Structural stability in a
minimization problem and applications to conductivity imaging

Nashed, M. Zuhair; Tamasan, Alexandru

doi:10.3934/ipi.2011.5.219

Cited by 18 publications

(19 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the recent years many authors have spent a significant effort to study weighted least gradient problems and further generalizations, due to its various applications to such areas as imaging conductivity problems, reduced models in superconductivity and superfluidity, models for a description of landsliding, and relaxed models in the theory of elasticity and in optimal design, among others. A list of important investigations in these directions can be found in [4,5,13,14,[17][18][19]21,23,[26][27][28][29][32][33][34][35][36][41][42][43]. In addition, the time dependent notion of total variation flow has proved to be useful in image processing including denoising and restoration, see for example [2,3,6,8,25].…”

Section: Introductionmentioning

confidence: 99%

Continuity of minimizers to weighted least gradient problems

Zuniga

2019

Nonlinear Analysis

View full text Add to dashboard Cite

We revisit the question of existence and regularity of minimizers to the weighted least gradient problem with Dirichlet boundary conditionwhere g ∈ C(∂Ω), and a ∈ C 2 (Ω) is a weight function that is bounded away from zero.Under suitable geometric conditions on the domain Ω ⊂ R n , we construct continuous solutions of the above problem for any dimension n ≥ 2, by extending the Sternberg-Williams-Ziemer technique [42] to this setting of inhomogeneous variations. We show that the level sets of the constructed minimizer are minimal surfaces in the conformal metric a 2/(n−1) In. This result complements the approach in [19] since it provides a continuous solution even in high dimensions where the possibility exists for level sets to develop singularities. The proof relies on an application of a strict maximum principle for sets with area minimizing boundary established by Leon Simon in [40]. MSC (2010). Primary: 49Q20; Secondary: 49J52, 49Q10, 49Q15.

show abstract

Section: Introductionmentioning

confidence: 99%

Continuity of minimizers to weighted least gradient problems

Zuniga

2019

Nonlinear Analysis

View full text Add to dashboard Cite

show abstract

“…According to classical results (see [20]), the resulting algorithm using (23) and (24) is globally convergent to the set:…”

Section: Descent Algorithm and Global Convergencementioning

confidence: 99%

“…The search criteria (23) and (24) can be extended to a finite number of directions. Given , , ∈ (0, 1), > , p ≥ 1, in steps A3) and A4), we look for p directions D k = (d k 1 , , d k p ) ∈ X p and p step lengths k = ( k1 , , kp ) ∈ p + , such that:…”

Section: Descent Algorithm and Global Convergencementioning

confidence: 99%

See 1 more Smart Citation

A Globally Convergent Algorithm for a PDE-Constrained Optimization Problem Arising in Electrical Impedance Tomography

Carrillo

Gómez

2015

Numerical Functional Analysis and Optimization

View full text Add to dashboard Cite

We study the convergence properties of an algorithm for the inverse problem of electrical impedance tomography, which can be reduced to a partial differential equation (PDE) constrained optimization problem. The direct problem consists of the potential equation div( u) = 0 in a circle, with Neumann condition describing the behavior of the electrostatic potential in a medium with conductivity given by the function (x, y). We suppose that at each time a current i is applied to the boundary of the circle (Neumann's data), and that it is possible to measure the corresponding potential i (Dirichlet data). The inverse problem is to find (x, y) given a finite number of Cauchy pairs measurements ( i , i ), i = 1, , N . The problem is formulated as a least squares problem, and the developed algorithm solves the continuous problem using descent iterations in its corresponding finite element approximations. Wolfe's conditions are used to ensure the global convergence of the optimization algorithm for the continuous problem. Although exact data are assumed, measurement errors in data and regularization methods shall be considered in a future work.

show abstract

“…Let n be sufficiently large so that a n ≤ 2 a . Recall the functional F δn (·; a n ) in (27) with δ = δ n and a = a n , and the induced norm on H 1 (Ω) in (28). We estimate min ǫ 2z , δ n 2 u n − h 2 1 ≤ F δn (u n − h; a n ) ≤ F δn (u n − h; a n ) + Ω a n |∇u n |dx = G δn (u n ; a n ) ≤ G δn (h; a n )…”

Section: Convergence Properties Of the Regularized Minimizing Sequencementioning

confidence: 99%

A regularized weighted least gradient problem for conductivity imaging

Tamasan¹,

Timonov²

2019

Inverse Problems

View full text Add to dashboard Cite

We propose and study a regularization method for recovering an approximate electrical conductivity solely from the magnitude of one interior current density field. Without some minimal knowledge of the boundary voltage potential, the problem has been recently shown to have nonunique solutions, thus recovering the exact conductivity is impossible. The method is based on solving a weighted least gradient problem in the subspace of functions of bounded variations with square integrable traces. The computational effectiveness of this method is demonstrated in numerical experiments.

show abstract

Structural stability in a minimization problem and applications to conductivity imaging

Cited by 18 publications

References 14 publications

Continuity of minimizers to weighted least gradient problems

Continuity of minimizers to weighted least gradient problems

A Globally Convergent Algorithm for a PDE-Constrained Optimization Problem Arising in Electrical Impedance Tomography

A regularized weighted least gradient problem for conductivity imaging

Contact Info

Product

Resources

About