POD-ROM methods: from a finite set of snapshots to continuous-in-time approximations
arxiv(2024)
摘要
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(Ω) 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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