2015/04/14 by Kevin Carlberg, Carlberg, Kevin, Matthew Barone +3 · 6 citations
Engineering · Mathematics · Physics and Astronomy · #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations #Power System Optimization and Stability
paper · pdf · doi:10.48550/arxiv.1504.03749
openalex publication_date 2015/04/14 · openalex created_date 2022/10/07 · openalex updated_date 2026/07/28
Least-squares Petrov--Galerkin (LSPG) model-reduction techniques such as the\nGauss--Newton with Approximated Tensors (GNAT) method have shown promise, as\nthey have generated stable, accurate solutions for large-scale turbulent,\ncompressible flow problems where standard Galerkin techniques have failed.\nHowever, there has been limited comparative analysis of the two approaches.\nThis is due in part to difficulties arising from the fact that Galerkin\ntechniques perform optimal projection associated with residual minimization at\nthe time-continuous level, while LSPG techniques do so at the time-discrete\nlevel. This work provides a detailed theoretical and computational comparison\nof the two techniques for two common classes of time integrators: linear\nmultistep schemes and Runge--Kutta schemes. We present a number of new\nfindings, including conditions under which the LSPG ROM has a time-continuous\nrepresentation, conditions under which the two techniques are equivalent, and\ntime-discrete error bounds for the two approaches. Perhaps most surprisingly,\nwe demonstrate both theoretically and computationally that decreasing the time\nstep does not necessarily decrease the error for the LSPG ROM; instead, the\ntime step should be `matched' to the spectral content of the reduced basis. In\nnumerical experiments carried out on a turbulent compressible-flow problem with\nover one million unknowns, we show that increasing the time step to an\nintermediate value decreases both the error and the simulation time of the LSPG\nreduced-order model by an order of magnitude.\n