This work deals with domain decomposition like methods for solving an inverse Cauchy problem governed by Stokes equation. As it is well known, this problem is one of highly ill-posed problems in the Hadamard’s sense (Hadamard 1953 Lectures on Cauchy’s Problem in Linear Partial Differential Equations (New York: Dover)). To solve this problem, we develop a technique based on its reformulation into a fixed point one involving a Steklov like operator. Firstly, we show the existence of its fixed point using the topological degrees of Leray–Schauder tools. Then, we propose a fixed point algorithm allowing us to reproduce the alternating one of Kozlov Mazya (1991 Comput. Math. Math. Phys. 31 45–52). So the proposed approach can be considered as a generalization of Kozlov–Mazya algorithm, since it offers the opportunities to exploit other domain decomposition algorithms for solving this inverse problem. Finally, we investigate the numerical approximation of this problem, using Robin–Robin domain decomposition algorithm and the finite element method. The obtained numerical results show the efficiency of the proposed approach.

This work describes a framework for solving support vector machine with kernel (SVMK). Recently, it has been proved that the use of non-smooth loss function for supervised learning problem gives more efficient results [1]. This gives the idea of solving the SVMK problem based on hinge loss function. However, the hinge loss function is non-differentiable (we can’t use the standard optimization methods to minimize the empirical risk). To overcome this difficulty, a special smoothing technique for the hinge loss is proposed. Thus, the obtained smooth problem combined with Tikhonov regularization is solved using a stochastic gradient descent method. Finally, some numerical experiments on academic and real-life datasets are presented to show the efficiency of the proposed approach.

This work deals with a geometric inverse source problem. It consists in recovering inclusion in a fixed domain based on boundary measurements. The inverse problem is solved via a shape optimization formulation. Two cost functions are investigated, namely, the least squares fitting, and the Kohn-Vogelius function. In this case, the existence of the shape derivative is given via the first order material derivative of the state problems. Furthermore, using the adjoint approach, the shape gradient of both cost functions is characterized. Then, the stability is investigated by proving the compactness of the Hessian at the critical shape for both considered cases. Finally, based on the gradient method, a steepest descent algorithm is developed, and some numerical experiments for non-parametric shapes are discussed.