An improved algorithm for the shallow water equations model reduction: Dynamic Mode Decomposition vs POD

SUMMARYThis article presents a non-intrusive reduced order model (NIROM) for general, dynamic partial differential equations. Based upon proper orthogonal decomposition (POD) and Smolyak sparse grid collocation, the method first projects the unknowns with full space and time coordinates onto a reduced POD basis. Then we introduce a new least squares fitting procedure to approximate the dynamical transition of the POD coefficients between subsequent time steps taking only a set of full model solution snapshots as the training data during the construction. Thus, neither the physical details nor further numerical simulations of the original PDE model is required by this methodology and the level of non-intrusiveness is improved compared to existing ROMs. Furthermore, we take adaptive measures to address the instability issue arising from reduced order iterations of the POD coefficients. This model can be applied to a wide range of physical and engineering scenarios and we test it on a couple problems in fluid dynamics. It is demonstrated that this reduced order approach captures the dominant features of the high fidelity models with reasonable accuracy while the computation complexity is reduced by several orders of magnitude.

Section: Accepted Articlementioning

confidence: 99%

“…reduced order model for each time step and reconstruct the LSROM-approximated solution on the physical mesh with expansion formula (7).…”

Section: Accepted Articlementioning

confidence: 99%

Non‐intrusive reduced order modelling with least squares fitting on a sparse grid

Lin

Xiao

Fang

et al. 2016

“…ROMs have been used to reduce the computational cost of scientific and engineering applications that are governed by relatively few recurrent dominant spatial structures. [1][2][3][4][5][6][7][8][9][10][11][12] In an off-line stage, full-order model (FOM) data on a given time interval are used to build the ROM. In an online stage, ROMs are repeatedly used for parameter settings and/or time intervals that are different from those used to build them.…”

Section: Introductionmentioning

confidence: 99%

“…where a is the vector of unknown ROM coefficients and A ∈ R r × r , B ∈ R r × r × r are ROM operators that are assembled in the off-line stage. Data-driven ROMs (DD-ROMs) (eg, sparse identification of nonlinear dynamics 14 and operator inference method 15,16 ) use a fundamentally different strategy: they first postulate a ROM ansatż a =Ã a + a ⊤B a, (4) and then, they choose the operatorsÃ andB to minimize the difference between the FOM and ansatz (4), 17,18 ie,…”

Section: Introductionmentioning

confidence: 99%

Physically constrained data‐driven correction for reduced‐order modeling of fluid flows

Mohebujjaman

Rebholz

Iliescu

2018

Summary We have recently proposed a data‐driven correction reduced‐order model (DDC‐ROM) framework for the numerical simulation of fluid flows, which can be formally written as follows. The new DDC‐ROM was constructed by using reduced‐order model spatial filtering and data‐driven reduced‐order model closure modeling (for the Correction term) and was successfully tested in the numerical simulation of a two‐dimensional channel flow past a circular cylinder at Reynolds numbers Re = 100,Re = 500, and Re = 1000. In this paper, we propose a physically constrained DDC‐ROM (CDDC‐ROM), which aims at improving the physical accuracy of the DDC‐ROM. The new physical constraints require that the CDDC‐ROM operators satisfy the same type of physical laws (ie, the Correction term's linear component should be dissipative, and the Correction term's nonlinear component should conserve energy) as those satisfied by the fluid flow equations. To implement these physical constraints, in the data‐driven modeling step, we replace the unconstrained least squares problem with a constrained least squares problem. We perform a numerical investigation of the new CDDC‐ROM and standard DDC‐ROM for a two‐dimensional channel flow past a circular cylinder at Reynolds numbers Re = 100,Re = 500, and Re = 1000. To this end, we consider a reproductive regime as well as a predictive (ie, cross‐validation) regime in which we use as little as 50% of the original training data. The numerical investigation clearly shows that the new CDDC‐ROM is significantly more accurate than the DDC‐ROM in both regimes.

“…A particular attention was given to reducing the computational complexity of the non‐linear reduced order models using discrete empirical interpolation method , tensorial POD , and non‐intrusive methods . Very important studies on the practical and theoretical aspects of dynamic mode decomposition have been recently published .…”

mentioning

confidence: 99%

Model reduction and inverse problems and data assimilation with geophysical applications. A special issue in honor of I. Michael Navon's 75th birthday

Ştefanescu

Noack

Sandu

2016

EDITORIALModel reduction and inverse problems and data assimilation with geophysical applications. A special issue in honor of I. Michael Navon's 75th birthdayProfessor Ionel Michael Navon retired in September 2014 from the Scientific Computing Department, Florida State University, Tallahassee, Florida, after a brilliant academic career. Since 1997, he is a fellow of the American Meteorological Society, and recently, he became an honorary member of the Academy of Romanian Scientists. His distinguished pioneering achievements in the domains of data assimilation, inverse problems, and reduced order modeling during the last few decades have greatly contributed to the establishment and development of these aforementioned fields. In particular, his rich interdisciplinary expertise allowed him to advance the science reduced order modeling and inverse modeling as core techniques for data-driven modeling. He generalized predictive computational modeling leading to fast novel solution approaches for real-world inverse problems of oceanography and weather forecast. At the same time, Professor Ionel Michael Navon has been an outstanding mentor, advisor, role model, and friend to his 9 PhD students and 11 postdoctoral scholars, with the large majority of them pursuing successful academic, scientific, and industry careers. This year Professor Ionel Michael Navon celebrates his 75th anniversary, and this special issue recognizes his accomplishments. The contributors to this special issue are former students, postdoctoral scholars, colleagues, close collaborators, and friends of Professor Ionel Michael Navon. The next three paragraphs describe his most important contributions to data assimilation and reduced order modeling fields in the last 10 years, while the last part of the editorial focuses on the special issue's contributions.Professor Navon's work addressed fundamental issues in both variational and statistical data assimilation with applications in fluid dynamics and atmospheric flows. Professor Navon extended and proposed frameworks to tackle non-differentiability in models and objective functions. For example, he formulated the maximum likelihood ensemble filter (MLEF) equations without the differentiability requirement for the prediction model and for the observation operators [1] and measured the impact of non-smooth observation operators on variational and sequential data assimilation [2]. Other efforts include coupling a Gaussian resampling method to generate more effective and efficient Particle Filter posterior analysis ensembles [3], improvement on the ensemble Kalman filters [4][5][6][7], and MLEF [8]. Professor Navon is also the co-author of a recent book [9] and several book chapters [10][11][12][13] describing the recent advancements and methodologies in variational data assimilation and non-linear sensitivity analysis. Goal-oriented adjoint sensitivity methods were proposed to guide to the adaptivity of finite element meshes [14,15]. Moreover, he developed different approaches to model error formulation [...