arXiv subjects

John R. Singler

Publications and source records attributed to John R. Singler.

17 recordsLinked to original sources

arXiv2025

Analysis of a hybridizable discontinuous Galerkin scheme for the tangential control of the Stokes system

We consider an unconstrained tangential Dirichlet boundary control problem for the Stokes equations with an $ L^2 $ penalty on the boundary control. The contribution of this paper is twofold. First, we obtain well-posedness and regularity results for the tangential Dirichlet control problem on a convex polygonal domain. The analysis contains new features not found in similar Dirichlet control problems for the Poisson equation; an interesting result is that the optimal control has higher local regularity on the individual edges of the domain compared to the global regularity on the entire boundary. Second, we propose and analyze a hybridizable discontinuous Galerkin (HDG) method to approximate the solution. For convex polygonal domains, our theoretical convergence rate for the control is optimal with respect to the global regularity on the entire boundary. We present numerical experiments to demonstrate the performance of the HDG method.

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arXiv2024

Incremental data compression for PDE-constrained optimization with a data assimilation application

We propose and analyze an inexact gradient method based on incremental proper orthogonal decomposition (iPOD) to address the data storage difficulty in time-dependent PDE-constrained optimization, particularly for a data assimilation problem as a detailed demonstration for the key ideas. The proposed method is proved robust by rigorous analysis. We first derive a sharp data compression error estimate of the iPOD with the help of Hilbert-Schmidt operators. Then we demonstrate a numerical PDE analysis to show how to properly choose the Hilbert space for the iPOD data compression so that the gradient error is under control. We further prove that for a convex problem with appropriately bounded gradient error, the inexact gradient method achieves the accuracy level of the optimal solution while not hurting the convergence rate compared with the usual gradient method. Finally, numerical experiments are provided to verify the theoretical results and validate the proposed method.

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arXiv2023

A New Proper Orthogonal Decomposition Method with Second Difference Quotients for the Wave Equation

Recently, researchers have investigated the relationship between proper orthogonal decomposition (POD), difference quotients (DQs), and pointwise in time error bounds for POD reduced order models of partial differential equations. In a recent work (Eskew and Singler, Adv. Comput. Math., 49, 2023, no. 2, Paper No. 13), a new approach to POD with DQs was developed that is more computationally efficient than the standard DQ POD approach and it also retains the guaranteed pointwise in time error bounds of the standard method. In this work, we extend this new DQ POD approach to the case of second difference quotients (DDQs). Specifically, a new POD method utilizing DDQs and only one snapshot and one DQ is developed and used to prove ROM error bounds for the damped wave equation. This new approach eliminates data redundancy in the standard DDQ POD approach that uses all of the snapshots, DQs, and DDQs. We show that this new DDQ approach also has pointwise in time data error bounds similar to DQ POD and use it to prove pointwise and energy ROM error bounds. We provide numerical results for the POD errors and ROM errors to demonstrate the theoretical results. We also explore an application of POD to simulating ROMs past the training interval for collecting the snapshot data for the standard POD approach and the DDQ POD method.

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arXiv2022

A New Global Divergence Free and Pressure-Robust HDG Method for Tangential Boundary Control of Stokes Equations

In [ESAIM: M2AN, 54(2020), 2229-2264], we proposed an HDG method to approximate the solution of a tangential boundary control problem for the Stokes equations and obtained an optimal convergence rate for the optimal control {that reflects its global regularity}. However, the error estimates depend on the pressure, and the velocity is not divergence free. The importance of pressure-robust numerical methods for fluids was addressed by John et al. [SIAM Review, 59(2017), 492-544]. In this work, we devise a new HDG method to approximate the solution of the Stokes tangential boundary control problem; the HDG method is also of independent interest for solving the Stokes equations. This scheme yields a $\mathbf{H}(\mathrm{div})$ conforming, globally divergence free, and pressure-robust solution. To the best of our knowledge, this is the first time such a numerical scheme has been obtained for an optimal boundary control problem for the Stokes equations. We also provide numerical experiments to show the performance of the new HDG method and the advantage over the non pressure-robust scheme.

