Description
- Abstract:
- This thesis presents several model reduction techniques that achieve a comparable accuracy with much less computational cost compared to that of high fidelity numerical simulation. In the first part, we briefly review the methodology of the reduced basis method. To demonstrate the performance, we adapt the method for a nontrivial application of electromagnetic scattering problems to compute the radar cross section, of importance for material and geometric design in real electromagnetic applications. A speedup of 10-15 times is achieved compared to the computational cost of the full model. In the second part, we introduce a high order multiscale finite element method for elliptic problems with highly oscillating coefficients. We describe how this method encodes the multiscale information into the multiscale bases, and examine the stability and convergence of the method through proofs and numerical examples. In addition, I describe how to parametrize the local problem and the construction of local oscillating test function, so the reduced basis method can be adapted. Furthermore, we describe how to extend the applicability of the reduced basis method to general unstructured meshes. In the third part, we introduce two empirical interpolation method (EIM) based numerical methods to improve the computational efficiency of model reduction for time dependent nonlinear partial differential equations. Finally, we explain how to use a reduced model as a coarse solver to stabilize and accelerate the Parareal algorithm and demonstrate the feasibility and performance through numerical examples.
- Notes:
- Thesis (Ph.D. -- Brown University (2013)
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Citation
Zhu, Xueyu,
"Reduced basis methods and their applications"
(2013).
Applied Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.7301/Z0HT2MNQ
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Applied Mathematics Theses and Dissertations
Theses and Dissertations for the Applied Mathematics department....