1,177 research outputs found
Incremental proper orthogonal decomposition for PDE simulation data: Algorithms and analysis
We propose an incremental algorithm to compute the proper orthogonal decomposition (POD) of simulation data for a partial differential equation. Specifically, we modify an incremental matrix SVD algorithm of Brand to accommodate data arising from Galerkin-type simulation methods for time dependent PDEs. We introduce an incremental SVD algorithm with respect to a weighted inner product to compute the proper orthogonal decomposition (POD). The algorithm is applicable to data generated by many numerical methods for PDEs, including finite element and discontinuous Galerkin methods. We also modify the algorithm to initialize and incrementally update both the SVDand an error bound during the time stepping in a PDE solver without storing the simulation data. We show the algorithm produces the exact SVD of an approximate data matrix, and the operator norm error between the approximate and exact data matrices is bounded above by the computed error bound. This error bound also allows us to bound the error in the incrementally computed singular values and singular vectors. We demonstrate the effectiveness of the algorithm using finite element computations for a 1D Burgers\u27 equation, a 1D FitzHugh-Nagumo PDE system, and a 2D Navier-Stokes problem --Abstract, page iv
A note on incremental POD algorithms for continuous time data
In our earlier work [Fareed et al., Comput. Math. Appl. 75 (2018), no. 6,
1942-1960], we developed an incremental approach to compute the proper
orthogonal decomposition (POD) of PDE simulation data. Specifically, we
developed an incremental algorithm for the SVD with respect to a weighted inner
product for the discrete time POD computations. For continuous time data, we
used an approximate approach to arrive at a discrete time POD problem and then
applied the incremental SVD algorithm. In this note, we analyze the continuous
time case with simulation data that is piecewise constant in time such that
each data snapshot is expanded in a finite collection of basis elements of a
Hilbert space. We first show that the POD is determined by the SVD of two
different data matrices with respect to weighted inner products. Next, we
develop incremental algorithms for approximating the two matrix SVDs with
respect to the different weighted inner products. Finally, we show neither
approximate SVD is more accurate than the other; specifically, we show the
incremental algorithms return equivalent results
Comparison of some Reduced Representation Approximations
In the field of numerical approximation, specialists considering highly
complex problems have recently proposed various ways to simplify their
underlying problems. In this field, depending on the problem they were tackling
and the community that are at work, different approaches have been developed
with some success and have even gained some maturity, the applications can now
be applied to information analysis or for numerical simulation of PDE's. At
this point, a crossed analysis and effort for understanding the similarities
and the differences between these approaches that found their starting points
in different backgrounds is of interest. It is the purpose of this paper to
contribute to this effort by comparing some constructive reduced
representations of complex functions. We present here in full details the
Adaptive Cross Approximation (ACA) and the Empirical Interpolation Method (EIM)
together with other approaches that enter in the same category
MORe DWR: Space-time goal-oriented error control for incremental POD-based ROM for time-averaged goal functionals
In this work, the dual-weighted residual (DWR) method is applied to obtain an error-controlled incremental proper orthogonal decomposition (POD) based reduced order model. A novel approach called MORe DWR (Model Order Reduction with Dual-Weighted Residual error estimates) is being introduced. It marries tensor-product space-time reduced-order modeling with time slabbing and an incremental POD basis generation with goal-oriented error control based on dual-weighted residual estimates. The error in the goal functional is being estimated during the simulation and the POD basis is being updated if the estimate exceeds a given threshold. This allows an adaptive enrichment of the POD basis in case of unforeseen changes in the solution behavior. Consequently, the offline phase can be skipped, the reduced-order model is being solved directly with the POD basis extracted from the solution on the first time slab and –if necessary– the POD basis is being enriched on-the-fly during the simulation with high-fidelity finite element solutions. Therefore, the full-order model solves can be reduced to a minimum, which is demonstrated on numerical tests for the heat equation and elastodynamics using time-averaged quantities of interest
MORe DWR: Space-time goal-oriented error control for incremental POD-based ROM
In this work, the dual-weighted residual (DWR) method is applied to obtain a
certified incremental proper orthogonal decomposition (POD) based reduced order
model. A novel approach called MORe DWR (Model Order Rduction with
Dual-Weighted Residual error estimates) is being introduced. It marries
tensor-product space-time reduced-order modeling with time slabbing and an
incremental POD basis generation with goal-oriented error control based on
dual-weighted residual estimates. The error in the goal functional is being
estimated during the simulation and the POD basis is being updated if the
estimate exceeds a given threshold. This allows an adaptive enrichment of the
POD basis in case of unforeseen changes in the solution behavior which is of
high interest in many real-world applications. Consequently, the offline phase
can be skipped, the reduced-order model is being solved directly with the POD
basis extracted from the solution on the first time slab and -- if necessary --
the POD basis is being enriched on-the-fly during the simulation with
high-fidelity finite element solutions. Therefore, the full-order model solves
can be reduced to a minimum, which is demonstrated on numerical tests for the
heat equation and elastodynamics.Comment: 42 pages, 13 figure
Proper general decomposition (PGD) for the resolution of Navier–Stokes equations
In this work, the PGD method will be considered for solving some problems of fluid mechanics by looking for the solution as a sum of tensor product functions. In the first stage, the equations of Stokes and Burgers will be solved. Then, we will solve the Navier–Stokes problem in the case of the lid-driven cavity for different Reynolds numbers (Re = 100, 1000 and 10,000). Finally, the PGD method will be compared to the standard resolution technique, both in terms of CPU time and accuracy.Région Poitou-Charente
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