method that employs one long-term and one short-term recurrence. Using this, we develop the fast flexible LSQR (FaFLSQR) algorithm, which offers comparable computational cost to flexible CGLS (FCGLS) while also supporting hybrid regularization like FLSQR. We analyze the properties of FaFLSQR and prove its mathematical equivalence to FCGLS. Numerical experiments demonstrate that in floating-point arithmetic, FaFLSQR outperforms both FCGLS and FLSQR in terms of computational efficiency.
