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Full-Text Articles in Physical Sciences and Mathematics
A New Approach To Proper Orthogonal Decomposition With Difference Quotients, Sarah Locke Eskew, John R. Singler
A New Approach To Proper Orthogonal Decomposition With Difference Quotients, Sarah Locke Eskew, John R. Singler
Mathematics and Statistics Faculty Research & Creative Works
In a Recent Work (Koc Et Al., SIAM J. Numer. Anal. 59(4), 2163–2196, 2021), the Authors Showed that Including Difference Quotients (DQs) is Necessary in Order to Prove Optimal Pointwise in Time Error Bounds for Proper Orthogonal Decomposition (POD) Reduced Order Models of the Heat Equation. in This Work, We Introduce a New Approach to Including DQs in the POD Procedure. Instead of Computing the POD Modes using All of the Snapshot Data and DQs, We Only Use the First Snapshot Along with All of the DQs and Special POD Weights. We Show that This Approach Retains All of the …
New Proper Orthogonal Decomposition Approximation Theory For Pde Solution Data, Sarah Locke, John R. Singler
New Proper Orthogonal Decomposition Approximation Theory For Pde Solution Data, Sarah Locke, John R. Singler
Mathematics and Statistics Faculty Research & Creative Works
In our previous work [J. R. Singler, SIAM J. Numer. Anal., 52 (2014), pp. 852- 876], we considered the proper orthogonal decomposition (POD) of time varying PDE solution data taking values in two different Hilbert spaces. We considered various POD projections of the data and obtained new results concerning POD projection errors and error bounds for POD reduced order models of PDEs. In this work, we improve on our earlier results concerning POD projections by extending to a more general framework that allows for nonorthogonal POD projections and seminorms. We obtain new exact error formulas and convergence results for POD …
A Higher-Order Ensemble/Proper Orthogonal Decomposition Method For The Nonstationary Navier-Stokes Equations, Max Gunzburger, Nan Jiang, Michael Schneier
A Higher-Order Ensemble/Proper Orthogonal Decomposition Method For The Nonstationary Navier-Stokes Equations, Max Gunzburger, Nan Jiang, Michael Schneier
Mathematics and Statistics Faculty Research & Creative Works
Partial differential equations (PDE) often involve parameters, such as viscosity or density. An analysis of the PDE may involve considering a large range of parameter values, as occurs in uncertainty quantification, control and optimization, inference, and several statistical techniques. The solution for even a single case may be quite expensive; whereas parallel computing may be applied, this reduces the total elapsed time but not the total computational effort. In the case of flows governed by the Navier-Stokes equations, a method has been devised for computing an ensemble of solutions. Recently, a reduced-order model derived from a proper orthogonal decomposition (POD) …