Patil, Prerna
(2022)
Real-time reduced order modeling of high-dimensional partial differential equations via time dependent subspaces.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
We present a new methodology for the real-time reduced-order modeling of stochastic partial differential equations (SPDEs) called the Dynamically/Bi-Orthonormal (DBO) decomposition. In this method, the stochastic fields are approximated by a low-rank decomposition to spatial and stochastic subspaces. Each of these subspaces is represented by a set of orthonormal time-dependent modes. We derive exact evolution equations of these time-dependent modes and the evolution of the factorization of the reduced covariance matrix. We show that DBO is equivalent to the Dynamically Orthogonal (DO) and Bi-Orthogonal (BO) decompositions via linear and invertible transformation matrices that connect DBO to DO and BO. We study the convergence properties of the method and compare it to the DO and BO methods. Overall we observe improvements in the numerical accuracy of DBO compared against DO and BO.
In the second part of this work we direct our attention towards solving SPDEs with nonhomogenous stochastic boundary conditions. A crucial question in this application is how do we determine the distribution of random boundary conditions among spatial bases. The DBO methodology is applied for determining the boundary conditions for time-dependent bases at no additional computational cost beyond that of solving similar SPDEs with homogeneous boundary conditions. The boundary conditions are determined by forming a variational principle whose minimization leads to the evolution of time-dependent bases at the boundary as well as the interior points. The formulation is applied for stochastic Dirichlet, Neumann and Robin boundary conditions and the performance of the method is assessed.
In the third part of this work, the focus is shifted towards application of this methodology and development of techniques to solve deterministic partial differentiation equations. A multi-dimensional variable is represented by a set of one-dimensional time-dependent orthonormal modes in each dimension and a core tensor. We derive evolution equations for these modes and the core tensor. Due to low rank representation of the solution at every time instant, the method also provides advantages in data storage for a large number of time steps. This advantage is notably evident in higher dimensions (d>2).
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Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
|
| Date: |
6 September 2022 |
| Date Type: |
Publication |
| Defense Date: |
6 June 2022 |
| Approval Date: |
6 September 2022 |
| Submission Date: |
2 August 2022 |
| Access Restriction: |
No restriction; Release the ETD for access worldwide immediately. |
| Number of Pages: |
181 |
| 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, time dependent basis, dynamically orthonormal, stochastic partial differential equation, uncertainty quantification |
| Date Deposited: |
06 Sep 2022 16:44 |
| Last Modified: |
06 Sep 2022 16:44 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/43426 |
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