On a generalization of the proper orthogonal decomposition and optimal construction of reduced order models

Abstract

In this paper, the popular proper orthogonal decomposition (POD) without the usual integral or inner product constraints is extended to general Hilbert spaces, such as Sobolev spaces, using functional analytic methods. It is shown that a particular tensor product space is dense in the Hilbert space where the partial differential equation (PDE) solution lives. This allows approximating the PDE solution by tensors to any desired accuracy. Optimal approximation by these tensors is shown to result in the POD using operator theoretic arguments. This is achieved by solving a nonlinear optimization problem where the PDE solution is approximated by operators of a prescribed finite rank in the corresponding trace class 2 norm. POD modes can then be computed by solving an infinite dimensional eigenvalue problem using Hilbert-Schmidt theory. Moreover, an optimal method in constructing reduced order models for the two-dimensional Burgers' equation subject to boundary control is presented and compared to the POD reduced models. A closed-loop feedback controller then designed using the reduced order model and then applied to the full order model.

Physics-informed cluster analysis and a priori efficiency criterion for the construction of local reduced-order bases

“…The POD is highly related to the theory of Hilbert-Schmidt operators. In [61,62], it is shown that the POD optimization problem is equivalent to finding the optimal approximation of a Hilbert-Schmidt operator related to u by a finite rank operator in the Hilbert-Schmidt norm. The POD basis functions can also be obtained from the eigenfunctions of the Hilbert-Schmidt integral operator [7] R u :…”

Temporal clustering for order reduction of nonlinear parabolic PDE systems with time‐dependent spatial domains: Application to a hydraulic fracturing process

“…[39][40][41] In this case, the use of global bases may fail to capture the nonlinear local dynamics since it assumes the solution belongs to an affine subspace trying to minimize the Euclidean distance, and thus leads to poor approximate solution. 42,43 The performance of the global eigenfunctions becomes even worse when the local dynamic behavior changes significantly owing to nonlinear process parameters that change with space and time. Based on this observation, in order to capture the local dynamics of a complex nonlinear system more effectively, the eigenfunctions must be tailored to capture the local behavior of every portion of the solution trajectory.…”

“…This could affect the accuracy and effectiveness of the developed low‐dimensional model, which might not be able to capture the local dynamics with a desired accuracy even with an unaffordable large number of eigenfunctions . In this case, the use of global bases may fail to capture the nonlinear local dynamics since it assumes the solution belongs to an affine subspace trying to minimize the Euclidean distance, and thus leads to poor approximate solution . The performance of the global eigenfunctions becomes even worse when the local dynamic behavior changes significantly owing to nonlinear process parameters that change with space and time.…”