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A. Leitao

Publications and source records attributed to A. Leitao.

31 records · Page 2Linked to original sources
arXiv2020

On inverse doping profile problems for the stationary voltage-current map

We consider the problem of identifying possibly discontinuous doping profiles in semiconductor devices from data obtained by\,stationary voltage-current maps. In particular, we focus on the so-called unipolar case, a system of PDE's derived directly from the drift diffusion equations. The related inverse problem corresponds to an inverse conductivity problem with partial data. The identification issue for this inverse problem is considered. In particular, for a discretized version of the problem, we derive a result connected to diffusion tomography theory. A numerical approach for the identification problem using level set methods is presented. Our method is compared with previous results in the literature, where Landweber-Kaczmarz type methods were used to solve a similar problem.

math.AP↗
arXiv2020

Regularization of systems of nonlinear ill-posed equations: I. Convergence Analysis

In this article we develop and analyze novel iterative regularization techniques for the solution of systems of nonlinear ill--posed operator equations. The basic idea consists in considering separately each equation of this system and incorporating a loping strategy. The first technique is a Kaczmarz-type method, equipped with a novel stopping criteria. The second method is obtained using an embedding strategy, and again a Kaczmarz-type approach. We prove well-posedness, stability and convergence of both methods.

math.NA↗
arXiv2020

Regularization of systems of nonlinear ill-posed equations: II. Applications

In part I we introduced modified Landweber-Kaczmarz methods and have established a convergence analysis. In the present work we investigate three applications: an inverse problem related to thermoacoustic tomography, a nonlinear inverse problem for semiconductor equations, and a nonlinear problem in Schlieren tomography. Each application is considered in the framework established in the previous part. The novel algorithms show robustness, stability, computational efficiency and high accuracy.

math.NA↗
arXiv2020

Mean value iterations for nonlinear elliptic Cauchy problems

We investigate the Cauchy problem for a class of nonlinear elliptic operators with $C^\infty$-coefficients at a regular set $Ω\subset R^n$. The Cauchy data are given at a manifold $Γ\subset \partialΩ$ and our goal is to reconstruct the trace of the $H^1(Ω)$ solution of a nonlinear elliptic equation at $\partial Ω/ Γ$. We propose two iterative methods based on the segmenting Mann iteration applied to fixed point equations, which are closely related to the original problem. The first approach consists in obtaining a corresponding linear Cauchy problem and analyzing a linear fixed point equation; a convergence proof is given and convergence rates are obtained. On the second approach a nonlinear fixed point equation is considered and a fully nonlinear iterative method is investigated; some preliminary convergence results are proven and a numerical analysis is provided.

math.NA↗
arXiv2020

A Mann iterative regularization method for elliptic Cauchy problems

We investigate the Cauchy problem for linear elliptic operators with $C^\infty$-coefficients at a regular set $Ω\subset R^2$, which is a classical example of an ill-posed problem. The Cauchy data are given at the manifold $Γ\subset \partialΩ$ and our goal is to reconstruct the trace of the $H^1(Ω)$ solution of an elliptic equation at $\partial Ω/ Γ$. The method proposed here composes the segmenting Mann iteration with a fixed point equation associated with the elliptic Cauchy problem. Our algorithm generalizes the iterative method developed by Maz'ya et al., who proposed a method based on solving successive well-posed mixed boundary value problems. We analyze the regularizing and convergence properties both theoretically and numerically.

math.NA↗
arXiv2020

On the relation between constraint regularization, level sets, and shape optimization

We consider regularization methods based on the coupling of Tikhonov regularization and projection strategies. From the resulting constraint regularization method we obtain level set methods in a straight forward way. Moreover, we show that this approach links the areas of asymptotic regularization to inverse problems theory, scale-space theory to computer vision, level set methods, and shape optimization.

math.OC↗
arXiv2020

Analysis of regularization methods for the solution of ill-posed problems involving discontinuous operators

We consider a regularization concept for the solution of ill--posed operator equations, where the operator is composed of a continuous and a discontinuous operator. A particular application is level set regularization, where we develop a novel concept of minimizers. The proposed level set regularization is capable of handling changing topologies. A functional analytic framework explaining the splitting of topologies is given. The asymptotic limit of the level set regularization method is an evolution process, which is implemented numerically and the quality of the proposed algorithm is demonstrated by solving an inverse source problem.

math.NA↗
arXiv2020

On a family of gradient type projection methods for nonlinear ill-posed problems

We propose and analyze a family of successive projection methods whose step direction is the same as Landweber method for solving nonlinear ill-posed problems that satisfy the Tangential Cone Condition (TCC). This family enconpasses Landweber method, the minimal error method, and the steepest descent method; thush providing an unified framework for the analysis of these methods. Moreover, we define in this family new methods which are convergent for the constant of the TCC in a range twice as large as the one required for the Landweber and other gradient type methods. The TCC is widely used in the analysis of iterative methods for solving nonlinear ill-posed problems. The key idea in this work is to use the TCC in order to construct special convex sets possessing a separation property, and to succesively project onto these sets. Numerical experiments are presented for a nonlinear 2D elliptic parameter identification problem, validating the efficiency of our method.

math.NA↗
arXiv2020

On projective Landweber-Kaczmarz methods for solving systems of nonlinear ill-posed equations

In this article we combine the projective Landweber method, recently proposed by the authors, with Kaczmarz's method for solving systems of non-linear ill-posed equations. The underlying assumption used in this work is the tangential cone condition. We show that the proposed iteration is a convergent regularization method. Numerical tests are presented for a non-linear inverse problem related to the Dirichlet-to-Neumann map, indicating a superior performance of the proposed method when compared with other well established iterations. Our preliminary investigation indicates that the resulting iteration is a promising alternative for computing stable solutions of large scale systems of nonlinear ill-posed equations.

math.NA↗
arXiv2020

Range-relaxed criteria for choosing the Lagrange multipliers in the Levenberg-Marquardt method

In this article we propose a novel strategy for choosing the Lagrange multipliers in the Levenberg-Marquardt method for solving ill-posed problems modeled by nonlinear operators acting between Hilbert spaces. Convergence analysis results are established for the proposed method, including: monotonicity of iteration error, geometrical decay of the residual, convergence for exact data, stability and semi-convergence for noisy data. Numerical experiments are presented for an elliptic parameter identification two-dimensional EIT problem. The performance of our strategy is compared with standard implementations of the Levenberg-Marquardt method (using a priori choice of the multipliers).

math.NA↗
arXiv2020

Range-relaxed criteria for choosing the Lagrange multipliers in nonstationary iterated Tikhonov method

In this article we propose a novel nonstationary iterated Tikhonov (NIT) type method for obtaining stable approximate solutions to ill-posed operator equations modeled by linear operators acting between Hilbert spaces. Geometrical properties of the problem are used to derive a new strategy for choosing the sequence of regularization parameters (Lagrange multipliers) for the NIT iteration. Convergence analysis for this new method is provided. Numerical experiments are presented for two distinct applications: I) A 2D elliptic parameter identification problem (Inverse Potential Problem); II) An image deblurring problem. The results obtained validate the efficiency of our method compared with standard implementations of the NIT method (where a geometrical choice is typically used for the sequence of Lagrange multipliers).

math.NA↗
arXiv2008

On Steepest-Descent-Kaczmarz Methods for Regularizing Systems of Nonlinear Ill-posed Equations

We investigate modified steepest descent methods coupled with a loping Kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill-posed operator equations. We show that the proposed method is a convergent regularization method. Numerical tests are presented for a linear problem related to photoacoustic tomography and a non-linear problem related to the testing of semiconductor devices.

math.NA↗