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Bosco García-Archilla, Volker John, Julia Novo
POD-ROM methods: from a finite
set of snapshots to continuous-in-time approximations, SIAM J. Numer. Anal.
63, 800 - 826, 2025 This paper studies discretization of time-dependent partial differential equations (PDEs) by proper orthogonal decomposition reduced order models (POD-ROMs). Most of the analysis in the literature has been performed on fully-discrete methods using first order methods in time, typically the implicit Euler time integrator. Our aim is to show which kind of error bounds can be obtained using any time integrator, both in the full order model (FOM), applied to compute the snapshots, and in the POD-ROM method. To this end, we analyze in this paper the continuous-in-time case for both the FOM and POD-ROM methods, although the POD basis is obtained from snapshots taken at a discrete (i.e., not continuous) set times. Two cases for the set of snapshots are considered: The case in which the snapshots are based on first order divided differences in time and the case in which they are based on temporal derivatives. Optimal pointwise-in-time error bounds {between the FOM and the POD-ROM solutions} are proved for the $L^2(\Omega)$ norm of the error for a semilinear reaction-diffusion model problem. The dependency of the errors on the distance in time between two consecutive snapshots and on the tail of the POD eigenvalues is tracked. Our detailed analysis allows to show that, in some situations, a small number of snapshots in a given time interval might be sufficient to accurately approximate the solution in the full interval. Numerical studies support the error analysis. |
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Bosco García-Archilla, Volker John, Sarah Katz, Julia Novo
POD-ROMs for incompressible flows including snapshots of the temporal derivative of the
full order solution: Error bounds for the pressure, Journal of Numerical Mathematics, 301 - 329, 2024 Reduced order methods (ROMs) for the incompressible Navier--Stokes equations, based on proper orthogonal decomposition (POD), are studied that include snapshots which approach the temporal derivative of the velocity from a full order mixed finite element method (FOM). In addition, the set of snapshots contains the mean velocity of the FOM. Both the FOM and the POD-ROM are equipped with a grad-div stabilization. A velocity error analysis for this method can be found already in the literature. The present paper studies two different procedures to compute approximations to the pressure and proves error bounds for the pressure that are independent of inverse powers of the viscosity. Numerical studies support the analytic results and compare both methods. |
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Bosco García-Archilla, Volker John, Julia Novo
POD-ROMs for incompressible flows including snapshots of the temporal derivative of the full order solution, SIAM J. Numer. Anal.
61, 1340 - 1368, 2023 In this paper we study the influence of including snapshots that approach the velocity time derivative in the numerical approximation of the incompressible Navier--Stokes equations by means of proper orthogonal decomposition (POD) methods. Our set of snapshots includes the velocity approximation at the initial time from a full order mixed finite element method (FOM) together with approximations to the time derivative at different times. The approximation at the initial velocity can be replaced by the mean value of the velocities at the different times so that implementing the method to the fluctuations, as done mostly in practice, only approximations to the time derivatives are included in the set of snapshots. For the POD method we study the differences between projecting onto $L^2$ and $H^1$. In both cases, pointwise in time error bounds are proved. Including grad-div stabilization both in the FOM and POD methods, error bounds with constants independent of inverse powers of the viscosity are obtained. |
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Bosco García-Archilla, Volker John, Julia Novo
Second order error bounds for POD-ROM methods based on first order divided differences, Appl. Math. Lett. 146, Article 108836, 2023
This note proves for the heat equation that using BDF2 as time stepping scheme in POD-ROM methods with snapshots based on difference quotients gives both the optimal second order error bound in time and pointwise estimates. |
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Volker John, Baptiste Moreau, Julia Novo
Error analysis of a SUPG-stabilized POD-ROM method for convection-diffusion-reaction equations, Computers and Mathematics with Applications 122, 48 - 60, 2022 A reduced order model (ROM) method based on proper orthogonal decomposition (POD) is analyzed for convection-diffusion-reaction equations. The streamline-upwind Petrov--Galerkin (SUPG) stabilization is used in the practically interesting case of dominant convection, both for the full order method (FOM) and the ROM simulations. The asymptotic choice of the stabilization parameter for the SUPG-ROM is done as proposed in the literature. This paper presents a finite element convergence analysis of the SUPG-ROM method for errors in different norms. The constants in the error bounds are uniform with respect to small diffusion coefficients. Numerical studies illustrate the performance of the SUPG-ROM method. |
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Swetlana Giere, Traian Iliescu, Volker John, David Wells
SUPG Reduced Order Models for
Convection-Dominated Convection-Diffusion-Reaction Equations,
Comput. Methods Appl. Mech. Engrg. 289, 454 - 474, 2015
This paper presents a Streamline-Upwind Petrov--Galerkin (SUPG) reduced order model (ROM) based on proper orthogonal decomposition (POD). This ROM is investigated theoretically and numerically for convection-dominated convection-diffusion-reaction equations. The SUPG finite element method was used on realistic meshes for computing the snapshots, leading to some noise in the POD data. Numerical analysis is used to propose the scaling of the stabilization parameter for the SUPG-ROM. Two approaches are used: One based on the underlying finite element discretization and the other one based on the POD truncation. The resulting SUPG-ROMs and the standard Galerkin ROM (G-ROM) are studied numerically. For many settings, the results obtained with the SUPG-ROMs are more accurate. Finally, one of the choices for the stabilization parameter is recommended.
Alfonso Caiazzo, Traian Iliescu, Volker John, Swetlana Schyschlowa
A Numerical Investigation of Velocity-Pressure Reduced Order Models
for Incompressible Flows, J. Comput. Phys. 259, 598 - 616 , 2014
| This report has two main goals. First, it numerically investigates three velocity-pressure reduced order models (ROMs) for incompressible flows. The proper orthogonal decomposition (POD) is used to generate the modes. One method computes the ROM pressure solely based on the velocity POD modes, whereas the other two ROMs use pressure modes as well. To the best of the authors' knowledge, one of the latter methods is novel. The second goal is to numerically investigate the impact of the snapshot accuracy on the results of the ROMs. Numerical studies are performed on a two-dimensional laminar flow past a circular obstacle. Comparing the results of the ROMs and of the simulations for computing the snapshots, it turns out that the latter results are generally well reproduced by the ROMs. This observation is made for snapshots of different accuracy. Both in terms of reproducing the results of the underlying simulations for obtaining the snapshots and of efficiency, the two ROMs that utilize pressure modes are superior to the ROM that uses only velocity modes. |