Erwin Hernández

Error Estimates for the Approximation of a Class of Optimal Control Systems Governed by Linear PDEs

This paper deals with a class of optimal control problems in which the system is governed by a linear partial differential equation and the control is distributed and with constraints. The problem is posed in the framework of the theory of optimal control of systems. A numerical method is proposed to approximate the optimal control. In this method, the state space as well as the convex set of admissible controls are discretized. An abstract error estimate for the optimal control problem is obtained that depends on both the approximation of the state equation and the space of controls. This theoretical result is illustrated by some numerical examples from the literature.

A locking-free scheme for the LQR control of a Timoshenko beam

In this paper we analyze a locking-free numerical scheme for the LQR control of a Timoshenko beam. We consider a non-conforming finite element discretization of the system dynamics and a control law constant in the spatial dimension. To solve the LQR problem we seek a feedback control which depends on the solution of an algebraic Riccati equation. An optimal error estimate for the feedback operator is proved in the framework of the approximation theory for control of infinite dimensional systems. This estimate is valid with constants that do not depend on the thickness of the beam, which leads to the conclusion that the method is locking-free. In order to assess the performance of the method, numerical tests are reported and discussed.

Numerical approximation of the LQR problem in a strongly damped wave equation

The aim of this work is to obtain optimal-order error estimates for the LQR (Linear-quadratic regulator) problem in a strongly damped 1-D wave equation. We consider a finite element discretization of the system dynamics and a control law constant in the spatial dimension, which is studied in both point and distributed case. To solve the LQR problem, we seek a feedback control which depends on the solution of an algebraic Riccati equation. Optimal error estimates are proved in the framework of the approximation theory for control of infinite-dimensional systems. Finally, numerical results are presented to illustrate that the optimal rates of convergence are achieved.

On the Error Estimates for the Finite Element Approximation of a Class of Boundary Optimal Control Systems

In this article, we consider an application of the abstract error estimate for a class of optimal control systems described by a linear partial differential equation (as stated in Numer. Funct. Anal. Optim. 2009; 30:523–547). The control is applied at the boundary and we consider both, Neumann and Dirichlet optimal control problems. Finite element methods are proposed to approximate the optimal control considering an approximation of the variational inequality resulting from the optimality conditions; this approach is known as classical one. We obtain optimal order error estimates for the control variable and numerical examples, taken from the literature, are included to illustrate the results.

A GPU implementation of an explicit compact FDTD algorithm with a digital impedance filter for room acoustics applications

In recent years, computational engineering has undergone great changes due to the development of the graphics processing unit (GPU) technology. For example, in room acoustics, the wave-based methods, that formerly were considered too expensive for 3-D impulse response simulations, are now chosen to exploit the parallel nature of GPU devices considerably reducing the execution time of the simulations. There exist contributions related to this topic that have explored the performance of different GPU algorithms; however, the computational analysis of a general explicit model that incorporates algorithms with different neighboring orders and a general frequency dependent impedance boundary model has not been properly developed. In this paper, we present a GPU implementation of a complete room acoustic model based on a family of explicit finite-difference time-domain (FDTD) algorithms. We first develop a strategy for implementing a frequency independent (FI) impedance model which is free from thread divergences and then, we extend the model adding a digital impedance filter (DIF) boundary subroutine able to compute the acoustic pressure of different nodes such as corners or edges without an additional performance penalty. Both implementations are validated and deeply analyzed by performing different 3-D numerical experiments. Finally, we define a performance metric which is able to objectively measure the computing throughput of a FDTD implementation using a simple number. The robustness of this metric allows us to compare algorithms even if these have been run in different GPU cards or have been formulated with other explicit models.

A two-level enriched finite element method for a mixed problem

The simplest pair of spaces is made inf-sup stable for the mixed form of the Darcy equation. The key ingredient is to enhance the finite element spaces inside a Petrov-Galerkin framework with functions satisfying element-wise local Darcy problems with right hand sides depending on the residuals over elements and edges. The enriched method is symmetric, locally mass conservative and keeps the degrees of freedom of the original interpolation spaces. First, we assume local enrichments exactly computed and we prove uniqueness and optimal error estimates in natural norms. Then, a low cost two-level finite element method is proposed to effectively obtain enhancing basis functions. The approach lays on a two-scale numerical analysis and shows that well-posedness and optimality is kept, despite the second level numerical approximation. Several numerical experiments validate the theoretical results and compares (favourably in some cases) our results with the classical Raviart-Thomas element

Numerical Absorbing Boundary Conditions Based on a Damped Wave Equation for Pseudospectral Time-Domain Acoustic Simulations

In the context of wave-like phenomena, Fourier pseudospectral time-domain (PSTD) algorithms are some of the most efficient time-domain numerical methods for engineering applications. One important drawback of these methods is the so-called Gibbs phenomenon. This error can be avoided by using absorbing boundary conditions (ABC) at the end of the simulations. However, there is an important lack of ABC using a PSTD methods on a wave equation. In this paper, we present an ABC model based on a PSTD damped wave equation with an absorption parameter that depends on the position. Some examples of optimum variation profiles are studied analytically and numerically. Finally, the results of this model are also compared to another ABC model based on an hybrid formulation of the scalar perfectly matched layer.

A robust numerical method for a control problem involving singularly perturbed equations

We consider an unconstrained linear-quadratic optimal control problem governed by a singularly perturbed convection-reaction-diffusion equation. We discretize the optimality system by using standard piecewise bilinear finite elements on the graded meshes introduced by Duran and Lombardi in (Duźan and Lombardi 2005, 2006). We prove convergence of this scheme. In addition, when the state equation is a singularly perturbed reaction-diffusion equation, we derive quasi-optimal a priori error estimates for the approximation error of the optimal variables on anisotropic meshes. We present several numerical experiments when the state equation is both a reaction-diffusion and a convection-reaction-diffusion equation. These numerical experiments reveal a competitive performance of the proposed solution technique.