VI PanAmerican Workshop

Îµ-Insensitizing Controls for the Heat Equation

Luz de Teresa
deteresa@matem.unam.mx

Instituto de Matemáticas, UNAM
Circuito Exterior, C.U.
Mexico, D.F., Mexico

Abstract: We consider a heat equation with partially known initial data. We define a functional on the solutions (the square LÂ² norm on an observation set) of the heat equation and acting with a control with support in an open subset of the domain try to obtain that the derivative of the functional , with respect to a small parameter of the perturbed initial data, is small. This problem is called the Îµ-insensitizing problem for the heat equation and can easily be transformed in an approximate control problem for a cascade system of heat equations. This problem can be transformed in a unique continuation problem for the adjoint of the cascade system that has a particular interest when the control and observation regions have empty intersection.

A Numerical Conjugate Gradient Method for ytt - yxx + y3 = 0

Carlos Barrón Romero
cbarron@correo.cua.uam.mx

UAM Unidad Cuajimalpa
División de Ciencias Naturales e Ingeniería
Departamento de Matemáticas Aplicadas y Sistemas
México, D. F., Mexico

Abstract: A Conjugate Gradient Control Method will be depicted and will present numerical results for the problem in the title, following the variational techniques of Glowinski and Lions.

Conjugate Gradient Methods for the Estimation of Hydrodynamic Vector Fields

L. Hector Juarez
hect@xanum.uam.mx

Marco A. Nunez
manp@xanum.uam.mx

Ciro F. Flores
ciro.flores@itesm.mx

Abstract: Diagnostic models in meteorology generate a wind field satisfying some time-independent physical constraints. Among these models, mass--consistent models play an important role, [1]. We study a variational method based on Sasaki's method [2], which provide an adjusted field by the minimization of a functional with a mass conservation constraint. Based on previous experience about the solution of Stokes-like problems in CFD [3], we introduce a methodology based on iterative conjugate gradient methods with preconditioning to solve the primitive problem (a minimization problem with a constraint).

Matrix Calculation of Perturbation Series for Self-Adjoint Operators

Marco A. Nuñez
manp@xanum.uam.mx

Gustavo Izquierdo B.
iubg@xanum.uam.mx

Departamento de Fisica
Universidad Autonoma Metropolitana Iztapalapa
Distrito Federal, Mexico

Abstract: A matrix approach to compute the perturbation series of perturbed self-adjoint operators is proposed. It is shown that the series can be computed by means of those corresponding to a matrix representation, when the zeroth order operator is perturbed by an operator relatively bounded by the former. This result includes Schroedinger operators of atomic and molecular systems, confined spatially in a box with impenetrable walls. The results are extended to zeroth order operators, for which the standard Rayleigh-Schroedinger perturbation theory does not work. In this case we have Schroedinger operators with a finite number of bounded states. The approach allows the calculation of perturbation series even when the eigenstates of the zeroth order operator are unknown. Numerical examples are given. A heuristic argument suggests that the approach can be applied to compute the perturbation series of singularly perturbed operators and numerical examples that support such a conjecture are given.



















	Mini Workshop: Numerical Computing, Optimization and Control of PDE-II This minisimposium intends to join researchers in the field of partial differential equations. Both novel applications as well as numerical and theoretical presentations on optimization, control and solution of direct problems on PDE. Îµ-Insensitizing Controls for the Heat Equation Luz de Teresa deteresa@matem.unam.mx Instituto de Matemáticas, UNAM Circuito Exterior, C.U. Mexico, D.F., Mexico Abstract: We consider a heat equation with partially known initial data. We define a functional on the solutions (the square LÂ² norm on an observation set) of the heat equation and acting with a control with support in an open subset of the domain try to obtain that the derivative of the functional , with respect to a small parameter of the perturbed initial data, is small. This problem is called the Îµ-insensitizing problem for the heat equation and can easily be transformed in an approximate control problem for a cascade system of heat equations. This problem can be transformed in a unique continuation problem for the adjoint of the cascade system that has a particular interest when the control and observation regions have empty intersection. A Numerical Conjugate Gradient Method for ytt - yxx + y3 = 0 Carlos Barrón Romero cbarron@correo.cua.uam.mx UAM Unidad Cuajimalpa División de Ciencias Naturales e Ingeniería Departamento de Matemáticas Aplicadas y Sistemas México, D. F., Mexico Abstract: A Conjugate Gradient Control Method will be depicted and will present numerical results for the problem in the title, following the variational techniques of Glowinski and Lions. Conjugate Gradient Methods for the Estimation of Hydrodynamic Vector Fields L. Hector Juarez hect@xanum.uam.mx Marco A. Nunez manp@xanum.uam.mx Ciro F. Flores ciro.flores@itesm.mx Abstract: Diagnostic models in meteorology generate a wind field satisfying some time-independent physical constraints. Among these models, mass--consistent models play an important role, [1]. We study a variational method based on Sasaki's method [2], which provide an adjusted field by the minimization of a functional with a mass conservation constraint. Based on previous experience about the solution of Stokes-like problems in CFD [3], we introduce a methodology based on iterative conjugate gradient methods with preconditioning to solve the primitive problem (a minimization problem with a constraint). Matrix Calculation of Perturbation Series for Self-Adjoint Operators Marco A. Nuñez manp@xanum.uam.mx Gustavo Izquierdo B. iubg@xanum.uam.mx Departamento de Fisica Universidad Autonoma Metropolitana Iztapalapa Distrito Federal, Mexico Abstract: A matrix approach to compute the perturbation series of perturbed self-adjoint operators is proposed. It is shown that the series can be computed by means of those corresponding to a matrix representation, when the zeroth order operator is perturbed by an operator relatively bounded by the former. This result includes Schroedinger operators of atomic and molecular systems, confined spatially in a box with impenetrable walls. The results are extended to zeroth order operators, for which the standard Rayleigh-Schroedinger perturbation theory does not work. In this case we have Schroedinger operators with a finite number of bounded states. The approach allows the calculation of perturbation series even when the eigenstates of the zeroth order operator are unknown. Numerical examples are given. A heuristic argument suggests that the approach can be applied to compute the perturbation series of singularly perturbed operators and numerical examples that support such a conjecture are given.