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In this paper, we deal with some shape optimization geometrical inverse spectral problems involving the first eigenvalue and eigenfunction of a p-Laplace operator, over a class of open domains with prescribed volume. We first briefly show the existence of the optimal shape design for the L p norm of the eigenfunctions. We carried out the shape derivative calculation of this shape optimization problem using deformation of domains by vector fields. Then we propose a numerical method using lagrangian functional, Hadamard's shape derivative and gradient method to determine the minimizers for this shape optimization problem. We investigate also numerically the problem of minimizing the first eigenvalue of the p-Laplacian-Dirichlet operator with volume-constraint on domains, using constrained and unconstrained shape optimization formulations. The resulting proposed algorithms of the optimization process are based on the inverse power algorithm (Biezuner et al. 2012) and the finite elements method performed to approximate the first eigenvalue and related eigenfunction. Numerical examples and illustrations are provided for different constrained and unconstrained shape optimization formulations and for various cost functionals to show the efficiency and practical suitability of the proposed approach. [ABSTRACT FROM AUTHOR]

In this work, we consider the inverse problem of identifying a Robin coefficient in a nonlinear elliptic equation with mixed boundary conditions. We firstly reformulate the inverse problem as a regularized optimal control one, where the functional cost is of type L 1 - L 2 ; then we prove the existence and uniqueness of a minimizer to the resulting optimization problem in a suitable functional space. Finally, we provide a primal-dual algorithm to solve the variational problem and give some numerical results that prove the accuracy of the proposed method in the identification of the Robin coefficient. [ABSTRACT FROM AUTHOR]

This paper is concerned by solving supervised machine learning problem as an inverse problem. Recently, many works have focused on defining a relationship between supervised learning and the well-known inverse problems. However, this connection between the learning problem and the inverse one has been done in the particular case where the inverse problem is reformulated as a minimization problem with a quadratic cost functional ( L 2 cost functional). Although, it is well known that the cost functional can be L 1 , L 2 or any positive function that measures the gap between the predicted data and the observed one. Indeed, the use of L 1 loss function for supervised learning problem gives more consistent results (see Rosasco et al. in Neural Comput 16:1063–1076, 2004). This strengthens the idea of reformulating the inverse problem, associated to machine learning problem, into a minimization problem using L 1 functional. However, the L 1 loss function is non-differentiable, which precludes the use of standard optimization tools. To overcome this difficulty, we propose in this paper a new technique of approximation based on the reformulation of the associated inverse problem into a minimizing one of a slanting cost functional Chen et al. (MIS Q Manag Inf Syst 36:1165–1188, 2012), which is solved using Tikhonov regularization and Newton's method. This approach leads to an efficient numerical algorithm allowing us to solve supervised learning problem in the most general framework. To confirm this, we present some numerical results showing the efficiency of the proposed approach. Furthermore, the numerical experiment validation is made through academic and real-life data. Thus, the comparison with existing methods and numerical stability of the algorithm is presented in order to show that our approach is better in terms of convergence speed and quality of predicted models. [ABSTRACT FROM AUTHOR]