The main purpose of this work is to study an inverse source problem for degenerate/singular parabolic equations with degeneracy and singularity occurring in the interior of the spatial domain. Using Carleman estimates, we prove a Lipschitz stability estimate for the source term provided that additional measurement data are given on a suitable interior subdomain. For the numerical solution, the reconstruction is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. The Fréchet differentiability of the Tikhonov functional and the Lipschitz continuity of the Fréchet gradient are proved. These properties allow us to apply gradient methods for numerical solution of the considered inverse source problem.

No abstract

In this article, we will determine the source term of the fractional diffusion equation (FDE). Our contribution to this work is the generalization of the common inverse diffusion equation issues and the inverse diffusion equation problems for fractional diffusion equations with energy source and using Caputo fractional derivatives in time and space. The problem is reformulated in a least-squares framework, which leads to a nonconvex minimization problem, which is solved using a Tikhonov regularization. By considering the direct problem with an implicit finite difference scheme (IFDS), the numerical inversions are performed for the source term in several approximate spaces. The inversion algorithm (IA) uniqueness is obtained. Furthermore, the effect of fractional order and regularization parameter on the inversion algorithm is carried out and shows that the inversion algorithm is effective. The order of fractional derivatives expresses the global property of the direct problem and also shows the badly posed nature of the inverted problem in question.

We propose a novel hybrid multiobjective (MO) immune algorithm for tackling continuous MO problems. Similarly to the nondominated neighbor immune algorithm (NNIA), it considers the characteristics of OM problems: based on the fitness values, the best individuals from the test population are selected and recombined to guide the rest of the individuals in the population to the Pareto front. But NNIA uses the simulated binary crossover (SBX), which uses the local search method. In our algorithm, the recombination is essentially inspired by the cross used in the backtracking search algorithm (BSA), but the adaptations are found in the immune algorithm. Thus, three variants are designed in this chapter, resulting in new recombination operators. They are evaluated through 10 benchmark tests. For the most advanced proposed variant, which is designed to have global search ability, results show that an improved convergence and a better diversity of the Pareto front are statistically achieved when compared with a basic immune algorithm having no recombination or to NNIA. Finally, the proposed new algorithm is demonstrated to be successful in approximating the Pareto front of the complex 10 bar truss structure MO problem.

No abstract