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A novel splitting method is presented for the `1-TV restoration of degraded images subject to impulsive noise. The functional is split into an `2-TV denoising and an `1-`2 deblurring part. The dual problem of the relaxed functional is smooth with convex constraints, and can be solved efficiently by applying an Arrow-Hurwicz type algorithm to the augmented(More)
We consider the initial boundary value problem for the homogeneous time-fractional diffusion equation ∂ t u − ∆u = 0 (0 < α < 1) with initial condition u(x, 0) = v(x) and a homogeneous Dirichlet boundary condition in a bounded polygonal domain Ω. We shall study two semidiscrete approximation schemes, i.e., Galerkin FEM and lumped mass Galerkin FEM, by using(More)
In this paper we develop a novel criterion for choosing regularization parameters for nonsmooth Tikhonov functionals. The proposed criterion is solely based on the value function, and thus applicable to a broad range of functionals. It is analytically compared with the local minimum criterion, and a posteriori error estimates are derived. An efficient(More)
In this paper, we study the inverse electromagnetic medium scattering problem of estimating the support and shape of medium scatterers from scattered electric/magnetic near-field data. We shall develop a novel direct sampling method based on an analysis of electromagnetic scattering and the behavior of the fundamental solution. It is applicable to a few(More)
In this work, we develop variational formulations of Petrov-Galerkin type for one-dimensional fractional boundary value problems involving either a Riemann-Liouville or Caputo derivative of order α ∈ (3/2, 2) in the leading term and both convection and potential terms. They arise in the mathematical modeling of asymmetric super-diffusion processes in(More)
We study multi-parameter regularization (multiple penalties) for solving linear inverse problems to promote simultaneously distinct features of the sought-for objects. We revisit a balancing principle for choosing regularization parameters from the viewpoint of augmented Tikhonov regularization, and derive a new parameter choice strategy called the balanced(More)
This paper investigates theoretical properties and efficient numerical algorithms for the so-called elastic-net regularization originating from statistics, which enforces simultaneously l 1 and l regularization. The stability of the minimizer and its consistency are studied, and convergence rates for both a priori and a posteriori parameter choice rules are(More)
This paper considers the numerical solution of inverse problems with a L1 data fitting term, which is challenging due to the lack of differentiability of the objective functional. Utilizing convex duality, the problem is reformulated as minimizing a smooth functional with pointwise constraints, which can be efficiently solved using a semismooth Newton(More)