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arXiv2021

A new approach to proper orthogonal decomposition with difference quotients

In a recent work [B. Koc et al., arXiv:2010.03750, SIAM J. Numer. Anal., to appear], 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 numerical analysis benefits of the standard POD DQ approach, while using a POD data set that has half the number of snapshots as the standard POD DQ approach, i.e., the new approach is more computationally efficient. We illustrate our theoretical results with numerical experiments.

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arXiv2020

On Optimal Pointwise in Time Error Bounds and Difference Quotients for the Proper Orthogonal Decomposition

In this paper, we resolve several long standing issues dealing with optimal pointwise in time error bounds for proper orthogonal decomposition (POD) reduced order modeling of the heat equation. In particular, we study the role played by difference quotients (DQs) in obtaining reduced order model (ROM) error bounds that are optimal with respect to both the time discretization error and the ROM discretization error. When the DQs are not used, we prove that both the ROM projection error and the ROM error are suboptimal. When the DQs are used, we prove that both the ROM projection error and the ROM error are optimal. The numerical results for the heat equation support the theoretical results.

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arXiv2018

A note on incremental POD algorithms for continuous time data

In our earlier work [Fareed et al., Comput. Math. Appl. 75 (2018), no. 6, 1942-1960], we developed an incremental approach to compute the proper orthogonal decomposition (POD) of PDE simulation data. Specifically, we developed an incremental algorithm for the SVD with respect to a weighted inner product for the discrete time POD computations. For continuous time data, we used an approximate approach to arrive at a discrete time POD problem and then applied the incremental SVD algorithm. In this note, we analyze the continuous time case with simulation data that is piecewise constant in time such that each data snapshot is expanded in a finite collection of basis elements of a Hilbert space. We first show that the POD is determined by the SVD of two different data matrices with respect to weighted inner products. Next, we develop incremental algorithms for approximating the two matrix SVDs with respect to the different weighted inner products. Finally, we show neither approximate SVD is more accurate than the other; specifically, we show the incremental algorithms return equivalent results.

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arXiv2018

Error Analysis of an Incremental POD Algorithm for PDE Simulation Data

In our earlier work [Fareed et al., Comput. Math. Appl. 75 (2018), no. 6, 1942-1960], we proposed an incremental SVD algorithm with respect to a weighted inner product to compute the proper orthogonal decomposition (POD) of a set of simulation data for a partial differential equation (PDE) without storing the data. In this work, we perform an error analysis of the incremental SVD algorithm. We also modify the algorithm to incrementally update both the SVD and an error bound when a new column of data is added. We show the algorithm produces the exact SVD of an approximate data matrix, and the operator norm error between the approximate and exact data matrices is bounded above by the computed error bound. This error bound also allows us to bound the error in the incrementally computed singular values and singular vectors. We illustrate our analysis with numerical results for three simulation data sets from a 1D FitzHugh-Nagumo PDE system with various choices of the algorithm truncation tolerances.

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arXiv2018

Balanced Truncation Model Reduction of a Nonlinear Cable-Mass PDE System with Interior Damping

We consider model order reduction of a nonlinear cable-mass system modeled by a 1D wave equation with interior damping and dynamic boundary conditions. The system is driven by a time dependent forcing input to a linear mass-spring system at one boundary. The goal of the model reduction is to produce a low order model that produces an accurate approximation to the displacement and velocity of the mass in the nonlinear mass-spring system at the opposite boundary. We first prove that the linearized and nonlinear unforced systems are well-posed and exponentially stable under certain conditions on the damping parameters, and then consider a balanced truncation method to generate the reduced order model (ROM) of the nonlinear input-output system. Little is known about model reduction of nonlinear input-output systems, and so we present detailed numerical experiments concerning the performance of the nonlinear ROM. We find that the ROM is accurate for many different combinations of model parameters.

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arXiv2018

An EDG Method for Distributed Optimal Control of Elliptic PDEs

We consider a distributed optimal control problem governed by an elliptic PDE, and propose an embedded discontinuous Galerkin (EDG) method to approximate the solution. We derive optimal a priori error estimates for the state, dual state, the optimal control, and suboptimal estimates for the fluxes. We present numerical experiments to confirm our theoretical results.

