In this paper, we consider an inverse problem related to a fractional diffusion equation. The model problem is governed by a nonlinear partial differential equation involving the fractional spectral Laplacian. This study is focused on the reconstruction of an unknown source term from a partial internal measured data. The considered ill-posed inverse problem is formulated as a minimization one. The existence, uniqueness, and stability of the solution are discussed. Some theoretical results are established. The numerical reconstruction of the unknown source term is investigated using an iterative process. The proposed method involves a denoising procedure at each iteration step and provides a sequence of source term approximations converging in norm to the actual solution of the minimization problem. Some numerical results are presented to show the efficiency and the accuracy of the proposed approach.

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The present article is dedicated to preliminary transient studies carried out for the analysis of the system overall behavior of the ASTRID (Advanced Sodium Technological Reactor for Industrial Demonstration) demonstrator developed in France by CEA and its industrial partners. ASTRID is foreseen to demonstrate the progress made in SFR technology at an industrial scale by qualifying innovative options, some of which still remain open in the areas requiring improvements, especially safety and operability. Among the innovative options, a gas power conversion systems (PCS) is envisaged. In this latter innovative PCS, the working gas is nitrogen whose flow rate delivers power to a turbine driving with the same shaft two compressors (low and high pressure) separated by an intercooler. The other part of the work delivered by the gas is used to drive the alternator that produces electricity. The main objective of such a PCS consists in avoiding physically the possibility of a sodium/water reaction with the secondary circuit but the impact of this PCS on the control of the accidents has also been studied. The main purpose of the studies presented in the paper is to assess the dynamic behavior of ASTRID including a gas PCS with the CATHARE2 code. The first transient presented deals with a loss of off-site power and has been calculated for the gas PCS but also for a classical steam/water PCS for comparison purpose. Then typical transients of gas system have been investigated. Several families of initiating events affecting the PCS are studied: breaches in the ternary circuit, a loss of power supply, a loss of main heat sink and the spurious

<p style='text-indent:20px;'>This paper is concerned with an inverse problem related to a fractional parabolic equation. We aim to reconstruct an unknown initial condition from noise measurement of the final time solution. It is a typical nonlinear and ill-posed inverse problem related to a nonlocal operator. The considered problem is motivated by a probabilistic framework when the initial condition represents the initial probability distribution of the position of a particle. We show the identifiability of this inverse problem by proving the existence of its unique solution with respect to the final observed data. The inverse problem is formulated as a regularized optimization one minimizing a least-squares type cost functional. In this work, we have discussed some theoretical and practical issues related to the considered problem. The existence, uniqueness, and stability of the optimization problem solution have been proved. The conjugate gradient method combined with Morozov's discrepancy principle are exploited for building an iterative reconstruction process. Some numerical examples are carried out showing the accuracy and efficiency of the proposed method.</p>