Efficient approximation of Sparse Jacobians for time‐implicit reduced order models

Ştefanescu, Răzvan; Sandu, Adrian

doi:10.1002/fld.4260

Cited by 14 publications

(13 citation statements)

References 87 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Furthermore, alternative approximations of the system tangent, e.g. [5,29,38], could be investigated. Yet some of these methods require additional high dimensional snapshots of the sparse system tangent which becomes computational infeasible rapidly.…”

Section: Resultsmentioning

confidence: 99%

A numerical study of different projection-based model reduction techniques applied to computational homogenisation

et al. 2017

View full text Add to dashboard Cite

Computing the macroscopic material response of a continuum body commonly involves the formulation of a phenomenological constitutive model. However, the response is mainly influenced by the heterogeneous microstructure. Computational homogenisation can be used to determine the constitutive behaviour on the macro-scale by solving a boundary value problem at the micro-scale for every so-called macroscopic material point within a nested solution scheme. Hence, this procedure requires the repeated solution of similar microscopic boundary value problems. To reduce the computational cost, model order reduction techniques can be applied. An important aspect thereby is the robustness of the obtained reduced model. Within this study reduced-order modelling (ROM) for the geometrically nonlinear case using hyperelastic materials is applied for the boundary value problem on the micro-scale. This involves the Proper Orthogonal Decomposition (POD) for the primary unknown and hyper-reduction methods for the arising nonlinearity. Therein three methods for hyper-reduction, differing in how the nonlinearity is approximated and the subsequent projection, are compared in terms of accuracy and robustness. Introducing interpolation or Gappy-POD based approximations may not preserve the symmetry of the system tangent, rendering the widely used Galerkin projection sub-optimal. Hence, a different projection related to a Gauss-Newton scheme (Gauss-Newton with Approximated Tensors-GNAT) is favoured to obtain an optimal projection and a robust reduced model.

show abstract

Section: Resultsmentioning

confidence: 99%

A numerical study of different projection-based model reduction techniques applied to computational homogenisation

et al. 2017

View full text Add to dashboard Cite

show abstract

“…We can refer to other works [23][24][25][26] for basic concepts in the POD method, [27][28][29][30][31][32] for using POD technique on numerical solution of PDEs, 33 for application of POD in PDE optimal control, 34-37 for error estimation of the POD method for optimal control problems and to previous studies [38][39][40][41][42][43][44] for other engineering applications. This method generates an optimal set of basis functions (so-called POD basis functions) where each of them has a global support and involves information about the system obtained out of a computational or experimental database (snapshots).…”

Section: Introductionmentioning

confidence: 99%

“…After this, the solution is obtained as a linear combination of the POD basis functions by means of Galerkin projection method. We can refer to other works [23][24][25][26] for basic concepts in the POD method, [27][28][29][30][31][32] for using POD technique on numerical solution of PDEs, 33 for application of POD in PDE optimal control, [34][35][36][37] for error estimation of the POD method for optimal control problems and to previous studies [38][39][40][41][42][43][44] for other engineering applications.…”

Section: Introductionmentioning

confidence: 99%

An indirect shooting method based on the POD/DEIM technique for distributed optimal control of the wave equation

Sabeh

Shamsi

Navon

2017

Numerical Methods in Fluids

View full text Add to dashboard Cite

Summary This paper presents a fast numerical method, based on the indirect shooting method and Proper Orthogonal Decomposition (POD) technique, for solving distributed optimal control of the wave equation. To solve this problem, we consider the first‐order optimality conditions and then by using finite element spatial discretization and shooting strategy, the solution of the optimality conditions is reduced to the solution of a series of initial value problems (IVPs). Generally, these IVPs are high‐order and thus their solution is time‐consuming. To overcome this drawback, we present a POD indirect shooting method, which uses the POD technique to approximate IVPs with smaller ones and faster run times. Moreover, in the presence of the nonlinear term, to reduce the order of the nonlinear calculations, a discrete empirical interpolation method (DEIM) is applied and a POD/DEIM indirect shooting method is developed. We investigate the performance and accuracy of the proposed methods by means of 4 numerical experiments. We show that the presented POD and POD/DEIM indirect shooting methods dramatically reduce the CPU time compared to the full indirect shooting method, whereas there is no significant difference between the accuracy of the reduced and full indirect shooting methods.

show abstract

“…Ballarin and Rozza make use of POD‐Galerkin and reduced basis methods to develop a novel reduced order modeling framework for fluid–structure interaction problems with prescribed boundary motion using efficient geometrical techniques. Stefanescu and Sandu propose a sparse matrix discrete interpolation method to efficiently compute matrix approximations in the reduced order modeling framework. Bistrian and Navon focus on a difficult task in hydrodynamic stability analysis by modeling the dynamics of swirl intense flows.…”

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

Numerical Methods in Fluids

Self Cite

View full text Add to dashboard Cite

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 [...

show abstract

Efficient approximation of Sparse Jacobians for time‐implicit reduced order models

Cited by 14 publications

References 87 publications

A numerical study of different projection-based model reduction techniques applied to computational homogenisation

A numerical study of different projection-based model reduction techniques applied to computational homogenisation

An indirect shooting method based on the POD/DEIM technique for distributed optimal control of the wave equation

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

Contact Info

Product

Resources

About