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arXiv2018

A New HDG Method for Dirichlet Boundary Control of Convection Diffusion PDEs I: High Regularity

We propose a new hybridizable discontinuous Galerkin (HDG) method to approximate the solution of a Dirichlet boundary control problem governed by an elliptic convection diffusion PDE. Even without a convection term, Dirichlet boundary control problems are well-known to be very challenging theoretically and numerically. Although there are many works in the literature on Dirichlet boundary control problems for the Poisson equation, the authors are not aware of any existing theoretical or numerical analysis works for convection diffusion Dirichlet control problems. We make two contributions. First, we obtain well-posedness and regularity results for the Dirichlet control problem. Second, under certain assumptions on the domain and the target state, we obtain optimal a priori error estimates in 2D for the control for the new HDG method. As far as the authors are aware, there are no existing comparable results in the literature. We present numerical experiments to demonstrate the performance of the HDG method.

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arXiv2018

A New HDG Method for Dirichlet Boundary Control of Convection Diffusion PDEs II: Low Regularity

In the first part of this work, we analyzed a Dirichlet boundary control problem for an elliptic convection diffusion PDE and proposed a new hybridizable discontinuous Galerkin (HDG) method to approximate the solution. For the case of a 2D polygonal domain, we also proved an optimal superlinear convergence rate for the control under certain assumptions on the domain and on the target state. In this work, we revisit the convergence analysis without these assumptions; in this case, the solution can have low regularity and we use a different analysis approach. We again prove an optimal convergence rate for the control, and present numerical results to illustrate the convergence theory.

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arXiv2017

An HDG Method for Distributed Control of Convection Diffusion PDEs

We propose a hybridizable discontinuous Galerkin (HDG) method to approximate the solution of a distributed optimal control problem governed by an elliptic convection diffusion PDE. We derive optimal a priori error estimates for the state, adjoint state, their fluxes, and the optimal control. We present 2D and 3D numerical experiments to illustrate our theoretical results.

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arXiv2017

HDG-POD Reduced Order Model of the Heat Equation

We propose a new hybridizable discontinuous Galerkin (HDG) model order reduction technique based on proper orthogonal decomposition (POD). We consider the heat equation as a test problem and prove error bounds that converge to zero as the number of POD modes increases. We present 2D and 3D numerical results to illustrate the convergence analysis.

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arXiv2017

An Optimal EDG Method for Distributed Control of Convection Diffusion PDEs

We propose an embedded discontinuous Galerkin (EDG) method to approximate the solution of a distributed control problem governed by convection diffusion PDEs, and obtain optimal a priori error estimates for the state, dual state, their fluxes, and the control. Moreover, we prove the optimize-then-discretize (OD) and discrtize-then-optimize (DO) approaches coincide. Numerical results confirm our theoretical results.

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arXiv2017

A Superconvergent Hybridizable Discontinuous Galerkin Method for Dirichlet Boundary Control of Elliptic PDEs

We begin an investigation of hybridizable discontinuous Galerkin (HDG) methods for approximating the solution of Dirichlet boundary control problems governed by elliptic PDEs. These problems can involve atypical variational formulations, and often have solutions with low regularity on polyhedral domains. These issues can provide challenges for numerical methods and the associated numerical analysis. We propose an HDG method for a Dirichlet boundary control problem for the Poisson equation, and obtain optimal a priori error estimates for the control. Specifically, under certain assumptions, for a 2D convex polygonal domain we show the control converges at a superlinear rate. We present 2D and 3D numerical experiments to demonstrate our theoretical results.

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arXiv2017

A Superconvergent HDG Method for Distributed Control of Convection Diffusion PDEs

We consider a distributed optimal control problem governed by an elliptic convection diffusion PDE, and propose a hybridizable discontinuous Galerkin (HDG) method to approximate the solution. We use polynomials of degree $k+1$ and $k \ge 0$ to approximate the state, dual state, and their fluxes, respectively. Moreover, we use polynomials of degree $k$ to approximate the numerical traces of the state and dual state on the faces, which are the only globally coupled unknowns. We prove optimal a priori error estimates for all variables when $ k > 0 $. Furthermore, from the point of view of the number of degrees of freedom of the globally coupled unknowns, this method achieves superconvergence for the state, dual state, and control when $k\geq 1$. We illustrate our convergence results with numerical experiments.

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