arXiv subjects

James Cheung

Publications and source records attributed to James Cheung.

8 recordsLinked to original sources

arXiv2025

Penalized Weighted Trace Minimization for Optimal Control Device Design and Placement

In this paper, we present a new analytical framework for determining the well-posedness of constrained optimization problems that arise in the study of optimal control device design and placement within the context of infinite dimensional linear quadratic control systems. We first prove the well-posedness of the newly minted "strong form" of the time-independent operator-valued Riccati equation. This form of the equation then enables the use of trace-class operator analysis and the Lagrange multiplier formalism to analyze operator-valued Riccati equation-constrained optimization problems. Using this fundamental result, we then determine the conditions under which there exists unique solutions to two important classes of penalized trace minimization problems for optimal control device placement and design.

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arXiv2023

Well-Posedness of the Bochner Integral Form of Operator-Valued Riccati Equations

In this short paper, we prove that the Bochner integral form of the operator-valued Riccati equation has a unique solution if and only if its mild form has a unique solution. This implies that the mild and Bochner integral forms of this equation are equivalent. The result is obtained through an operator representation argument.

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arXiv2023

On the Approximation of Operator-Valued Riccati Equations in Hilbert Spaces

In this work, we present an abstract theory for the approximation of operator-valued Riccati equations posed on Hilbert spaces. It is demonstrated here that the error of the approximate solution to the operator-valued Riccati equation is bounded above by the approximation error of the governing semigroup, under the assumption of boundedness on the semigroup and compactness on the coefficient operators. One significant outcome of this result is the correct prediction of optimal convergence for finite element approximations of the operator-valued Riccati equations for when the governing semigroup involves parabolic, as well as hyperbolic processes. We derive the abstract theory for the time-dependent and time-independent operator-valued Riccati equations in the first part of this work. In the second part, we derive optimal error estimates for the finite element approximation of the functional gain associated with model weakly damped wave and thermal LQR control systems. These theoretical claims are then corroborated with computational evidence.

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arXiv2023

Enforcing Neumann Boundary Conditions with Polynomial Extension Operators to Acheive Optimal Convergence Rates on Polytopial Meshes in the Finite Element Method

In \cite{cheung2019optimally}, the authors presented two finite element methods for approximating second order boundary value problems on polytopial meshes with optimal accuracy without having to utilize curvilinear mappings. This was done by enforcing the boundary conditions through judiciously chosen polynomial extension operators. The $H^1$ error estimates were proven to be optimal for the solutions of both the Dirichlet and Neumann boundary value problems. It was also proven that the Dirichlet problem approximation converges optimally in $L^2$. However, optimality of the Neumann approximation in the $L^2$ norm was left as an open problem. In this work, we seek to close this problem by presenting new analysis that proves optimal error estimates for the Neumann approximation in the $W^1_\infty$ and $L^2$ norms.

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arXiv2022

Approximation of Algebraic Riccati Equations with Generators of Noncompact Semigroups

In this work, we demonstrate that the Bochner integral representation of the Algebraic Riccati Equations (ARE) are well-posed without any compactness assumptions on the coefficient and semigroup operators. From this result, we then are able to determine that, under some assumptions, the solution to the Galerkin approximations to these equations are convergent to the infinite dimensional solution. Going further, we apply this general result to demonstrate that the finite element approximation to the ARE are optimal for weakly damped wave semigroup processes in the $H^1(\Omega) \times L^2(\Omega)$ norm. Optimal convergence rates of the functional gain for a weakly damped wave optimal control system in both the $H^1(\Omega) \times L^2(\Omega)$ and $L^2(\Omega)\times L^2(\Omega)$ norms are demonstrated in the numerical examples.

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arXiv2017

Optimally accurate higher-order finite element methods on polytopial approximations of domains with smooth boundaries

Meshing of geometric domains having curved boundaries by affine simplices produces a polytopial approximation of those domains. The resulting error in the representation of the domain limits the accuracy of finite element methods based on such meshes. On the other hand, the simplicity of affine meshes makes them a desirable modeling tool in many applications. In this paper, we develop and analyze higher-order accurate finite element methods that remain stable and optimally accurate on polytopial approximations of domains with smooth boundaries. This is achieved by constraining a judiciously chosen extension of the finite element solution on the polytopial domain to weakly match the prescribed boundary condition on the true geometric boundary. We provide numerical examples that highlight key properties of the new method and that illustrate the optimal $H^1$ and $L^2$-norm convergence rates.

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arXiv2017

A Coupling Approach for Linear Elasticity Problems with Spatially Noncoincident Interfaces

We present a new formulation based on the classical Dirichlet-Neumann formulation for interface coupling problems in linearized elasticity. By using Taylor series expansions, we derive a new set of interface conditions that allow our formulation to pass the linear consistency test. In addition, we propose an iterative method to determine the solution of our formulation. We demonstrate in our numerical results that we may achieve the desired piecewise linear finite element error bounds for both nonoverlapping domain decomposition problems as well as for interface coupling problems where the Lam\'e parameters of the structures differ.

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arXiv2017

An Optimally Convergent Coupling Approach for Interface Problems Approximated with Higher-Order Finite Elements

In this paper, we present a new numerical method for determining the numerical solution of interface problems to optimal accuracy with respect to the polynomial order of the Lagrangian finite element space on polytopial meshes. We introduce the notion of a virtual interface, and on this virtual interface we enforce that "extended" interface conditions are satisfied in the sense of a Dirichlet--Neumann coupling. The virtual interface framework serves to bypass geometric variational crimes incurred by the classical finite element method. Further, this approach does not require that the geometric interfaces are spatially matching. Our analysis indicates that this approach is well--posed and optimally convergent in $H^1$. Numerical experiments indicate that optimal $H^1$ and $L^2$ convergence is achieved.

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