Log In
Or create an account ->
Imperial Library
Home
About
News
Upload
Forum
Help
Login/SignUp
Index
Title Page
Copyright
Contents
Preface to the second volume of Model Order Reduction
1 Basic ideas and tools for projection-based model reduction of parametric partial differential equations
Acknowledgement
Introduction
1.1 Basic notions and tools
1.1.1 Parameterized partial differential equations
1.1.2 Parameterized variational formulation
1.1.3 Model reduction basic concepts
1.1.4 Error bounds
1.2 Geometrical parameterization for shapes and domains
1.2.1 Free form deformation
1.2.2 Radial basis function interpolation
1.2.3 Inverse distance weighting interpolation
1.3 Beyond affinity assumptions: parametric interpolation
1.3.1 Nonaffine problems
1.3.2 The empirical interpolation method
1.3.3 The discrete empirical interpolation method
1.3.4 Further options
1.3.5 Some examples
1.4 Advanced tools: reduction in parameter spaces
1.4.1 Active subspaces property and its applications
1.4.2 Active subspaces definition
1.4.3 Some examples
1.4.4 Active subspaces as preprocessing tool to enhance model reduction
1.4.5 About nonlinear dimensionality reduction
1.5 Conclusion and outlook
Bibliography
2 Model order reduction by proper orthogonal decomposition
2.1 Introduction
2.2 POD
2.2.1 The POD method
2.2.2 SVD and POD
2.2.3 The POD method for nonlinear evolution problems
2.2.4 The POD method with snapshots generated by spatially adaptive finite element methods
2.3 The POD-Galerkin procedure
2.3.1 The POD-Galerkin procedure
2.3.2 Time-discrete reduced-order model
2.3.3 Discussion of the computation of the nonlinear term
2.3.4 Expressing the POD solution in the full spatial domain
2.4 Certification with a priori and a posteriori error estimates
2.5 Optimal snapshot location for computing POD basis functions
2.6 Optimal control with POD surrogate models
2.7 Miscellaneous
Bibliography
Notes
3 Proper generalized decomposition
Acknowledgement
3.1 PGD: fundamentals
3.1.1 Principles
3.1.2 Different types of separated representations
3.1.3 Illustrating the simplest separated representation constructor
3.1.4 Convergence properties
3.1.5 Verification
3.1.6 Limits
3.2 PGD for nonlinear time-dependent problems
3.2.1 State of the art
3.2.2 The LATIN-PGD computation method
3.2.3 Illustration
3.2.4 Additional reduction or interpolation
3.2.5 Extensions
3.3 Parametric solutions
3.3.1 Model parameters as extra-coordinates
3.3.2 Boundary conditions as extra-coordinates
3.3.3 Parametric domains
3.3.4 Related works
3.4 Space separated representations
3.4.1 Heat transfer in laminates
3.4.2 Three-dimensional Resin Transfer Moulding
3.4.3 The elastic problem defined in plate domains
3.4.4 Three-dimensional elastic problem in a shell domain
3.4.5 Squeeze flow in composite laminates
3.4.6 Electromagnetic models in composite laminates
3.5 Conclusions
Bibliography
4 Reduced basis methods
Acknowledgement
4.1 Motivation
4.2 Parameterized PDEs
4.2.1 Weak form
4.2.2 Justification for reduction
4.2.3 Extensions
4.3 Projection
4.3.1 Elliptic problems
4.3.2 Extensions
4.4 Approximation spaces
4.4.1 Elliptic problems: weak greedy method
4.4.2 Optimality of the weak greedy method
4.4.3 Extensions
Bibliography
Notes
5 Computational bottlenecks for PROMs: precomputation and hyperreduction
Acknowledgement
5.1 Introduction
5.1.1 Computational bottlenecks
5.1.2 Solution approaches
5.1.3 Chapter organization
5.2 Global and pointwise ROBs for parametric PROMs
5.3 Exact precomputation-based methodologies
5.3.1 Linear FOMs and efficient parameter-affine representation
5.3.2 Nonlinear FOMs with polynomial dependence on the generalized coordinates
5.4 Approximate reconstruction methodologies
5.4.1 Database of linear PROMs and interpolation on matrix manifolds
5.4.2 Hyperreduction
5.5 Applications
5.5.1 Hyperreduction of a parametric Helmholtz-elasticity model
5.5.2 Hyperreduction of a parametric PDE-ODE wildfire model
5.5.3 Hyperreduction of nonlinear structural dynamics models
5.6 Summary and conclusions
Bibliography
6 Localized model reduction for parameterized problems
Acknowledgement
6.1 Introduction
6.2 Parameterized partial differential equations and localization
Definition 6.1 (Parameterized coercive problem in variational form).
Example 6.2 (Parametric elliptic multiscale problems).
Example 6.3 (Incompressible fluid flow).
Example 6.4 (Linear elasticity).
Definition 6.5 (Nonoverlapping domain decomposition).
Definition 6.6 (Localizing space decomposition).
Definition 6.7 (Locally decomposed full order model (FOM)).
Definition 6.8 (Locally decomposed reduced-order model (ROM)).
6.3 Coupling local approximation spaces
6.3.1 Conforming approach
6.3.2 Nonconforming approach
6.4 Preparation of local approximation spaces
6.4.1 Polynomial-based local approximation spaces
6.4.2 Local approximation spaces based on empirical training
6.5 A posteriori error estimation
6.5.1 Residual-based a posteriori error estimation
6.5.2 Local flux reconstruction-based error estimation
6.6 Basis enrichment and online adaptivity
6.7 Computational complexity
6.7.1 Online efficiency
6.7.2 Offline costs and parallelization
6.8 Applications and numerical experiments
6.8.1 Multiscale problems
6.8.2 Fluid dynamics
6.9 Further perspectives
6.9.1 Parabolic problems
6.9.2 Nonaffine parameter dependence and nonlinear problems
6.10 Conclusion
Bibliography
Notes
7 Data-driven methods for reduced-order modeling
1 Introduction
2 Data-driven reductions
2.1 Singular value decomposition
2.2 Dynamic mode decomposition
2.3 Koopman theory and observable selection
2.4 Time-delay embeddings for Koopman embeddings
3 Data-driven model discovery
3.1 SINDy: sparse identification of nonlinear dynamics
3.2 Model discovery for PDEs
4 Data-driven ROMs
4.1 Application of DMD and Koopman to ROM models
4.2 Application of SINDy for ROMs
4.3 Application of time-delay embeddings for ROMs
5 Conclusion and outlook
Bibliography
Subject Index
← Prev
Back
Next →
← Prev
Back
Next →