Cauchy problem for Laplace equation: An observer based approach

Majeed, Muhammad Usman; Zayane-Aissa, Chadia; Laleg‐Kirati, Taous‐Meriem

doi:10.1109/icosc.2013.6750929

Cited by 3 publications

(3 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Equation ( 54) contains second order derivative in variable y. To discretize this second derivative using second order accurate centered finite difference discretization scheme on Γ B , there needs to be a fictitious point [29] further outside the boundary Γ B as shown in figure 2.…”

Section: A Boundary Condition On γ Bmentioning

confidence: 99%

Iterative observer for boundary estimation for elliptic equations

Majeed¹,

Laleg‐Kirati²

2014

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper we propose the design of an iterative observer using space as a time-like variable and prove its convergence. The iterative observer algorithm solves boundary estimation problem for a steady-state elliptic equation system namely Cauchy problem for Laplace equation. The Laplace equation is formulated as a first order state space-like system in one of the space variables and an iterative observer is developed that sweeps over the whole domain to recover the unknown data on the boundary. State operator matrix is proved to generate strongly continuous semigroup under certain conditions and the system is shown to be observable. Convergence results of proposed algorithm are established using semigroup theory and concepts of observability for distributed parameter systems. The algorithm is implemented using finite difference discretization schemes and numerical implementation is detailed. Further, the simulation results are presented towards the end to show efficiency of the algorithm. M. U. Majeed and T. M. Laleg-Kirati are with Computer Electrical and Mathematical Sciences and Engineering (CEMSE) Division at

show abstract

Section: A Boundary Condition On γ Bmentioning

confidence: 99%

Iterative observer for boundary estimation for elliptic equations

Majeed¹,

Laleg‐Kirati²

2014

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Second the performance of these techniques for noisy data is not well studied. Another kind of observer based technique to solve Cauchy problem for Laplace equation for only smooth Cauchy data is introduced in [14], where the idea is to use one of the space variables as a time-like variable to reduce numerical computation cost.…”

Section: Introductionmentioning

confidence: 99%

“…This paper is an extension of the observer based technique presented in [14] to non-smooth and noisy data. Cauchy problem for the Laplace equation is presented as a first order system and an optimal mean square error (MSE) minimizer algorithm is developed to solve the problem.…”

Section: Introductionmentioning

confidence: 99%

An optimal iterative algorithm to solve Cauchy problem for Laplace equation

Majeed

Laleg‐Kirati

2015

2015 3rd International Conference on Control, Engineering &Amp; Information Technology (CEIT)

Self Cite

View full text Add to dashboard Cite

Abstract-An optimal mean square error minimizer algorithm is developed to solve severely ill-posed Cauchy problem for Laplace equation on an annulus domain. The mathematical problem is presented as a first order state space-like system and an optimal iterative algorithm is developed that minimizes the mean square error in states. Finite difference discretization schemes are used to discretize first order system. After numerical discretization algorithm equations are derived taking inspiration from Kalman filter however using one of the space variables as a time-like variable. Given Dirichlet and Neumann boundary conditions are used on the Cauchy data boundary and fictitious points are introduced on the unknown solution boundary. The algorithm is run for a number of iterations using the solution of previous iteration as a guess for the next one. The method developed happens to be highly robust to noise in Cauchy data and numerically efficient results are illustrated.

show abstract

Localization of Point Sources for Poisson Equation using State Observers

Majeed

Laleg‐Kirati

2016

IFAC-PapersOnLine

View full text Add to dashboard Cite

Cauchy problem for Laplace equation: An observer based approach

Cited by 3 publications

References 13 publications

Iterative observer for boundary estimation for elliptic equations

Iterative observer for boundary estimation for elliptic equations

An optimal iterative algorithm to solve Cauchy problem for Laplace equation

Localization of Point Sources for Poisson Equation using State Observers

Contact Info

Product

Resources

About