Numerical solution method for the dbar-equation in the plane

Knudsen, Kim; Mueller, Jennifer L.; Siltanen, Samuli

doi:10.1016/j.jcp.2004.01.028

Cited by 73 publications

(119 citation statements)

References 30 publications

Supporting

Mentioning

118

Contrasting

Order By: Relevance

“…A numerical solution method for (1) was developed and implemented in [18]. The approach taken was to express the fundamental equation (1), normalized so that ψ = 1 + v, with v = O (1/|z|) as |z| → ∞, in terms of the operator ∂ −1 , which is a well-known singular integral operator in C. The integral equation takes the [18], with k replaced by z and T (k ) replaced by Q(z ).)…”

Section: Brief Description Of the Numerical Methods Developed By Knudsmentioning

confidence: 99%

“…The approach taken was to express the fundamental equation (1), normalized so that ψ = 1 + v, with v = O (1/|z|) as |z| → ∞, in terms of the operator ∂ −1 , which is a well-known singular integral operator in C. The integral equation takes the [18], with k replaced by z and T (k ) replaced by Q(z ).) Because of the application to the inverse conductivity problem, they considered functions Q of compact support.…”

Section: Brief Description Of the Numerical Methods Developed By Knudsmentioning

confidence: 99%

“…These approaches are described briefly in [18], and we quote from that article, replacing citations with our own citation numbering scheme as needed (equation (1.1) in the quotation below is ∂v(…”

Section: Brief Description Of the Numerical Methods Developed By Knudsmentioning

confidence: 99%

See 2 more Smart Citations

Spectral Approach to D‐bar Problems

Klein

McLaughlin

2017

Comm Pure Appl Math

View full text Add to dashboard Cite

Abstract. We present the first numerical approach to D-bar problems having spectral convergence for real analytic rapidly decreasing potentials. The proposed method starts from a formulation of the problem in terms of an integral equation which is solved with Fourier techniques. The singular integrand is regularized analytically. The resulting integral equation is approximated via a discrete system which is solved with Krylov methods. As an example, the D-bar problem for the Davey-Stewartson II equations is solved. The result is used to test direct numerical solutions of the PDE.

show abstract

Section: Brief Description Of the Numerical Methods Developed By Knudsmentioning

confidence: 99%

Section: Brief Description Of the Numerical Methods Developed By Knudsmentioning

confidence: 99%

See 1 more Smart Citation

Spectral Approach to D‐bar Problems

Klein

McLaughlin

2017

Comm Pure Appl Math

View full text Add to dashboard Cite

show abstract

“…This result was complemented by constructive steps and numerical implementation by Knudsen and Tamasan [KT04,Knu02,Knu03]; see also [KMS04]. Francini [Fra00] extended the uniqueness proof to complex conductivities whose real and imaginary parts are twice differentiable, and her approach was subsequently implemented in [HHMV12,Ham12,HM13,Her12].…”

Section: (B)mentioning

confidence: 99%

Nonlinear Inversion from Partial EIT Data:Computational Experiments

Hamilton¹,

Siltanen²

2014

Inverse Problems and Applications

Self Cite

View full text Add to dashboard Cite

Abstract. Electrical impedance tomography (EIT) is a non-invasive imaging method in which an unknown physical body is probed with electric currents applied on the boundary, and the internal conductivity distribution is recovered from the measured boundary voltage data. The reconstruction task is a nonlinear and ill-posed inverse problem, whose solution calls for special regularized algorithms, such as D-bar methods which are based on complex geometrical optics solutions (CGOs). In many applications of EIT, such as monitoring the heart and lungs of unconscious intensive care patients or locating the focus of an epileptic seizure, data acquisition on the entire boundary of the body is impractical, restricting the boundary area available for EIT measurements. An extension of the D-bar method to the case when data is collected only on a subset of the boundary is studied by computational simulation. The approach is based on solving a boundary integral equation for the traces of the CGOs using localized basis functions (Haar wavelets). The numerical evidence suggests that the D-bar method can be applied to partial-boundary data in dimension two and that the traces of the partial data CGOs approximate the full data CGO solutions on the available portion of the boundary, for the necessary small k frequencies.

show abstract

“…These assumptions are realistic in several applications; see [8,6,2] and the references therein. For the Beltrami equation and its applications, see [1,5].…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of the $\mathbb R$-linear Beltrami equation

Huhtanen

Perämäki

2012

Math. Comp.

View full text Add to dashboard Cite

Abstract. The R-linear Beltrami equation appears in applications, such as the inverse problem of recovering the electrical conductivity distribution in the plane. In this paper, a new way to discretize the R-linear Beltrami equation is considered. This gives rise to large and dense R-linear systems of equations with structure. For their iterative solution, norm minimizing Krylov subspace methods are devised. In the numerical experiments, these improvements combined are shown to lead to speed-ups of almost two orders of magnitude in the electrical conductivity problem.

show abstract

Numerical solution method for the dbar-equation in the plane

Cited by 73 publications

References 30 publications

Spectral Approach to D‐bar Problems

Spectral Approach to D‐bar Problems

Nonlinear Inversion from Partial EIT Data:Computational Experiments

Numerical solution of the $\mathbb R$-linear Beltrami equation

Contact Info

Product

Resources

About