Donello, Michael
(2023)
Real-Time Reduced Order Modeling Using Time Dependent Bases: Applications and Advances.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
In the first part, we present a reduced order modeling (ROM) strategy for computing finite time sensitivities in evolutionary systems, called the forced optimally time dependent} (f-OTD) decomposition. The approach is used for low-rank approximation of sensitivities governed by forced linear differential equations. The sensitivity fields are approximated using time dependent bases (TDB), that are evolved via closed form evolution equations. We demonstrate the accuracy of f-OTD for computing sensitivities in a variety evolutionary systems.
In the second part, we extend f-OTD to approximate nonlinear sensitivities, which we call NL-fOTD. Unlike solving a linearized system that assumes infinitesimal perturbations around a base trajectory, this framework allows for finite perturbations as nonlinear interactions are considered. Similar to f-OTD, we solve low-rank evolution equations that leverage TDB by extracting correlations between sensitivities on-the-fly. The resulting equations are Jacobian-free and leverage the same nonlinear solver that is used to compute the evolution of the base state. For nonlinear sensitivities with arbitrarily time dependent base state, we demonstrate that low-rank structure often exists, and can be accurately extracted in real time directly from the governing equations.
In the third part, we address some of the outstanding challenges of TDB based ROMs, like f-OTD and NL-fOTD. In particular, TDB ROMs are (i) inefficient for solving general nonlinear equations, (ii) intrusive to implement, and (iii) ill-conditioned in the presence of small singular values. Since these challenges can arise regardless of the governing equations, we develop a new TDB ROM method for solving general nonlinear matrix differential equations (MDEs) that is computationally efficient, minimally intrusive, robust in the presence of small singular values, and rank-adaptive. The new method is based on a sparse sampling strategy for the low-rank approximation of a time discrete MDE. Guided by the discrete empirical interpolation method (DEIM), a low-rank approximation is computed at each iteration of the time stepping scheme. The new method is coined TDB-CUR, since the resulting low-rank approximation is equivalent to a CUR matrix factorization.
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Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
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| ETD Committee: |
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| Date: |
14 September 2023 |
| Date Type: |
Publication |
| Defense Date: |
20 July 2023 |
| Approval Date: |
14 September 2023 |
| Submission Date: |
28 June 2023 |
| Access Restriction: |
No restriction; Release the ETD for access worldwide immediately. |
| Number of Pages: |
115 |
| Institution: |
University of Pittsburgh |
| Schools and Programs: |
Swanson School of Engineering > Computational Modeling and Simulation |
| Degree: |
PhD - Doctor of Philosophy |
| Thesis Type: |
Doctoral Dissertation |
| Refereed: |
Yes |
| Uncontrolled Keywords: |
reduced order modeling, sensitivity analysis, uncertainty quantification, time dependent basis |
| Date Deposited: |
14 Sep 2023 13:39 |
| Last Modified: |
14 Sep 2023 13:39 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/44962 |
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