Charla y cursillo de matematica en el aniversario de la facultad de ciencias
Charla:
Aproximación Numérica de ONDAS
Cursillo:
TEORIA MATEMATICA DEL CONTROL: MOTOR DEL DESARROLLO CIENTIFICO, TECNOLOGICO Y SOCIAL
Abordaremos algunos aspectos de la Teoría Matemática del Control,
comenzando con algunas consideraciones históricas generales sobre sus orígenes y evolución.
Mas adelante, describimos algunos elementos centrales de la Teoría y diversos avances recientes que se caracterizan tanto por su interés Matemático como por su transcendencia desde un punto de vista social, tecnológico e industrial. Por último mencionamos algunos problemas abiertos y los retos que se plantean en esta disciplina para un futuro inmediato.
Audiencia:
Público interesados en las aplicaciones de la matemática. En particular en teroría de Control, Computación y Análisis Numérico
Patrocinantes:
Vicerrectorado Académico de la ULA
Decanato de la Facultad de Ciencias
Tópicos:
-Teoría Matemática del Control
-Análisis Numérico
Expositores:
Dr. Enrique ZUAZUA
Departamento de Matemáticas
Facultad de Ciencias
Universidad Autónoma de Madrid
Comité Organizador:
Decanato de la Facultad de Ciencias
Departamento de Matemática
Grupo de Matemáticas Aplicada
Lugar y Fecha:
Fecha de Realización:
24 de abril de 2006
al 27 de abril de 2006
Horario:
Cursillo: Desde Martes 25 al jueves 27. Horario a fijarse el día Lunes 24
Charla: Lunes 24, 10am
Dirección:
Auditorio A-10
Información e Inscripciones:
Costo:
Gratuito
Información de Contacto:
Giovanni Calderón
E-mail: giovanni@ula.ve
Coordinador del Grupo de Matemáticas Aplicada (GMA)
Información Adicional:
Sobre la Charla:
In this lecture we shall discuss several topics related with numerical approximation of waves.
Control Theory is by now and old subject, ubiquitous in many areas of Science and Technology. There is a quite well-established finite-dimensional theory and many progresses have been done also in the context of PDE (Partial
Differential Equations). But gluing these two pieces together is often a hard task from a mathematical point of view.
This is not a merely mathematical problem since it affect modelling and computational issues. In particular, the following two questions arise: Are finite-dimensional and infinite-dimensional models equally efficient from a control theoretical point of view? Are controls built for finite-dimensional numerical schemes efficient at the continuous level?
In this talk we shall briefly analyze these issues for the wave equation as a model example of propagation without damping. We shallshow that high frequency spurious oscillations may produce the divergence of the most natural numerical schemes. This confirms the fact that finite and infinite-dimensional modelling may give completely different results from the point ofview of control. We shall then discuss some remedies like filtering of high frequencies, multi-grid techniques and numerical viscosity.
Similar questions arise when building numerical approximation schemes for nonlinear Schrodinger equations.
We first consider finite-difference space semi-discretizations and show that the standard conservative scheme does not reproduce at the discrete level the properties of the continuous Schrodinger equation.
This is due to high frequency numerical spurious solutions.
In order to damp out these high-frequencies and to reflect the properties of the continuous problem we add a suitable extra numerical viscosity term at a convenient scale.
We prove that the dispersive properties of this viscous scheme are uniform when the mesh-size tends to zero. Finally we prove the convergence of this viscous numerical scheme for a class of nonlinear Schrodinger equations with nonlinearities that may not be handeled by standard energy methods and that require the so-called Strichartz inequalities. Finally, we show that similarconvergence results may be obtained by a two-grid algorithm based on the idea of resolving on a fine grid slow oscillations of the initial data and nonlinearity. This is a joint work with Liviu Ignat.
Mas información...