Optimization and numerical analysis of PDE-constrained optimization problems with applications to Maxwell’s equations, bounded variation and neural networks

Abstract

The analysis and numerical discretization of a variety of problems related to electromagnetism, neural network architectures motivated by fractional time derivatives, and an image denoising problem based on some nonlocal differential operators are the focus of this thesis. Some of these optimization problems fall in the category of inverse problems and others are optimization problems with partial differential equations (PDEs) as constraints. A common thread linking them is the fact that they are usually ill-posed. ☐ This thesis is divided into three chapters. In the first chapter, we consider a boundary control problem for Maxwell’s equations in the frequency domain. The Wirtinger derivative is used in order to find the optimality conditions because the cost functional is not complex differentiable. The Rao-Wilton-Glisson basis and high-order Nédélec elements are used forthe numerical discretization of the control and state equation, respectively. A modified version of the BFGS method is used for the numerical optimization. ☐ In the second chapter, we consider a type of network architecture based on the discretization of a fractional time derivative. We consider a scaling factor for the activation functions, which is based on an adaptive time-stepping method for ODEs. This method may be used to remove unnecessary layers and help with the vanishing gradient problem. We also include several numerical experiments that support and illustrate our theoretical findings. ☐ Finally, in the third chapter, we proposed two fractional bounded variation (BV ) spaces based on the Riesz and Gagliardo gradients. We demonstrate that analogous properties of the BV space, such as lower semicontinuity and Sobolev embeddings, are still valid for the fractional case. However, the relationship with the space Wα,1 is different. This framework is used to create a fractional version of the total variation denoising model.