Mach Nguyet Minh

Enhancing residual-based techniques with shape reconstruction features in electrical impedance tomography

In electrical impedance tomography, algorithms based on minimizing a linearized residual functional have been widely used due to their flexibility and good performance in practice. However, no rigorous convergence results have been available in the literature yet, and reconstructions tend to contain ringing artifacts. In this work, we shall minimize the linearized residual functional under a linear constraint defined by a monotonicity test, which plays a role of a special regularizer. Global convergence is then established to guarantee that this method is stable under the effects of noise. Moreover, numerical results show that this method yields good shape reconstructions under high levels of noise without appearance of artifacts.

Reconstruction of the electric field of the Helmholtz equation in three dimensions

In this paper, we rigorously investigate the truncation method for the Cauchy problem of Helmholtz equations which is widely used to model propagation phenomena in physical applications. The truncation method is a well-known approach to the regularization of several types of ill-posed problems, including the model postulated by Reginska and Reginski (2006). Under certain specific assumptions, we examine the ill-posedness of the non-homogeneous problem by exploring the representation of solutions based on Fourier mode. Then the so-called regularized solution is established with respect to a frequency bounded by an appropriate regularization parameter. Furthermore, we provide a short analysis of the nonlinear forcing term. The main results show the stability as well as the strong convergence confirmed by the error estimates in L 2 -norm of such regularized solutions. Besides, the regularization parameters are formulated properly. Finally, some illustrative examples are provided to corroborate our qualitative analysis.

Backward heat equations with locally lipschitz source

Let be in and let , be a locally Lipschitz function with respect to the variable and . In this article, we consider the following backward heat problem with a nonlinear heat source The problem is ill-posed in the sense of Hadamard. Hence a regularization is in order. We use the -dimensional Fourier transform to construct an approximate solution for this problem. We also prove error estimates for our problem and give comments on similar problems.

Hölder-Type Approximation for the Spatial Source Term of a Backward Heat Equation

We consider the problem of determining a pair of functions (u, f) satisfying the two-dimensional backward heat equation with a homogeneous Cauchy boundary condition, where ϕ and g are given approximately. The problem is severely ill-posed. Using an interpolation method and the truncated Fourier series, we construct a regularized solution for the source term f and provide Hölder-type error estimates in both L 2 and H 1 norms. Numerical experiments are provided.

BV solutions constructed using the epsilon-neighborhood method

We study a certain class of weak solutions to rate-independent systems, which is constructed by using the local minimality in a small neighborhood of order e and then taking the limit e → 0. We show that the resulting solution satisfies both the weak local stability and the new energy-dissipation balance, similarly to the BV solutions constructed by vanishing viscosity introduced recently by Mielke et al. [A. Mielke, R. Rossi and G. Savare, Discrete Contin. Dyn. Syst. 2 (2010) 585–615; ESAIM: COCV 18 (2012) 36–80; To appear in J. Eur. Math. Soc. (2016)].

Regularity of weak solutions to rate-independent systems in one-dimension

We show that under some appropriate assumptions, every weak solution (e.g. energetic solution) to a given rate-independent system is of class SBV, or has finite jumps, or is even piecewise C1. Our assumption is essentially imposed on the energy functional, but not convexity is required.